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Comprehensive uncertainty quantification in composites manufacturing processes

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Comprehensive uncertainty quantification in composites manufacturing processes

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Content COMPREHENSIVE UNCERTAINTY QUANTIFICATION IN COMPOSITES MANUFACTURING PROCESSES by Ziad Georges Ghauch A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Civil Engineering) May 2019 Copyright 2019 Ziad Georges Ghauch Acknowledgements My first thanks and deepest gratitude goes to my advisor, Prof. Roger Ghanem, for the wonderful guidance, hands-on mentorship, and caring support over the past years, for being an inspiration on both professional and personal levels, and for making me ever increasingly enthralled by research within the realm of uncertainty quantification and computational stochastic mechanics. Moreover, I would like to thank each and everyone of my committee members, namely Prof. Aiichiro Nakano & Prof. Sami Masri for being part of this work. My next acknowl- edgment goes to our project partners and for the insightful discussions that we have had on a weekly basis. In particular, thanks to Venkat Aitharaju and William Rodgers from General Motors Company’s R&D Center for providing practical insights that validated our stochastic analysis in addition to providing data. Also, thanks to Praveen Pasupuleti and Arnaud Dereims from ESI Group, for their support on the composite solvers. I would like to further thank my colleagues here within the uncertainty quantification group of Prof. Roger Ghanem; special thanks to Charan, Panos, Loujaine, Zhiheng, & Xiaoshu. Thanks also to Prof. Erik Johnson and Dr. Qiming Wang for providing constructive comments on this work over the past years. I would also like to acknowledge the financial support provided by the Office of En- ergy Efficiency and Renewable Energy within the U.S. Department of Energy (DoE) over the past two years. The project received a Distinguished Achievement award prior to completion from the U.S. DoE. ii Finally, I would like to dedicate this work to each and everyone of my wonderful family. Ziad Georges Ghauch, October 2018 Los Angeles, CA iii Table of Contents Acknowledgements ii List Of Tables vi List Of Figures vii Abstract x Chapter 1: Introduction 1 1.1 Overview of Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . 1 1.2 Light-weight Composites Applications . . . . . . . . . . . . . . . . . . . . . 3 1.3 Integrated Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Manuscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: Composite Manufacturing Models 9 2.1 Mechanical Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Flow in Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Resin Curing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Mechanical Distortion & Residual Stresses . . . . . . . . . . . . . . . . . . 14 Chapter 3: Polynomial Chaos & Dimension Reduction 17 3.1 Homogeneous Chaos Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Polynomial Chaos Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Homogeneous Chaos Basis Adaptation . . . . . . . . . . . . . . . . . . . . . 21 Chapter 4: Integrated Forward RTM Problem 25 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Stochastic RTM Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 Computational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2 Parametrization of Random Input (74d) . . . . . . . . . . . . . . . . . 31 4.2.3 Random Fiber Orientation Fields . . . . . . . . . . . . . . . . . . . . 35 4.2.4 Local Sheared Permeability . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.5 Quantities of Interest (QoI) . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Results & Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.1 Observations of Physical Behavior . . . . . . . . . . . . . . . . . . . . 41 4.3.2 First Order Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3.3 Higher-order Adapted PCE . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 Findings & Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 iv Chapter 5: Flow in Random Heterogeneous Media 58 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2.2 Polynomial Chaos Surrogates . . . . . . . . . . . . . . . . . . . . . . 63 5.2.3 Parametric Kernel Optimization . . . . . . . . . . . . . . . . . . . . . 64 5.2.3.1 Autocorrelation Approximation . . . . . . . . . . . . . . . . . 64 5.2.3.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.4 Mesh Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 Application: Fabric Forming . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.3.0.1 Physics of Fabric Deformation . . . . . . . . . . . . . . . . . 72 5.3.1 Computational FE Model . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.3.2 Stochastic Forming Input . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3.3 Discretization of Sheared Permeability Process . . . . . . . . . . . . 76 5.4 Results & Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4.2 Kernel Optimization & Performance . . . . . . . . . . . . . . . . . . . 83 5.5 Conclusions & Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 6: Adapted Polynomial Chaos UQ Workflow 91 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Review of UQ Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.3 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.4 UQ Workflow Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.5 Complexity, Error Formulation, & Convergence . . . . . . . . . . . . . . . . 100 Chapter 7: Conclusions 103 Appendix A Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.1 Demos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.2 Integrated RTM Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Appendix B Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 v List Of Tables 3.1 First seven one-dimensional Hermite Polynomial Chaos . . . . . . . . . . . 22 3.2 Correspondance of the continuous random variables with the Wiener- Askey polynomial chaos [52] . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.1 Probabilistic models of random variables from draping, injection, curing, and distortion, corresponding to sets 1 ; 2 ; 3 ;and 4 , respectively . . . . . 33 4.2 Convergence of different reduced probabilistic models for each QoI. . . . . 51 4.3 Number of forward evaluations (i.e. quadrature points) required for obtain- ing a PDF representation of the QoI in a d = 74 random physical system with three QoI’s using Smolyak sparse quadrature for classical, full PCE, as well as the adapted PCE. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.1 Some stationary isotropic covariance kernels ( =jx yj); y p= q ! 2 0 + 2 =4. . 66 5.2 Random input parameters of the 6-D fabric forming model (CoV =5%). . . 76 5.3 Optimal kernel parameters. y D:(x;y)2[0;L x ][0;L y ]. . . . . . . . . . . . . . 85 vi List Of Figures 2.1 (a) Definition of fiber directions for a woven isotropic fabric where ! 1 (e) and! 2 (e) describe the warp and weft directions within the aforementioned RVE-based multi-scale coupling approach for a given homogenized shell element e within the setD p e , and ! 0 1 (e) and ! 0 2 (e) represent the distorted warp and weft directions, each perturbed by 1 (e); 2 (e), respectively. (b) local shearing of fabric and corresponding permeability tensor. When an elemente in the fabric domainD p e is sheared by an angle(e) with respect to weft directione 2 , principal components of permeability tensorK 1 (e) and K 2 (e) rotated by(e) with respect to warp directione 1 . . . . . . . . . . . . . 12 4.1 Truncated square pyramid model geometry of the woven fabric domain. . 31 4.2 Illustration of the parametrization scheme of the manufacturing problem, where (x) represent the shear angle field, c(x) the curing field, and T(x) the temperature field along the domainD. . . . . . . . . . . . . . . . . . . . 34 4.3 Illustration of the proposed approach for computing locally sheared per- meability tensor of a random elemente,8e 2 D f e . . . . . . . . . . . . . . . 39 4.4 Contour of vertical displacement field,d z (D p ) and corresponding shearing angle field(D p ) for a random realization withCoV =12% at different form- ing time steps for d z (D p ) at (a) 0.25P f , (b) 0.5P f , (c) 0.75P f , and (d) 1.0P f and for (D p ) at (e) 0.25P f , (f) 0.5P f , (g) 0.75P f , and (h) 1.0P f , where P f is the final forming load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.5 Histogram of (a) shearing angle field(D p ) for the same realization, and (b) corresponding sheared permeability field, and (c) histogram of (D) field, representing the direction of the principal permeability tensor with respect to the warp direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.6 Contours of filling factor within the resin injection simulation at different time steps for a random realization witht f = 58.60 sec. . . . . . . . . . . . 43 4.7 Normalized first order PCE coefficients oft f ,t c , r for CoV=4% . . . . . . . 47 vii 4.8 PCE order convergence based on level 2 sparse grid quadrature for CoV=12%. The matrix plot corresponds to PDF sampled from PCE of order P=2 and P=3 for each of the predefined QoIs at the first three projection reduced subspaces, whereby each QoI along column, and projection reduced di- mension along rows. Thet f QoI of the realization used to generate previous plots that correspond to the quadrature point with highest filling time from the adapted solution for a 3-d space is marked with a star point . . . . . 49 4.9 First three rows of the change-of-basis adaptation rotationA t f ,A t c , andA r that correspond to each QoI,t f ,t c , and r . . . . . . . . . . . . . . . . . . . 50 4.10PDF using samples from 2 nd order adapted PCE for different inputCoV = 4% for QoI (a)t f , (c)t c , and (e) r , andCoV =12% for QoI (b)t f , (d)t c , and (f) r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.11Comparison of PDF obtained from the proposed adapted PCE with respect to 10 3 Latin Hypercube Samples for each QoI: (a) fill timet f , (b) cure time t c , and (c) maximum residual stress r . . . . . . . . . . . . . . . . . . . . . . 54 4.12PDF oft f based on a 3-D reduced space for second and fifth PCE orders with level two and four sparse grid quadrature, respectively. . . . . . . . . 55 5.1 Probabilistic distribution of resin filling time for different representations of the fabric permeability field . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 (a) Fabric media discretized with 2-D shell elements, (b) fabric media en- closed within truncated square pyramid molds. . . . . . . . . . . . . . . . . 74 5.3 Interpolated1 st order PCE coefficients for each stochastic dimension of the forward forming problem (a) E f 1 ;t , (b) E f 2 ;t , (c) f 1 , (d) f 2 , (e) E f 1 ;b , (f) E f 2 ;b ; PCE-based (g) mean and (h) variance of process across the fabric domain. 79 5.4 (a) PDF of local fabric shearing2cos 2 ( 4 (x) 2 ) at five random discrete points, and (b) corresponding Q-Q plots with respect to a lognormal distribution. 81 5.5 (a) Eigenvalues in log scale of the high-order, PCE-based covariance matrix, and (b) corresponding first four eigenfunctions. . . . . . . . . . . . . . . . . 82 5.6 PCE surrogate covariance with respect to (a) a random point in the lower truncated portion, (b) a random point in the upper flange; (c) autocorrela- tion along random paths with respect to a random source point. . . . . . . 83 5.7 Kernel optimization results; Kernel evaluated at optimal parameters for (a) Exponential, (b) Gaussian, (c) Rational Quadratic, (d) AR2, and (e) Mat´ ern functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.8 Error in autocorrelation function approximation for different kernels at op- timal parameters, for (a) Exponential, (b) Gaussian, (c) R. Quad, (d) Matern, and (e) AR2 models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 viii 5.9 (a) Eigenvalues in log scale for each kernel at optimal parameters, and (b) corresponding contribution of ranked eigenvalues to the total variance of the process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.10Select few eigenmodes alongD corresponding to each covariance kernel evaluated at optimal parameters. . . . . . . . . . . . . . . . . . . . . . . . . 88 6.1 High-level flow of information in a stochastic simulation. . . . . . . . . . . 96 6.2 Details of maps from standard input variables (germ) to quantities of in- terest (QoI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 A.1 Number of basis terms (log scale) in polynomial chaos expansion . . . . . 113 A.2 1d Hermite polynomials of order up to 16 evaluated using Monte Carlo samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 A.3 Quadrature points following the Gauss-Hermite rule (sparse) for a 2-D problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 ix Abstract Light-weight composites are increasingly being used in both structural as well as non- structural applications, as the benefits such composite components offer when com- pared to conventional metallic alloys are understood and applied. Composite usage spans across many industries, ranging from aircraft and spacecraft applications within the aerospace industry, to automotive applications within the transportation industry, in addition to energy, defense, and marine industries. While the usage of composites is consistently growing across industries, one challenge that needs to be solved is develop- ing consistent composite design standards. Such standards would ideally be anchored in a comprehensive assessment of manufacturing of composites, a task that is often complex, that spans across multiple physics and scales, and that presents a high level of inherent variability. We address the problem of performing integrated uncertainty quantification in mul- tiphysics and multiscale composite manufacturing models for lightweight composite ap- plications. We formulate and adopt an integrated uncertainty propagation framework, which encompasses all manufacturing phases following the RTM process, namely (1) fabric forming, (2) resin injection through the deformed fabric domain, (3) resin curing, and (4) mechanical ply distortion. The integrated framework, tailored for multiscale and multiphysics applications, is centered on polynomial chaos methods. Further- more, an adaptation scheme of the Gaussian basis of the chaos expansions is adopted such that the proposed polynomial chaos integrated UQ framework scales well with x high-dimensional composite manufacturing problem at hand. All the variables involved across different scales, physics, and manufacturing phases of the RTM process are parametrized as part of an integrated, high-dimensional, forward problem formulation. The latter is fully automated and embedded within a polynomial chaos UQ workflow with dimension reduction via an adaptation of the PCE basis; which allowed us to efficiently develop surrogate models of predefined QoI along the RTM manufacturing sequence. Proper uncertainty quantification in heterogeneous media is premised on an accurate representation of spatially varying material properties as random fields. Even though uncertainties in predictions are oftentimes sensitive to the parameters of the underly- ing material processes that characterize heterogeneous media, research methodologies for accurately characterizing such material processes are lacking. We propose, for that matter, a computational recipe for learning, via kernel functions, the correlation struc- ture of permeability fields in random fabric media within composite manufacturing. We numerically optimize a selection of covariance kernels with respect to the correlation structure of the random fabric media obtained from a higher-order Polynomial Chaos (PC) surrogate model. Moreover, for discovering the correlation structure on curved domains in space, we propose a geodesic subroutine such that geodesic distances be- tween vertices on curved surfaces are used in the kernel functions. Within a learning setting, the performance of each kernel is evaluated within the fabric forming model. We observed extreme uncertainty in the local permeability field along the sheared fabric from the PC surrogate. Even at optimal parameters, kernels only partially captured the correlation structure of the fabric media. Our last effort was devoted to proposing, within and beyond the scope of the RTM composite manufacturing application, a generic, in-house, UQ scientific workflow soft- ware, tailored for high-dimensional, mutiphysical random systems, and based on the xi aforementioned polynomial chaos and basis adaptation machinery. We provide an im- plementation of the polynomial chaos and adaptation algorithms within a light software, scalable from personal computers to HPC servers. xii Chapter 1 Introduction 1.1 Overview of Uncertainty Quantification Uncertainty is ubiquitous across all aspects of life, from engineering systems to phys- ical or biological sciences, to name a few. To a varying extent, scientists, engineers, and the general public, are increasingly understanding the importance of first under- standing and then attempting to quantify such uncertainty and also incorporating such randomess into model predictions. Our older mechanistic view of the world as a predictable, ordered machine, is be- ing shattered by evolving awareness of the central role of randomness in all aspects of life. With our ever increasing computational capabilities that allow us to acquire a deep understanding of the dynamics involved, some systems that we once though of as pre- dictable and controllable turned out to be highly unpredictable and sometimes chaotic. The task of uncertainty management in such systems with inherent unpredictability becomes particularly challenging. And the tools that we mastered in the past for fore- casting purposes might not be appropriate enough when doing uncertainty analysis for systems with high inherent uncertainty. Even though the field of Uncertainty Quantification (UQ) has been in existence for a while, a growing recent understanding across the industry spectrum of the importance 1 of incorporating uncertainty into models and predictions has resulted in a fast-growing UQ field of research, whereby constantly emerging research efforts are pushing the frontiers in UQ research and practice. Our increasing knowledge and familiarity with uncertainty has further cemented the assertion that simply disregarding the uncer- tainty in an analysis could be potentially misleading, particularly for certain critical applications, where a probabilistic uncertainty assessment becomes imperative for any serious attempt at understanding a given system and reaching robust conclusions. From a UQ perspective, factors that generate uncertainty, whether epistemic or aleatoric, in any random system, can be classified into well-known categories. Sources of uncertainty, for that matter, could be essentially classified into: (1) from a mathe- matical perspective, inability of the theoretical model to fully capture the response of a given system, (2) from a computational perspective, any simplifying assumptions in the numerical model, or discretization errors, among other (3) from an experimental per- spective, data bias, error in measuring instruments, etc. All the aforementioned factors are premised on the assumption that we have an apriori understanding of the presence of inherent uncertainty and error. In some instances, however, uncertainty may be hid- den, whereby engineers or scientists, could end up ignoring any uncertainty that they do not know of its existence in the first place. Such attempts for quantifying uncertainty, depending on the level of sophistication, can range from simple methods such as error bound analysis, to more advanced uncer- tainty assessment using elaborate statistical approaches [93]. There is ever increasing demand from agencies and practitioners for computational and stochastic models to provide accurate enough predictions that can ultimately be used as surrogates and bypass the need to explore the experimental domain from ob- servations based on the physical reality. This push partially explains why the fields of 2 UQ and computational stochastic mechanics are progressing fast, as new computation- ally efficient methods are constantly being introduced by researchers. 1.2 Light-weight Composites Applications The advantages of traditional composites have been well-understood and applied for decades, which resulted in composites permeating through a wide spectrum of indus- tries, ranging from aerospace to automotive applications. As more research and devel- opment is further invested into light-weight and ultralightweight composites, the usage of composites will likely grow as new industries take note of the benefits offered by light-weight composites, whether for structural or non-structural applications. Aerospace The aerospace industry has recognized early on the benefits of compos- ites as compared to metallic alloys. Such comparative benefits include a similar or even higher structural strength offered by composites, at a significantly lower weight; which translates among other into aircraft fuel savings and better aircraft performance. This explains the consistent widespread and growing usage of composite components, fiberglass mainly, in aircraft structures. The Boeing 787 aircraft (i.e. Dreamliner) for instance consists of around 50% of composite based materials [1, 2]. Automotive For automotive applications, over the past decades, composites compo- nents were essentially limited to non-structural and semi-structural applications [4], and essentially lagged behind the aerospace industry where adoption of composites was more widespread. That being said, recently, with the continuous shift towards ligh- weight vehicles, the automotive industry has witnesses increased usage of composite parts in structural automotive components. For light duty vehicles, the body structure consists of 23-28% of the weight [5]; the DoE has set 50% reduction in weight target 3 for vehicle body and chassis structures [6]. The automotive industry is following this trend accordingly; for instance, one commercial car manufacturer released a low-volume electric vehicle with essentially a carbon fiber body [7, 8]. For composite automotive applications, the promise of increased vehicle fuel effi- ciency associated with weight reduction, among other, has pushed the usage of compos- ites. While previously, composites components in automotive applications were mostly limited to non-structural parts, structural components are increasingly being manu- facturing using composites, and replacing conventional metallic alloys for structural parts. In [9], it was reported that for conventional internal combustion engines, up to 8% increase in fuel efficiency is associated with a 10% vehicle weight reduction. Defense & Space Spacecraft vehicles are another well-researched and promising in- dustry application of light-weight and ultra-lightweight composites components. The benefits offered by such composites in reducing the weight of spacecraft vehicles among other are beneficial for space agencies, particularly for deep space exploration [17, 18, 19]. With the consistent practice among space agencies of using lightweight composites in spacecraft vehicles, such agencies were among the first to propose solvers for the analysis of composite structures [11, 12, 13, 14, 15, 16]. Use of light-weight composites across various applications in the defense industry has also been well underlined [10]. Such light-weight composite components could potentially result in improved performance, fuel savings, among other, for different ap- plications with the defense industry. 4 1.3 Integrated Methodology Widespread adoption of composites has been adversely affected by the absence of con- sistent composite design standards across different applications [3]. Increasing growth of composites in the future is premised on the ability of the composites community to provide consistent standards for composite design. An integrated analysis framework, for that matter, would constitute a first milestone towards such consistent composite design standards and ultimately to continued increase in composite usage across many vital industries. A critical milestone towards achieving consistent composite design standards in- volves, in some form or another, performing integrated analysis of composite materials following a comprehensive methodology that takes into considerations all the factors involved in composite manufacturing. Early design philosophies were based on the premise that systems, irrespective of how inherently complex, could be analyzed by breaking down such systems into a set of components, and analyzing each component separately. However, for certain iter- connected complex systems, the behavior of the integrated system is more than the summation of the behavior of each component evaluated separately. For such com- plex, interconnected systems, a holistic analysis is required for truly understanding the response of the system. Such a holistic assessment of all the aspects involved in composite applications encompasses a multifaceted evaluation of such composite com- ponents, ranging from structural and performance criteria to economical and safety considerations. 5 With increasing computational capabilities, the cost of adopting an integrated method- ology is reduced. Moreover, more elaborate physical models that span all the underly- ing scales and physics involved in the manufacturing processes, while still being cost- efficient in providing a strong alternative to potentially costly experimental schemes. Once an integrated analysis framework is adopted, common tasks such as uncer- tainty quantification, design exploration, optimization, become more insightful and in- formative given the integrated nature of the research methodology. Moreover, consistent composite design standards are further premised on proper un- certainty management, given the high level of inherent variability in composites and the presence of several sources of uncertainty across the manufacturing process. Uncer- tainty quantification using a comprehensive framework becomes imperative. 1.4 Manuscripts This work presented herein is based on the following publications and conference pre- sentations: • Ghauch, Z., Aitharaju, V., Rodgers, W., Pasupuleti, P., Dereims, A., Ghanem, R., (2019). Integrated stochastic analysis of fiber composites manufacturing using adapted polynomial chaos expansions, Composites Part A, vol 118, pp. 179-193. • Ghauch, Z., Aitharaju, V., Rodgers, W., Pasupuleti, P., Dereims, A., Ghanem, R., (2019). Covariance learning of permeability fields in fabric media, International Journal of Material Forming. • Ghauch, Z., Ghanem, R., Tsilifis, P. (2019). Polynomial chaos workflow for uncer- tainty management in multiphysics problems, AIAA Journal. 6 • Ghauch, Z., Ghanem, R., Aitharaju, V., Rodgers, W., Papsuleti, P., Dereims, A., (2019). Comparison of multiscale and kernel-based correlations for stochastic per- meability models of composites manufacturing,34 th Annual meeting of the American Society for Composites, Atlanta, Georgia. • Ghauch, Z., Aitharaju, V., Rodgers, W., Papsuleti, P., and Dereims, A., Ghanem, R. (2018). Manufacturing to performance: a comprehensive multiscale stochastic predictive model for composites, 33 rd Annual meeting of the American Society for Composites, Seattle. • Mehrez, L., Ghauch, Z., Aitharaju, V., Rodgers, W., Papsuleti, P., and Dereims, A., Ghanem, R. (2018). Statistical machine learning and sampling for composite manufacture and performance, 33 rd Annual meeting of the American Society for Composites, Seattle. 1.5 Outline This dissertation is structured as follows. In Chapter 2, a detailed review of the physical models within each phase of the RTM composite manufacuring process is presented. The review includes a background on (1) mechanical forming of the fabric material, (2) resin flow in porous media and (3) resin curing coupled with heat transfer processes, and (4) mechanical ply distortion. These phases constitute the backbone of the popular RTM composite manufacturing process, which constitutes the focus of this work. The mathematical background of the proposed statistical research methodology for forward uncertainty propagation is presented in Chapter 3. The research methodology is centered on a combination of conventional generalized Polynomial Chaos expansions, 7 a well-known uncertainty quantification method, along with state-of-the-art dimension- ality reduction methods revolving around an adaptation of the basis of the Homogeneous chaos. The mathematical formulations of both polynomial chaos and basis adaptation are reviewed. In Chapter 4, a detailed formulation of the forward RTM composite manufacturing problem is provided. This includes a description of the comprehensive parametrization of the forward integrated problem, encompassing all the aforementioned, interconnected phases of the RTM process. Within the multiscale and multiphysics formulation of the RTM problem, the computational models at evey scale and for every physical process of the manufacturing process are described in detail Once the integrated forward RTM problem is established, we shift our attention in Chapter 5 to understanding how forming induced deformations affect subsequent resin flow in random heterogeneous media of the deformed fabric. For that matter, we formu- late a research framework tailored for characterizing random fields in deformed media and use the proposed framework for better understanding the impact of forming defor- mations on fabric permeability fields. Finally, in Chapter 6, an in-house scientific UQ workflow, based on adapted poly- nomial chaos, is presented. A thorough description of the features of the proposed UQ software is provided, including complexity and error analysis, computational effi- ciency within multiphysical forward systems, advanced variable mapping, and efficient quadrature allocation within multiphysical forward random problems. 8 Chapter 2 Composite Manufacturing Models We present, in this chapter, a review of the physical processes relevant to composite manufacturing that are part of our integrated computational modeling and uncertainty analysis framework. Focus is placed on the popular RTM composite manufacturing methodology, and a formulation for each physical process within the RTM method is presented, namely: (1) dry fabric forming, (2) resin injection within deformed fabric, (3) resin curing, and (4) ply distortion. Given our integrated approach for uncertainty propagation through the aforementioned four physical processes of the RTM method, a thorough review of the RTM mathematical formulation through this chapter provides a clear perception of the workflow of the integrated RTM problem. In other words, this chapter should provide a clear understanding, for instance, of how different differential equations are coupled together, or connected sequentially, or how scalar or vector field solutions from one differential equation affect subsequent models. The comprehensive mathematical review in this chapter would ideally set the integrated tone of our research approach, whereby the reader will be able to understand the mathematical context of all the random variables of interest that we use subsequently for forward uncertainty propagation and sensitivity analysis, among other. In this chapter, a review of the aforementioned four constituent physical processes of the forward RTM problem is presented, consisting essentially of mechanical forming 9 of the dry fabric, resin injection through the deformed fabric domain, resin curing, and finally mechanical ply distortion and residual stress generation. 2.1 Mechanical Forming In an initial pre-injection phase, the fabric is subjected to a draping load within a mold enclosure following a predefined configuration. The deformation of the fabric within the enclosure depends, among other, on the micro-mechanical constitutive properties of the fibers. Several FE formulations of the dry fabric draping problem have been proposed, with varying levels on complexity. In [34, 35], an FE formulation of dry fabric draping was proposed whereby a Representative Volume Element (RVE) was defined to couple the microscale fabric structure to the macroscale domain. Such homogenization is needed since the fabric does not form a continuum, and as such, classical continuum mechan- ics approaches are not applicable. The adopted RVE for multiscale coupling consists of a four-node shell element coupled with five beam elements that represent the warp and weft fibers and their interaction. In a different approach, the homogenized behavior of a continuum ply is considered without explicitly modeling the microscale properties. This FE framework for solving the multiscale draping problem was adopted in the current work. Figure 2.1a shows the definition of warp and weft fiber directions,! 1 (e) and! 2 (e) for an arbitrary elemente2D p e , whereD p denotes the physical domain occupied by the material during the draping process. An important measure of local deformation associated with draping is the shearing angle, denoted by (x) within the domainD p . Figure 2.1b illustrates the definitions of fiber orientation and shearing angle at a pointx2D p for a woven isotropic fabric. 10 Preform deformation during draping is approximated as the deformations of an ef- fective, homogenized shell structure. The local elasticity tensor for this shell varies in space, reflecting both heterogeneity of spatial fiber layout within the fabric and irregu- larities in fiber geometries due to the natural waviness of fiber bundles. The principal components of the sheared permeability tensor K (s) f in the filling stage inherit perturbations to their spatial distribution from the preform draping deformation. Several models for predicting the permeability tensor of sheared fabrics can be found in the literature. One model [36, 37, 38, 39] relates the principal values of the in-plane sheared permeability tensor K (s) f to those of the unsheared tensor K (0) f and shear angle as follows 0 B B B B B B B B B B @ K (s) 1 (x) K (s) 2 (x) 1 C C C C C C C C C C A = 0 B B B B B B B B B B @ K (0) 1 (x) K (0) 2 (x) 1 C C C C C C C C C C A 2 0 B B B B B B B B B B @ cos 2 (45(x)=2) cos 2 (45+(x)=2) 1 C C C C C C C C C C A : (2.1) When a fabric elemente is sheared by an angle(e) with respect to the weft, the direction of the principal permeability tensor is rotated by(e) with respect to the warp direction, as shown in Figure 2.1b. For an isotropic woven fabric, the principal direction (e) of the sheared permeability tensor can be expressed as (e)= 0 (e)=2; e2D p , where 0 represents the direction of the unsheared permeability tensor [92]. 2.2 Flow in Porous Medium LetD denote the spatial domain occupied by the fabric following the draping process. This is the domain through which resin injection, curing, and distortion take place. In the RTM filling step, a pressure difference is created such that resin is injected and flows through a fibrous reinforcement preform that acts as a porous medium. Resin flow can be modeled by Darcy’s law, wherein the resin velocityV r is governed by the permeability tensor K (0) f (x) of the porous medium (i.e. unsheared fabric), viscosity r of the resin and 11 Figure 2.1: (a) Definition of fiber directions for a woven isotropic fabric where ! 1 (e) and! 2 (e) describe the warp and weft directions within the aforementioned RVE-based multi-scale coupling approach for a given homogenized shell element e within the set D p e , and! 0 1 (e) and! 0 2 (e) represent the distorted warp and weft directions, each perturbed by 1 (e); 2 (e), respectively. (b) local shearing of fabric and corresponding permeability tensor. When an element e in the fabric domainD p e is sheared by an angle (e) with respect to weft directione 2 , principal components of permeability tensorK 1 (e) andK 2 (e) rotated by(e) with respect to warp directione 1 . pressure gradientrp(x). For an unsheared fabric, the corresponding Darcy’s equation over a domainD is given by V r (x)= K (0) f (x) r rp(x); x2D: (2.2) Assuming the resin fluid to be incompressible, the continuity condition yields the dif- ferential equationr:V r (x)=0 over the domainD, supplemented by suitable conditions along the boundary @D. The dependence of the resin viscosity on temperature is de- scribed through a constitutive law. In the present work an exponential model of the form r =C 1 e C 2 T is used, whereC 1 andC 2 are fitting parameters that can be parametrized as detailed later. 12 2.3 Resin Curing Following the injection phase, the resin gradually solidifies in a curing stage whereby the polymer changes from liquid to rubbery to glassy regimes with corresponding mod- ifications in thermal, chemical, and mechanical properties. The kinetics of resin poly- merization, for a resin withN components, can be described using the Kamal-Sourour model [40]. For each component j, c j (x) and dc j (x)=dt, which represent the degree and rate of cure, respectively, at a pointx2D, can be expressed as dc j (x) dt =K j (x)c j (x) m j (Bc j (x)) n j ; j =1;:::;N; x2D (2.3) subject to the condition, c(x)= N X j=1 Q j c j (x); x2D; (2.4) wherem j andn j represent model constants describing the sensitivity of the autocatalytic reaction, and Q j designates the coefficient corresponding to the j th component. The quantitiesK j are defined via Arrhenius law, K j (x)=A j e E j =(RT(x)) j =1;:::;N; x2D (2.5) with prefactor A j and activation energy E j . Further, R denotes the universal gas con- stant, andT(x) is the temperature at a pointx2D. 2.4 Heat Transfer Resin flow and curing are strongly dependent on the heat transfer processes between different components. The viscosity of the resin and the rate of resin polymerization in the curing stage are both temperature-dependent. 13 Mathematical models of heat transfer include three thermal processes: first, heat transfer by convection during the resin flow stage, second, heat transfer by conduction between the resin and fibrous reinforcement, and third, heat generation via exothermic resin polymerization. The governing equation of the temperature field T(x) subject to these three thermal processes can be expressed as C p @T(x;t) @t + r C p;r V r (x;t):rT(x;t)=r:[krT(x;t)] r h @c(x;t) @t ; x2D (2.6) where is the density of the composite, evaluated using the rule of mixtures from the resin and fabric densities, C p is the composite specific heat, also computed using the rule of mixtures based on the resin and fabric specific heat, r is the resin density, C p;r is the resin specific heat, V r (x;t) is Darcy’s velocity of the resin, k is the thermal conductivity tensor of the composite, evaluated using the rule of mixtures based on the thermal conductivity tensors of the resin and fabric components, h is the total enthalpy for resin polymerization, and @c(x;t) @t is the resin curing rate. Equation (2.6) is subject to temperature (T(x;t) =T 0 ), heat flux (n:krT(x;t) =q(x;t)), and heat convection boundary conditions (n:krT(x;t)=h[T(x;t)T 1 ]), where n denotes the surface vector at the boundaries. 2.5 Mechanical Distortion & Residual Stresses Residual stresses associated with the manufacturing process of composites have an impact on the performance and quality of the finished composite product. It follows that an accurate quantification of stresses following the curing stage is an essential part of building composite performance prediction models. 14 For a linear isotropic viscoelastic material following the standard linear solid model, the stress-strain relationship can be expressed using the relaxation modulusE(t) in the form (t)= Z t 0 E(t) @" @ d ; (2.7) where the relaxation modulus E(t) can be decomposed into instantaneous and time- dependent parts. Equation (2.7) can be reformulated in a three-dimensional setting, for x2D as (x;t)= 8 > > > > > < > > > > > : C r "+ Z t t (C g C r ) @" @ d forT(x)<T g (c(x)); C r " forT(x)T g (c(x)); (2.8) where C r and C g represent the stiffness matrices under rubbery and glassy conditions, respectively, and T g is the glass transition temperature, function of the degree of cure c(x) at point x2D [106]. The glass transition temperature was defined following Di Benedetto’s equation T g (x)T g0 T g1 T g0 = c(x) 1(1)c(x) ; (2.9) whereT g1 andT g0 represent the glass transition temperature for a completely cured and uncured material, respectively [105]. Models for residual stresses generated after curing of the constrained composite can be traced, among other, to incremental strains, ", generated both by thermal expan- sion and chemical shrinkage under glassy and rubbery conditions. Shrinkage occurs, during curing, from chemical shrinkage associated with cross-linking (for thermoset composites) as well as from thermal shrinkage associated with the drop from cure to room temperature. The level of thermal shrinkage is controlled, in addition to temper- ature gradient, by the coefficient of thermal expansion matrix. The latter has different principal components under rubbery and glassy conditions, denoted respectively by r;ply and g;ply , respectively. Similarly, the level of chemical shrinkage is controlled, by 15 the curing gradient, and by the coefficient of chemical shrinkage matrix. The latter also has different principal components under rubbery and glassy conditions, denoted by r;ply and g;ply . No stresses are assumed to occur as long as the resin is in liquid state. Following resin polymerization in the curing stage, the homogenized ply is subject to constraints along the boundary@D of the domain. Such constraints generate residual stress fields throughout the domainD. The residual stress tensors generated along the homogenized ply in the distortion phase were initially computed in the principal material direction, and subsequently transformed based on the fiber angle distortion fields 1 (x) and 2 (x) into global coordinates. At the end of the curing stage, upon demolding, residual stresses across the macro-scale domain are released, thereby inducing shape distortion. 16 Chapter 3 Polynomial Chaos & Dimension Reduction A review of the polynomial chaos methodology, which constitutes the backbone of the stochastic surrogate model used in our analysis, together with the adaptation proce- dure used to enable these expansions to be reformulated in a lower-dimensional sub- space are provided in this chapter. The use of polynomial chaos methods for uncer- tainty quantification purposes is not new, and we therefore only provide a brief review of polynomial chaos theory. Focus is placed on the mathematical formulation of the adaptation scheme of the basis of the PCE, which essentially consists of projecting a high-dimensional model into a low dimensional subspace that captures the dominant statistical content with respect to a given QoI. Through this chapter, the reader should form a clear understanding of the proposed statistical method, centered on an adap- tation of the Gaussian germs of the generalized Polynomial Chaos Expansions. The potential implication of this polynomial chaos dimensionality reduction metric are only discussed in the next chapter, within the context of addressing the curse of dimension- ality problem that conventional polynomial chaos methods suffer from when applied in high-dimensional settings. 17 3.1 Homogeneous Chaos Theory Consider a random variableX defined on a probability space (;F;P), consisting of a probability measureP , an appropriate-algebraF , and a sample space. The expec- tation ofX can be expressed as E[X]:= Z X()P()dP(): (3.1) Consider a Hilbert spaceL 2 (;F;P) defined as follows L 2 (;F;P)=fX :!R: Z jXj 2 dP()g; (3.2) and a linear subspaceGL 2 (;F;P), referred to as a Gaussian linear space, consisting of standard normal Gaussian random variables. The closure ofG inL 2 (;F;P), denoted byG, is a Gaussian Hilbert space. Let P n (G) denote a linear subspace of L 2 (;F;P), defined for n 0. For a d-variate polynomialp of degree up ton,P n (G) can be formulated as P n (G)=p( 1 ;:::; d ); d2N; (3.3) wheref i g d i=1 . The spacesH n can be thus defined as follows H n =P n (G)\P n1 (G) ? ; (3.4) whereH 0 =P 0 (G). Thus, we have d M 0 H k =P n (G); (3.5) 18 where L denotes the orthogonal summation of linear spaces. The spacesfG n g 1 n=0 form a sequence of closed, pairwise orthogonal linear subspaces ofL 2 (;F;P) such that 1 M 0 G n =L 2 (;(G);P); (3.6) which is known as the Cameron-Martin theorem [104]. 3.2 Polynomial Chaos Methods Polynomial Chaos Expansions (PCE) represent a well established area of UQ that has received considerable attention starting with the original formulation of Homogeneous Chaos [41] to formulation in computational science [42, 43] and several applications across science and engineering [52, 44, 45]. In the following we present a brief review of the steps involved in a PCE construction for equations with stochastic parameters. LetQ represent a scalar Quantity of Interest (QoI) that can be expressed as a func- tion of ad-dimensional vector=f 1 ;:::; d g of uncorrelated standard Gaussian random variables. ExpressingQ as a function of in the formQ=q(), and representingq() in an orthogonal polynomial expansion with respect to results in, Q=q()= 1 X jj=0 q (); (3.7) where =f i g d i=1 are multi-indices with modulusjj = 1 +:::+ d , andf g represent normalized multivariate Hermite polynomials that can be expressed in terms of their univariate counterparts using the following notation, ()= d Y j=1 j ( j )= d Y j=1 h j ( j ) p j ! : (3.8) 19 Here, h j represents the one-dimensional Hermite polynomial of order j , as shown in table 3.1. The corresponding Polynomial Chaos of order j can be expressed as h j ()=(1) j @ j @ e T 2 : (3.9) The collection of the multivariate polynomials forms an orthogonal set with respect to the multivariate Gaussian density function. Table 3.2 shows the correspondence of the Wiener-Askey polynomial chaos with the type of random variables. Equation (3.7), known as the PCE of Q, can be truncated in a number of ways to provide an approximation of Q. Truncation, for instance, at polynomials of order P results in the approximation q()q P ()= X 2I d P q (); (3.10) whereI d P =fjj Pg represents the set of indices associated with the ddimensional polynomials of order at most equal toP . Other truncation schemes are also possible by limiting to any specified indexing setI d . The truncated form in equation (3.10) is an expression withN q basis terms where N q = d+P P ! : (3.11) Given the orthnormality off g, the common statistical moments ofQ can be expressed as E[Q]=q 0 ; Var[Q]= X 2I q 2 ; and Cov[Q 1 Q 2 ]= X 2I q 1; q 2; ; (3.12) 20 whereq 1; andq 2; refer to the coefficients in the expansion of two quantities of interest Q 1 andQ 2 , respectively. In addition, the coefficients in the expansion can be expressed as, q =E[Q ]; 2I ; (3.13) which involves a multidimensional integral over the d-dimensional space with respect to the Gaussian density function. Typically, this integral is approximated using a nu- merical quadrature resulting in the expression, q nq X r=1 q( (r) ) ( (r) )w r ; 2I ; (3.14) where the summation is over the nq points in the quadrature rule, (r) and w r denote, respectively, the r th coordinate and the associated weight in that rule. The numerical accuracy of the approximation of the integral shown in equation (3.13) is dictated by the quadrature level, whereby higher levels generally increase the fidelity of the surrogate PCE model. Clearly, the efficiency of the PCE approximation depends on the quadrature rule adopted. Efficient sparse grid quadrature rules have been developed specifically to this purpose [46]. In particular, a Smolyak quadrature rule is used in this study. Irrespective of the sparse grid quadrature rule adopted, its computational cost increases significantly with dimension. This motivates the development in the following section where, prior to numerical quadrature, a procedure for dimension reduction through basis adaptation is described. 3.3 Homogeneous Chaos Basis Adaptation The basis adaptation scheme consists of applying a rotation to the initial random vari- ables, followed by a polynomial expansion of a particular QoI with respect to the first few 21 Homogeneous i th Hermite Polynomial Chaos Order j Chaosh j 0 1 1 1 2 2 1 -1 3 3 1 - 3 1 4 4 1 - 6 2 1 + 3 5 5 1 - 10 3 1 + 15 1 6 6 1 - 15 4 1 + 45 2 1 - 15 7 7 1 - 21 5 1 + 105 3 1 - 105 1 Table 3.1: First seven one-dimensional Hermite Polynomial Chaos Random Variable Wiener-Askey Chaos () Domain Gaussian Hermite chaos (1,1) Gamma Laguerre chaos [0,1) Uniform Legendre chaos [a,b] Beta Jacobi chaos [a, b] Table 3.2: Correspondance of the continuous random variables with the Wiener-Askey polynomial chaos [52] rotated variables [33]. A key aspect of the methodology involves evaluating a rotation matrix that permits substantial concentration of the uncertainty around the first few variables. A methodology, expanded in this subsection, which evaluates this rotation matrix for each QoI and which capitalizes on the QoI being a polynomial in Gaussian variables has proven to be robust with wide range of applicability. Thus we introduce a rotation A:R d 7!R d such that, =A: (3.15) Given the Gaussian distribution of, is also standard Gaussian, and the QoIQ can be thus expanded with respect to either or. For the change-of-basis scheme, we define q A () as the representation of Q in terms of. Thus we have q A ()=q() or, for a PCE with a polynomial orderP , X jjP q A ()= X jjP q (); (3.16) 22 thus q A = X jjP q h ; A i;jjP; (3.17) where A ()= (A) andhu;vi denotes the inner product of u and v as functions of which is equal to the correlation betweenu andv. Denoting byV I the span off A g 2I , the projection of Q onV I can be expressed as q A;I ()= X 2I X 2I P < ; A > (): (3.18) A number of procedures have been proposed for the construction of the rotation A [33, 47]. In this study, we use the so-called Gaussian adaptation procedure in light of its simplicity. Using the Gaussian components (i.e. the components of the 1 st order approximation) as the first row of a matrix ˆ A, this matrix is completed by placing 1 on the diagonals and zeros elsewhere. The rotation matrix A is then obtained from ˆ A through a Gram-Schmidt orthogonalization scheme that keeps the first row invariant. Another procedure for completing ˆ A consists of first ranking the random variables according to the magnitude of the coefficients in the linear PCE for the QoI. Call these ranked variables ˆ . The first row of ˆ A is then constructed as before, featuring the coef- ficients in the linear expansion. The second row and beyond, however, thei th row (i>1) has zeros everywhere except at the location associated with the ˆ i1 . A Gram-Schmidt procedure is then applied to transform ˆ A into a rotation matrix A, keeping its first row invariant. Once matrix A has been constructed, we can write, q A ()=q A 0 +q A 1 1 + X 2I 1<jjP q A () + X <I 1<jjP q A (); (3.19) 23 and q A;I ()=q A 0 + X 2I q A (); (3.20) where 1 is the first component of vector. The last term in equation (3.19) represents the error associated with the rotation and projection scheme represented in equation (3.20). Typically, the projectionI in (3.20) is designed to yield a high-order polynomial representation of the first few components of. The procedure developed above relies on the basic random variables being Gaus- sian. For the important situations where this is not the case, a mapping from the distribution of the variables to the Gaussian distribution is first implemented, thus transforming all the input variables into Gaussians. In the numerical examples detailed subsequently in the study, the inverse CDF method is adopted to map independent beta random variables into corresponding independent Gaussian variables. 24 Chapter 4 Integrated Forward RTM Problem The manufacturing of fiber composite materials involves a set of complex, intercon- nected processes that span across multiple physics and scales. The characterization of uncertainty in composite manufacturing predictions is a challenging task that involves high-dimensional, multiscale, multiphysics stochastic models. We demonstrate, in this chapter, the use of a basis adaptation scheme within a polynomial chaos representation that permits the incorporation of a large number of stochastic variables in the analy- sis. We use the proposed PCE-based workflow to analyze the interplay of uncertainty through all the fiber composite manufacturing stages that comprise the Resin Transfer Molding (RTM) process. The proposed framework is centered on an integrated assess- ment of uncertainty in composite structures using probabilistic surrogate models for predefined QoI. 4.1 Background Over the past decades, the aerospace and automotive industries have spearheaded and advanced the use of composites for both structural and non-structural applications. Depending on the application, benefits offered by composite components, as compared to metallic parts, include weight reduction, higher fatigue life and durability, lower main- tenance cost, higher corrosion resistance and thermal stability, to cite only a few, are 25 well known and documented in the literature [20]. As the scientific principles explaining these advantages and clarifying their limitations are better understood, an increasing number of critical structural applications are relying on composites. One popular com- posite manufacturing technique is Liquid Composite Molding (LCM), which constitutes a superclass of Resin Transfer Molding (RTM), the focus of this study. The RTM process consists of a sequence of manufacturing steps, starting with an initial textile draping step that precedes resin injection, whereby the draped textile is enclosed in a mold and liquid resin is injected. The filling step is followed by a curing of the resin, where the latter is subject to increased heat and pressure to promote curing and consolidation. Predictability of the RTM process is particularly challenging as it involves a sequence of complex, inter-dependent processes that span a multitude of spatial and temporal scales. Further, several of the physical phenomena involved in RTM exhibit nonlinear behavior either in their equations of state or as a consequence of large deformation. More specifically the RTM problem, as considered in the present study, involves the fol- lowing physical processes: (1) the mechanical deformation of the fabric preform in the draping stage, (2) resin flow in porous media with heat transfer and chemical curing, and (3) viscoelastic relaxation of residual stress. The RTM problem is further complicated by the large number of parameters corresponding to all the relevant physical models, thus introducing significant uncertainty in the associated predictions. In light of the complex nature of the problem, this uncertainty is manifested in both statistical vari- ations of the parameters and in the form of modeling errors, and is typically modeled by embedding the problem into a parametric space of sufficiently high dimension, thus further exacerbating the computational burden. Finally, the complexity of the RTM problem is further exacerbated by the expense of securing experimental observations for calibration, verification, and validation. 26 The above challenges to the predictability of the composites manufacturing process have clear implications on the reliability of the performance of structural components made of these composites. Modeling and simulation (M&S) provides an alternative framework for exploring design and parameter space more efficiently and economically, thus significantly accelerating the time-to-insertion of new technologies. While the M&S framework is conceptually simple, it is challenged by a potentially prohibitive computa- tional burden, in particular when parametric and modeling uncertainties are accounted for in the design optimization step. The fabric preform usually undergoes large deformations in the initial draping stage. These deformations of the textile alter, to varying extents, certain properties such as fiber orientation and volume fraction, thus affecting the remainder of the manufacturing process. It is well documented that the properties and behavior of the finished composite structure is sensitive to the pre-injection manufacturing deformations. This underlines the need for an integrated framework for the analysis of RTM-based composite materials. Numerical procedures are promising in this context, as they provide the option to explore design performance prior to committing particular manufacturing schedules. Previous studies have tackled the RTM modeling problem both deterministically [21, 22, 23] and from a probabilistic approach. In the latter, some studies were focused on the experimental assessment of the variability in flow properties in porous media [24, 25]. Other studies examined the RTM problem from a stochastic perspective using Monte Carlo-based methods [26, 27, 28, 29, 30] and polynomial chaos (PC) methods [31]. From a physical perspective, the essential sources of uncertainty in the RTM process are associated with either material or process boundary conditions [32]. For the former, uncertainty emanates from a spectrum of pre-manufacturing processes including pro- duction, storage, transportation of both preform, or dry textiles, or pre-pregs as well as 27 resin materials. Additional sources of uncertainty emerge once the RTM problem is ex- amined from a computational perspective whereby additional uncertainty is introduced once physical models are transformed into computational models that involve assump- tions, computational domain discretizations, as well as upscaling and homogenization of constitutive material properties across scales. A comprehensive, integrated stochastic methodology for propagating uncertainty across a series of multiphysics/multiscale within a high-dimensional setting is lacking, and is the main focus of the present study. Uncertainty in constitutive material properties is found across all scales, and the quantification of uncertainty in such systems needs to consider material uncertainty across scales. We considered, in this work, all successive stages of the RTM process, namely dry fabric forming, resin injection, thermo-chemical curing, and non-linear mechanical ply distortion. To address the aforementioned technical challenges, we work in parallel on two main axes. First, the proposed framework for uncertainty propagation through the RTM pro- cess involves a sequence of physical models. Some manufacturing computational mod- els are oftentimes computationally intensive, particularly for high-fidelity predictions. In this regard, previous studies were essentially limited to a single physical model, most commonly the resin injection step, while ignoring the remaining steps in the RTM pro- cess. This reduction has significant implications on the fidelity of the predictions, es- pecially on the structural performance, where the nucleation of damage is predicated on characterizing manufacturing-induced fluctuations and defects in the materials. We develop in this study the workflow required to establish an integrated framework, in- cluding numerical models across all RTM stages, communication of deterministic and statistical parameters and I/O across scales and between physical models, in such a way so that the RTM workflow could be combined with non-intrusive forward propagation techniques. 28 Our second effort centers on tackling the curse of dimensionality that is critical for a credible uncertainty quantification approach. We address this issue through a basis adaptation scheme that builds on the polynomial chaos decomposition [33]. The dimen- sionality reduction is based on identifying a lower dimensional polynomial expansion that captures the dominant statistical content of particular quantities of interest (QoI). This reduction is achieved by projecting the original high-dimensional model onto a sub- space chosen so as to satisfy certain criteria, to be elucidated in the study. This reduced representation enables a reformulation of the problem into a lower dimensional one, with controlled accuracy. This model reduction has clear implications on design opti- mization under uncertainty. In order to verify the robustness of the proposed stochastic reduction and workflow, our forward model was based on ranges of statistical scatter of the RTM random input that could somewhat be higher than experimental observations. The tangible implications of the proposed integrated framework for the stochastic analysis and optimization of composite components are numerous. In particular, an in- tegrated framework that incorporates uncertainty across scales and physics constitutes a milestone towards a rigorous design methodology, rigorous standards and guidelines for the design of composites that would result in a more predictable performance that takes uncertainty into consideration. The ability to accurately incorporate uncertainty from micro-to-meso-to-macro scales and across all physical models of composite man- ufacturing and propagate the uncertainty forwards leads to more confident, precise predictions. Increased confidence in predictions of composite design provides a rational for the reduction and rigorous management of various factors of safety that permeate the design process. 29 4.2 Stochastic RTM Models 4.2.1 Computational Model In the initial forming stage, a homogenized preform fabric consisting of fibers aligned in two orthogonal directions is enclosed within truncated pyramid molds. The homog- enized ply inD p is modeled using 6,618 shell elements, denoted by setD p e , while the mold parts are assumed to be rigid. The upper mold moves downwards at a constant draping velocityv d forcing the preform fabric into the cavity of the lower mold and thus giving the preform fabric a truncated pyramid topology. The laminae of the woven fabric initially have isotropic properties. Subsequent to fabric draping, resin injection and curing phases follow. For this step, a 3-D model of a single-ply, truncated square pyramid, denoted byD, is developed. Let D f e denote a discretization of the truncated square model in the filling phase into 82,896 three-dimensional tetrahedral elements. Radial basis functions are used to interpolate the shearing angle field from the coarser forming meshD p e onto the finer resin injection computational domainD f e . The computational model of this study is composed of a single-ply woven fabric draped over a truncated square pyramid geometry. Figure 4.1 shows the geometry of the truncated square pyramid model with central resin injection. Outputs from filling and curing simulations, including spatial temperature fields T and spatial degree-of- cure fieldsc computed at all nodal locations and stored in external files. The distortion analysis phase follows, where the truncated domainD was discretized with hexahe- dral elements, denoted by setD d e . Residual stresses across the domain are generated based on temperatureT and degree-of-curec fields interpolated from resin injection dis- cretizationD f e onto hexahedral elementsD d e . The PAM-COMPOSITES™computational environment for draping, resin injection, curing, and distortion phases was used [92]. 30 Figure 4.1: Truncated square pyramid model geometry of the woven fabric domain. 4.2.2 Parametrization of Random Input (74d) Let 1 , 2 , 3 , and 4 denote the sets of random input parameters associated with pre- form draping, resin injection, curing, and distortion stages, respectively. Table 4.1 lists the specific random variables associated with each stage together with specifics of their probabilistic parameters for 1 through 4 . Figure 4.2 shows a diagram of the parametrization scheme of the RTM workflow and the corresponding allocation of the stochastic input. Input to the forward physical model at any stage along the manufac- turing process is dependent, explicitly or implicitly, on all previous random inputs up to and including the current stage. In other words, the forward model at the k th stage in the manufacturing sequence is dependent on random input from 1 k . Beta distri- butions, with shape parameters i and i and support [L i ;U i ], are used for the random inputs such that Y i B(L i ;U i ; i ; i ) forY i 2f 1 4 g : (4.1) 31 The ability of the Beta distribution to shape the probability non-uniformly over a bounded domain was deemed consistent with expert opinion and its interpretation of experimen- tal scatter of the uncertain variables. Indeed, the bounded support property eliminates otherwise spurious simulations associated with negative tails (e.g. Gaussian models) while tapering of the density function around the edges of the support is also consistent with experimental observations. Two problems are considered; in the first problem, all variables have a coefficient of variation (CoV) equal to 4% except for the second activation energy of the curing model (E 2 ) and the boundary temperature at top/bottom of the mold T bt for which a CoV of 2% is specified. In the second problem, all coefficients have a CoV of 12% except forE 2 and T bt which again have a CoV of 2%. The lower CoV on these two random variables is required given the very large sensitivity, discovered following an extensive parametric study, of the numerical solution to their values. For the distortion phase, we considered uncertainty in both the microscale resin/- fiber properties as well as the homogenized, upscaled, ply-level constitutive properties. The homogenized properties were computed using an upscaling algorithm internal to the distortion solver. The multiscale character of the RTM physics reflects the signifi- cance of the elastic properties and geometric configuration of fibers and the orientation of plies on the eventual distribution of residual stresses. This character is captured in our formulation as we develop a functional form expressing the explicit dependence of residual stresses on all these properties of the forming phase as well as on all other properties influencing the solution of the governing equations. The rationale for introducing uncertainty at both microscale (resin/fiber) and up- scaled homogenized properties is justified as follows: (1) the uncertainty in the micro- scale parameters (resin/fiber) is associated with uncertainty in mechanical properties 32 Parameters Probabilistic Model Parameters Probabilistic Model β(59.904, 80.096, 6, 6) β(55.882, 74.718, 6, 6) β(59.904, 80.096, 6, 6) β(55.882, 74.718, 6, 6) β(42.789, 57.211, 6, 6) β(6.350, 8.490, 6, 6) β(42.789, 57.211, 6, 6) β(0.022, 0.030, 6, 6) β(41.755, 48.245, 6, 6) β(0.236, 0.316, 6, 6) β(131.106, 138.894, 6, 6) β(0.236, 0.316, 6, 6) β(0.446, 0.554, 6, 6) β(2.696, 3.604, 6, 6) β(1.712, 2.288, 6, 6) β(2.276, 3.044, 6, 6) β(0.428, 0.572, 6, 6) β(2.276, 3.044, 6, 6) β(0.428, 0.572, 6, 6) β(1.780, 2.380, 6, 6) β(607.602, 812.398, 6, 6) β(1.780, 2.380, 6, 6) β(1454.823, 1945.177, 6, 6) β(3.420, 4.580, 6, 6) β(0.094, 0.126, 6, 6) β(0.0010, 0.0013, 6, 6) β(0.094, 0.126, 6, 6) β(0.0010, 0.0013, 6, 6) β(0.094, 0.126, 6, 6) β(0.0133, 0.0178, 6, 6) β(1031.212, 1378.788, 6, 6) β(53.058, 70.942, 6, 6) β(941.356, 1258.644, 6, 6) β(53.058, 70.942, 6, 6) β(285.102, 381.198, 6, 6) β(1.968, 2.632, 6, 6) β(322.727, 431.503, 6, 6) β(0.009, 0.011, 6, 6) β(86.712, 115.938, 6, 6) β(0.424, 0.500, 6, 6) β(1.583, 2.117, 6, 6) β(0.424, 0.500, 6, 6) β(1.583, 2.117, 6, 6) β(33.589, 44.911,6 , 6) β(1.734, 1.986, 6, 6) β(34.180, 45.700, 6, 6) β(1.734, 1.986, 6, 6) β(34.180, 45.700, 6, 6) β(0.337, 0.451, 6, 6) β(1.095, 1.465, 6, 6) β(0.0642, 0.0858, 6, 6) β(1.095, 1.465, 6, 6) β(0.101, 0.135, 6, 6) β(1.155, 1.545, 6, 6) β(2.901, 3.879, 6, 6) β(0.0003, 0.0004, 6, 6) m β(0.391, 0.523, 6, 6) β(0.0003, 0.0004, 6, 6) n β(1.027, 1.373, 6, 6) β(0.0201, 0.0269, 6, 6) B β(0.992, 1.000, 5, 2) β(0.599, 0.801, 6, 6) β(3971.451, 4268.549, 6, 6) β(190.967, 255.333, 6, 6) β(7755.197, 7924.803, 6, 6) β(331.588, 443.352, 6, 6) β(400.243, 406.057, 6, 6) λ β(0.757, 1.012, 6, 6) h (J/kg) β(299.522, 400.478, 6, 6) Γ 1 (Draping)) E f1,t (GPa) Γ 4 (Distortion) E 11,g,ply (GPa) E f2,t (GPa) E 22,g,ply (GPa) E f1,b (Mpa) E 33.g,ply (GPa) E f2,b (MPa) ν 12,g,ply θ f1 (deg)) ν 13,g,ply θ f2 (deg)) ν 23,g,ply Γ 2 (Filling)) V f G 12,g,ply (GPa) k f,1 (W/m.K) G 12,g,ply (GPa) k f,2 (W/m.K) G 23,g,ply (GPa) k f,3 (W/m.K) α 11,g,ply (K -1 )(E-06) C p,f (J/kg).K) α 22,g,ply (K -1 )(E-06) ρ f (kg/m 3 ) α 33,g,ply (K -1 )(E-05) k r,1 (W/m.K) Φ 11,g,ply k r,2 (W/m.K) Φ 22,g,ply k r,3 (W/m.K) Φ 33,g,ply C p,r (J/kg).K) E 11,r,ply (GPa) ρ r (kg/m 3 ) E 22,r,ply (GPa) T r (C) E 33.r,ply (GPa) P I (Kpa) ν 12,r,ply P V (Pa) ν 13,r,ply K rt,1 (m 2 ) E-08 ν 23,r,ply K rt,2 (m 2 ) E-08 G 12,r,ply (GPa) δ K1 (m 2 ) E-11 G 13,r,ply (GPa) δ K2 (m 2 ) E-11 G 23,r,ply (GPa) c 1 α 11,r,ply (K -1 )(E-06) c 2 α 22,r,ply (K -1 )(E-06) Γ 3 (Curing)) A 1 α 33,r,ply (K -1 )(E-05) A 2 (E+06) Φ 11,r,ply Φ 22,r,ply Φ 33,r,ply α g E 1 T g0 (K) E 2 T g ∞ (K) T top/bot (K) Table 4.1: Probabilistic models of random variables from draping, injection, curing, and distortion, corresponding to sets 1 ; 2 ; 3 ;and 4 , respectively 33 Figure 4.2: Illustration of the parametrization scheme of the manufacturing problem, where(x) represent the shear angle field,c(x) the curing field, andT(x) the temperature field along the domainD. of constituents, (2) the uncertainty in the homogenized distortion properties is associ- ated with error in the upscaling model used for homogenization. As such, the set of random inputs 4 for the distortion phase can be decomposed into a subset of material input properties that are upscaled from lower scales via a homogenization scheme in addition to material and process input parameters that are introduced at the distortion phase and are not part of the upscaling algorithm. Although, clearly, the upscaled pa- rameters also depend on microscale properties, we presently assume that the statistical fluctuations in properties of the micro-constituents and of the homogenized systems are independent. The mean values for the forward model (i.e. the forward model whereby all 74d ran- dom input take the expected value) for resin injection and curing phases were calibrated against experimental measurements of injection time t f and curing time t c conducted 34 at GM R&D. The propagation of uncertainty through the manufacturing sequence was performed using an in-house toolbox for UQ centered on PCE and basis adaptation. 4.2.3 Random Fiber Orientation Fields Using the aforementioned RVE-based mutliscaling approach, the microscale warp and weft fiber directions are parametrized using two independent spatial stochastic pro- cesses, with fluctuations within the plane of the preform. These two stationary pro- cesses, denoted by 1 (x) and 2 (x), capture the stochastic fluctuations of the local warp and weft directions of the fibers. Given the microstructure of the woven fabric repre- sented by the effective shell, each of these directions corresponds to the behavior and fluctuations of fibers aligned along that direction. Letfe 1 (x);e 2 (x)g define the local elasticity tensor orientation vectors (warp and weft directions) for a pointx2D p in the initial undistorted state. It is assumed that initially fe 1 (x);e 2 (x)g are pointing in the global warp and weft directions. We definefe 0 1 (x);e 0 2 (x)g as the corresponding local elasticity tensor orientation vectors in the distorted state. Letf 1 (x); 2 (x)g represent the distortion angle corresponding to the angular measure of the local elasticity tensor orientation vector in distorted and natural configurations, respectively. The random distortion fields were each discretized along the domainD p with a Karhunen- Loeve (KL) expansion [43]. Each of the two stochastic processes was discretized with a M=5-term KL expansion with two-directional correlation lengths (c x ;c y ), withc x c y for 1 (x) and c y c x for 2 (x). For each process, the shorter correlation length was set to 35 one element diameter. An exponential covariance model was used. Truncating the spec- tral expansion after the M th term yields a spectral expansion of each of the stochastic processes as follows 1 (x)= (0) 1 (x)+ 1 M X i=1 f (1) i (x) 1;i q (1) i ; 2 (x)= (0) 2 (x)+ 2 M X i=1 f (1) i (x) 2;i q (1) i ; (4.2) whereE[ 1 (x)]= (0) 1 (x)=0 ,E[ 2 (x)]= (0) 2 (x)=0 , 1 = 2 =3 ,f 1;i g M i=0 ,f 2;i g M i=0 represent a set of uncorrelated random variables, andf i g M i=0 andff i g M i=0 denote the eigenvalue and eigenvectors of the covariance kernel of the process. The rational for truncating the KL expansion at M=5 was not based on capturing a certain threshold of the variance of the full eigenvalue problem, but rather to use the first five modes to form the backbone of the spatial distribution of initially-orthogonal wavy fibers across the computational domainD p . Given the spatial variability of the fiber orientation in the forming phase, which was modeled as a random field, and the integrated nature of our approach, spatial fluctuations from forming simulations are propagated to all subsequent phases of the RTM process. We describe in the following section how the local permeability tensor of the fabric in the resin injection phase is affected by spatial fluctuations from previous forming phases. 4.2.4 Local Sheared Permeability In the resin injection phase, each principal component of the permeability tensor field inherits the spatial fluctuations of the shearing angle field from the initial dry fabric forming step. Thus, the spatial variability of both the shearing angle and the local 36 sheared permeability tensor fields are dependent among other, on the random fiber orientation field, modeled using an exponential covariance kernel, as well as the forming parameters such as mold geometry, and fiber constitutive properties. Shearing angle fields from draping simulations were mapped onto filling domain using radial basis functions (RBF), according to (x)= X i2D p e i f(x;x i ); 8x2D ; (4.3) wheref(:;x i ) is the RBF centered atx i [48]. Models for computing sheared permeability tensors from unsheared tensors were found to show poor agreement with experimental measurements that themselves exhibit large statistical variations [49]. This underlines the need to appropriately account for uncertainty in local sheared permeability predic- tions. To address this issue, we introduce uncertainty in the characterization of the sheared permeability principal components, denoted by K (0) 1 and K (0) 2 . The random permeability field in each of the in-plane principal directions is then expressed as 0 B B B B B B B B B B @ K (s) 1 (x) K (s) 2 (x) 1 C C C C C C C C C C A =2 0 B B B B B B B B B B @ K (0) 1 K (0) 2 1 C C C C C C C C C C A 0 B B B B B B B B B B @ cos 2 (45(x)=2) cos 2 (45+(x)=2) 1 C C C C C C C C C C A + 0 B B B B B B B B B B @ K (0) 1 K (0) 2 1 C C C C C C C C C C A (4.4) where K (0) 1 and K (0) 2 are independent random variables with the following beta distribu- tions, such that K (0) i B(rK (0) i ;rK (0) i ; i ; i ), for i =f1;2g. The value of r that defines the support in the previous two equations is selected in such a way as to satisfy a target CoV for the local uncertainty in the sheared permeability model, while satisfying the 37 constraint of a strictly positive perturbed permeability field across the domain. The latter constraint can be formally expressed, for all realizations, as, min x2D p K (s) 1 (x) >0; min x2D p K (0) 1 2cos 2 (45+(x)=2)+ K (0) 1 >0: (4.5) The spatial distribution of the shearing angle field(x) across the domain , is obviously dependent, among other, on the fabric draping configuration. For the racetracking disturbances along the domain edge@D, the equivalent in-plane principal components of the permeability tensor along the domain boundary @D, de- noted byK rt;1 ;K rt;2 , of the woven fabric is determined empirically based on the channel cross-sectional dimensions and is assumed to remain in an unsheared state. Values of K rt;1 ;K rt;2 along the domain edge @D were essentially three orders of magnitude higher than the corresponding permeability field of the bulk domain. For an element e2D f e , a 3-D transformation T e () of the sheared principal perme- ability tensor K (s) f , defined as the rotation by an angle about an axis n e =(n e;x ;n e;y ;n e;z ). The rotated permeability tensor K (s); f (e) can be computed from the original permeabil- ity tensor K (s) f (e) using the above 3-D transformation matrix, and can be expressed as K (s); f (e)=T e ()K (s) f (e)T T e (). The proposed procedure for computing the local permeability tensor over the com- putational domain is described next. Initially, we compute the element normal vector as the cross product of the non-rotated original permeability tensor, in an unsheared state. This element normal constitutes the axis about which 3-D rotation of the perme- ability tensor occurs. Next, we interpolated the local shearing angle at every element in question from the forming simulations using Radial Basis Functions (RBF). The local shearing angle is used to (1) scale down the magnitude of the principal component, and 38 (2) calculate the 3-D rotation matrix, along with the normal vector. The scaled down permeability tensor is finally rotated using the 3-D rotation matrix. Figure 4.3 shows an illustration of this approach for computing the local sheared permeability tensor at every point in the domain. Figure 4.3: Illustration of the proposed approach for computing locally sheared perme- ability tensor of a random elemente,8e 2 D f e While several models have been proposed in the literature that relate the local vol- ume fraction to the local shearing angle, accounting for such a relationship would have entailed significant changes to the commercial PAM-COMPOSITES solvers. Moreover, the effects of the draping phase on the local permeability values was assumed to be dominated by fiber volume fractionV f , thus implying that the local permeability is only reduced with shearing deformation. A previous study [32] documented the fact that the opposite could take effect when the reorientation of the principal permeability tensor dominates over volume fraction effects. In all cases, it is clear that shear deformations have a considerable impact on the subsequent flow behavior, particularly in terms of amplifying the effect of any pre-existing defects and heterogeneity present in the fabric. In the subsequent resin flow phase, the shearing deformations from the draping phase affect the flow pattern in terms of volume fractionV f , and permeability anisotropy ratio, among other [50, 83, 49]. As the fabric is sheared in the initial draping phase, voids within the fabric are compressed, and the volume fraction of fibers V f increases, thus 39 causing the permeability of the fabric to drop. However, in [50], it was noted that the op- posite effect could occur in some instances wherein (1) the fabric is subject to significant local reorientation of the permeability tensor and fibers and (2) the local compression of the fabric during draping reduces the volume of voids in highly sheared local zones, resulting in a reduced amount of resin needed to reach local saturation and thus an ac- celerated flow front. These observations further motivate the introduction of the model error which we parameterize, as shown in equation (4.4) 4.2.5 Quantities of Interest (QoI) The RTM manufacturing processes essentially involve three different QoI for design pur- poses: filling time t f at the end of resin injection, curing time t c at the end of resin polymerization, and maximum residual stress r in the distortion phase. For the latter, we compute the maximum residual stress over all integration pointsfx g g within the set D d e based on Von Mises expression r = max x g 2D d e VM . The filling time t f ( 1 ; 2 ), curing time t c ( 1 ::: 3 ), and residual stress r ( 1 ::: 4 ) QoI’s can be respectively approximated via PCE in the original random space that consists of f 1 g2R 11 ,f 2 ; 3 g2R 30 , andf 4 g2R 33 . LetA t f ,A t c , andA r represent the change-of-basis rotation with respect to each QoI. In the reducedspace, the PCE of each QoI can be formulated as t f () X jjP c (t f ) (A t f ); t c () X jjP c (t c ) (A t c ); and r () X jjP c ( r ) (A r ): (4.6) In the polynomial chaos methodology, the PDF can be obtained simply by sampling from the random domain after a functional representation of the QoI is obtained. 40 4.3 Results & Discussions 4.3.1 Observations of Physical Behavior In this subsection we first observe noteworthy behaviors from a single realization. Figure 4.4 shows the contours of vertical displacement and the corresponding shearing angle field for a realization corresponding to the quadrature point of the second order, 3-D adapted PCE solution with CoV = 12% that has the highest injection time t f . Results are shown at successive loading steps along the forming simulation. The majority of shearing occurs on the upper flanges of the truncated square pyramid domain. (a) (b) (c) (d) (e) (f) (g) (h) Figure 4.4: Contour of vertical displacement field, d z (D p ) and corresponding shearing angle field(D p ) for a random realization withCoV =12% at different forming time steps ford z (D p ) at (a) 0.25P f , (b) 0.5P f , (c) 0.75P f , and (d) 1.0P f and for(D p ) at (e) 0.25P f , (f) 0.5P f , (g) 0.75P f , and (h) 1.0P f , whereP f is the final forming load. Figure 4.5 shows histograms of the shearing angle random field(D) and the corre- sponding local sheared permeability random field K I (D) for the same realization shown in Figure 4.4. Each histogram corresponds to same discrete step along the forming 41 simulation as in figure 4.4. It should be noted that although these histograms show a large range of variation for the shearing angle and permeability, it should be noted that these variations occur within the same realization, highlighting the significance of spatial heterogeneity in characterizing these material properties. It is important to reiterate that our procedure deduces the spatial variations of these two properties from (1) knowledge of the initial configuration prior to forming, (2) knowledge of the statisti- cal information about the fiber moduli, and (3) knowledge of the draping process that induces the local shearing deformations. 0 400 800 1200 0 400 800 1200 0 250 500 750 1000 −30 −20 −10 0 10 20 30 α(x,θ ) 0 100 200 300 (a) 0 20000 40000 0 10000 20000 0 4000 8000 12000 1.0 1.2 1.4 1.6 1.8 K I (x,θ) 1e−11 0 1000 2000 3000 (b) 0 400 800 1200 0 400 800 1200 0 250 500 750 1000 30 35 40 45 50 55 60 β(x,θ ) 0 100 200 300 (c) Figure 4.5: Histogram of (a) shearing angle field(D p ) for the same realization, and (b) corresponding sheared permeability field, and (c) histogram of (D) field, representing the direction of the principal permeability tensor with respect to the warp direction. 42 Figure 4.6 shows contour plots of the filling factor at different discrete time steps along the resin injection stage. Filling contour plots correspond to the same realization of Figure 4.4, with CoV = 12%. The filling factor ranges from 0. to 1., representing unfilled and completely filled nodes, respectively. The filling contour plots correspond to the realization based on the quadrature point of the second order, 3-D adapted PCE solution withCoV =12% that shows the highest resin injection time. At the last discrete filling time step, all nodes within the computational domainD were completely filled (i.e. filling factor equal to 1.0,8x2D). for the same random realization of previous figure with CoV =12%. . The selected realization corresponds to the quadrature point of the second order, 3-D adapted PCE solution with CoV =12% that shows the highest resin injection time. (a) 0.05t f (b) 0.16t f (c) 0.27t f (d) 0.38t f (e) 0.49t f (f) 0.60t f (g) 0.71t f (h) 0.94t f Figure 4.6: Contours of filling factor within the resin injection simulation at different time steps for a random realization witht f = 58.60 sec. For dry spot formation, the termination criterion for all realizations was set to full saturation of all nodes within the computational domainD. Thus, at the last resin 43 injection step, dry spots are absent wherein all nodes have a ubiquitous filling factor of 1.0. The formation of dry spots depends, among other factors, on the geometry of the formed fabric, the configuration of both vent and injection nodes, racetrack distur- bances along the domain edge@D, as well as the resin properties. A detailed analysis of dry spot formation is beyond the scope of this study. Suffices it to say that dry spots are observed along the upper flanges of the truncated square pyramid geometry. In addi- tion, the contribution of edge racetrack disturbances as a catalyst of dry spot formation is clearly highlighted in the filling contours. The resin flow front is dependent on the spatial fluctuations of the underlying perme- ability field across the 3-D computational domain. Given the aforementioned parametriza- tion and spatial discretization of each and every stage in the RTM process sequence, several factors potentially contribute to the spatial variability of the permeability field. The latter depends, first, on the KL discretization of the initial fiber angle field, and second, on the local shearing of the permeability tensor at the injection stage based on shearing angles induced by pre-injection forming. These two factors contribute to the spatial correlation and fluctuations of the random permeability field; fluctuations that control the configuration of resin injection contours. This can be visualized in figure 4.6 wherein the resin flow front follows a preferential path close to the -45 o as compared to +45 o . The major reason behind the rotated flow front is that the realization in question corresponds to a highly anisotropic permeability ratio. The aforementioned observation of the resin flow front following a preferential path along the -45 o as compared to +45 o can be explained by the fact that the particular re- alization has stronger fiber properties that corresponds to the -45 o as compared to the fiber that corresponds to the +45 o direction. A fiber with stronger mechanical proper- ties deforms less under the same conditions than a fiber with weaker properties, more 44 deformation implies more shearing, and more shearing implies greater reduction in the local sheared permeability tensor for the zone in question. 4.3.2 First Order Solution Initially, uncertainty was propagated through the manufacturing sequence using a first order PCE expansion. Figure 4.7 shows the corresponding PCE coefficients of each QoI for each of the 74d variables that constitute the random space. These are the coeffi- cients of the first order expansion and can be considered to represent the sensitivities of the QoI to the various uncertain parameters. We should emphasize that the PCE coefficients represent sensitivity to the random variables. Sensitivity with respect to the parameter itself can be approximated by multiplying the corresponding coefficient with the standard deviation of the parameter. The following observations can be made from Figure 4.7: (1) the highest predictor of uncertainty in filling timet f is the uncertainty in volume fraction V f of the fabric, followed by the fiber elastic properties and stacking config- uration in the pre-injection draping stage. Furthermore, uncertainty in racetracking disturbances K rt;1 ;K rt;2 and distortion angle random field KL had limited to no impact on the filling timet f or any other QoI. The small effect of racetracking uncertainty was expected since the mere presence of racetracking strongly favors flow along these tracks, and small perturbations to their permeability has very little effect on the enhanced mo- bility within the tracks. (2) the uncertainty in curing timet c was found to depend only on the input of the cur- ing stage and essentially independent of any previous forming or filling random input. Uncertainties in the second activation energyE 2 and the boundary temperature at the top and bottom platesT bt were the highest contributors to the curing time uncertainty. 45 (3) the uncertainty in residual stress r was mostly dependent on uncertainty in distortion-level parameters, but also affected by the previous curing and filling results. The highest contributors to uncertainty in residual stresses include g ,T ginf , 33 ,E 33;ply . Not every microscale property for the resin and fiber phases was part of the input of the computational model. Some microscale properties were therefore only parametrized via a dependent random input derived from the finer scales. In particular, the microscale fiber properties related to fiber shape, diameter, spacing were not part of the input do- main, and the only property related to the microscale fiber properties is the Volume fraction V f of the fabric, which is a direct reflection of these microscale properties in- cluding fiber diameter, spacing. It follows that the uncertainty in V f is inherited from uncertainties in each of these underlying microscale fiber properties that controlV f but that were parameterized within our computational model. Moreover, it could be the case that the homogenized computational domain in the forming simulation has a shell element discretization that is not fine enough to cap- ture small variability in some microscale parameters. For instance, slight variations in fiber angles, on a local element level, might require a comparatively much finer mesh discretization so that slight variations of the order of a couple of degrees in fiber angles could be adequately captured by the forward model. 4.3.3 Higher-order Adapted PCE In figure 4.8, convergence PDF plots with respect to the order of the PCE are presented for each QoI based on level 2 sparse grid quadrature and for a CoV=12%. Figure 4.9 shows the first three rows of rotation matrices A t f , A t c , and A r respectively, for CV=4% and 12%. For each case, the transformed random vector is expressed as a linear transformation of. For 4.9(a), the1 st associated with the first row is clearly dominated by the coefficient of volume fractionV f indexed at j=7 . The individual influence of each 46 0.0 0.2 0.4 0.6 0.8 1.0 1.2 q (QoI) α 1 E f1,t E f2,t E f1,b E f2,b θ f1 θ f2 V f k f,1 k f,2 k f,3 C p,f ρ f k r,1 k r,2 k r,3 C p,r ρ r T r P I P V A 1 A 2 m n B E 1 E 2 T bt h E 11r,ply E 22r,ply E 33r,ply ν 12r,ply ν 13r,ply G 12r,ply G 13r,ply G 23r,ply α 11r α 22r α 33r φ 11r φ 22r φ 33r α g T g0 T ginf λ E 11g,ply E 22g,ply E 33g,ply ν 12g,ply ν 13g,ply G 12g,ply G 13g,ply G 23g,ply α 11g α 22g α 33g φ 11g φ 22g φ 33g δ K 1 δ K 2 c 1 c 2 K rt,1 K rt,1 K rt,2 K rt,2 γ KL,1 γ KL,2 γ KL,3 γ KL,4 γ KL,5 t f t c σ r Figure 4.7: Normalized first order PCE coefficients oft f ,t c , r for CoV=4% 47 of the other coefficients is also included, and reflected in the higher order terms. For figure 4.9(c) that corresponds to the adapted basis with respect to the curing time QoI (t c ). In this case, and as previously noted, the only parameters with non-negligible coefficients are those that pertain specifically to the curing simulation. All other non- curing j ’s have zero coefficients in the curing adaptation matrix. It is noted that in all cases, orthogonality between the rows of the adaptation matrices is achieved by slight changes in the contributions from the various random components. For figure 4.9(e), that corresponds to the adapted basis with respect to the maximum residual stress r QoI. As observed previously, the residual stress r QoI was dependent on random input than spans the whole manufacturing sequence. Figures 4.9(b),(d), and (f) show the corresponding first three rows of the rotation matrices for CV=12%. Figure 4.10 shows the adapted PDF’s from samples synthesized from2 nd order PCE of t f ,t c , and r , adapted each with respect to itself, corresponding to a reduced stochastic set of dimensions 1, 2, and 3. Table 4.2 shows the first two statistical moments and the corresponding CoV of the QoIs computed from the 2 nd and 3 rd order PCE at the first three reduced dimensions. 48 30 35 40 45 50 55 60 t f (sec) 0.0 0.1 0.2 0.3 0.4 1dη PDFoft f 0 1000 2000 3000 t c (sec) 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 PDFoft c od=2 od=3 5 10 15 20 25 30 35 40 45 50 σ r (MPa) 0.00 0.02 0.04 0.06 0.08 0.10 PDFofσ r 30 35 40 45 50 55 60 t f (sec) 0.00 0.05 0.10 0.15 0.20 2d η 0 1000 2000 3000 t c (sec) 0.000 0.001 0.002 0.003 0.004 0.005 5 10 15 20 25 30 35 40 45 50 σ r (MPa) 0.00 0.02 0.04 0.06 0.08 0.10 30 35 40 45 50 55 60 t f (sec) 0.00 0.05 0.10 0.15 0.20 3d η 0 1000 2000 3000 t c (sec) 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 5 10 15 20 25 30 35 40 45 50 σ r (MPa) 0.00 0.02 0.04 0.06 0.08 0.10 Figure 4.8: PCE order convergence based on level 2 sparse grid quadrature for CoV=12%. The matrix plot corresponds to PDF sampled from PCE of order P=2 and P=3 for each of the predefined QoIs at the first three projection reduced subspaces, whereby each QoI along column, and projection reduced dimension along rows. Thet f QoI of the realization used to generate previous plots that correspond to the quadrature point with highest filling time from the adapted solution for a 3-d space is marked with a star point 49 0.0 0.5 1.0 |q (tf) α 1 | η 1 0.0 0.5 1.0 |q (tf) α 1 | η 2 0 25 50 75 0.0 0.5 1.0 |q (tf) α 1 | η 3 (a) 0 25 50 75 (b) 0.0 0.5 |q (t c) α 1 | η 1 0.0 0.5 |q (t c) α 1 | η 2 0 25 50 75 0.0 0.5 1.0 |q (t c) α 1 | η 3 (c) 0 25 50 75 (d) 0.0 0.5 |q (σ r) α 1 | η 1 0.0 0.5 |q (σ r) α 1 | η 2 0 25 50 75 ξ j 0.0 0.5 1.0 |q (σ r) α 1 | η 3 (e) 0 25 50 75 ξ j (f) Figure 4.9: First three rows of the change-of-basis adaptation rotation A t f , A t c , and A r that correspond to each QoI,t f ,t c , and r 50 CoV (%) P QoI=q E[q] Var[q] CoV q (%) 1-D 37.24 0.94 2.60 4 2 t f 2-D 37.25 1.05 2.75 3-D 37.22 0.78 2.38 1-D 617.09 4065.27 10.32 4 2 t c 2-D 609.48 3998.35 10.39 3-D 602.03 6085.62 12.96 1-D 24.95 5.06 9.05 4 2 r 2-D 24.68 4.94 9.03 3-D 24.51 5.30 9.40 Table 4.2: Convergence of different reduced probabilistic models for each QoI. Using equations (3.12), we computed Pearsons’s correlation coefficient among each pair of aforementioned QoIs. The correlation coefficients were found to be t f ;t c =0:0084, t f ; r =0:47, and t c ; r =0:0037. This highlights a correlation between filling timet f and the residual maximum stress r . Moreover, we confirm again that the curing part was a comparatively isolated step with limited interconnection with previous or subsequent steps. This is consistent with the physical processes used to model the curing stage, and which involve phenomena and parameterizations that are essentially independent of fabric forming and resin flow during injection. Figure 4.10 shows the corresponding PDFs constructed using samples from the 2 nd order PCE of each QoI for the first three reduced dimensions. For all three QoIs, a re- duced dimensionality of in a 3-D space is essentially converged and the correspond- ing PDF captures with enough accuracy the probabilistic response of the corresponding QoI. Table 3 shows a comparative assessment of the computational cost of the chaos adaptation scheme with respect to conventional statistical methods. 51 30 35 40 45 50 55 60 t f (sec) 0.0 0.1 0.2 0.3 0.4 0.5 PDFoft f 1dη 2dη 3dη (a) 30 35 40 45 50 55 60 t f (sec) 0.0 0.1 0.2 0.3 0.4 0.5 PDFoft f 1dη 2dη 3dη (b) 0 1000 2000 3000 t c (sec) 0.000 0.002 0.004 0.006 0.008 PDFoft c 1dη 2dη 3dη (c) 0 1000 2000 3000 t c (sec) 0.000 0.002 0.004 0.006 0.008 PDFoft c 1dη 2dη 3dη (d) 5 10 15 20 25 30 35 40 45 σ r (MPa) 0.00 0.05 0.10 0.15 0.20 PDFofσ r 1dη 2dη 3dη (e) 5 10 15 20 25 30 35 40 45 σ r (MPa) 0.00 0.05 0.10 0.15 0.20 PDFofσ r 1dη 2dη 3dη (f) Figure 4.10: PDF using samples from 2 nd order adapted PCE for different inputCoV = 4% for QoI (a)t f , (c)t c , and (e) r , andCoV =12% for QoI (b)t f , (d)t c , and (f) r The accuracy of the adapted PCE scheme was assessed with respect to a conven- tional statistical method, namely Latin Hypercube Sampling (LHS). Figure 4.11 shows a 52 Reduced Quadrature Full Adapted Dimension Level PCE PCE 1 149 149 1-D 2 11,397 176 3 594,813 176 1 149 149 2-D 2 11,397 212 3 594,813 284 1 149 149 3-D 2 11,397 260 3 594,813 500 Table 4.3: Number of forward evaluations (i.e. quadrature points) required for obtaining a PDF representation of the QoI in a d = 74 random physical system with three QoI’s using Smolyak sparse quadrature for classical, full PCE, as well as the adapted PCE. PDF plot for each QoI for the adapted PCE and LHS models. The LHS effort was limited to 1,000 samples, as compared to 260 samples for a converged adapted PCE expansion (3D at quadrature level 2). It is noted that the PDF from LHS as synthesized from 1,000 samples is still rough and is likely to change with additional samples. More importantly, it should be noted that a PCE representation, unlike sampling-based procedures, pro- vides a functional relationship between the QoI and the basic random variables that describes the input parameters. This functional dependence is significant in assessing sensitivities and for integration of stochastic models into reliability-based optimization schemes. 53 32 34 36 38 40 42 44 t f (sec) 0.0 0.1 0.2 0.3 0.4 PDF of t f (a) PCE-BA LHS 300 400 500 600 700 800 900 1000 t c (sec) 0.000 0.002 0.004 0.006 0.008 PDF of t c (b) PCE-BA LHS 15 20 25 30 35 σ r (MPa) 0.00 0.04 0.08 0.12 0.16 0.20 PDFofσ r (c) PCE-BA LHS Figure 4.11: Comparison of PDF obtained from the proposed adapted PCE with respect to 10 3 Latin Hypercube Samples for each QoI: (a) fill time t f , (b) cure time t c , and (c) maximum residual stress r . 54 For completeness, we evaluated a high-fidelity solution of the first QoI, t f , in order to examine the effect of PCE order on the probabilistic distribution of the QoI. Figure 4.12 shows the corresponding PDF oft f for the2 nd order PCE, with level two sparse grid quadrature versus a5 th order , level four sparse quadrature. It can be observed that the low-order solution gives an accurate enough approximation of the corresponding higher order expansion, while still at a significantly lower computational cost. While the error in PCE approximation can be addressed by increasing PCE order (while avoiding over- fitting), this implies that there is no strong justification to switching into higher order solution at a higher computational cost. 32 34 36 38 40 42 44 t f (sec) 0.0 0.1 0.2 0.3 0.4 PDF of t f od=2 od=5 Figure 4.12: PDF oft f based on a 3-D reduced space for second and fifth PCE orders with level two and four sparse grid quadrature, respectively. 4.4 Findings & Conclusion We developed an integrated framework for forward propagation of uncertainty in com- posite RTM manufacturing models while taking into consideration the multiscale, mul- tiphysics, and high-dimensional nature of the RTM problem. We used an adapted PCE to compute the stochastic representation of predefined QoI’s along the RTM process 55 within a high-dimensional setting; we used the low-dimensional adapted PCE to formu- late surrogate models of QoI’s of interest in RTM design problems. We used in this study a 3-D truncated square pyramid model for developing the un- certainty propagation framework. Future work will be centered on applying the frame- work to full-scale computational models of fiber composite structures within automotive applications. The proposed integrated approach allows us to identify material parameters and pro- cess conditions that are critical to the whole design process. Within a first order sen- sitivity analysis, we were able to rank the contribution of each of the parameters in the 74d random input on the set of predefined QoIs. In particular, we observed that the strongest predictor of uncertainty filling time t f is the fabric’s volume fraction V f followed by the fiber microscale properties. Given the significance of the fabric’s vol- ume fractionV f on the characterization of filltimet f , assessing the impact of its spatial fluctuations on each QoI is an important task that was not addressed in the present work. The curing timet c , was most sensitive to the boundary temperatureT bt of the bottom and top plates followed by second activation energy E 2 of the curing model. Moreover, we observed that the curing time t c is contributed to by parameters exclusively from the curing stage. For the residual stresses along the subsequent distortion phase, the strongest predictors of r were found to be the degree of cure at gelation g , the final glass transition temperatureT g;inf , and the out-of-plane properties of chemical shrinkage 33 and elasticity tensorE 33 . In order to gain more detailed insight into the distributions of the QoIs and to capture the effect of nonlinear sensitivities, we constructed higher order PCE representations and associated probability density functions. These were made possible through a change of variable that numerically constructs approximation bases that are adapted to each of the QoIs. With the computational burden thus alleviated, 56 rigorous uncertainty quantification becomes a viable tool for credible validation, design optimization, and resource allocation. Further, it is observed that the probability density functions of the QoI, including their shape and their support, change with the level of uncertainty (going from 4% to 12% CoV). Even for small uncertainties, the shapes of these density functions deviate significantly from the Gaussian distribution. This suggests that design guidelines and failure assessments based on Gaussian approximations may not be consistent with experimental evidence and prevalent physics principles. 57 Chapter 5 Flow in Random Heterogeneous Media Proper uncertainty quantification in heterogeneous media is premised on an accurate representation of spatially varying material properties as random fields. Even though uncertainties in predictions are oftentimes sensitive to the parameters of the underly- ing material processes that characterize heterogeneous media, research methodologies for accurately characterizing such material processes are lacking. We propose, in this chapter, a computational recipe for learning, via kernel functions, the correlation struc- ture of permeability fields in random fabric media within composite manufacturing. We numerically optimize a selection of covariance kernels with respect to the correlation structure of the random fabric media obtained from a higher-order Polynomial Chaos (PC) surrogate model. Moreover, for discovering the correlation structure on curved domains in space, we propose a geodesic subroutine such that geodesic distances be- tween vertices on curved surfaces are used in the kernel functions. Within a learning setting, the performance of each kernel is evaluated within the fabric forming model. We observed extreme uncertainty in the local permeability field along the sheared fabric from the PC surrogate. Even at optimal parameters, kernels only partially captured the correlation structure of the fabric media. 58 5.1 Introduction Stochastic analysis of fiber composite materials requires, among other, an adequate probabilistic representation of the physical models input. In particular, resin flow through the fabric porous media is dictated to a varying extent by the permeability tensor of fabrics. For the latter, different stochastic representations lead to different response surfaces and ultimately to different designs and behaviors of the end composite com- ponent. One representation of permeability fields involves modeling the process as a random field with a covariance kernel with predefined spatial correlation. While ran- dom permeability field models capture the spatial correlation introduced within the pre-injection forming phase, an accurate characterization of the spatial correlation pa- rameters is lacking. Over the past couple of decades, research tackling stochastic partial differential equations with spatially varying constitutive material properties has underlined the im- portance of adopting proper random fields models to accurately capture the underlying physics of heterogeneous media. For such models, the task of uncertainty propagation is sensitive to the parameters of the random field used to model the underlying material spatial heterogeneity. While the importance of accurately characterizing such random fields within the context of uncertainty quantification has been recognized, recipes for evaluating such random fields are lacking. We propose a methodology for learning co- variance kernels of random permeability field models of the fabric media subject to forming deformations. Fabric forming, within composite manufacturing, consists of a multiphysical, mul- tiscale, problem, involving a sequence of interconnected set of processes, where the random permeability field in the resin injection phase is essentially locally inherited from the initial material deformation phase known as fabric forming. Fabric forming 59 computational models constitute, for that matter, advanced models whereby multiscale approaches need to be implemented in order to couple the micostructural behavior with the macroscale domain. The spatial correlations of the permeability field, which are usually done via classical covariance functions such as Exponential or Gaussian, for instance, require an a-priori knowledge of the correlation length for the covariance function. The response of the QoI is dependent, to a varying extent, on the choice of covariance kernel as well as spatial correlation lengths. As such, an adequate quantification of the covariance kernel parameters is needed in order to provide sound input models of random permeability fields into physical models of resin flow in porous media. Otherwise, this would lead to composite design scenarios that are overconservative whereby high factors of safety are adopted to compensate for the lack of appropriate characterization of the knowledge in spatial correlation of permeability fields. For flow in random heterogeneous media, it is well documented in the literature that the probabilistic distribution and uncertainty of a given QoI is strongly dependent on the correlation length of the permeability process. Figure 5.1 shows the probabilistic distribution of the filling time of the resin injection phase based on different spatial representations of the permeability across the domain: random variable (i.e. infinite correlation) and random field models. Research is needed for better understanding and characterizing random material fields. Within that context, a theoretical and computa- tional framework for efficiently evaluating the correlation structure of random material fields in heterogeneous media constitutes an important milestone to achieve. The goals of this chapter is to establish a methodology for learning covariance ker- nels of fabric forming models involving stochastic partial differential equations with heterogeneous constitutive material properties. The manuscript is organized as follows: 60 first, the research methodology and its corresponding mathematical framework are pre- sented in detail, followed by the application of the proposed framework to fabric forming. Results and discussions are presented afterwards. 40 45 50 55 60 65 70 75 80 Filltime, t c (sec) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Probability Density K 1 ∼ r.v. K 1 ∼ r.p. Figure 5.1: Probabilistic distribution of resin filling time for different representations of the fabric permeability field 5.2 Research Methodology We formulate, in this section, the mathematical background of the proposed research methodology. 5.2.1 Mathematical Formulation Let us consider a random physical system whose governing equation can be expressed in operator notation in the form L(h(x;);u(x;))=0; x2D ; (5.1) 61 whereP is a probability measure, is the sample space, u(x;) :D!R denotes the stochastic solution to be explored,DR d ;d2f1;2;3g the domain, and h(x;) de- notes a stochastic process with square-integrable continuous covariance function such that h(x;)2H, whereH denotes a Hilbert space. The mean ( ¯ h(x)) and covariance ( h ) functions ofh(x;) are defined as ¯ h(x)=E[h(x;)]= Z h(x;)dP() ; 2 ; (5.2) and h (x;y)=E[h(x;);h(y;)]; 8x;y2D : (5.3) A decomposition, commonly known as the Karhunen-Loeve Expansion, ofh(x;) con- sists of expanding with respect to the eigenfunctions of the covariance matrix ofh(x;) such that h(x;)= ¯ h(x)+ 2 1 X i=1 p i i () i (x); (5.4) where ¯ h(x) refers to the expected value of the stochastic process,f i g andf i g denote the eigenvalues and eigenfunctions of the covariance matrix h of the process, which can be evaluated from Z D C k (x;y)(y)dy=(x); x2D ; (5.5) whereC k denotes the autocorrelation function corresponding to a kernel of typek. Let ' k denote a kernel function of type k. For a real-valued process, it can be shown that, by definition, the kernel function' k is symmetrical, ' k (x;y)=' k (y;x); (5.6) 62 positive-definite such that X m X n m n ' k (x m ;x n )0; (5.7) and satisfies the Cauchy-Schwartz inequality ' 2 k (x;y)' k (x;x)' k (y;y): (5.8) We present in this work, a framework for accurately approximating the covariance h of a stochastic process h(x;) that represents a spatially-varying constitutive material property. 5.2.2 Polynomial Chaos Surrogates Letq(;x) represent tensor Quantities of Interest (QoI), which admits a Polynomial Chaos Expansion (PCE) as follows q(;x)= 1 X jj=0 q (x) () X 2I d P q (x) (); (5.9) where q represent the PCE coefficients of the QoI. Let (x;y) denote the covariance matrices for a domain , where x;y2 . Using the functional representation in equation (5.9), the covariance matrix can be computed respectively as (x;y)= X i1 q i (x)q i (y) 8x;y2 : (5.10) Let C t (x;y) denote the corresponding autocorrelation function, which can be evaluated as C t (x;y)= (x;y) p x y ; (5.11) 63 where 2 x = X i q 2 i (x); 2 y = X i q 2 i (y): (5.12) 5.2.3 Parametric Kernel Optimization 5.2.3.1 Autocorrelation Approximation In this section, the parametric optimization procedure for approximating the covariance kernel properties with respect to the target autocorrelation function obtained from a high order PCE surrogate model based on material deformation is presented. We start by building a forward problem for propagating uncertainty through material deformation models. As such, we build a surrogate PCE model of the QoI, from which we compute a covariance matrix that accounts for all material deformations. Such a covariance matrix carries the spatial deformation information that characterizes the deformed domain , and thus constitutes the target of the optimization subroutine. We aim to parametrically optimize off-the-shelve covariance kernels in such a way so that a maximum of information on the deformed domain is retained. 5.2.3.2 Objective Function Given a predefined covariance kernel described in terms of it’s kernel typek and initial value of parameters (i.e. b k ), we build a simple optimization subroutine to backcalculate the optimal parameters of the covariance kernel. Let Σ(x 0 ;y 0 ) denote the covariance matrix, calculated using equation (5.10) based on the higher-order PCE-based material deformation surrogate model, andΣ k (x;y;b k ) the corresponding parametric covariance kernel of typek, and whose parameters, namely correlation length, is denoted by b k . We define accordingly a minimization problem, denoted byP, with objective function consisting of the error involved in approximating the surrogate-based autocorrelation 64 function. Given an autocorrelation function C k (x;y;b k ) corresponding to a kernel type k, the covariance matrixΣ k can be formulated as k;ij = C k (x i ;y j ;b k ); (5.13) where (x i ) and (y j ) represent the spatial coordinates of nodesi andj respectively,8i;j2 N. Let C t (x 0 ;y 0 ) denote the autocorrelation function obtained from the PCE-surrogate model, corresponding toΣ k . Following optimization problemP based on the autocorrelation function of the pro- cess, the optimal kernel parameters can be evaluated following an objective function of the form b k = arg min b k 8 > > > < > > > : N X j;i=1 [C t (x i ;y j ) C k (x i ;y j ;b k )] 2 + k 9 > > > = > > > ; ; x i ;y j 2 ; (5.14) where N denotes the nodes of the mesh corresponding to a given discretization of the domain , the last term k represents a penalization function for adjusting the influence of points farther in the process. For any two nodes x i ;y j 2 , the relative error can be computed following e ij = [C t (x i ;y j ) C k (x i ;y j ;b k )] 2 [C t (x i ;y j )] 2 ; (5.15) while the total relative error across the domain , for a given kernel with parameters b k , can be computed as N X j;i=1 e ij = N X j;i=1 [C t (x i ;y j ) C k (x i ;y j ;b k )] 2 [C t (x i ;y j )] 2 : (5.16) Algorithm 1 summarizes the proposed methodology for computing covariance ker- nel parameters based on a target covariance matrix formulated using a higher-order 65 PCE-based material deformation surrogate model, formulated as a minimization prob- lem. Table 5.1 summarizes popular types of stationary covariance kernels, denoted by k, found in the literature. Stationary covariance kernels can be broadly classified into isotropic or anisotropic. For the isotropic case, the process is assumed to be indepen- dent of the direction, and only depends on the magnitude of the distance between two points in the process (i.e. (jx yj)). For an anisotropic process, the correlation between two points in the process is assumed to be dependent on both the direction and the magnitude of the distance between two points (x y). Kernel Covariance Model Exponential ' E (;b)=e b Gaussian ' G (;b)=e ( b ) 2 Rational Quadratic ' RQ (;b)=1 2 2 +b Mat´ ern ' M (;b;)= 2 1 () ( p 2 b ) K ( p 2 b ) AR(2) y ' AR2 (;! 0 ;)=e 1 2 h cos(p)+ 2p sin(p) i Table 5.1: Some stationary isotropic covariance kernels ( =jx yj); y p= q ! 2 0 + 2 =4. Algorithm 1 Proposed optimization subroutine of covariance kernel parameters follow- ingP Input: kernel typek; initial correlation length b k Step 1: Compute PCE coefficientsq (x 0 ) from a higher order PCE surrogate model, 8x 0 2 Step 2: Construct PCE-based covariance matrix, of the form (x 0 ;y 0 ) P q (x 0 ):q (y 0 ) T ; (x 0 ;y 0 )2 Step 3: Construct covariance matrix k (x;y;b k ) Step 4: Evaluate optimal correlation such that b k arg min b k n P N j;i=1 [C t (x i ;y j ) C k (x i ;y j ;b k )] 2 + k o Return: b k While the importance of characterizing spatially-varying constitutive material prop- erties with random fields models has been highlighted over the past couple of decades, recent research is pinpointing to the fact that uncertainty quantification in predictions 66 with underlying stochastic material processes is dependent on the corresponding cor- relation lengths of these processes. To our knowledge, a methodology for efficiently characterizing covariance kernel functions and corresponding correlation lengths that are tailored to the physical material deformation problem in question is lacking. This leaves researchers using predefined kernels with assumed parameters that are agnostic to the specific physics of the random physical system at hand, and specifically to the material deformation phase. The proposed framework provides a solution to the aforementioned problem, by pro- viding a computational recipe for identifying optimal covariance kernel parameters. The proposed recipe is computationally efficient due to the fact that it is based on propa- gating uncertainty using PCE-based surrogate models, which are comparatively cheap to evaluate, particularly for systems with low stochastic dimensions. In the research methodology highlighted above, the target covariance matrix (i.e. ) used in the optimization subroutine was computed using a higher order PCE surrogate model. That being said, other stochastic surrogate models can be used to efficiently approximate the covariance matrix of the process. 5.2.4 Mesh Geodesics Traditional kernel functions are agnostic to the manifold on which the solution needs to be evaluated. For discovering the correlation structure of curved domain in space such as composites, computation of geodesics distances for covariance kernels becomes essential. We propose and validate a subroutine for evaluating geodesics on curved polyhedral surfaces in space within the context of learning the correlation structure of curved domain. One of the main challenges encountered in the proposed methodology resides in learning the correlation structure of domains with a complex topology with non-trivial 67 geometry. In particular, the correlation between two points(x;y)2D in the process must follow the part topology and geodesic paths and distances need to be computed in order to have accurate covariance models mapped onto curved geometries in space. The study of covariance functions of random field models is an vibrant and active area of research. In [109], a methodology for simulating random fields on curved surfaces in space was proposed. A diffusion maps methodology was proposed in [108] in order to compute distances for covariance functions evaluation in high-dimensional settings. An accurate assessment of the correlation structure of curved domains is premised on having accurate measures of geodesic paths and distances on discrete polyhedral surfaces of complex geometries. In particular, an accurate correlation measure of the process can be obtained if the geodesic paths and distances between pairs of spatial points x;y2D are accurately computed and provided as input to the covariance kernel models. The problem of computing geodesic paths and distances on discrete surfaces in three- dimensional space has received considerable attention within the realm of differential and computational geometry. Several methods based on different formulations have proposed algorithms for computing Geodesic paths and distances, and to a varying de- gree, such algorithms have been implemented in various applications such as robotics, computer graphics, transportation, among many other. In a general sense, algorithms for computing geodesic distance and paths on discrete polyhedral surfaces have poly- nomial time complexity. Studies related to formulating algorithms and corresponding implementations for computing geodesic paths and distances on polyhedral surfaces in space can be broadly classified into two categories: (1) discrete differential geometry [53], [54], [55], [56], [57], [58], [59], and (2) graph theory [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70]. For the former, given that geodesic paths and distances are computed on 68 discrete surface in three-dimensional space, such surfaces are not differentiable, and it follows that methods from differential geometry cannot be used. For the latter, discrete surfaces can be considered as a graph, and shortest path finding methods from graph theory can be used appropriately. A detailed review of literature on geodesic paths in three-dimensional space is beyond the scope of this work, keeping in mind that ample research has been devoted in this regard for computing geodesics on surfaces in space. LetV represent a surface in three-dimensional space, discretized via polyhedronP N V with N vertices. Let v2P N V denote a random vertex in the polyhedron, and e v denote the influence distance for that vertex. Let SSSP(v;:) denote an single source shortest path problem whereby the geodesic distance of a source vertex v with respect to all other vertices in polyhedron P N V within influence zone e v . Algorithm 2 highlights how the geodesic distance subroutine is integrated into the proposed framework. Algorithm 2 Geodesic path subroutine 1: Input: P N V ;e v 2: for vertex v2P N V do 3: P e v Identify vertices in neighborhoode v 4: SSSP(v;P e v ) 5: end for Within the context of uncertainty quantification, computing geodesic paths and dis- tances for covariance kernels of random fields of constitutive material parameters is a step that precedes the forward problem. As such, geodesics are computed on curved domains in space only once, initially, for all realizations. Mesh geodesics are ubiquitous for all realizations, and depend only on the topology of the curved domain in space onto which the correlation structure needs to be discovered, assuming that any potential ma- terial deformations will not have a significant impact on the topology of the polyhedral surface in question. If, on the contrary, different realizations have variability in material deformation in such a way that some realizations experience significant material deformation and 69 shape distortion that has a measurable impact on the topology of the polyhedral sur- face, then the task of learning the process autocorrelation becomes dependent on each realization, and the geodesics computations are no longer ubiquitous across all realiza- tions. 5.3 Application: Fabric Forming The performance of the aforementioned methodology for random fields characterization in deformed heterogeneous media is evaluated on the problem of fabric forming, part of fiber composite manufacturing. In the fiber composite manufacturing process, such as the popular Resin Transfer Molding, a fibrous reinforcement fabric is draped within a mold enclosure in an initial pre-injection stage. As the latter deforms along the mold under the forming load, the microscale mechanical and geometric properties of the fabric are transformed. Material deformations induced within the forming phase on a microscale level across the fibrous preform reinforcement have an impact on the permeability of the sheared fabric during the subsequent resin injection phase. Adequate predictions models are conditioned on the premise of having accurate stochastic process permeability models. An adequate characterization of the kernel types and parameters that best capture the deformation information from forming is required, such that subsequent models of resin flow in random heterogeneous media incorporate local fabric material deformation. This constitutes the focus of this part, as we investigate how kernel parameters of random permeability fields in resin flow in heterogeneous fabric media is impacted by the forming processes. Previous studies have examined the effect of local fabric shearing on the permeability tensor of fabrics both using experimental and numerical methodologies [72], [73], [74], 70 [75], [30], [76], [77], [78], [79], [80], [81], [25], [85], [36], [86]. One study investigated the effect of fabric shearing under preform draping on the dual-scale permeability ten- sor (i.e. fiber and tow scales), tow saturation, and resin filling time for different physical shearing models and highlighted the importance of capturing reduced permeability ten- sors associated with high locally sheared regions in order to ensure full saturation of fiber tows (longer filling times) in the final composite part [82]. Another study examined the impact of shearing deformation on the permeability of deformed woven fabrics using 3D Finite Element (FE) analysis with mesoscale RVE models [83]. In [84] an experimen- tal methodology for characterizing the in-plane permeability tensor from unidirectional injection experiments and a corresponding mathematical framework for computing the principal permeability tensor of sheared fabrics was proposed. Challenges that stand in the way of characterizing permeability tensor fields in sheared fabrics include the fact that an accurate experimental measurement of permeability in fabrics is a difficult task that involves a high level of uncertainty. This is coupled with the observation that local shearing of the fabric during forming further adds to the level of uncertainty in local sheared permeability tensors. Several studies related to characterizing flow in random porous media can be found in the literature [42], [87], [88], [89], [90]. In [88], an implementation roadmap of the Stochastic Galerkin (i.e. Spectral Stochastic Finite Element Method) was established. In [42] and [87], a UQ framework for flow in random porous medium was proposed where the spatial flow properties were characterized as random processes, and discretized via KL expansions. Another study used the aforementioned Stochastic Galerkin framework to charac- terize the principal components of the permeability tensor as random processes for resin injection simulations [89]. A recent study conducted one-dimensional analytical and two-dimensional numerical analysis of the filling stage in the RTM problem and 71 observed that the expected value of the filling time is independent of the correlation length of the stationary process of the hydraulic conductivity, but that the variance of the filling time is dependent on the correlation length of the stationary process, which highlights the importance of capturing the spatial variability of constitutive properties and/or the selection of adequate schemes for spatial homogenization [90]. 5.3.0.1 Physics of Fabric Deformation Within the framework of the current application, all the material deformations of the fabric as it is subjected to a forming load within the rigid mold enclosure. The tensor QoI represent, in the forming application, the sheared local permeability tensors of the fabric in resin injection in random heterogeneous media. Let K(x;) denote the corresponding fabric sheared permeability tensor field. Before proceeding further, we review briefly the main assumptions adopted in the application of fabric forming within composite manufacturing: (1) even though nonlin- ear geometry could occur due to large deformations in the forming stage, all microscale constitutive properties of the fabric consist of linear elastic material properties, with absence of plasticity or damage at all stages of fabric deformation. (2) kernel parame- ters of the random permeability fields K(x 0 ) in the deformed sheared configuration are unknown, as the central theme of the proposed methodology is to provide an efficient framework for characterizing such kernel parameters. (3) several sources that might cause material deformation subsequent to fabric forming are ignored; that is, we fo- cus exclusively on material deformation that occurs in the forming phase before heated resin injection. 72 5.3.1 Computational FE Model Classical continuum mechanics cannot be applied for the case of a woven fabric material that includes voids and thus constitutes a non-continuum media. One approach in [34], [35] proposed a multiscale coupling for anisotropic continuum model such as woven fabric medium, whereby an RVE-based multiscaling approach of the dry woven fabric media at every gauss point of a four-node shell element to couple the microscale material constitutive properties of the fibers with the macroscale domain. In another approach, which was adopted in the current paper, a homogenized fabric ply domain was used in the computational forming model. In [91], the constitutive equations of woven fabric forming were formulated, following the small deformation regime assumption, for a prescribed tensile strain along the weft (f 1 ) direction. Following that approach, the stress-strain relationship can be expressed as follows 0 B B B B B B B B B B B B B B B B B B B @ xx yy xy 1 C C C C C C C C C C C C C C C C C C C A = 0 B B B B B B B B B B B B B B B B B B B @ E f 1 bc +( a h )( a 2 c ) ( a h )( b 2 c ) ( a h )( ab c ) ( b a )( a 2 c ) ( b a )( b 2 c )+ E f 2 ac ( b a )( ab c ) ( b h )( a 2 c ) ( b h )( b 2 c ) ( b h )( ab c ) 1 C C C C C C C C C C C C C C C C C C C A 0 B B B B B B B B B B B B B B B B B B B @ " xx " yy " xy 1 C C C C C C C C C C C C C C C C C C C A (5.17) where a, b, c, and h characterize the dimensions of the unit cell of the fabric domain, and =A f 2 E f 2 = p a 2 +b 2 . The computational model of fabric forming is shown in Figure (5.2). Material deformation in the context of the forming application, refers to the local shearing of the fabric as the forming load is incrementally applied and the fabric is forced into the mold enclosure. One common metric of fabric material deformation is the shearing angle, denoted by (n) (x;), which refers to a scalar that quantifies the distortion of the angle between initially orthogonal fibers due to deformation across the n th -ply. Since material fabric deformation is a local process, the shearing angle is thus 73 (a) (b) Figure 5.2: (a) Fabric media discretized with 2-D shell elements, (b) fabric media en- closed within truncated square pyramid molds. represented as a spatial field, defined at x;8x2D, and it is possible to obtain a functional representation as follows (n) (x;)= X (n) (x) (); x2D : (5.18) Given a one-to-one mapping of the shearing angle field (n) (x;) onto the local sheared permeability tensor fieldK (n) (x;), the corresponding full PCE of the permeability tensor field can be expressed as K (n) (x;)= X K (n) (x) (); x2D ; (5.19) from which, the corresponding covariance matrix of the permeability random field can be expressed for any plyn2N p along the stacking thickness using equation (5.10). Several models for quantifying the impact of local fabric deformation in the forming stage on the subsequent flow in porous media parameters can be found in the literature. 74 In particular, one model [36], [37], [38], [39] relates the in-plane sheared permeability tensor K (n) (x;) to the local shear angle(x;), can be formulated as 0 B B B B B B B B B B @ K (n) 1 (x) K (n) 2 (x) 1 C C C C C C C C C C A = 2 0 B B B B B B B B B B @ K (n) 0;1 (x) K (n) 0;2 (x) 1 C C C C C C C C C C A 0 B B B B B B B B B B @ cos 2 ( 4 (n) (x) 2 ) cos 2 ( 4 + (n) (x) 2 ) 1 C C C C C C C C C C A ; x2D ; (5.20) which was used to characterize the impact of forming-related fabric deformations on the principal components of the local permeability tensor at any point x and for each plyn2N p across the thickness. 5.3.2 Stochastic Forming Input The FE forming model described above was parametrized with 6D stochastic dimen- sions. The draping parameters that were part of the random input include: (1) fiber stacking angle, denoted by (n) f 1 and (n) f 2 , (2) tensile modulus , denoted byE (n) f 1 ;t , andE (n) f 2 ;t , and (3) bending modulus, denoted byE (n) f 1 ;b , andE (n) f 2 ;b for each of the initially orthogonal fibers. Subscripts f 1 , f 2 refer to fibers in the warp and weft direction, respectively. The computational forming model was developed using the PAM-COMPOSITES TM solvers [92]; involving the forming of a 45=-45 o preform fabric into mold enclosures following a truncated square pyramid geometry [107]. Table 5.2 shows the input parameters of the forming model that were part of the ran- dom space, and the corresponding probabilistic representation for each, which involves a mix of Uniform probabilistic distributions,U(L;U), and Beta distributionsB(L;U;;), whereL;U represent the bounds for each probabilistic model. 75 Parameter Probabilistic Model E (n) f 1 ;t (GPa) B(57:38;82:62;6;6) E (n) f 2 ;t (GPa) B(57:38;82:62;6;6) E (n) f 1 ;b (MPa) B(40:98;59:01;6;6) E (n) f 2 ;b (MPa) B(40:98;59:01;6;6) (n) f 1 (deg) U(43:00;47:00) (n) f 2 (deg) U(133:00;137:00) Table 5.2: Random input parameters of the 6-D fabric forming model (CoV =5%). 5.3.3 Discretization of Sheared Permeability Process At a point x2D, along a laminae n, a local PCE expansion of the permeability process takes the form k (n) t ()= X j k (n) t;j j (); (5.21) where t denotes a material deformation level at which the high-order PCE surrogate model is obtained. For thek (n) t () QoI, we have k (n) t ()= X i i t;i p t;i ; (5.22) wheref t;i g N i=1 ,f t;i g N i=1 denote, respectively, the eigenfunctions and eigenvalues of the PCE-based surrogate correlation matrix of the form C t;kk t;i = t;i t;i ; (5.23) such that C t;kk = X i1 k (n) t;i k (n) t;i T : (5.24) 76 Solving for i in equation (5.22) results in i = T t;i k (n) t () p t;i = 1 p t;i X j T t;i k (n) t;j j (): (5.25) whereby h i j i= ij ;h i i=0; 8i;j =f1;::;Ng: (5.26) Equation (5.25) can expressed in matrix form as =Φ t Δ 1=2 t k t : (5.27) Once i are evaluated, we can compute realizations of the permeability process based on the optimal kernels. Consider a generalized eigenvalue problem, formulated with respect to a kernel of typek, of the form, C k e k;i = k;i e k;i ; (5.28) once a covarianceC k of the permeability process in the sheared fabric domain is approx- imated using kernel functions, a KL decomposition can be used to discretize following a log-normal process, such that logK 1 (x;)= ¯ K 1 (x)+ K 1 M X i=1 e () k;i (x) i q () k;i ; (5.29) where e () k and () k denote the eigenfunction and eigenvalues of the covariance matrix C k evaluated at optimal kernel parameters b k , and i can be evaluated using equation (5.25). 77 5.4 Results & Discussions 5.4.1 Preliminary Results Uncertainty was propagated following the aforementioned methodology to first obtain PCE surrogate models of shearing angle field (x;) and the corresponding sheared permeability tensor fieldK(x;), followed by the corresponding covariance of the sheared permeability process. After conducting parameteric studies with respect to chaos order and coressponding sparse quadrature leve, a PCE surrogate model was developed using 4 th order polynomial chaos with level four sparse grid quadrature for integration. Figures 5.3(a-f) show the countour plot of the1 st order PCE coefficients of the sheared permeability field, interpolated across the computational domainD e using shape func- tions, following q(x)= X j2D e q j (x;x j ); x2D : (5.30) The following observations can be put forth: (1) the 1 st order PCE coefficients show similar magnitudes for all six random variables, namely E f 1 ;t , E f 2 ;t , E f 1 ;b , E f 2 ;b , f 1 , f 2 ; which highlights that all the forming variables are an integral part of the forward prob- lem, as measured in the sheared permeability field QoI, and (2) to a varying extent, the interpolated PCE coefficients across the domain reflect the actual material deformation pattern, particularly forE f 1 ;t ,E f 2 ;t . Figures 5.3(g,h) show the mean and variance of the sheared permeability process based on the PCE coefficients. The variance of the pro- cess acrossD has higher magnitude in the zones of discontinuity and corners in the sloping sides and upper flanges of the truncated square pyramid domain. Such extreme variance within the parts in the fabric most directly impacted by forming, which corre- sponds to coefficients of variation of up to 180% suggest that fabric forming constitutes 78 (a) (b) (c) (d) (e) (f) (g) (h) Figure 5.3: Interpolated 1 st order PCE coefficients for each stochastic dimension of the forward forming problem (a)E f 1 ;t , (b)E f 2 ;t , (c) f 1 , (d) f 2 , (e)E f 1 ;b , (f)E f 2 ;b ; PCE-based (g) mean and (h) variance of process across the fabric domain. extreme loading that amplifies preexisting uncertainties in the fabric as measured by local transformations of the permeability process along the sheared fabric. Using the 4 th order PCE surrogate model, samples were generated at select few dis- crete points in the computational domainD e . Figure 5.4(a) shows the Probability Den- sity Function (PDF) of local fabric shearing at random points inD e following equation (5.20). A level of fabric shearing equal to unity implies that the local fabric element is unsheared. The farther away from unity the local fabric shearing is, the higher the 79 amount of local material deformation. It can be clearly observed that uncertainty in lo- cal fabric material shearing varies considerably across the fabric domainD. Moreover, it can be seen that the PDF of the sheared local permeability at there five random points is not close to lognormal distribution, which is oftentimes the assumption of probabilis- tic permeability models in porous media. The Q-Q plots of the samples at each of the five discrete points in the domain are shown in figure 5.4(b) with respect to a lognormal distribution. It can be observed that the PDF of local fabric shearing at these five ran- dom points inD deviate rather significantly from the theoretical lognormal distribution. This leads us to reexamine the traditional assertion of adopting a lognormal proba- bilistic model for problems of flow in porous media, whereby for the fabric forming, the actual PDF from the PCE surrogate deviate from the lognormal assumption. Figures 5.6(a,b) show the 4 th -order PCE surrogate covariance matrix (N 2 ;N = 6600) evaluated at two random discrete points in the domainD at the end of forming. It can be observed from figure (5.6b) that negative correlation is found across the domainD with respect a point located in the upper flange of the truncated square pyramid (dark spot shown in figure), as compared to figure (5.6a), where correlation is evaluated with respect to a point located in the lower portion of the truncated pyramid that displaces during forming. In figure (5.5), the eigenvalues and the first four eigenfunctions of the corresponding covariance matrix are plotted along the domainD e . Figure 5.6(c) shows the autocorrelation measured along four random paths and with respect a random source point at a distance. The following observations can be put forth: (1) negative correlation is present in all random paths and at all ranges of distance to source, (2) For points in the process close to the source (i.e. small), autocorrela- tion decays relatively quickly, and (3) points in the process that are relatively far from each other (i.e. large) still show some significant level of autocorrelation. 80 0.0 0.5 1.0 1.5 2.0 2cos 2 ( π 4 ± α (x) 2 ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PDF (x,y)=(0.02L x ,0.30L y ) (x,y)=(0.18L x ,0.18L y ) (x,y)=(0.05L x ,0.55L y ) (x,y)=(0.14L x ,0.94L y ) (x,y)=(0.32L x ,0.62L y ) (a) (b) Figure 5.4: (a) PDF of local fabric shearing 2cos 2 ( 4 (x) 2 ) at five random discrete points, and (b) corresponding Q-Q plots with respect to a lognormal distribution. 81 0 50 100 150 200 250 300 Rank of eigenvalue 10 −1 10 0 10 1 10 2 Eigenvalue (a) (b) Figure 5.5: (a) Eigenvalues in log scale of the high-order, PCE-based covariance matrix, and (b) corresponding first four eigenfunctions. Negative correlations for points in the proces cannot be captured by strictly positive functions such as the Exponential, Gaussian, and Rational Quadratic kernels. That is not the case for the AR2 kernel, that can capture negative autocorrelation at points in the process. This leads us to expect a better performance of the AR2 kernel in approximating the autocorrelation of the process obtained from PCE surrogates. 82 (a) (b) 0.0 0.5 1.0 C t (x i ,x i +ΔΔ 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.1 0.2 0.3 Δ 0.0 0.5 1.0 (c) Figure 5.6: PCE surrogate covariance with respect to (a) a random point in the lower truncated portion, (b) a random point in the upper flange; (c) autocorrelation along random paths with respect to a random source point. 5.4.2 Kernel Optimization & Performance Figures 5.7 shows optimization results for one and two-parameter kernels respectively using the Basin-Hopping stochastic global optimization algorithm for solving the afore- mentioned minimization problem based on autocorrelation objection functions. 83 −0.2 0.0 0.2 |x−y| 0.0 0.5 1.0 Σ E (|x−y|;b * E ) (a) −0.2 0.0 0.2 |x−y| 0.0 0.5 1.0 Σ G (|x−y|;b * G ) (b) −0.2 0.0 0.2 |x−y| 0.0 0.5 1.0 Σ RQ (|x−y|;b * RQ ) (c) −0.2 −0.1 0.0 0.1 0.2 |x−y| 0.00 0.25 0.50 0.75 1.00 Σ AR2 (|x−y|;b * AR2 ) (d) −0.2 −0.1 0.0 0.1 0.2 |x−y| 0.00 0.25 0.50 0.75 1.00 Σ M (|x−y|;b * M ) (e) Figure 5.7: Kernel optimization results; Kernel evaluated at optimal parameters for (a) Exponential, (b) Gaussian, (c) Rational Quadratic, (d) AR2, and (e) Mat´ ern functions. For the one-parameter kernel functions in figures (5.7a-c) evaluated at optimal pa- rameters, the Exponential kernel shows linear behavior at the origin, while the Rational quadratic and Gaussian kernels have essentially parabolic behavior at the origin. Fig- ures (5.7d,e) show the corresponding two-parameter kernel optimization results for the AR2 (3120 iterations) and Mat´ ern (6884 iterations) kernels, respectively. At optimal parameters, the AR2 kernel is the only function to show negative autocorrelation. 84 Table 5.3 shows optimal kernel parameters and corresponding error in terms of ob- jective function evaluated at optimal kernel parameters. The relative error was calcu- lated using equation (5.16). It can be observed that the two-parameter kernel perform significantly better than the one-parameters kernels in terms of error minimization at optimal parameters. In particular, the AR2 kernel shows comparatively the lowest er- ror at optimal parameters, which suggests that the AR2 kernel best approximates the autocorrelation structure of the fabric domain following forming experiment, as mea- sured by a PCE surrogate model. Moreover, for the one-parameter kernel, it can be observed that optimal correlation length is not ubiquitous across all kernel functions investigated, which leads to the assertion that such values should be considered as kernel parameters and not as physical properties of the process. Domain Kernel Optimal Abs. Error Rel. Error Par. (b k ) @b k @b k Exponential fb E g= 0.0193L 3806.52 1.25 Gaussian fb G g= 0.0267L 3677.10 1.21 D y R. Quadratic fb RQ g= 0.0017L 3176.97 1.04 Mat´ ern fb M ;g= 0.0188L, 10.0 3029.05 0.99 AR(2) f! 0 ;g= 0.01, 201.89 2766.16 0.91 Table 5.3: Optimal kernel parameters. y D:(x;y)2[0;L x ][0;L y ]. Figure (5.8) shows the error in approximating the autocorrelation function for each kernel at optimal parameters. The two-parameter kernels (i.e. Mat´ ern and AR2), as com- pared to the one-parameter kernels, better approximates the target correlation structure throught the domain. Moreover, the error in the approximation for the one-parameter kernels seems to be concentrated near the diagonal of the autocorrelation matrix. Once the optimal parameters b k are obtained for each kernel type, the generalized eigenproblem can be solved accordingly. We further obtain the KL decomposition of the zero-mean sheared peremeability field along the fabric domainD following equation (5.29) whereby M=100, K 1 =1:8E-12, and ¯ K 1 (x)=1:8E-11. 85 (a) (b) (c) (d) (e) Figure 5.8: Error in autocorrelation function approximation for different kernels at op- timal parameters, for (a) Exponential, (b) Gaussian, (c) R. Quad, (d) Matern, and (e) AR2 models. Figure 5.9(a) and (b) show the eigenvalues in log scale and the corresponding contri- bution of the eigenvalue, respectively, for each kernel evaluated at optimal parameters. For the AR2 kernel, the eigenvalues decay at a much faster rate than all the other inves- tigated kernels. The first eigenvalue of the AR2 kernel is significantly larger than that of the other optimal kernels, which explains why the select few eigenvalues of the AR2 kernel already capture more than 80% of the variance of the process. Figure 5.10 shows a select few eigenfunctions of each kernel evaluated at optimal parameters. Our main observation from that figure is that, for the AR2 kernel, the first 86 0 200 400 600 800 1000 Rank of eigenvalue 10 8 10 6 10 4 10 2 Eigenvalue Exponential Gaussian R. Quadratic Matern AR2 (a) 0 100 200 300 400 500 Rank of eigenvalue 0.2 0.4 0.6 0.8 1.0 Contribution of λ Exponential Gaussian R. Quadratic Matern AR2 (b) Figure 5.9: (a) Eigenvalues in log scale for each kernel at optimal parameters, and (b) corresponding contribution of ranked eigenvalues to the total variance of the process. eigenmode already exhibits the high fluctuations that are only provided by the other kernels at high enough modes. 5.5 Conclusions & Findings We proposed a methodology for learning kernel functions of spatially varying constitutive material processes, and examined the methodology on the problem of fabric forming within fiber composite manufacturing. 87 Figure 5.10: Select few eigenmodes alongD corresponding to each covariance kernel evaluated at optimal parameters. A spectrum of classical kernel functions were investigated and we identified which kernels best describe random sheared permeability fields for resin flow in random porous fabric media. Within a learning setting, optimal kernel parameters were evaluated us- ing a minimization problem based on a target autocorrelation function obtained from a high-order PCE surrogate model. Of the examined kernel functions, the AR2 process best captures the autocorrela- tion structure of the local sheared permeability process when compared to classical kernels such as Exponential and Gaussian. We observed significant negative correla- tion for points in the sheared permeability process, which partly explains the better performance of the AR2 kernel in terms of accommodating for negative correlation. 88 For solving the generalized eigenproblem on curved domains in space, a geodesic distance subroutine was proposed and embedded within the kernel functions modules. As such, for domains that follow curved polyhedral surfaces in space, the geodesic distance between two points in the process can be evaluated while taking into account the shape of the manifold in question. For all the investigated kernel functions, a relatively high level of relative error was observed, which suggests that, for the fabric forming application in question, kernel functions, even at optimal parameters, still do not fully capture the autocorrelation structure of the sheared fabric medium, as measured via a high-order PCE surrogate model. This is attributed among other to the fact that most kernel functions, with the exception of AR2 kernels, cannot accommodate negative correlation between points in the process, which was quite the case in physical fabric forming problem. The global minization problem was defined with respect to both the full fabric domain as well as a local window of reduced variance. At optimal parameters, the AR2 kernel captures the variance with only a select few eigenmodes, which is not the case for the other kernels. Such an AR2 kernel would entail tremendous computational savings within a stochastic setting whereby such eigenmodes translate into stochastic dimensions. For the forming application, based on the PC surrogate, extreme coefficients of vari- ation, up to 180%, were observed along of the imprint of the mold onto the fabric. Such high levels of uncertainty rendered the kernel optimization particualrly challeng- ing, whereby high relative errors were observed at the end of kernel optimization to a varying extent for all kernels. This suggests that for such fabric forming problems where high uncertainty is introduced with material deformations, discovering the correlation structure of the permeability field using kernels involves error, and the approach is in- herently suboptimal; in other words, kernels at optimal parameters still only capture partially the true correlation structure of the fabric media. That being said, for other, 89 less severe forming applications, errors associated with learning kernels would presum- ably be low, and kernels can be used more effectively to learn with enough accuracy the correlation structure of the media. 90 Chapter 6 Adapted Polynomial Chaos UQ Workflow In this chapter, a scientific workflow for uncertainty propagation in high-dimensional multiphysical problems based on adapted polynomial chaos formulation is presented. The disctinctive features of the proposed UQ workflow are threefold: (1) a basis adap- tation scheme of the homogeneous chaos such that low-dimensional subspaces within which different QoI are embedded are identified, whereby PCE surrogate models are only developed in the adapted, low-dimensional space, and (2) a scheme for identifying redundant quadrature within multiphysical forward problems such that the number of evaluations of partial differential equations along the forward problem is minimized, and (3) an advanced allocation scheme of germs along the multiphysical forward problem. The proposed workflow is ideal for problems that span across multiple physics and that require an integrated uncertainty management approach in a high-dimensional space. The performance of the proposed workflow is validated on a spectrum of practical prob- lems of varying complexity. 6.1 Introduction The use of PCEs within forward uncertainty propagation problems is not new. A spec- trum of commercial and open-source UQ libraries have been developed for the forward uncertainty propagation using PCE surrogate models. For the latter, the advantages of 91 stochastic expansions in general and PCE are well know: first, obtaining a functional dependence of the QoI on the uncertainty in input parameters in the form of a poly- nomial projection onto a basis set, second, having closed-form analytical expression of the statistical moments, and third, having faster convergence as compared to Monte Carlo methods. That being said, for high-dimensional stochastic systems, conventional PCE-based approaches show inherently low convergence rates and are challenged with what is popularly know as the curse of dimensionality. We propose a scientific workflow for uncertainty management, centered on PCE methods, and tailored specifically to multi-physical random systems with inherent high- dimensionality. To achieve these goals in an efficient computational setting, three dif- ferent innovations are proposed: (1) a stochastic dimensionality reduction based on a basis adaptation scheme of the PCE based on the lower-order Gaussian germs, that includes delineating a subspace in parameter space around which a specific QoI is con- centrated and obtaining a probabilistic representation of the solution only in a lower dimensional subspace, (2) a scheme for identifying redundant quadrature in forward evaluations of multiphysical problems involving a sequence of partial different equa- tions, and (3) an advanced scheme for allocating germs from quadrature to different physical models of the forward problems, which allows for a highly flexible assigning of germs to corresponding random physical variables at any phase of the multiphysical forward problem. As such, for integrated, high-dimensional problems, the proposed UQ workflow al- lows for efficient and accurate predictions of different QoI at a fraction of the compu- tational cost of conventional PCE-based methods or MC/QMC sampling. The proposed workflow provides a valuable tool for performing integrated uncertainty propagation whereby uncertainty needs to be assessed in a high-dimensional setting, over a se- quence of physical models. The proposed workflow could be ultimately used, beyond 92 forward uncertainty propagation, for inverse problems, design exploration, optimiza- tion under uncertainty, among other. 6.2 Review of UQ Libraries A detailed review of each commercial or open-source UQ library is beyond the scope of this work and can be found in [93], in which a detailed and comprehensive assessment of UQ libraries was provided. We only briefly review the main public or commercial UQ libraries. Major UQ libraries that have PCE-based intrusive or non-intrusive forward uncertainty propagation functionalities are presented next [94, 95, 96, 97]. It should be noted that UQ libraries are in ever constant state of evolution, and the following review of common UQ libraries is limited by the scope of the time at which this manuscript is prepared as the features of UQ libraries mentioned below might evolve in the future. DAKOTA, developed at the Sandia National Laboratories, provides comprehensive ca- pabilities for forward uncertainty propagation, in addition to sensitivity analysis, op- timization under uncertainty, design exploration, among other; it is bundled within a framework scalable to HPC platforms [94]. UQTk proposes a set of functionalities related to forward uncertainty propagation, sensitivity analysis, and surrogate model building using PCE-based surrogate models among other [95]. In the OpenTURNS software, a com- prehensive collection of probabilistic tools for uncertainty propagation is proposed inC++ libraries and python modules [96]. Stokhos provides a framework for intrusive uncer- tainty quantification, including PCE-based stochastic Galerkin methods; it is scalable to HPC platforms as it leverages Trilinos’ large-scale libraries [97]. QUESO focuses on solving large-scale, parallelized, computationally demanding mod- els inverse problems based on Bayesian Inference formulation [98]. A similar flexible python-based package tailored for Bayesian inference and MCMC is proposed in [103]. 93 In the PSUADE software, non-intrusive UQ, optimization under uncertainty, sensitivity analysis, Bayesian inference, among other utilities, are provided [99]. In NESSUS, a prob- abilistic FE analysis software platform is introduced, including capabilities for design of experiments, and an interface to commercial FE codes, among other [100]. In COSSAN, a general purpose computational framework is provided, designed for risk assessment and mitigation and uncertainty management [101]. SIMLAB provides a comprehensive software for global sensitivity analysis; it is implemented in the R environment [102]. What is proposed in this work builds on the limitations of the aforementioned UQ libraries as follows. First, to tackle the curse of dimensionality, a dimension reduction scheme via adaptation of the Gaussian basis of the PCE is proposed such that each QoI is only formulated in a low-dimensional appropriate subspace. Second, to reduce the computational cost of forward evaluations, for multiphysical forward problems where redundant quadrature may appear, a method for identifying and pruning out unecessary quadrature is proposed. Third, to provide an adequate probabilistic formulation of the forward problem, an advanced allocation scheme of quadrature germs is proposed such that random physical variables across physical models in a sequential forward problem can easily share germs; alternatively, subsets of germs could be allocated to different random variables along any physical model of the forward problem. In [110], an intrusive approach for quadrature reduction in network coupled multi- physical systems was proposed in such as way so that a surrogate model can be con- structed with less forward PDE evaluations. In [111, 112], a dimension reduction via adaptation of the Karhunen-Loeve decomposition was proposed within coupled multi- physical random systems. Through this chapter, our objective is describe an adapted PCE-based UQ workflow tailored for high-dimensional, multiphysical forward problems. Such cases whereby 94 physical problems in question span across different physics and scales and are com- posed of interconnected physical models and thus require holistic uncertainty assess- ment are quite common and frequently encountered in a spectrum of disciplines and practical applications; and we aim to propose a scientific workflow is ideally suited for uncertainty propagation in such applications. Moreover, one goal of the proposed UQ workflow is to be as abstract and problem-independent as possible in such a way so as to be applicable on a wide array of physical problems. A generic description of the formulation of the PCE-based UQ workflow is provided, without delving into the impl- mentation, which is beyond the scope od this work. 6.3 Description We propose a scientific workflow for uncertainty management, centered on PCE meth- ods, and tailored specifically to multi-physical random systems with inherent high- dimensionality. To achieve this, two different schemes are proposed: (1) a stochastic dimensionality reduction based on an adaptation of the lower-order Gaussian basis, that includes delineating a subspace in parameter space around which a specific QoI is concentrated and obtaining a probabilistic representation of the solution only in a lower dimensional subspace [33], and (2) a scheme for identifying and pruning out redundant quadrature in multiphysical problems involving a sequence of partial differential equa- tions. As such, for integrated, high-dimensional problems, the proposed UQ workflow allows for computationally-efficient predictions as compared to conventional PCE. The proposed workflow provides a valuable tool for performing integrated uncertainty propagation whereby uncertainty needs to be assessed in a high-dimensional setting, over a sequence of physical models. While the proposed workflow is centered on an efficient formulation of the forward problem, it could be ultimately used, beyond forward 95 uncertainty propagation, for inverse problems, design exploration, optimization under uncertainty, among other UQ-related tasks. A high-level description of the in-house UQ workflow software in a typical stochastic analysis is shown in Figure (6.1). Data is obtained from physical experiments. Con- straints are synthesized from some data components. Together, data and constraints are used to infer the probability distributions of model parameters. The constraints typ- ically serve to introduce statistical dependence between these parameters. The stochas- tic parameters are then pushed through the model to obtain a statistical characteriza- tion of the QoI. A key challenge in this description is that a statistical characterization of parameters and QoI requires significant computational effort that quickly becomes prohibitive as the number of parameters increases. This is particularly true for prob- lems where the tail of the QoI is of interest (such is the case for failure events). This UQ software addresses this challenge, using the basis adaptation feature of polynomial chaos approximations. MODEL PARAMETERS QUANTITIES OF INTEREST (QOI) DATA CONSTRAINTS Figure 6.1: High-level flow of information in a stochastic simulation. The adapted PCE approach describes all QoI as low-dimensional mappings from a set of Gaussian germ. This germ is a vector of Gaussian random variables used to shape the distribution of all the QoI. The shaping is done using the simulation tools that embody the governing equations and additional expert knowledge. At first, the germ is mapped to another germ following equation (3.15), whereA is a rotation, so that is a standard orthogonal Gaussian vector. The rotationA is designed so that very few of the new germ are sufficient to characterize the QoI. In the workflow, the germ 96 is thus and the numerical quadrature is carried out in a lower-dimensional setting. A transformation A 1 is used to obtain (q) for each realization of (q) . The physical parameters for theq th quadrature node are obtained by mapping (q) , and theq th QoI are obtained by mapping these parameters. The dimension of the reduced germ is increased until sufficient accuracy is achieved (small change between successive iterations). In the limit, when the reduced germ is equal to the whole vector, the original problem is recovered exactly, with no computational savings. 6.4 UQ Workflow Formulation In figure (6.2), a detailed illustration of the proposed UQ workflow is shown, consisting of a mapping from the germ to the QoI. A step-by-step description of the proposed UQ workflow is provided next. MODEL m 1 m 1 m 3 m 3 m 2 m 2 f 1 f 1 f 2 f 2 f 3 f 3 f 4 f 4 INPUT PARAMETERS INPUT FILES ··· PCE GERM ··· QOI ··· ⇠ n ···⇠ n+r ⇠ 1 ⇠ m ···⇠ m+s p 1 p 2 p 3 (x) Q 1 Q 2 Q l ⇠ 2 ,···⇠ 2+d (p 4 ,··· ,p 4+j ) Figure 6.2: Details of maps from standard input variables (germ) to quantities of interest (QoI). 97 Germ-to-Input parameters The germs are first mapped into the physical parameters required by the simulation codes. The mappings from the germs to these parameters are designed to satisfy various requirements on these parameters. Specifically, the mapping from the PCE Germs to the Parameters can be designed to meet the following potential requirements: 1. one of the parameters is specified through its univariate probability density func- tion. In this case, denoting the CDF of the scalar parameter byF P (p) and the CDF of the univariate Gaussian by(), the map from top is computed as P =F 1 P () (6.1) where the inverse CDFF 1 P is evaluated numerically. In this case, one of the PCE germ variables is allocated to this particular parameterP . 2. several parameters (e.g. s) are specified as joint random variables through a joint probability distribution. In this case the Rosenblatt transform is used to map the vector of s random variablesP to a vector of s independent normalized Gaussian variables. A subset of s variables from the PCE germ set is allocated for these s parametersP . 3. one or several parametersP (eg. s) are specified through a polynomial chaos ex- pansion (PCE) which was obtained through some other mechanism. The PCE is expressed as a polynomial in d independent standard Gaussian variables (d not necessarily equal tos) resulting in P = X P (): (6.2) 98 Each coefficientP in this expansion is a vector of dimensions. Complete specifi- cation of this option comes with two files. One file contains the numerical values of the coefficientsP and another file contains the multi-indices. In this case, a subset ofd variables from the PCE germ set is allocated for the random vectorP . 4. one of the parameters is specified as stochastic process via its Karhunen-Loeve expansion. A subset of the PCE germ variables is allocated to the random variables appearing in the KL expansion. Thus assume that the KL expansion of random processp(x) has the form, p(x)= ¯ p(x)+ N X i=1 i e i (x) (6.3) wheref i g is a set of uncorrelated random variables andfe i g is a set of orthogonal determnistic functions. In this case, the functionsfe i g would be known and avail- able through a file or a script. Likewise, the joint density function of the random variablesf i g would be known (as part of specifying the KL expansion). A Rosen- blatt transform or a PCE is then used to map a subset of the Gaussian germ onto variablesf i g in accordance with the joint CDF of these variables. The user would then provide a script that synthesizes the summation in the KL expansion for each realization of the variablesf i g. Each of these realizations must then be draped over the spatial discretization as specified by the analysis code. The details of this specification is dependent on the analysis code as it requires access to specific properties at the element level. 5. one of the parameters is specified as stochastic process via its PC expansion. The expansion of the random process looks then as follows, p(x)= X p (x) (): (6.4) 99 In this case, the coefficientsp are provided in a file, with the same discretization as p(x), while the multi-indices used in the expansion are provided in a separate file. A subset of the PCE germ is again allocated to this stochastic process. As with the case of the KL expansion, the details of this implementation is case-dependent as it requires knowledge of and access to the specific discretization scheme. In the above specifications, a probability distribution can be replaced by data, thus requiring an additional intermediate step to compute statistics from the data; these statistics are then used to constrain the PCE representation. Input parameters-to-Input files Each of the random input parameters is associated with an input file for one of the simulation codes. The files are denoted by f i in figure (6.2). Several parameters may be mapped into a single file. This association between parameters and files is controlled in the input deck user.i, as explained below. Input file-to-Simulation code The association from input files to simulation code is implicit in the calling sequence to each analysis code and does not require any particular attention from the UQ workflow. Simulation code-to-QoI The mapping from the simulation code to QoI is the respon- sibility of the user who should provide a script that outputs one or several QoI following each simulation. 6.5 Complexity, Error Formulation, & Convergence In this section, a complexity analysis on the PCE adaptation scheme if presented first, followed by error formulation and convergence analysis. 100 Complexity Analysis Let d denote the original stochastic dimension of the random system at hand, and d o the corresponding reduced dimensionality following the adap- tation operation, and p the highest order of the adapted PCE. The worst-case complexity of the adaptation operation of the homogeneous chaos of the PCEcan be measured by the total number of basis terms required to obtain a PCE. By neglecting the last term of equation (3.19) corresponding to the setf:1<jjP :<Ig, the worst-case complexity of the adaptation formulation can be expressed as O h q A () i =O 2 6 6 6 6 6 6 6 6 6 6 6 4 q A 0 +q A 1 1 + X 2I 1<jjP q A () 3 7 7 7 7 7 7 7 7 7 7 7 5 (6.5) =O " d+1 d !# +O " d o + p d o !# =O " (d+1)! d! # +O " (d o + p)! d o !p! # =O[d+1]+O " 1 p! (d o + p)(d o + p1):::(d o +1) # O[d]+O " 1 p! (d o ;d o ;:::;d o ) # =O[d]+O d p1 o : From the aforementioned description of the adaptation problem, it is evident that once the Gaussian (i.e. linear) solution is obtained, the higher-order adapted solution in the low-dimensional subspace with respect to a specific QoI is completely independent and thus can be embarrassingly parallelized. Error Formulation Let q I =fq I ; 2Ig denote the projection of q ontoV I , where q = fq I ;2I p g. The error associated with the project onV I can be formulated as qq I =[ICC T ]q (6.6) 101 whereC, the Gramian, can be evaluated asC =< ; A >, or expectance with respect to the Gaussian measure. For the set of orthonormal Hermite polynomials that form the basis of the PC ex- pansion, it is the case that for homogeneous chaos with even degree, the corresponding polynomial chaos are not centered, as shown in table 3.1 and Figure A.2. This is not the case for homogeneous chaos with odd degree. Within the basis adaptation context, the fact that the even-degree homogeneous chaos are non-centered creates leftover terms within the projection scheme when higher dimensional terms are zeroed out. Convergence Analysis For a given QoIq for a given physical problem, such a QoI can be embedded in its own lower-dimensional subspace of dimensionk, denoted by k . In the current work, as in the literature, the lower dimensional subspace k for each QoI q is evaluated in an iterative scheme, starting with a 1-D-space (i.e. 1 )and incremen- tally going into higher subspaces until convergence via some metric is achieved. Given the reduced dimension d o , one convergence metric is based on the l 2 norm of the PCE coefficientsq of the adapted solution at two consecutive iterations, as follows q (k)= jjq (k) q (k1) jj 2 jjq (k1) jj 2 ; fork2f1::dg; (6.7) that is, k represents the dimension of the adapted PCE in the -space, which also de- notes the index of the adaptive iteration. Convergence in relative l 2 error between the adapted, low-dimensional chaos coefficients was obtained after mapping the chaos co- efficients from the reduced-space back to the original-space wherel 2 relative error was computed. 102 Chapter 7 Conclusions In this work, we tackle the problem of performing integrated uncertainty assessment in multiphysics and multiscale RTM composite manufacturing models. The latter en- compasses essentially all the phases of the RTM process, namely fabric forming, resin injection, curing, and ply distortion. We formulate and adopt an integrated uncertainty propagation framework, tailored for multiscale and multiphysics applications. The in- tegrated uncertainty propagation framework is centered on Polynomial Chaos methods, as well as an adaptation scheme of the Gaussian basis of the chaos expansions such that the proposed integrated UQ framework scales well with high-dimensional prob- lems such as the RTM problem at hand. All the variables involved across different scales, physics, and manufacturing phases of the RTM process are parametrized as part of an integrated, high-dimensional, forward problem formulation. The latter is fully automated and embedded within a polynomial chaos UQ workflow with dimension reduction via an adaptation of the PCE basis; which allowed us to develop surrogate models of predefined QoI along the RTM manufacturing sequence. Comprehensive un- certainty assessment allowed us to evaluate the contribution of each variable along the RTM manufacturing phases and across multiple scales and physics to the uncertainty in the QoI. 103 Our second effort revolved around proposing a generic stochastic framework for char- acterizing random material fields in deformed heterogeneous media in multiphysical set- tings. We used, for that matter, the proposed framework to understand the role of spatial random material fields within the RTM process, namely the random permeability field in the resin injection phase. This originated from the observation that the probabilistic distribution of different QoI are sensitive to the underlying parameters of the covariance kernel. As such, adequate uncertainty propagation in the RTM problem necessitates appropriate models of random material fields, in particular for the random permeabil- ity fields in resin injection following fabric forming. For the latter, we computed optimal parameters for off-the-shelve covariance kernels of the permeability random field with respect to the covariance matrix obtained from a PCE surrogate model of fabric forming and local material deformation. Our last effort was devoted to proposing, within and outside the scope of the RTM composite manufacturing application, an in-house, generic UQ scientific workflow, tai- lored for high-dimensional, mutiphysical random systems, and based on the aforemen- tioned polynomial chaos and basis adaptation machinery. We proposed a light software, for that matter, easily deployable and scalable from personal computers to HPC servers. We solved a particularly challenging problem of integrated uncertainty management in the multiscale, multiphysics RTM composite manufacturing processes. Within the RTM application, potential implications of the findings highlighted in this work include: (1) proposing an efficient Modeling & Simulation scheme for exploring parameter and de- sign spaces, (2) establishing a milestone towards a comprehensive design methodology for fiber composites, (3) increasing performance predictability whereby uncertainty is taken into account, thus reducing the potential for overdesign and high safety factors to 104 account for poor uncertainty assessment, (4) higher confidence in computational manu- facturing predictions, (5) and finally, paving the way for widespread usage of lightweight composite components in different applications. 105 Reference List [1] J. Hawk, The Boeing 787 Dreamliner: More Than an Airplane. AIAA/AAAF Aircraft Noise and Emissions Reduction Symposium, American Institute of Aeronautics & Astronautics, 2005. [2] The Boeing Company, 787 Dreamliner Program Fact Sheet. [3] D. Brosius, Can we achieve global standards for composites?, CompositesWorld, 2018. [4] Laboratory for Energy & the Environment, On the road in 2035. MIT, 2008. [5] Vehicle Technologies Office, Light-duty vehicles technical requirements and gaps for lightweight and propulsion materials, US DoE, 2013. [6] Vehicle Technologies Office, Materials technologies goals, strategies, and top accom- plishments, US DoE, 2010. [7] CompositesWorld, BMW formally launches i3 manufacture and assembly, 2013. [8] G. Warwick, Automotive Industry Embraces Composites; Can Aerospace Benefit?, Avi- ation Week, 2015. [9] Quarterly report on the first quadrennial technology review. U.S. Department of En- ergy, 2011. [10] National Research Council, Application of lightweighing technology to military air- craft, vessels, and vehicles, 2012. [11] Glenn Research Center, Integrated composite analyzer users guide, NASA. [12] P. Murthy, C. Ginty, J. Sanfeliz, Second generation integrated composite analyzer (ICAN) computer code, Lewis Research Center, NASA, 1993. [13] C.C. Chamis, Probabilistic simulation of multi-scale composite behavior, Theoretical and Applied Fracture Mechanics, 41, p. 51-61, 2004. [14] C.C. Chamis, Probabilistic simulation of multi-scale composite behavior, Glenn Re- search Center, NASA, 2006. [15] J. Lemonds, V. Kumar, Probabilistic analysis of large-scale composite structures us- ing the IPACS code, Lewis Research Center, NASA, 1995. [16] L. Sobel, C. Buttitta, J. Suarez, Probabilistic and structural reliability analysis of laminated composite structures based on the IPACS code, NASA, 1993. [17] K. Damadeo NASA looking to tiny technology for big payoffs, Langley Research Cen- ter, 2018. [18] J. Buck, C. Rink, NASA selects proposals for ultra-lightweight materials for journey to mars and beyond, 2015. 106 [19] ExecutiveGov NASA scientists seek to leverage nanoscale materials to facilitate spacecraft flights, 2018. [20] F.L. Mathews , R.D. Rawlings, Composite materials: engineering and science. Wood- head Publishing, 2006. [21] K.T. Hsiao, S.G. Advani, Flow sensing and control strategies to address racetracking disturbances in RTM Part I: design and algorithm development. Composites: Part A, 35, 1149 - 59, 2004. [22] D.R. Nieslon, R. Pitchumani, Control of flow in RTM with real-time preform perme- ability estimation. Polymer Composites, 23, 1087-1110, 2002. [23] D.H. Lee , W.I. Lee , M.K. Kang, Analysis and minimization of void formation during resin transfer molding process. Compos Sci Technol, 66, 3281-9, 2006. [24] K. Hoesa , D. Dinescu , H. Sol , V. Marleen , R. Parnas , Y. Luo , I. Verpoest, New set-up for measurement of permeability properties of fibrous reinforcements for RTM. Composites: Part A, 33, 959 - 69, 2002. [25] A. Endruweit , P. McGregor , A.C. Long , M.S. Johnson, Influence of the fabric archi- tecture on the variations in experimentally determined in-plane permeability values. Compos Sci Technol, 66, 1778 - 92, 2006. [26] B. Markicevic , D. Heider , S.G. Advani, S. Walsh, Stochastic modeling of preform heterogeneity to address dry spots formation in the VARTM process. Composites: Part A, 36, 851 - 8, 2005. [27] S.K. Padmanabhan , R. Pitchumani, Stochastic modelling of nonisothermal flow dur- ing resin transfer molding. Int J Heat Mass Transfer, 42, 3057 - 70, 1999. [28] B. Verleye , S.V. Lomov , A. Long , I. Verpoest , D. Roose, Permeability prediction for the meso-macro coupling in the simulation of the impregnation stage of RTM. Compos- ites: Part A, 41, 29 - 35, 2010. [29] J. Li , C. Zhang , R. Liang , B. Wang, Permeability prediction for the meso-macro coupling in the simulation of the impregnation stage of RTM. Composites: Part A, 36, 564 - 80, 2005. [30] A.A. Skordos , M.P.F. Sutcliffe, Stochastic simulation of woven composites forming. Compos Sci Technol, 68, 283 - 96, 2008. [31] F. Zhang , B. Cosson , S. Comas-Cardona , C. Binetruy, Efficient stochastic simula- tion approach for RTM process with random fibrous permeability. Compos Sci Technol, 71, 1478-85, 2011. [32] T.S. Mesogitis , A.A. Skordos , A.C. Long, Uncertainty in the manufacturing of fibrous thermosetting composites: a review. Composites: Part A, 57, 67 - 75, 2014. [33] R. Tipireddy , R. Ghanem, Basis adaptation in homogeneous chaos spaces. Journal of Computational Physics, 259, 304 - 317, 2014. [34] B. Kaiser , E. Haug , T. Pyttel , F. Duddeck, A coupled multi-scale approach for the simulation of textile membranes. 7th Int Conf Textile Compos, Barcelona, Spain, 2015. 107 [35] B. Kaiser , T. Pyttel , E. Haug , F. Duddeck, Draping simulation with a new finite element formulation involving an internal unit cell. Proc 20th Int ESAFORM Conf Mat Form, Dublin, Ireland, 2017. [36] S. Aranda , D.C. Berg , M. Dickert , M. Drechsel , G. Ziegmann, Influence of shear on the permeability tensor and compaction behaviour of a non-crimp fabric. Composites Part B, 65, 158 - 163, 2014. [37] P. Smith , C.D. Rudd , A.C. Long, The effect of shear deformation on the processing and mechanical properties of aligned reinforcements. Composites Science and Tech- nology, 57(3), 327 - 344, 1997. [38] X. Zeng , A. Endruweit , L.P. Brown , A.C. Long, Numerical prediction of in-plane per- meability for multilayer woven fabrics with manufacture-induced deformation. Com- posites Part A, 77, 266 - 274, 2015. [39] A.C. Long , C.D. Rudd , M. Blagdon , P. Smith, Characterizing the processing and performance of aligned reinforcements during preform manufacture. Composites Part A, 27(4), 247 - 253, 1996. [40] M.R. Kamal , S. Sourour, Kinetics of thermal characterization of thermoset cure. Polymer Eng Sci, 13, 59 - 64, 1973. [41] N. Wiener, The homogeneous chaos. American Journal of Mathematics, 60, 897 - 936, 1938. [42] R. Ghanem, Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resources Research, 34(9), 2123 - 2136, 1998. [43] R. Ghanem , P.D. Spanos, Stochastic finite element: a spectral approach. Dover Publications,2003. [44] H.N. Najm, Uncertainty quantification and polynomial chaos techniques in computa- tional fluid dynamics. Annual Review of Fluid Mechanics, 41(1), 35-52, 2009. [45] M.T. Reagan , H.N. Najm , R. Ghanem , O.M. Knio, Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection. Combustion and Flame, 132(3), 545 - 555, 2003. [46] O. LeMaitre , O. Knio, Spectral methods for uncertainty quantification. Springer, 2010. [47] P. Tsilifis , R. Ghanem, Reduced Wiener chaos representation of random fields via basis adaptation and projection. Journal of Computational Physics, 341, 102 - 120, 2017. [48] M. Buhmann, Radial basis functions: theory and implementations. Cambridge Uni- versity Press 341, 2003. [49] A. Endruweit , P. Ermanni, The in-plane permeability of sheared textiles. Experimen- tal observations and a predictive conversion model. Composites Part A, 35(4), 439 - 451, 2004. [50] S. Bickerton , P. Simacek , S.E. Guglielmi , S.G. Advani, Investigation of draping and its effects on the mold filling process during manufacturing of a compound curved composite part. Composites Part A, 28(9-10), 801 - 816, 1997. 108 [51] R. Ghanem , C. Soize, Remarks on stochastic properties of materials through finite deformations. Int J Multiscale Computational Eng, 13(4), 367 - 374, 2015. [52] X. Dongbin , G. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM Journal on Scientific Computing, 24(2), 619-644, 2002. [53] D. Kirsanov, Minimal discrete curves and surfaces, PhD thesis, Applied Mathemat- ics, Harvard University, 2001. [54] R. Kimmel , K. Nahum, Finding the shortest paths on surfaces by fast global approx- imation and precise local refinement, International Journal of Pattern Recognition and Artificial Intelligence, 10(6), 643-656, 1996. [55] R. Kimmel , J.A. Sethian, Computing geodesic paths on manifolds, National Academy of Sciences of the USA 95, 8431 - 8435, 1998. [56] J.A. Sethian , A. Vladimirsky, Fast methods for the Eikonal and related Hamilton– Jacobi equations on unstructured meshes, Proceedings of the National Academy of Sciences 97, 5699 - 5703, 2000. [57] F. Memoli , G. Sapiro, Distance functions and geodesics on point clouds, Technical Report 1902, University of Minnesota, Institute of Mathematics and its Applications, 2002. [58] A. Spira , R. Kimmel, An efficient solution to the eikonal equation on parametric manifolds, Interfaces and Free Boundaries, 6(4), 315 - 327, 2004. [59] O. Weber , Y. Devir , A. Bronstein , M. Bronstein , R. Kimmel, Parallel algorithms for approximation of distance maps on parametric surfaces, ACM Transactions on Graphics, 27(4), 1 - 16, 2008. [60] M. Sharir , A. Schorr, On shortest paths in polyhedral spaces, SIAM Journal on Computing, 15, 193 - 215, 1986. [61] J. Hershberger , S. Suri, Practical methods for approximating shortest paths on a convex polytope in R3, Computational Geometry Theory and Applications, 10(1), 31 - 46, 1998. [62] P. Agarwal , S. Har-Peled , M. Sharir , K. Varadarajan, Approximating shortest paths on a convex polytope in three dimensions, Journal of the ACM, 44(4), 567 - 584, 1997. [63] J. Mitchell , D. Mount , C. Papadimitriou, The discrete geodesic problem, SIAM Journal on Computing, 16, 647 - 668, 1987. [64] Y. Schreiber , M. Sharir, An optimal-time algorithm for shortest paths on a convex polytope in three dimensions, Discrete and Computational Geometry, 39, 500 - 579, 2008. [65] J. Rourke , S. Suri , H. Booth, Shortest paths on polyhedral surfaces, Symposium on Theoretical Aspects in Computer Science, 243 - 254, 1985. [66] V. Surazhsky , T. Surazhsky , D. Kirsanov , S.J. Gortler , H. Hoppe, Fast exact and approximate geodesics on meshes, ACM Transactions on Graphics, 24(3), 553 - 560, 2005. 109 [67] C. Jindong , Y. Han, Shortest paths on a polyhedron; Part I: computing shortest paths, International Journal of Computational Geometry and Applications, 6(2), 127 - 144, 1996. [68] K. Varadarajan , P. Agarwal, Approximating shortest paths on an nonconvex polyhe- dron, SIAM Journal on Computing, 30(4), 1321 - 1340, 2000. [69] J. Mitchell , C. Papadimitriou, The weighted region problem: finding shortest paths through a weighted planar subdivision, Journal of the ACM, 38(1), 18 - 73, 1991. [70] M. Lanthier , A. Maheshwari , J.R. Sack, Approximating weighted shortest paths on polyhedral surfaces, Algorithmica, 30(4), 527 - 562, 2001. [71] R. Ghanem , J. Red-Horse , A. Benjamin , A. Doostan , Y. Zou, Stochastic process model for material properties under incomplete information, 48th AIAA Structural, Structural Dynamics, and Materials Conference, Hawaii, 2007. [72] A. Endruweit , A.C. Long, Influence of stochastic variations in the fibre spacing on the permeability of bi-directional textile fabrics, Composites: Part A, 37, 679 - 694, 2006. [73] Q. Govignon , S. Bickerton , P.A. Kelly, Simulation of the reinforcement compaction and resin flow during the complete resin infusion process, Composites: Part A, 41, 45 - 57, 2010. [74] M. Grujicic , K.M. Chittajallu , S. Walsh, Effect of shear, compaction and nesting on permeability of the orthogonal plain-weave fabric preforms, Mat Chem and Phys, 86, 358 - 369, 2004. [75] B. Martin , S. Comas-Cardona , C. Binetruy , N. Billon , J.L. Bouvard, Influence of fabrics design parameters on the morphology and 3D permeability tensor of quasi- unidirectional non-crimp fabrics, Composites: Part A, 90, 470 - 479, 2016. [76] M.W. Tahir , F. Stig , M. Akermo , S. Hallstrom, Influence of fabrics design parame- ters on the morphology and 3D permeability tensor of quasi-unidirectional non-crimp fabrics, Composites: Part A, 74, 18 - 25, 2015. [77] M. Yun , T. Carella , P. Simacek , S. Advani, Stochastic modeling of through the thickness permeability variation in a fabric and its effect on void formation during vacuum assisted resin transfer molding, Compos Sci Tech, 149, 100 - 107, 2017. [78] M. Grujicic , K.M. Chittajallu , S. Walsh, Lattice boltzmann method based compu- tation of the permeability of the orthogonal plain-weave fabric preforms, J Mater Sci, 41, 7989 - 8000, 2006. [79] M. Nordlund , T.S. Lundstrom, Numerical study of the local permeability of noncrimp fabrics, J Comp Mat, 39(10), 929 - 947, 2004. [80] F. Zhou , N. Kuentzer , P. Simacek , S. Advani , S. Walsh, Analytical characterization of the permeability of dual-scale fibrous porous media, Compos Sci Tech, 66, 2795 - 2803, 2006. [81] M. Louis , U. Huber, Investigation of shearing effects on the permeability of woven fabrics and implementation into LCM simulation, Compos Sci Tech, 63, 2081 - 2088, 2003. 110 [82] J. Walther , P. Simacek , S.G. Advani, The effect of fabric and fiber tow shear on the dual scale flow and fiber bundle saturation during liquid molding of textile composites, Int J Mater Form, 5, 83 - 97, 2012. [83] F. Loix , P. Badel , L. Orgeas , C. Geindreau , P. Boisse, Woven fabric permeability: from textile deformation to fluid flow mesoscale simulations, Compos Sci Tech, 68, 1624 - 1630, 2008. [84] C. Demaria , E. Ruiz , F. Trochu In-plane anisotropic permeability characterization of deformed woven fabrics by uni-directional injection. Part II: prediction models and numerical simulations, Pol Compos, 28(6), 797 - 811, 2007. [85] G. Rieber , J. Jiang , C. Deter , N. Chen , P. Mitschang, Influence of textile parameters on the in-plane permeability, Composites: Part A, 52(6), 89 - 98, 2013. [86] F.D. Dungan , M.T. Senoguz , A.M. Sastry , D.A. Faillaci, On the use of darcy per- meability in sheared fabrics. J Reinforced Plast Comp, 18(5), 472 - 484, 1999. [87] R. Ghanem , S. Dham, Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media, Transport in Porous Media, 32(3), 239–262, 1998. [88] R. Ghanem, Ingredients for a general purpose stochastic finite elements implemen- tation, Computer Methods in Applied Mechanics and Engineering, 168(1), 19 - 34, 1999. [89] F. Zhang , B. Cosson , S. Comas-Cardona , C. Binetruy, Efficient stochastic simula- tion approach for RTM process with random fibrous permeability, Compos Sci Technol, 71, 1478 - 85, 2011. [90] M. Park , V. Tretyakov, Stochastic resin transfer molding process, SIAM/ASA J Un- certainty Quantification, 168(1), 1110 - 1135, 2017. [91] W.R. Yu , F. Pourboghrat , K. Chung , M. Zampaloni , T.J. Kang, Non-orthogonal constitutive equation for woven fabric reinforced thermoplastic composites, Composites Part A, 33, 1095 - 1105, 2002. [92] ESI GROUP. PAM-RTM users guide & tutorials, version 2018, ESI Group, Paris, 2018. [93] R. Ghanem , D. Higdon , H. Ohwadi, Handbook of Uncertainty Quantification. Springer, 2017. [94] B.M. Adams , L.E. Bauman , W.J. Bohnhoff , K.R. Dalbey , J.P. Eddy. , M.S. Ebeida , M.S. Eldred , P.D. Hough , K.T. Hu , J.D. Jakeman , L.P. Swiler , D.M. Vigil, DAKOTA: A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis, Sandia Technical Report SAND2010-2183, 2013. [95] B. Debusschere , K. Sargsyan , C. Safta , K. Chowdhary, UQ toolkit, Sandia, 2015. [96] A. Dutfoy , I. Dutka-Malen , A. Pasanisi , R. Lebrun , F. Mangeant , J.S. Gupta , M. Pendola , T. Yalamas, OpenTURNS, an open source initiative to treat uncertain- ties, Risks and statistics in a structured industrial approach, Journees de Statistique, Bordeaux, 2009. [97] E.T. Phipps, Stokhos, embedded uncertainty quantification methods, , Trilinos, 2015. 111 [98] K.C. Estacio-Hiroms , E.E. Prudencio, QUESO, Quantification of uncertainty for es- timation, simulation, and optimization , Users Manual, 2015. [99] Lawrence Livermore Nat Lab, PSUADE, Problem solving environment for uncertainty analysis and design exploration, 2018. [100] Southwest Research Institute, NESSUS, Numerical evaluation of stochastic struc- tures under stress, 2018. [101] E. Patelli , H.M. Panayirci , M. Broggi , B. Goller , P.B.H.J. Pradlwarter , G.I. SchuÃńller, COSSAN, General purpose software for efficient uncertainty management of large finite element models, Finite Elem. Anal. Des, 51, 31 - 48, 2012. [102] European Commission, Joint Research Centre, SIMLAB: Sensitivity analysis soft- ware, 2015. [103] J. Salvatier , T.V. Wiecki , C. Fonnesbeck, Probabilistic programming in Python using PyMC3, PeerJ Computer Science, 2, 2016. [104] O.G. Ernst, A. Mugler, H.J.Starkloff, E. Ullmann, On the convergence of general- ized polynomial chaos expansions, ESIAM: Mathematical Modelling and Numerical Analysis, 46(2), pp. 317 - 339, 2012. [105] A.T. DiBenedetto, Prediction of the glass transition temperature of polymers: A model based on the principle of corresponding states, Journal of Polymer Science: Part B, 25, pp. 949 - 1969, 1987. [106] J.M. Svanberg, Predictions of manufacturing induced shape distortions, PhD Dis- sertation, Lulea University of Technology, Sweden. [107] Z. Ghauch, V. Aitharaju, W. Rodgers, P. Pasupuletti, A. Dereims, R. Ghanem, Integrated stochastic analysis of fiber composites manufacturing using adapted poly- nomial chaos expansions Composites: Part A, 118, 179 - 193, 2019. [108] C. Thimmisetty, A. Khodabakhshnejad, N. Jabbari, F. Aminzadeh, R. Ghanem, K. Rose, J. Bauer, C. Disenhof Multiscale stochastic representation in high-dimensional data using gaussian processes with implicit diffusion metrics Dynamic Data-Driven Env. Systems Sci., 157 - 166, 2015. [109] C. Scarth, S. Adhikari, P.H. Cabral, G. Silva, A.P. Prado Random field simulation over curved surfaces: applications to computational structural mechanics Comput. Methods Appl. Mech. Engrg., 345, 283 - 307, 2019. [110] P. Constantine, E. Phipps, T. Wildey Efficient uncertainty propagation for network multiphysics systems Int. J. Num. Meth. in Eng., 99, 183 - 202, 2014. [111] M. Arnst, R. Ghanem, E. Phipps, J. Red-Horse Dimension reduction in stochastic modeling of coupled problems Int. J. Num. Meth. in Eng., 92(11), 940 - 968, 2012. [112] M. Arnst, R. Ghanem, E. Phipps, J. Red-Horse Measure transformation and effi- cient quadrature in reduced-dimensional stochastic modeling of coupled problems Int. J. Num. Meth. in Eng., 92(12), 1044 - 1080, 2012. 112 Appendix A Software A.1 Demos Figure (A.1) shows the number of basis terms in logarithmic scale of a polynomial chaos expansion followingN q = d+P P . Figure (A.2) shows the 1d Hermite polynomials of order up to 16, evaluated on Monde Carlo samples. Figure (A.3) shows the quadrature points following the Gauss-Hermite rule (sparse) for the first four levels for a 2-D problem. 2 4 6 8 10 Chaos Order 20 40 60 80 100 Stochastic dimension 4.0 8.0 12.0 16.0 20.0 24.0 28.0 No. basis terms (logscale) Figure A.1: Number of basis terms (log scale) in polynomial chaos expansion 113 Figure A.2: 1d Hermite polynomials of order up to 16 evaluated using Monte Carlo samples 114 Figure A.3: Quadrature points following the Gauss-Hermite rule (sparse) for a 2-D prob- lem A.2 Integrated RTM Problem The following listing shows the input file corresponding to the 74-D forward RTM prob- lem using the proposed, in-house, UQ workflow software centered on polynomial chaos with basis adaptation of the homogeneous chaos. 115 ------------------------------------------------------ * STOCHASTIC_INPUT f1, $PWD/main/model/VFORM_TP_Satin_45_20Feb2017.pc f2, $PWD/main/model/TruncatedPyramid_RTM_5mmMesh.dtf f3, $PWD/main/model/TruncatedPyramid_CURING.dtf f4, $PWD/main/model/TruncatedPyramid_Distortion.pc f5, $PWD/main/model/TruncatedPyramid_Distortion-STAGE2.pc f6, $PWD/main/model/field2.py f7, $PWD/main/model/write_visco.py f8, $PWD/main/model/racetrack.m f9, $PWD/main/model/racetrack2.m f10, $PWD/main/model/field1.py ------------------------------------------------------ * ANALYSIS forward ------------------------------------------------------ * INIT_DIMENSION 74 ------------------------------------------------------ * QUAD_LEVEL, QUAD_RULE 3, GH, sparse ------------------------------------------------------ * CHAOS_ORDER 3 ------------------------------------------------------ * RESTART_ANALYSIS no ------------------------------------------------------ * SOLVER_PATHS $PWD/main/model/job.sh ------------------------------------------------------ * RAND_VAR VAR1, 59.90E+09, 80.10E+09, 6, 6, [0], f1, pdf_beta VAR2, 59.90E+09, 80.10E+09, 6, 6, [1], f1, pdf_beta VAR3, 42.789E+07, 57.211E+07, 6, 6, [2], f1, pdf_beta VAR4, 42.789E+07, 57.211E+07, 6, 6, [3], f1, pdf_beta VAR5, 41.755, 48.245, 6, 6, [4], f1, pdf_beta VAR6, 131.106, 138.894, 6, 6, [5], f1, pdf_beta VAR7, 0.446, 0.554, 6, 6, [6], f2 f3, pdf_beta VAR8, 1.712, 2.288, 6, 6, [7], f2 f3, pdf_beta VAR9, 0.428, 0.572, 6, 6, [8], f2 f3, pdf_beta VAR10, 0.428, 0.572, 6, 6, [9], f2 f3, pdf_beta VAR11, 607.602, 812.398, 6, 6, [10], f2 f3, pdf_beta VAR12, 1454.823, 1945.177, 6, 6, [11], f2 f3, pdf_beta VAR13, 0.094, 0.126, 6, 6, [12], f2 f3, pdf_beta VAR14, 0.094, 0.126, 6, 6, [13], f2 f3, pdf_beta VAR15, 0.094, 0.126, 6, 6, [14], f2 f3, pdf_beta VAR16, 1031.212, 1378.788, 6, 6, [15], f2 f3, pdf_beta VAR17, 941.356, 1258.644, 6, 6, [16], f2 f3, pdf_beta VAR18, 285.102, 381.198, 6, 6, [17], f2, pdf_beta 116 VAR19, 322.727, 431.503, 6, 6, [18], f2, pdf_beta VAR20, 86712, 115938, 6, 6, [19], f2, pdf_beta VAR21, 0.101, 0.135, 6, 6, [20], f2 f3, pdf_beta VAR22, 2.901E+06, 3.879E+06, 6, 6, [21], f2 f3, pdf_beta VAR23, 0.391, 0.523, 6, 6, [22], f2 f3, pdf_beta VAR24, 1.027, 1.373, 6, 6, [23], f2 f3, pdf_beta VAR25, 0.992, 1.000, 5, 2, [24], f2 f3, pdf_beta VAR26, 3971.451, 4268.549, 6, 6, [25], f2 f3, pdf_beta VAR27, 7755.197, 7924.803, 6, 6, [26], f2 f3, pdf_beta VAR28, 400.243, 406.057, 6, 6, [27], f2 f3, pdf_beta VAR29, 299522, 400478, 6, 6, [28], f2 f3, pdf_beta VAR30, 53.058E+09, 70.942E+09, 6, 6, [29], f4 f5, pdf_beta VAR31, 53.058E+09, 70.942E+09, 6, 6, [30], f4 f5, pdf_beta VAR32, 1.968E+09, 2.632E+09, 6, 6, [31], f4 f5, pdf_beta VAR33, 0.009, 0.011, 6, 6, [32], f4 f5, pdf_beta VAR34, 0.424, 0.500, 6, 6, [33], f4 f5, pdf_beta VAR35, 33.589E+06, 44.910E+06, 6, 6, [34], f4 f5, pdf_beta VAR36, 34.179E+06, 45.700E+06, 6, 6, [35], f4 f5, pdf_beta VAR37, 34.179E+06, 45.700E+06, 6, 6, [36], f4 f5, pdf_beta VAR38, 1.095E-06, 1.465E-06, 6, 6, [37], f4 f5, pdf_beta VAR39, 1.095E-06, 1.465E-06, 6, 6, [38], f4 f5, pdf_beta VAR40, 1.155E-06, 1.545E-06, 6, 6, [39], f4 f5, pdf_beta VAR41, 0.0003, 0.0004, 6, 6, [40], f4 f5, pdf_beta VAR42, 0.0003, 0.0004, 6, 6, [41], f4 f5, pdf_beta VAR43, 0.0201, 0.0269, 6, 6, [42], f4 f5, pdf_beta VAR44, 0.599, 0.801, 6, 6, [43], f4 f5, pdf_beta VAR45, 190.967, 255.333, 6, 6, [44], f4 f5, pdf_beta VAR46, 331.588, 443.352, 6, 6, [45], f4 f5, pdf_beta VAR47, 0.757, 1.012, 6, 6, [46], f4 f5, pdf_beta VAR48, 55.882E+09, 74.718E+09, 6, 6, [47], f4 f5, pdf_beta VAR49, 55.882E+09, 74.718E+09, 6, 6, [48], f4 f5, pdf_beta VAR50, 6.350E+09, 8.490E+09, 6, 6, [49], f4 f5, pdf_beta VAR51, 0.022, 0.030, 6, 6, [50], f4 f5, pdf_beta VAR52, 0.236, 0.316, 6, 6, [51], f4 f5, pdf_beta VAR53, 26.957E+06, 36.043E+06, 6, 6, [52], f4 f5, pdf_beta VAR54, 22.763E+06, 30.436E+06, 6, 6, [53], f4 f5, pdf_beta VAR55, 22.763E+06, 30.436E+06, 6, 6, [54], f4 f5, pdf_beta VAR56, 1.780E-06, 2.380E-06, 6, 6, [55], f4 f5, pdf_beta VAR57, 1.780E-06, 2.380E-06, 6, 6, [56], f4 f5, pdf_beta VAR58, 0.342E-06, 0.458E-06, 6, 6, [57], f4 f5, pdf_beta VAR59, 0.0010, 0.0013, 6, 6, [58], f4 f5, pdf_beta VAR60, 0.0010, 0.0013, 6, 6, [59], f4 f5, pdf_beta VAR61, 0.0133, 0.0178, 6, 6, [60], f4 f5, pdf_beta VAR62, 1.717E-11, 1.983E-11, 4, 4, [61], f6 f6, pdf_beta VAR63, 1.717E-11, 1.983E-11, 4, 4, [62], f6 f6, pdf_beta VAR64, 0.337, 0.451, 6, 6, [63], f7 f7, pdf_beta VAR65, 0.0642, 0.0858, 6, 6, [64], f7 f7, pdf_beta VAR66, 1.717E-08, 1.983E-08, 6, 6, [65], f8 f8, pdf_beta VAR67, 1.717E-08, 1.983E-08, 6, 6, [66], f8 f8, pdf_beta VAR68, 1.717E-08, 1.983E-08, 6, 6, [67], f9 f9, pdf_beta VAR69, 1.717E-08, 1.983E-08, 6, 6, [68], f9 f9, pdf_beta 117 VAR70, .0, 1., [69], f10, pdf_gaussian VAR71, .0, 1., [70], f10, pdf_gaussian VAR72, .0, 1., [71], f10, pdf_gaussian VAR73, .0, 1., [72], f10, pdf_gaussian VAR74, .0, 1., [73], f10, pdf_gaussian ------------------------------------------------------ * DESIGN_VECS ------------------------------------------------------ * PRE_PROCESS ------------------------------------------------------ * ISOM, ADAPT_TOL, ADAPT_MAXTIER 3, 0.01, 3 * QOI_ADAPT, SAMPLES 1, 1e5 ------------------------------------------------------ * EXEC_QUEUES, NP 1, 2 * DEL 0 * EXEC_ENV slurm , ------------------------------------------------------ * PROJECT_NAME Truncated_UQ ------------------------------------------------------ 118 Appendix B Nomenclature Abbreviations UQ: uncertainty quantification PCE: polynomial chaos expansion RTM: resin transfer molding FE: finite element RVE: representative volume element KL: karhunen-loeve RV : random variable Probability & Chaos Theory : sample space P: probability measure S P : -algebra H: Hilbert space Σ h : covariance matrix of stochastic processh(x;) 0 : undeformed, natural domain : deformed domain G: Gaussian subspace : multivariate hermite polynomials, normalized d: problem dimensionality, initial P c : highest polynomial chaos order Y 1 ;Y 2 : Pearson’s correlation coefficient for RVY 1 andY 2 E: mathematical expectation : vector of uncorrelated, gaussian RV L : orthogonal summation of linear spaces Composite Manufacturing D p : domain of preform D: domain of truncated-square pyramid part D p e : set of elements in draping domainD p D f e : set of elements for the filling/curing discretization within domainD D d e : set of elements for the distortion discretization within domainD @D: boundary of domainD j : stacking angle ofj th fiber E j;t : elastic tensile moduli ofj th fiber E j;b : elastic bending moduli ofj th fiber (x): distortion angle field8x2D p 119 e 1 (x);e 2 (x): fiber orientation fields in warp and weft directions8x2D p V f : fiber volume fraction k f : fiber preform thermal conductivity tensor C p;f : fiber preform specific heat f : carbon-fiber preform density k r : resin thermal conductivity tensor P I : inlet pressure C p;r : resin specific heat r : resin density T r : resin temperature P V : vent pressure K (0) f : fiber preform unsheared permeability tensor K (s) f : fiber preform sheared permeability tensor K rt : racetrack permeability tensor r : resin viscosity t f : filling time h: specific enthalpy c(x): resin degree of cure field8x2D A 1 ;A 2 : prefactor parameters of curing model E 1 ;E 2 : activation energy of curing model B: maximum degree of cure of curing model m;n: curing model parameters T bt : boundary temperature of top/bottom molds t c : curing time E r;ply : laminae elasticity tensor under rubbery conditions E g;ply : laminae elasticity tensor under glassy conditions r;ply : poisson’s ratio of laminae under rubbery conditions g;ply : poisson’s ratio of laminae under glassy conditions G r;ply : shearing tensor of laminae under rubbery conditions G g;ply : shearing tensor of laminae under glassy conditions r;ply : coefficient of thermal expansion of laminae under rubbery conditions g;ply : coefficient of thermal expansion of laminae under glassy conditions r;ply : coefficient of chemical shrinkage of laminae under rubbery conditions g;ply : coefficient of chemical shrinkage of laminae under glassy conditions g : degree of cure at gelation T g0 : initial glass transition temperature T g1 : final glass transition temperature : material curing constant r : maximum residual von mises stress C r , C g : stiffness matrix under rubbery and glassy conditions (x): shearing angle field 120

Abstract (if available)

Abstract Light-weight composites are increasingly being used in both structural as well as non-structural applications, as the benefits such composite components offer when compared to conventional metallic alloys are understood and applied. Composite usage spans across many industries, ranging from aircraft and spacecraft applications within the aerospace industry, to automotive applications within the transportation industry, in addition to energy, defense, and marine industries. While the usage of composites is consistently growing across industries, one challenge that needs to be solved is developing consistent composite design standards. Such standards would ideally be anchored in a comprehensive assessment of manufacturing of composites, a task that is often complex, that spans across multiple physics and scales, and that presents a high level of inherent variability. ❧ We address the problem of performing integrated uncertainty quantification in multiphysics and multiscale composite manufacturing models for lightweight composite applications. We formulate and adopt an integrated uncertainty propagation framework, which encompasses all manufacturing phases following the RTM process, namely (1) fabric forming, (2) resin injection through the deformed fabric domain, (3) resin curing, and (4) mechanical ply distortion. The integrated framework, tailored for multiscale and multiphysics applications, is centered on polynomial chaos methods. Furthermore, an adaptation scheme of the Gaussian basis of the chaos expansions is adopted such that the proposed polynomial chaos integrated UQ framework scales well with high-dimensional composite manufacturing problem at hand. All the variables involved across different scales, physics, and manufacturing phases of the RTM process are parametrized as part of an integrated, high-dimensional, forward problem formulation. The latter is fully automated and embedded within a polynomial chaos UQ workflow with dimension reduction via an adaptation of the PCE basis

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Creator Ghauch, Ziad Georges (author)

Core Title Comprehensive uncertainty quantification in composites manufacturing processes

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 03/26/2020

Defense Date 11/27/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag covariance kernel,fiber composite manufacturing,geodesics,integrated stochastic analysis,model reduction,OAI-PMH Harvest,polynomial chaos expansion,random heterogeneous media,resin transfer molding,uncertainty quantification

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ghanem, Roger (committee chair), Masri, Sami (committee member), Nakano, Aiichiro (committee member)

Creator Email ghauch@usc.edu,zdghaouche@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-136327

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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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