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This dissertation contains work that addresses both theoretical and numerical aspects of reduced order models (ROMs). In an under-resolved regime, the classical Galerkin reduced order model (G-ROM) fails to yield accurate approximations. Thus, we propose a new ROM, the data-driven variational multiscale ROM (DD-VMS-ROM) built by adding a closure term to the G-ROM, aiming to increase the numerical accuracy of the ROM approximation without decreasing the computational efficiency. The closure term is constructed based on the variational multiscale framework. To model the closure term, we use data-driven modeling. In other words, by using the available data, we find ROM operators that approximate the closure term. To present the closure term's effect on the ROMs, we numerically compare the DD-VMS-ROM with other standard ROMs. In numerical experiments, we show that the DD-VMS-ROM is significantly more accurate than the standard ROMs. Furthermore, to understand the closure term's physical role, we present a theoretical and numerical investigation of the closure term's role in long-time integration. We theoretically prove and numerically show that there is energy exchange from the most energetic modes to the least energetic modes in closure terms in a long time averaging. One of the promising contributions of this dissertation is providing the numerical analysis of the data-driven closure model, which has not been studied before. At both the theoretical and the numerical levels, we investigate what conditions guarantee that the small difference between the data-driven closure model and the full order model (FOM) closure term implies that the approximated solution is close to the FOM solution. In other words, we perform theoretical and numerical investigations to show that the data-driven model is verifiable. Apart from studying the ROM closure problem, we also investigate the setting in which the G-ROM converges optimality. We explore the ROM error bounds' optimality by considering the difference quotients (DQs). We theoretically prove and numerically illustrate that both the ROM projection error and the ROM error are suboptimal without the DQs, and optimal if the DQs are used. / Doctor of Philosophy / In many realistic applications, obtaining an accurate approximation to a given problem can require a tremendous number of degrees of freedom. Solving these large systems of equations can take days or even weeks on standard computational platforms. Thus, lower-dimensional models, i.e., reduced order models (ROMs), are often used instead. The ROMs are computationally efficient and accurate when the underlying system has dominant and recurrent spatial structures. Our contribution to reduced order modeling is adding a data-driven correction term, which carries important information and yields better ROM approximations. This dissertation's theoretical and numerical results show that the new ROM equipped with a closure term yields more accurate approximations than the standard ROM.

Ultra-high temperature ceramics (UHTCs) are a class of materials that maintain their structural integrity at high temperatures, e.g. 2000 °C. They have been limited in their aerospace applications because of their relatively high density and the difficulty involved in forming them into complex shapes, like leading edges and inlets. Recent advanced processing techniques have made significant headway in addressing these challenges, where the introduction of multiscale porosity has resulted in lightweight UHTCs dubbed multiscale porous UHTCs. The effect of multiscale porosity on material properties must be characterized to enable design, but doing so experimentally can be costly, especially when attempting to replicate hypersonic flight conditions for relevant testing of selected candidate samples. As such, this dissertation seeks to computationally characterize the thermomechanical properties of multiscale porous UHTCs, specifically titanium diboride, and validate those results against experimental results so as to build confidence in the model. An implicit quasi-static variant of the Material Point Method (MPM) is developed, whose capabilities include intrinsic treatment of large deformations and contact which are needed to capture the complex material behavior of the as-simulated porous UHTC microstructures. It is found that the MPM can successfully obtain the elastic thermomechanical properties of multiscale porous UHTCs over a wide range of temperatures. Furthermore, characterizations of post-elastic behavior are found to be qualitatively consistent with data obtained from uniaxial compression experiments and Brazilian disk experiments. / Doctor of Philosophy / This dissertation explores a class of materials called ultra-high temperature ceramics (UHTCs). These materials can sustain very high temperatures without degrading, and thus have the potential to be used on hypersonic aircraft which routinely experience high temperatures during flight. In lieu of performing experiments on physical UHTC specimens, one can perform a series of computer simulations to figure out how UHTCs behave under various conditions. This is done here, with a particular focus what happens when pores are introduced into UHTCs, thus rendering them more like a sponge than a solid block of material. Doing computer simulations instead of physical experiments is attractive because of the flexibility one has in a computational environment, as well as the significantly decreased cost associated with running a simulation vs. setting up and performing an experiment. This is especially true when considering challenging operating environments like those experienced by high-speed aircraft. The ultimate goal with this research is to develop a computational tool than can be used to design the ideal distribution of pores in UHTCs so that they can best perform their intended functions.

Cuttlebone, the endoskeleton of cuttlefish, offers an intriguing biological structural model for designing low-density cellular ceramics with high stiffness and damage tolerance. Cuttlebone is highly porous (porosity ~93%) and lightweight (density less than 20% of seawater), constructed mainly by brittle aragonite (95 wt%), but capable of sustaining hydrostatic water pressures over 20 atmospheres and exhibits energy dissipation capability under compression comparable to many metallic foams (~4.4 kJ/kg). Here we computationally investigate how such a remarkable mechanical efficiency is enabled by the multiscale structure of cuttlebone. Using the common cuttlefish, Sepia Officinalis, as a model system, we first conducted high-resolution synchrotron micro-computed tomography (µ-CT) and quantified the cuttlebone's multiscale geometry, including the 3D asymmetric shape of individual walls, the wall assembly patterns, and the long-range structural gradient of walls across the entire cuttlebone (ca. 40 chambers). The acquired 3D structural information enables systematic finite-element simulations, which further reveal the multiscale mechanical design of cuttlebone: at the wall level, wall asymmetry provides optimized energy dissipation while maintaining high structural stiffness; at the chamber level, variation of walls (number, pattern, and waviness amplitude) contributes to progressive damage; at the entire skeletal level, the gradient of chamber heights tailors the local mechanical anisotropy of the cuttlebone for reduced stress concentration. Our results provide integrated insights into understanding the cuttlebone's multiscale mechanical design and provide useful knowledge for the designs of lightweight cellular ceramics. Upon the prior curvature analysis of the cuttlebone walls, we discovered that the walls were primarily "saddle-shaped". Thus, the characterization of different curvatures, varying between flat, domed, saddled, or cylindrical surfaces, were explored. A mathematical model was utilized to generate multiple walls with different curvature characteristics. We observed the mechanical performance of these walls via finite-element analysis and formulated different techniques for designing effective ceramic structures through incorporation of curvature. / Master of Science / The cuttlefish is a marine species that instead of having an inflatable swim bladder like fish, is a mollusk capable of swimming by utilizing their skeleton, called the cuttlebone. The cuttlefish can freely traverse the waters by controlling the flow of water in and out of their brittle skeletons, changing their buoyancy. For this reason, the cuttlebone must be very porous yet strong to withstand the deep-water pressures, enticing an interest for closer observation of the structure which may be useful in engineering applications involving ceramic structures. In this study, we examined an actual cuttlebone structure to better visualize its features with high-resolution synchrotron micro-computed tomography (µ-CT) and tabulated its mechanical performance through a variety of tests using computational software. The skeletal design of the cuttlebone consists of multiple layered chambers supported by wavy, pillar-like walls. It was revealed that the cuttlebone is remarkable due to its multiscale design: the asymmetric geometry of the walls are designed to tolerate considerable amounts of energy while a stiff construction; at the chamber level, variation of walls (number, pattern, and waviness amplitude) helps avoid complete destruction of the structure in the event of an excessive force; at the entire skeletal level, various of chamber heights reduces inflicted stress in concentrated regions of the cuttlebone. The wavy walls were also observed to retain a saddle-shaped curviness, versus simple flat, domed, or cylindrical shaped walls. This created an incentive to explore the effects of curvature on the structural integrity of brittle ceramic structures. We developed an effective way for generating walls with different curvatures and observed the mechanical performance of each wall by crushing them in computer simulations. It was identified that adding curvature to brittle walls prolonged the failure period significantly. While the cylindrical walls were found to be rather stiff, saddle-shaped walls, although not capable of withstanding as much force as flat or cylindrical walls, has a more progressive failure behavior meanwhile maintaining high energy absorption, hence the saddled walls of the cuttlebone to allow maintenance and self-repair in damaged regions.

<p dir="ltr">Modeling and simulation play a pivotal role in engineering and research, enabling cost effective solutions for design, manufacturing, and failure analysis, especially where physical testing is infeasible. This work explores numerical methods for multi-scale domains, where structures span diverse length scales, presenting unique challenges in meshing and accuracy. Advanced approaches such as domain decomposition and global-local methods are discussed, with an emphasis on their application in heterogeneous integration (HI) for advanced packaging. HI, which addresses the limitations of Moore’s Law, integrates diverse components into 2.5D and 3D architectures but introduces complex mechanical and thermo-mechanical challenges. This research addresses gaps in multi-scale numerical frameworks, proposing novel methods to handle non-linear physical evolution while maintaining compatibility with existing tools. A non-intrusive global-local inspired methodology that couples the local subdomain back to the global subdomain was implemented to increase the accuracy in non-linear multi-scale simulations involving evolution at local scale. The developed framework was then generalized to solve rate dependent and rate independent phenomenon. The work further extends into numerical methods for design of HI packages as well. Unlike detailed analysis, the design stage analysis prioritizes speed of computation with a first order accuracy of results. This is achieved using machine learning techniques for efficient design space exploration in HI. The study overall aims to advance computational frameworks tailored for accuracy in reliability analysis and speed in design stages, focusing on semiconductors and advanced packaging applications.</p>

A multiscale model is proposed to study the influence of interfacial interactions at the nanoscale in carbon nanotube(CNT)-polymer nanocomposites on the macroscale bulk elastic material properties. The efficiency of CNT reinforcement in terms of interfacial load transferring is assessed for the non-functionalized and functionalized interfaces between the CNTs and polymer matrix using force field based molecular dynamic simulations at the nanoscale. Polyethylene (PE) as a thermoplastic material is adopted and studied first because of its simplicity. Characterization of the nanoscale load transfer has been done through the identification of representative nanoscale interface elements for unfunctionalized CNT-PE interface models which are studied parametrically in terms of the length of the PE chains, the number of the PE chains and the "grip" position. Referring to the non-functionalized interface, CNTs interact with surrounding polymer only through weakly nonbonded van der Waals (vdW) forces in our study. Once appropriate values of these parameters are deemed to yield sufficiently converged results, the representative interface elements are subjected to normal and sliding mode simulations in order to obtain the force-separation responses at 100K and 300K for unfunctionalized CNT-PE interfaces. To study the functionalization effects, atomistic interface representative elements for functionalized CNT-PE interface are built based on non-functionalized interface models by grafting functional groups between the PE matrix and the graphene sheet. This introduces covalent bonding forces in addition to the non-bonded vdW forces. A modified consistent covalent force field (CVFF) and adaptive intermolecular reactive empirical bond order (AIREBO) potentials, both of which account for bond breaking, are applied to investigate the interfacial characteristic of functionalized CNT-PE interface in terms of the force-separation responses at 100K in both normal opening and sliding mode separations. In these studies, the focus has been on the influence of the functionalization density on the load transfer at the nanoscale interface. As an important engineering material, Epon 862/DETDA epoxy polymer,a thermoset plastic, has also been used as the polymer matrix material in order to see the difference in interfacial load transfer between a network structured polymer and the amorphous entangled structure of the PE matrix. As for thermoset epoxy polymer, emphasis has been put on investigating the effects of the crosslink density of the epoxy network on the interfacial load transfer ability for both non-functionalized and functionalized CNT-Epoxy interface at different temperatures(100K and 300K) and on the functionalization effect influenceing the interfacial interactions at the functionalized CNT-Epoxy interface. Cohesive zone traction-displacement laws are developed based on the force-separation responses obtained from the MD simulations for both non-functionalied and functionalized CNT-PE/epoxy interfaces. Using the cohesive zone laws, the influence of the interface on the effective elastic material properties of the nanocomposites are observed and determined in continuum level models using analytic and computational micromechanics approaches, allowing for the assessment of the improvement in reinforcement efficiency of CNTs due to the functionalization. It is found that the inclusion of the nanoscale interface in place of the perfectly bonded interface results in effective elastic properties which are dependent on the applied strain and temperature in accordance with the interface sensitivity to those effects, and which are significantly diminished from those obtained under the perfect interface assumption for non-functionalized nanocomposites. Better reinforcement efficiency of CNTs are also observed for the nanocomposites with the functionalized interface between CNTs and polymer matrix, which results in large increasing for the effective elastic material properties relative to the non-functionalized nanocomposites with pristine CNTs. Such observations indicates that trough controlling the degree of functionalization, i.e. the number and distribution of covalent bonds between the embedded CNTs and the enveloping polymer, one can tailor to some degree the interfacial load transfer and hence, the effective mechanical properties. The multiscale model developed in this study bridges the atomistic modeling and micromechanics approaches with cohesive zone models, which demonstrates to deepen the understanding of the nanoscale load transfer mechanism at the interface and its effects on the effective mechanical properties of the nanocomposites. It is anticipated that the results can offer insights about how to engineer the interface and improve the design of nanocomposites. / Ph. D.

Cellular mechanical metamaterials (CMMs) are a special class of materials that consist of microstructural architectures of macroscopic hierarchical frameworks that can have extraordinary properties. These properties largely depend on the topology and arrangement of the unit cells constituting the microstructure. The material hierarchy facilitates the synthesis and design of CMMs on the micro-scale to achieve enhanced properties (i.e., improved strength, toughness, low density) on the component (macro)-scale. However, designing on-demand cellular metamaterials usually requires solving a challenging inverse problem to explore the complex structure-property relations. The first part of this study (Ch. 3) proposes an experience-free and systematic design methodology for microstructures of CMMs using an advanced stochastic searching algorithm called micro-genetic algorithm (μGA). Locally, this algorithm minimizes the computational expense of the genetic algorithm (GA) with a small population size and a conditionally reduced parameter space. Globally, the algorithm employs a new search strategy to avoid local convergence induced by the small population size and the complexity of the parameter space. What's more, inspired by natural evolution in the GA, this study applies the inverse design method with the standard GA (sGA) as a sampling algorithm for intuitively mapping material-property spaces of CMMs, which requires the selection of objective properties and stochastic search of property points within the property space. The mapping methodology utilizing the sGA is proposed in the second part of the study (Ch. 4). This methodology involves a robust strategy that is shown to identify more comprehensive property spaces than traditional mapping approaches. The resulting property space allows designers to acknowledge the limitations of material performance, and select an appropriate class of CMMs based on the difficulty of the realization and fabrication of their microstructures. During the fabrication process, manufacturing defects cause uncertainty in the microstructures, and thus the structural properties. The third part of the study (Ch. 5) investigates the effects of the uncertainty stemming from manufacturing defects on the material property space. To accelerate the uncertainty quantification (UQ) via the Monte Carlo method, this study utilizes a machine learning technique to bypass the expensive simulations to compute properties. In addition to reducing the computational expense of the simulations, the deep learning method has been proven to be practical to accomplish non-intuitive design tasks. Due to the numerous combinations of properties and complex underlying geometries of metamaterials, it is numerically intractable to obtain optimal material designs that satisfy multiple user-defined performance criteria at the same time. Nevertheless, a deep learning method called conditional generative adversarial networks (CGANs) is capable of solving this many-to-many inverse problem. The fourth part of the study (Ch. 6) proposes a new inverse design framework using CGANs to overcome this challenge. Given combinations of target properties, the framework can generate a group of geometric patterns providing these target properties. Therefore, the proposed strategy provides alternative solutions to satisfy on-demand requirements while increasing the freedom in the fabrication process. Besides, with the advances in additive manufacturing (AM), the design space of an engineering material can be further enlarged by multi-scale topology optimization. As the interplay between microstructure and macrostructure drives the overall mechanical performance of engineering materials, it is necessary to develop a multi-scale design framework to optimize structural features in these two scales simultaneously. The final part of the study (Ch. 7) presents a concurrent multi-scale topology optimization method of CMMs. Structures in micro and macro scales are optimized concurrently by utilizing sequential quadratic programming (SQP) with the Solid Isotropic Material with Penalization (SIMP) method and a numerical homogenization approach. / Doctor of Philosophy / Cellular materials widely exist in natural biological systems such as honeycombs, bones, and wood. Recent advances in additive manufacturing have enabled us to fabricate these materials with high precision. Inspired by architectures in nature, cellular mechanical metamaterials (CMMs) have been introduced recently as a new class of architected systems. The materials are formed by hierarchical microstructural topologies, which have a decisive influence on the structural performance at the macro-scale. Therefore, the design of these materials primarily focuses on the geometric arrangement of their microstructures rather than the chemical composition of their base material. Tailoring the microstructures of these materials can lead to several outstanding features, such as high stiffness and strength, low density, and high energy absorption. However, it is challenging to design microstructures that satisfy user-defined requirements for properties and material costs. This is mainly due to the trade-off between the accuracy and computing times of the optimization process. In the first part of this study (Ch. 3), a design framework is proposed to overcome this issue. The framework employs a global search algorithm called the genetic algorithm (GA). With a newly designed search algorithm, the framework reduces errors between target and optimized material properties while improving computational efficiency. Inspired by the algorithm behind the GA, the second part of the study (Ch. 4) employs a similar algorithm to identify a material property chart demonstrating all possible combinations of mechanical properties of CMMs. Each axis of the material property chart corresponds to a selected mechanical property, such as Young's modulus or Poisson's ratio, along different directions. The boundary of the property space helps designers understand material performance limitations and make informed decisions in engineering practices. In the fabrication process, unexpected material properties might be achieved due to defects and tolerances in additive manufacturing (AM), such as uneven surfaces, shrinkage of pores, etc. The third part of the study (Ch. 5) investigates the uncertainty propagation on mechanical properties as a result of these manufacturing defects. To investigate the uncertainty propagation problem efficiently, the study uses a deep learning method to predict the variations (stochasticity) of properties. Consequently, the material property space boundary also varies with the uncertainty of properties. In addition to their computational efficiency, deep learning methods are beneficial for solving many-to-many inverse design problems. Traditionally, the global and local search/optimization methods retrieve alternative optimal solutions in their Pareto front set, where each solution is considered to be equally good. A deep learning method called conditional generative adversarial networks (CGANs) can bypass the property calculation to accelerate the simulation process while obtaining a group of candidates with on-demand properties. The fourth part of the study (Ch. 6) employs CGANs to build a new inverse design framework to increase flexibility in the fabrication process by generating alternative solutions for the microstructures of CMMs. Besides, as fabrication technologies have advanced, designing engineering systems has become increasingly complex. Material design is now not only focused on meeting micro-scale requirements but also addressing needs at multiple scales. The interaction between the microstructure (small-scale) and macrostructure (large-scale) significantly influences the overall performance of engineering systems. To optimize structures effectively, there is a need for a design framework that considers these two scales simultaneously. Thus, the final part of the study (Ch. 7) introduces a method called concurrent multi-scale topology optimization. To obtain the extreme performance of a multi-scale structure, this approach optimizes its structure at both micro- and macro-scales concurrently, using gradient-based optimization algorithms with density-based property determination methods in the two scales.

[pt] Compósitos cimentícios estão ganhando cada vez mais relevância na indústria da construção civil. No entanto, as diretrizes para o projeto do material compósito e dos seus elementos estruturais são ainda incipientes, pois mecanismos de ponte de transferência de forças providos pelas fibras ainda estão sob investigação. Este trabalho apresenta uma estratégia de modelagem de elementos finitos que leva em consideração a estrutura de nível mesoestrutural do material cimentício reforçado com fibras. Desta forma, quatro fases do material são consideradas no modelo numérico: agregados graúdos, argamassa, zona de transição interfacial (ZTI) e fibras. A argamassa e os agregados são modelados usando elementos contínuos triangulares com comportamento linear-elástico. As fibras são incluídas usando elementos de treliça unidimensionais acopladas a elementos bidimensionais contínuos. Uma técnica de fragmentação de malha é usada para introduzir elementos de interface nas arestas dos elementos de argamassa e na interface entre agregados e argamassa para representar a ZTI. O método Take-and-Place, proposto por Wriggers e Moftah (2006), foi adotado neste estudo para incluir agregados no modelo. Primeiro, os agregados são gerados seguindo uma curva de Fuller, que define um empacotamento entre os agregados perfeitos. Na segunda fase, os agregados são introduzidos no modelo garantindo a não sobreposição entre eles. Finalmente, as fibras são adicionadas. Para validar a metodologia proposta, testes experimentais foram simulados com sucesso em um framework de simulação numérica – GeMA. Por fim, o trabalho explora a influência do empacotamento fibra-agregado na resposta mecânica e nos padrões de fraturamento de compósitos cimentícios fibrosos. / [en] Fiber Reinforced Concrete (FRC) materials are gaining more relevance in the construction industry. However, the guidelines for the design of the composite material and of structural elements thereof are incipient and the stress bridging mechanisms are still under investigation. This work presents a finite element modelling strategy that takes into account the material meso-level structure. Four phases of the FRC material are considered in the model: coarse aggregates, mortar, interfacial transition zone (ITZ), and fibers. The mortar and aggregates are modelled using triangular linear elements with linear–elastic behavior. Fibers are included using one-dimensional truss elements which are coupled to the matrix through the technique proposed by Congro (2021). Zero-thickness interface elements are introduced at the interface between mortar elements, and at the interface between aggregates and mortar to represent the ITZ. The Take-and-Place method, obtained from Wriggers and Moftah (2006), was adopted in this study to include aggregates in the model. First, the aggregates are generated following a Fuller s curve that means a perfect aggregate package. In the second phase, the aggregates are placed in the model without overlapping. Finally, fibers were added. A mesh fragmentation technique is used to introduced zero-thickness interface elements at the interface between mortar elements, and at the interface between aggregates and mortar to represent the ITZ. To validate the proposed methodology, direct tensile test models were successfully reproduced in finite element analyses performed in an in-house framework – GeMA. Based on the obtained results, the authors could explore the influence of the fibers-aggregate packing in the mechanical response of the composite material.

In the contribution at hand, a new material modelling approach is introduced. This formulation is based upon the Principle of Multiscale Virtual Power and consideration of micromechanically motivated assumptions. Consequently, the evolution of dissipative phenomena depends on the chosen microstructure. Therefore, a strong anisotropy, which is induced by damage, is represented even with isotropic material formulations. This phenomenon is present in concrete. Furthermore, the modelling approach is validated by different material tests. Tensile-tensile and compression-tensile tests are used for validation of the proposed description. Some material tests are taken from the existing literature, while others are presented in the contribution at hand. Furthermore, the capabilities of the proposed formulation to capture different amounts of textile reinforcement in concrete are shown by additional experiments from the literature. Subsequently, the consistent linearisation of the proposed model is verified based on numerical analyses.