Materials play a critical role in the behavior and functionality of natural and engineered systems. For example, the use of cast-iron and steel led to dramatically increased bridge spans per material volume with the move from compression-dominant arch bridges to tensilecapable truss, suspension, and cable-stayed bridges; materials underlie many of the major technological advancements in the auto and aerospace industries that have made cars and airplanes increasingly light, strong, and damage tolerant; and the great diversity of biological materials and bio-composites enable complex biological and mechanical functions in nature. Topology optimization is a computational design method that simultaneously enhances efficiency and design freedom of engineered parts, but is often limited to a single, solid, isotropic, linear-elastic material. To understand how the material space can be tailored to enhance design freedom and/or promote desired mechanical behavior, several topology optimization problems are explored in this dissertation in which the space of available materials is either relaxed or restricted. Specifically, in a discrete topology optimization setting defined by 1D (truss) elements, tension-only systems are targeted by restricting the material space to that of a tension-only material and tailoring a formulation to handle the associated nonlinear mechanics. The discrete setting is then enhanced to handle 2D (beam) elements in pursuit of cloaking devices that hide the effect of a hole or defect on the elastostatic response of lattice systems. In this case the material space is relaxed to allow for a continuous range of stiffness and the objective is formulated as a weighted least squares function in which the physically-motivated weights promote global stiffness matching between the cloaked and undisturbed systems. Continuous 2D and 3D structures are also explored in a density-based topology optimization setting in which the material space is relaxed to accommodate an arbitrary number of candidate materials in a general continuum mechanics framework that can handle material anisotropy. The theoretical and physical relevance of such framework is highlighted via a continuous embedding scheme that enables manufacturing in the relaxed (or restricted) design space of lattice-based microstructural-materials. Implications of varying the material design space on the mechanics, mathematics, and computations needed for topology optimization are discussed in detail.

Origami has unfolded engineering applications in various fields, such as electrical, civil, aerospace, biomedical, and materials engineering. Those applications take advantage of the origami shape change capabilities to create tunable, deployable, and multifunctional systems. Although origami has catalyzed innovative solutions for such systems, its feasibility is challenged by pervasive pragmatic aspects. Thus, this thesis focuses on practical aspects that must be addressed for multifunctional origami applications, such as geometric imperfections, manufacturing, multiphysics considerations, and actuation strategies across scales. Specifically, it provides an in-depth study of geometric imperfections that may occur during the fabrication or service of origami systems and investigates how such inevitable imperfections impact both geometric and mechanical properties of origami patterns. Regarding manufacturing, we bring origami to the micro-scale and create architected metamaterials with remarkable mechanical properties, e.g., stiffness and Poisson’s ratio tunable anisotropy, a significant degree of shape recoverability, and reversible auxeticity. On the multiphysics front, we examine the coupling of mechanical and electromagnetic fields by using origami to fabricate spatial filters – frequency selective surfaces with dipole resonant elements placed across the pattern fold lines. The electrical length of the dipole elements changes as the pattern changes folding states, facilitating tunable frequency responses. Finally, we propose an untethered actuation solution with direct applications to origami robotics. Our solution couples geometric bi-stability and magnetic-responsive materials, allowing for instantaneous shape locking and local/distributed actuation with controllable speed, which can be as fast as a tenth of a second. The proposed actuation leads to direct application to robots capable of shape-changing, computing, and sensing.

Topology optimization is a computational design method used to find the optimized geometry of materials or structures meeting some performance criteria while satisfying constraints applied either globally (as usual) or locally (a focus of this work). Topology optimization can be used, for instance, to find lightweight structures that safely carry loads without failing. All you need is a design objective (e.g., minimize the weight) and constraints (e.g., material strength) and, through a nonlinear programming technique, the computer explores the solution space to find the optimized design. Despite the design freedoms afforded by topology optimization, its widespread adoption has primarily been hindered by the inability of current formulations to efficiently handle problems involving, for instance, multi-physics, multiple materials, and local material failure constraints. Thus, this thesis contributes to theoretical formulations, computer algorithms, and numerical implementations for topology optimization with an emphasis on problems subjected to either global constraints (e.g., energy-type constraints) or local constraints (e.g., material failure constraints), and for applications involving single or multiple physical phenomena and single- or multi-material designs. This work can be divided into two parts. In the first part, we present a general multi-material formulation that can handle an arbitrary number of materials and volume constraints (i.e., global-type constraints), and any type of objective function. To handle problems with such generality, we adopt a special linearization of the original optimization problem using a non-monotonous convex approximation of the objective function written in terms of positive and negative components of its gradient. The outcome is a scheme that updates the design variables associated with one constraint independently of the others, leading to an efficient, parallelizable formulation. The new update scheme allows us to design multi-phase viscoelastic microstructures, thermoelastic structures, and structures subjected to general dynamic loading. In the second part of this thesis, we introduce an augmented Lagrangian formulation to solve problems with local stress constraints correctly—a dilemma that has been unresolved thus far. First, we create a formulation to solve stress-constrained problems both for linear and nonlinear structures and provide an educational open-source code aiming to bridge the gap between research and education. Next, to extend the range of applications to structures that can be made of materials other than ductile metals, we introduce a function that unifies several classical strength criteria to predict the failure of a wide spectrum of materials, including either ductile metals or pressure-dependent materials, and use it to solve topology optimization problems with local stress constraints. We then extend the framework to time-dependent problems and address stress-constrained problems for structures subjected to general dynamic loading, in which the stress constraints are satisfied both in space (i.e., locally at every point of the discretized domain) and time (i.e., throughout the duration of the dynamic event). Unlike most work in the literature, this augmented Lagrangian framework leads to a scalable formulation that solves the optimization problem consistently with the local definition of stress and handles thousands or even millions of constraints efficiently. In summary, all components of this work are aimed to address critical challenges that have prevented topology optimization from being embraced as a practical design tool for industry-relevant applications.

Multi-functional structural systems are ubiquitous in nature, with potential applications across scales: from deployable outer space structures, to transformable multi-role robots, and to microstructures of metamaterials. To achieve the desired functionality, the system has to be able to change its behavior on demand, which usually involves programmable physical states, such as geometry, and stress distribution. Compared to other reconfigurable and programmable structural systems, such as membranes and truss frames, the present understanding of origami and tensegrity is incipient and thus there is room for further investigation and great creativity – this is the focus of this thesis. Both origami and tensegrity are deeply rooted in art, and are found to abound in nature under various forms, implying their exclusive performance as multi-functional platforms. Thus, we study the mechanics and physics of origami and tensegrity while emphasizing their subtle artistic connection. We explore their potential applications to reconfigurable structures and programmable metamaterials by means of examples of informative and illuminative designs. For instance, we demonstrate that by harnessing rigid and non-rigid folding of origami, we can generate a globally smooth hyperbolic paraboloid surfaces by folding a flat sheet; we can design metamaterials with arbitrary Poisson's ratio; and we can obtain programmable multi-stable structures and metamaterials. We also show that the mechanical properties of origami assemblages can be very sensitive to geometric imperfections. Moreover, by taking advantage of the prestress within tensegrity systems, we can deploy a stable structural platform of desired geometry from an unstable and compact assembly; we can create metamaterials whose elastostatic and elastodynamic properties are responsively tunable to changing prestress level, which provides a new dimension of programmability beyond geometry. The aforementioned findings open new avenues enabling their exploration beyond the realm of this thesis, while laying the path to unanticipated interdisciplinary discoveries.

Due to its unique and intriguing properties, polygonal and polyhedral discretization is an emerging field in computational mechanics. This thesis deals with developing finite element and virtual element formulations for computational mechanics problems on general polygonal and polyhedral discretizations.

The construction of finite element approximation on polygonal/polyhedral meshes relies on generalized barycentric coordinates, which are non-polynomial (e.g. rational) functions. Thus, the existing numerical integration schemes, typically designed to integrate polynomial functions, will lead to persistent consistency errors that do not vanish with mesh refinement. To overcome the limitation, this thesis presents a general gradient correction scheme, which restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field, and applies it to formulate both lower- and higher-order polygonal finite elements for finite elasticity problems. With the gradient correction scheme, the optimal convergence is recovered in finite elasticity problems.

The Virtual Element Method (VEM) was recently proposed as an attractive framework to handle unstructured polygonal/polyhedral discretizations and beyond (e.g., arbitrary non-convex shapes). The VEM is inspired by the mimetic methods, which mimics fundamental properties of mathematical and physical systems (e.g., exact mathematical identities of tensor calculus). Unlike the Finite Element Method (FEM), there are no explicit shape functions in the VEM, which is a unique feature that leads to flexible definitions of the local VEM spaces. This thesis also develops novel VEM formulations for several classes of computational mechanics problems. First, to study soft materials, we present a general VEM framework for finite elasticity. The framework features a nonlinear stabilization scheme, which evolves with deformation; and a local mathematical displacement space, which can effectively handle any element shape, including concave elements or ones with non-planar faces. We verify the convergence and accuracy of the proposed virtual elements by means of examples using unique element shapes inspired by Escher (the Dutch artist famous for his so-called impossible constructions). Second, to fully realize the potential of VEM in mesh adaptation (i.e., refinement, coarsening and local re-meshing), we develop a gradient recovery scheme and a posteriori error estimator for VEM of arbitrary order for linear elasticity problems. The a posteriori error estimator is simple to implement yet has been shown to be effective through theoretical and numerical analyses. Finally, from a design viewpoint, we present an efficient topology optimization framework on general polyhedral discretizations by synergistically incorporating the VEM and its mathematical/numerical features in the underlining formulation. As a result, the tailored VEM space naturally leads to a continuous material density field interpolated from nodal design variables. This approach yields a mixed virtual element with an enhanced density field. We present examples that explore the aforementioned features of our VEM-based topology optimization framework and contrast our results with the traditional FEM-based approaches that dominate the technical literature.

Topology optimization is a practical tool that allows for improved structural designs. This thesis focuses on developing both theoretical foundations and computational algorithms for topology optimization to effectively and efficiently handle many materials, many constraints, and many load cases. Most work in topology optimization is restricted to linear material with limited constraint settings for multiple materials. To address these issues, we propose a general multi-material topology optimization formulation with material nonlinearity. This formulation handles an arbitrary number of materials with flexible properties, features freely specified material layers, and includes a generalized volume constraint setting. To efficiently handle such arbitrary constraints, we derive an update scheme, named ZPR, that performs robust updates of design variables associated with each constraint independently. The derivation is based on the separable feature of the dual problem of the convex approximated primal subproblem with respect to the Lagrange multipliers, and thus the update of design variables in each constraint only depends on the corresponding Lagrange multiplier. This thesis also presents an efficient filtering scheme, with reduced-order modeling, and demonstrates its application to 2D and 3D topology optimization of truss networks. The proposed filtering scheme extracts valid structures, yields the displacement field without artificial stiffness, and improve convergence, leading to drastically improved computational performance. To obtain designs under many load cases, we present a randomized approach that efficiently optimizes structures under hundreds of load cases. This approach only uses 5 or 6 stochastic sample load cases, instead of hundreds, to obtain similar optimized designs (for both continuum and truss approaches). Through examples using Ogden-based, bilinear, and linear materials, we demonstrate that proposed topology optimization frameworks with the new multi-material formulation, update scheme, and discrete filtering lead to a design tool that not only finds the optimal topology but also selects the proper type and amount of material with drastically reduced computational cost.

Origami has gained popularity in science and engineering because a compactly stowed system can be folded into a transformable 3D structure with increased functionality. Origami can also be reconfigured and programmed to change shape, function, and mechanical properties. In this thesis, we explore origami from structural and stiffness perspectives, and in particular we study how geometry affects origami behavior and characteristics. Understanding origami from a structural standpoint can allow for conceptualizing and designing feasible applications in all scales and disciplines of engineering.

We improve, verify, and test a bar and hinge model that can analyze the elastic stiffness, and estimate deformed shapes of origami. The model simulates three distinct behaviors: stretching and shearing of thin sheet panels; bending of the flat panels; and bending along prescribed fold lines. We explore the influence of panel geometry on origami stiffness, and provide a study on fold line stiffness characteristics. The model formulation incorporates material characteristics and provides scalable, and isotopic behavior. It is useful for practical problems such as optimization and parametrization of geometric origami variations.

We explore the stiffness of tubular origami structures based on the Miura-ori folding pattern. A unique orientation for zipper coupling of rigidly foldable origami tubes substantially increases stiffness in higher order modes and permits only one flexible motion through which the structure can deploy. Deployment is permitted by localized bending along folds lines, however other deformations are over-constrained and engage the origami sheets in tension and compression. Furthermore, we couple compatible origami tubes into a variety of cellular assemblages that can enhance mechanical characteristics and geometric versatility. Practical applications such as deployable slabs, roofs, and arches are also explored.

Finally, we introduce origami tubes with polygonal cross-sections that can reconfigure into numerous geometries. The tubular structures satisfy the mathematical definitions for flat and rigid foldability, meaning that they can fully unfold from a flattened state with deformations occurring only at the fold lines. From a global viewpoint, the tubes do not need to be straight, and can be constructed to follow a non-linear curved line when deployed. From a local viewpoint, their cross-sections and kinematics can be reprogrammed by changing the direction of folding at some folds.

Structural optimization aims to provide structural designs that allow for the best performance while satisfying given design constraints. Among various applications of structural optimization, topology optimization based on mathematical programming and finite element analysis has recently gained great attention in research community as well as in applied structural engineering fields. One of the most fundamental requirements for building structures is to withstand various uncertain loads such as earthquake ground motions, wind loads and ocean waves. The design of structures, therefore, needs to ensure safe and reliable operations of structures over a prolonged period of time during which they may be exposed to various randomness of excitations caused by hazardous events. As such, significant amount of time and financial resources are invested to control the dynamic response of a structure under random vibrations caused by natural hazards or operations of non-structural components. In this regard, topology optimization of structures with stochastic response constraints is of great importance and consideration in industrial applications. This thesis discusses the development of structural optimization frameworks for a wide spectrum of deterministic and probabilistic constraints in engineering and investigate numerical applications.

First, the efficient optimization framework for statics and dynamics problems is investigated. In many incidences, expensive computational cost and labor hours are so prohibitive that optimization processes become impractical or inapplicable. To alleviate the computational burden in dynamic topology optimization, the multiresolution topology optimization approach is adopted. Based on the polygonal finite element method and multiresolution topology optimization techniques, a method of polygonal multiresolution topology optimization for statics and dynamics problems is developed. This development provides methods to discretize complicated geometries and reduce computational cost to obtain topology results of high-resolutions.

Despite rapid technological advances, incorporating stochastic response of structures into topology optimization is considered a relatively new field of research mainly due to computational challenges. In order to overcome such technical challenges in this field, a new method is introduced for incorporating random vibration theories into topology optimization using a discrete representation method for stochastic processes. Furthermore, a novel formulation is developed for sensitivity analysis of stochastic responses to use gradient-based optimization algorithms for the proposed topology optimization employing the discrete representation method.

To assess the reliability of a structure subjected to random excitations, the probability of the occurrence of at least one failure event over a time interval, i.e. the first-passage probability, often needs to be evaluated. In this thesis, a new method is proposed to incorporate probabilistic constraints on the first-passage probability into structural design and topology optimization. To obtain the first-passage probability effectively during each iteration, the failure event is described as a series system event consisting of failure events defined at discrete time points, and the system failure probability is obtained with the sequential compounding method. A new sensitivity formulation is developed employing the sequential compounding method to facilitate the use of gradient-based optimizers for the proposed method.

Finally, the conventional filter effects are investigated in reliability-based topology optimization using the elastic formulation of the ground structure method. In addition, an optimization scheme employing the discrete filter is proposed to ensure that optimized solutions satisfy the probabilistic constraints and global equilibrium. Moreover, the single-loop approach is incorporated to enhance the computational efficiency of the proposed RBTO method.

This research addresses hydraulic fracturing or hydro-fracking, i.e. fracture propagation process in rocks through the injection of a fluid under pressure, which generates cracks in the rock that propagate according to the amount of fluid injected. This technique leads to an increase of the hydraulic transmissivity of the rock mass and, consequently, improves oil production. Several analytical and numerical models have been proposed to study this fracture mechanism, generally based in continuum mechanics or using interface elements through a known propagation path. In this work, the crack propagation is simulated using the PPR potential-based cohesive zone model[72] by means of an extrinsic implementation. Thus, interface cohesive elements are adaptively inserted in the mesh to capture the softening fracture process. The fluid pressure is simulated using the lattice Boltzmann model[84] through an iterative procedure. The boundaries of the crack, computed using the finite element method, are transferred to the lattice Bolztmann model as boundary conditions, where the fluid pressure (or fluid forces) applied on these boundaries, caused by the fluid injected, can be calculated. These forces are then used in the finite element model as external forces applied on the faces of the crack. The new position of the crack faces is then calculated and transferred to the lattice-Boltzmann model to update the boundary conditions. This feedback-loop for fluid-structure interaction allows modeling of hydraulic fracturing processes for irregular path propagation.

Material failure pervades the fields of materials science and engineering; it occurs at various scales and in various contexts. Understanding the mechanisms by which a material fails can lead to advancements in the way we design and build the world around us. For example, in structural engineering, understanding the fracture of concrete and steel can lead to improved structural systems and safer designs; in geological engineering, understanding the fracture of rock can lead to increased efficiency in oil and gas extraction; and in biological engineering, understanding the fracture of bone can lead to improvements in the design of bio-composites and medical implants. In this thesis, we numerically investigate a wide spectrum of failure behavior; in soft and quasi-brittle materials with nonhomogeneous microstructures considering a statistical distribution of material properties.

The first topic we investigate considers the influence of interfacial interactions on the macroscopic constitutive response of particle reinforced elastomers. When a particle is embedded into an elastomer, the polymer chains in the elastomer tend to adsorb (or anchor) onto the surface of the particle; creating a region in the vicinity of each particle (often referred to as an interphase) with distinct properties from those in the bulk elastomer. This interphasial region has been known to exist for many decades, but is primarily omitted in computational investigations of such composites. In this thesis, we present an investigation into the influence of interphases on the macroscopic constitutive response of particle filled elastomers undergoing large deformations. In addition, at large deformations, a localized region of failure tends to accumulate around inclusions. To capture this localized region of failure (often referred to as interfacial debonding), we use cohesive zone elements which follow the Park-Paulino-Roesler traction-separation relation. To account for friction, we present a new, coupled cohesive-friction relation and detail its formulation and implementation. In the process of this investigation, we developed a small library of cohesive elements for use with a commercially available finite element analysis software package.

Additionally, in this thesis, we present a series of methods for reducing mesh dependency in two-dimensional dynamic cohesive fracture simulations of quasi-brittle materials. In this setting, cracks are only permitted to propagate along element facets, thus a poorly designed discretization of the problem domain can introduce artifacts into the fracture behavior discretization of the problem domain can introduce artifacts into the fracture behavior. To reduce mesh induced artifacts, we consider unstructured polygonal finite elements. A randomly-seeded polygonal mesh leads to an isotropic discretization of the problem domain, which does not bias the direction of crack propagation. However, polygonal meshes tend to limit the possible directions a crack may travel at each node, making this discretization a poor candidate for dynamic cohesive fracture simulations. To alleviate this problem, we propose two new topological operators. The first operator we propose is adaptive element splitting, and the second is adaptive mesh refinement. Both operators are designed to improve the ability of unstructured polygonal meshes to capture crack patterns in dynamic cohesive fracture simulations. However, we demonstrate that element-splitting is more suited to pervasive fracture problems, whereas, adaptive refinement is more suited to problems exhibiting a dominant crack.

Finally, we investigate the use of geometric and constitutive design features to regularize pervasive fragmentation behavior in three-dimensions. Throughout pervasive fracture simulations, many cracks initiate, propagate, branch and coalesce simultaneously. Because of the cohesive element method's unique framework, this behavior can be captured in a regularized manner. In this investigation, unstructuring techniques are used to introduce randomness into a numerical model. The behavior of quasi-brittle materials undergoing pervasive fracture and fragmentation is then examined using three examples. The examples are selected to investigate some of the significant factors influencing pervasive fracture and fragmentation behavior; including, geometric features, loading conditions, and material gradation.

As engineers and scientists, we have a host of reasons to understand how structural systems fail. We may be able to improve the safety of buildings during natural disaster by designing more fracture resistant connectors, to lengthen the life span on industrial machinery by designing it to sustain very large deformation at high temperatures, or prepare evacuation procedures for populated areas in high seismic zones in the event of rupture in the earth's crust. In order to achieve a better understanding of how any of these structures fail, experimental, theoretical, and computational advances must be made. In this dissertation we will focus on computational simulation by means of the finite element method and will investigate topological and physical aspects of adaptive remeshing for two types of structural systems: quasi-brittle and ductile. For ductile systems, we are interested in modeling the large deformations that occur before rupture of the material. The deformations can be so large that element distortion can cause lack of numerical convergence. Thus, we present a remeshing and internal state variable mapping technique to enable large deformation modeling and alleviate mesh distortion. We perform detailed studies on the Lie-group interpolation and variational recovery scheme and conclude that the approach results in very limited numerical diffusions and is applicable for modeling systems with significant ductile distortion. For quasi brittle systems mesh adaptivity is the central theme as it is for the work on ductile systems. We investigate two- and three-dimensional problems on CPU and GPU systems with the main goals of either improving computational efficiency or fidelity of the final solution. We investigate quasi-brittle fracture by means of the inter-element extrinsic cohesive zone model approach in which interface elements capable of separating are adaptively inserted at bulk element facets when and where they are needed throughout the numerical simulation. The inter-element cohesive zone model approach is known to suffer from mesh bias. Thus, we utilize polygonal element meshes with adaptive splitting to improve the capability of the mesh to represent experimentally obtained fracture patterns. The fact that we utilize the efficient linear polygonal elements and only apply the adaptive element splitting where needed means that we also achieve improved computational efficiency with this approach. In the last half of the dissertation, we depart from the use of unstructured meshes and focus on the development of hierarchical mesh refinement and coarsening schemes on the structured 4k mesh in two and three dimensions. In three-dimensions, the size of the problem increases so rapidly that mesh adaptivity is critical to enable the simulation of large-scale systems. Thus, we develop the topological and physical aspects of the mesh refinement and coarsening scheme. The scheme is rigorously tested on two benchmark problems; both of which shows significant speed up over a uniform mesh implementation and demonstrate physically meaningful results. To achieve greater speed up, the adaptive mesh refinement and coarsening scheme on the 2D 4k mesh is mapped to a GPU architecture. Considerations for the numerical implementation on the massively parallel system are detailed. Further, a study on the impact of the parallelization of the dynamic fracture code is performed on a benchmark problem, and a statistical investigation reveals the validity of the approach. Finally, the benchmark example is extended to such that the speicmen dimensions matches that of the original experimental system. The speedup provided by the GPU allows us to model this large system in a pratical amount of time and ultimately allows us to investigate differences between the commonly used reduced-scale model and the actual experimental scale. This dissertation concludes with a summary of contribution and comments on potential future research directions. Appendices featuring scripts and codes are also included for the interested reader.

This work focuses on optimal structural systems, which can be modeled using discrete elements (e.g. slender columns and beams), continuum elements (e.g. walls or slabs), or combinations of these. Optimization problems become meaningful only after the objective function, or benchmark, that evaluates a given design has been defined. Thus, it is logical to explore a variety of objectives, with emphasis on the ones that yield distinct results. The design may include constraints in response to performance or habitability, which must be included in the optimization to yield feasible designs. Structural optimization can be used to improve structural designs by giving cheaper, stronger, lighter and safer structures. Gradient--based optimization is the preferred approach in this work, for it consciously improves a design using the gradient information, as opposed to making random guesses. The optimization problem has an internal dependency on structural analysis, which may require modifications or careful analysis, in order to obtain meaningful gradient information. Simple problems composed solely of discrete elements are of particular interest to engineers in practice. The design of lateral bracing systems falls into this category. A novel discrete element topology optimization algorithm is proposed, and to facilitate the adoption by industry and academia, the implementation is also provided. Discrete element topology optimization has the potential to aid in the discovery of new closed--form solutions for common problems in structural engineering. These closed--form solutions, while often impractical to build, give insight into the physics of the optimal structural system. This information can be used to steer civil structural projects towards more efficient load transfer systems. The manufacturing of optimal structures often lags behind our ability to analyse and design them. Additive manufacturing presents itself as the (much sought) final stage required for a complete structural optimization design process. A clean and streamlined methodology for manufacturing optimal structures is proposed. This includes optimal structures obtained from density--based methods as well as the ground structure method. The goal of this work is to improve the current sequential design process of civil structures. It does so by facilitating the integration of optimization techniques into existing design processes, in addition to extending optimization algorithms to address a wider variety of problems. Despite being centered primarily on civil structures, this work has the potential to impact other disciplines. In particular, an example that incorporates optimization techniques into the medical field is shown.

Topology optimization refers to the optimum distribution of materials, so as to achieve certain prescribed design objectives while simultaneously satisfying constraints. Engineering applications often require unstructured meshes to capture the domain and boundary conditions accurately and to ensure reliable solutions. Hence, unstructured polyhedral elements are becoming increasingly popular. Since the pioneering work of Wachspress, many interpolants for polytopes have come forth; such as, mean value coordinates, natural neighbor-based coordinates, metric coordinate method and maximum entropy shape functions. The extension of the shape functions to three-dimensions, however, has been relatively slow partly due to the fact that these interpolants are subject to restrictions on the topology of admissible elements (e.g., convexity, maximum valence count) and can be sensitive to geometric degeneracies. More importantly, calculating these functions and their gradients are in general computationally expensive. Numerical evaluation of weak form integrals with sufficient accuracy poses yet another challenge due to the non-polynomial nature of these functions as well as the arbitrary domain of integration. Virtual Element Method (VEM), which has evolved from Mimetic Finite Difference methods, addresses both the issues of accuracy and efficiency. In this work, a VEM framework for three-dimensional elasticity is presented. Even though VEM is a conforming Galerkin formulation, it differs from tradition finite element methods in the fact that it does not require explicit computation of approximation spaces. In VEM, the deformation states of an element are kinematically decomposed into rigid body, linear and higher order modes. The discrete bilinear form is constructed to capture the linear deformations exactly which ensures that the displacement patch test is passed and optimum convergence is achieved. The present work focuses on first-order VEM with degrees of freedom associated with the vertices of the elements. Construction of the stiffness matrix reduces to the evaluation of surface integrals, in contrast to the volume integrals encountered in the conventional finite element method (FEM), thus reducing the overall computational cost. By means of the aforementioned approach, a framework for three-dimensional topology optimization is developed for polyhedral meshes. In the literature, topology optimization problems are typically solved with either tetrahedral or brick meshes. Numerical anomalies, such as checkerboard patterns and one-node connections, are present in such formulations. Constraints in the geometrical features of spatial discretization can also result in mesh dependent sub-optimal designs. In the current work, polyhedral meshes are proposed as a means to address the geometric features of the domain discretization. Polyhedral meshes not only provide greater flexibility in discretizing complicated domains but also alleviate the aforementioned numerical anomalies. For topology optimization problems, many approaches are available; which can mainly be classified as density-based methods and differential equation-driven methods (further subclassified as level-set and phase-field methods). Before choosing density-based methods for polyhedral topology optimization, a couple of differential equation-driven methods; which are representative of the literature, are exhaustively analyzed in two-dimensions. Finally, we also investigate aesthetics in topology optimization designs. In this work, two-dimensional topology optimization on tessellations is investigated as a means to coalesce art and engineering. M.C. Escher's tessellations using recognizable figures are mainly utilized. The aforementioned Mimetic Finite Difference-inspired approach (VEM) facilitates accurate numerical analysis on any non self-intersecting closed polygons such as tessellations.

The contribution of this work centers on the establishment of a novel topology optimization framework targeted specifically towards the needs of the structural engineering industry. Topology optimization can be used to minimize the material consumption in a structure, while at the same time providing a tool to generate design alternatives integrating architectural and structural engineering concepts. This tool can be an initial step towards the creation of efficient designs and provides an interactive, rational process for a project where architects and engineers can more effectively incorporate each other’s ideas. Through the selection of layout constraints, the objection function, and other metrics that might fit the problem being studied, the engineer can then present the architect with a spectrum of solutions based on these parametric studies. This selection process has been shown to provide new ways to look at designs, which in turn inspires the overall design of the structure. To streamline and simplify the design process, the computational framework described throughout this thesis is based on an integrated topology optimization approach involving the concurrent optimization of both continuum (e.g. Q4, polygonal) and discrete (e.g. beam, truss) finite elements to design the structural systems of high-rise buildings. For instance, after the overall shape and location of the perimeter columns of the building are known, topology optimization can be used to design the internal structural system, while concurrently sizing the members. Moreover, while typical topology optimization problems are based on a single objective function (i.e. minimum compliance), in the context of buildings it is important to evaluate and account for potential geometric instabilities as well. Thus, multi-objective optimization, including linearized buckling, has been studied in this context. To handle the large amounts of data associated with a high-rise, this new framework has been written to take advantage of a topological data structure together with object-oriented programming concepts to handle a variety of finite element problems, in an efficient, but generic fashion as demonstrated in this work. Several practical examples and case studies of high-rise buildings and other architectural structures are given to show the importance and relevance of this approach to the structural design industry. Finally, to better understand the geometries derived throughout the thesis, optimal structures are explored in more detail using the notions of graphic statics and reciprocal diagrams. The advantage to using graphic statics for this class of optimal problems is that it provides all of the information needed to determine the total load path in a graphical manner, allowing the engineer and/or architect to gain valuable insight to the problem at hand. Moreover, using the reciprocal form and force diagrams, we describe how in the course of finding one minimum load path structure, a second minimum load path structure is also found. These analytical studies parallel several of the numerical examples derived throughout the thesis to verify the resulting topologies from a different perspective.

This dissertation deals with problems of shape and topology optimization in which the goal is to find the most efficient shape of a physical system. The behavior of this system is captured by the solution to a boundary value problem that in turn depends on the given shape. As such, optimal shape design can be viewed as a form of optimal control in which the control is the shape or domain of the governing state equation. The resulting methodologies have found applications in many areas of engineering, ranging from conceptual layout of high-rise buildings to the design of patient-tailored craniofacial bone replacements. Optimal shape problems and more generally PDE-constrained inverse problems, however, pose several fundamental challenges. For example, these problems are often ill-posed in that they do not admit solutions in the classical sense. The basic compliance minimization problem in structural design, wherein one aims to find the stiffest arrangement of a fixed volume of material, favors non-convergent sequences of shapes that exhibit progressively finer features. To address the ill-posedness, one either enlarges the admissible design space allowing for generalized micro-perforated shapes, an approach known as “relaxation,” or alternatively places additional constraints to limit the complexity of the admissible shapes, a strategy commonly referred to as “restriction.” We discuss the issue of existence of solutions in detail and outline the key elements of a well-posed restriction formulation for both density and implicit function parametrizations of the shapes. In the latter case, we demonstrate both mathematically and numerically that without an additional “transversality” condition, the usual smearing of the Heaviside map (which links the implicit functions to the governing state equation), no matter how small, will transform the problem into the so-called variable thickness problem, whose theoretical optimal solutions do not have a clearly-defined boundary. Within the restriction setting, we also analyze and provide a justification for the so-called Ersatz approximation in structural optimization where the void regions are filled by a compliant material in order to facilitate the numerical implementation. Another critically important but challenging aspect of optimal shape design is dealing with the resulting large-scale non-convex optimization systems which contain many local minima and require expensive function evaluations and gradient calculations. As such, conventional nonlinear programming methods may not be adequately efficient or robust. We develop a simple and tailored optimization algorithm for solving structural topology optimization problems with an additive regularization term and subject only to a set of box constraints. The proposed splitting algorithm matches the structure of the problem and allows for separate treatment of the cost function, the regularizer, and the constraints. Though our mathematical and numerical investigation is mainly focused on Tikhonov regularization, one important feature of the splitting framework is that it can accommodate nonsmooth regularization schemes such as total variation penalization. We also investigate the use of isoparametric polygonal finite elements for the discretization of the design and response fields in two-dimensional topology optimization problems. We show that these elements, unlike their low-order Lagrangian counterparts, are not susceptible to certain grid-scale instabilities (e.g., checkerboard patterns) that may appear as a result of inaccurate analysis of the design response. The better performance of polygonal discretizations is attributed to the enhanced approximation characteristics of these elements, which also alleviate shear and volumetric locking phenomena. In regards to the latter property, we demonstrate that low-order finite element spaces obtained from polygonal discretizations satisfy the well-known Babuska-Brezzi condition required for stability of the mixed variational formulation of incompressible elasticity and Stokes flow problems. Conceptually, polygonal finite elements are the natural extension of commonly used linear triangles and bilinear quads to all convex n-gons. To facilitate their use, we present a simple but robust meshing algorithm that utilizes Voronoi diagrams to generate convex polygonal discretizations of implicit geometries. Finally, we provide a self-contained discretization and analysis Matlab code using polygonal elements, along with a general framework for topology optimization.

Structural optimization methods have been developed and applied to a variety of engineering practices. This study aims to overcome technical challenges in applying design and topology optimization techniques to large-scale structural systems with uncertainties. The specific goals of this dissertation are: (1) to develop an efficient scheme for topology optimization; (2) to introduce an efficient and accurate system reliability-based design optimization (SRBDO) procedure; and (3) to investigate the reliability-based topology optimization (RBTO) problem. First, it is noted that the material distribution method often requires a large number of design variables, especially in three-dimensional applications, which makes topology optimization computationally expensive. A multiresolution topology optimization (MTOP) scheme is thus developed to obtain high-resolution optimal topologies with relatively low computational cost by introducing distinct resolution levels to displacement, density and design variable fields: the finite element analysis is performed on a relatively coarse mesh; the optimization is performed on a moderately fine mesh for design variables; and the density is defined on a relatively fine mesh for material distribution. Second, it is challenging to deal with system events in reliability-based design optimization (RBDO) due to the complexity of system reliability analysis. A new single-loop system RBDO approach is developed by using the matrix-based system reliability (MSR) method. The SRBDO/MSR approach utilizes matrix calculations to evaluate the system failure probability and its parameter sensitivities accurately and efficiently. The approach is applicable to general system events consisting of statistically dependent component events. Third, existing RBDO approaches employing first-order reliability method (FORM) can induce significant error for highly nonlinear problems. To enhance the accuracy of component and system RBDO approaches, algorithms based on the second-order reliability method (SORM), termed as SORM-based RBDO, are proposed. These technical advances enable us to perform RBTO of large-scale structures efficiently. The proposed algorithms and approaches are tested and demonstrated by various numerical examples. The efficient and accurate approaches developed for design and topology optimization can be applied to large-scale problems in engineering design practices.

Asphalt concrete pavements are inherently graded viscoelastic structures. Oxidative aging of asphalt binder and temperature cycling due to climatic conditions being the major cause of non-homogeneity. Current pavement analysis and simulation procedures dwell on the use of layered approach to account for these non-homogeneities. The conventional finite-element modeling (FEM) technique discretizes the problem domain into smaller elements, each with a unique constitutive property. However the assignment of unique material property description to an element in the FEM approach makes it an unattractive choice for simulation of problems with material non-homogeneities. Specialized elements such as “graded elements” allow for non-homogenous material property definitions within an element. This dissertation describes the development of graded viscoelastic finite element analysis method and its application for analysis of asphalt concrete pavements. Results show that the present research improves efficiency and accuracy of simulations for asphalt pavement systems. Some of the practical implications of this work include the new technique’s capability for accurate analysis and design of asphalt pavements and overlay systems and for the determination of pavement performance with varying climatic conditions and amount of in-service age. Other application areas include simulation of functionally graded fiber-reinforced concrete, geotechnical materials, metal and metal composites at high temperatures, polymers, and several other naturally existing and engineered materials.

The characterization of nonlinear constitutive relationships along fracture surfaces is a fundamental issue in mixed-mode cohesive fracture simulations. A generalized potential-based constitutive theory of mixed-mode fracture is proposed in conjunction with physical quantities such as fracture energy, cohesive strength and shape of cohesive interactions. The potential-based model is verified and validated by investigating quasi-static fracture, dynamic fracture, branching and fragmentation. For quasi-static fracture problems, intrinsic cohesive surface element approaches are utilized to investigate microstructural particle/debonding process within a multiscale approach. Macroscopic constitutive relationship of materials with microstructure is estimated by means of an integrated approach involving micromechanics and the computational model. For dynamic fracture, branching and fragmentation problems, extrinsic cohesive surface element approaches are employed, which allow adaptive insertion of cohesive surface elements whenever and wherever they are needed. Nodal perturbation and edge-swap operators are used to reduce mesh bias and to improve crack path geometry represented by a finite element mesh. Adaptive mesh refinement and coarsening schemes are systematically developed in conjunction with edge-split and vertex-removal operators to reduce computational cost. Computational results demonstrate that the potential-based constitutive model with such adaptive operators leads to an effective and efficient computational framework to simulate physical phenomena associated with fracture. In addition, the virtual internal bond model is utilized for the investigation of quasi-brittle material fracture behavior. All the computational models have been developed in conjunction with verification and/or validation procedures.

A novel four-layer functionally graded fiber-reinforced cementitious composite (FGFRCC) as a beam component has been fabricated using extrusion and pressing techniques. The FGFRCC features a linear gradation of fiber volume fraction through the beam depth. The bending test shows the enhanced bending strength of the FGFRCC without delamination at layer interface. Microstructure investigation verifies the fiber gradation and the smooth transition between homogeneous layers. The remaining part of the study is the development of a hybrid technique for the extraction of mode I cohesive zone model (CZM). First, a full-field digital image correlation (DIC) technique has been adopted to compute the two-dimensional displacement fields. Such displacement fields are used as the input to the finite element (FE) formulation of an inverse problem for computing the CZM. The CZM is parameterized using flexible splines without assumption of the model shape. The Nelder-Mead optimization method is used to solve the ill-posed nonlinear inverse problem. Barrier and regularization terms are incorporated in the objective function for the inverse problem to assist optimization. Numerical tests show the robustness of the technique and the tolerance to experimental noise. The techniques are then applied to plastics and homogeneous FRCCs to demonstrate its broader application.

Topology optimization is a powerful tool for global and multiscale design of structures, microstructures, and materials. The computational bottleneck of topology optimization is the solution of a large number of extremely ill-conditioned linear systems arising in the finite element analysis. Adaptive mesh refinement (AMR) is one efficient way to reduce the computational cost. We propose a new AMR scheme for topology optimization that results in more robust and efficient solutions. For large sparse symmetric linear systems arising in topology optimization, Krylov subspace methods are required. The convergence rate of a Krylov subspace method for a symmetric linear system depends on the spectrum of the system matrix. We address the ill-conditioning in the linear systems in three ways, namely rescaling, recycling, and preconditioning. First, we show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with homogeneous density. Second, the changes in the linear system from one optimization step to the next are relatively small. Therefore, recycling a subspace of the Krylov subspace and using it to solve the next system can improve the convergence rate significantly. We propose a minimum residual method with recycling (RMINRES) that preserves the short-term recurrence and reduces the cost of recycle space selection by exploiting the symmetry. Numerical results show that this method significantly reduces the total number of iterations over all linear systems and the overall computational cost (compared with the MINRES method which is optimal for a single symmetric system). We also investigate the recycling method for adaptive meshes. Third, we propose a multilevel sparse approximate inverse (MSPAI) preconditioner for adaptive mesh refinement. It significantly improves the conditioning of the linear systems by approximating the global modes with multilevel techniques, while remaining cheap to update and apply, especially when the mesh changes. For convection-diffusion problems, it achieves a level-independent convergence rate. We then make a few changes in the MSPAI preconditioner for topology optimization problems. With these extensions, the MSPAI preconditioner achieves a nearly level-independent convergence rate. Although for small to moderate size problems the incomplete Cholesky preconditioner is faster in time, the multilevel sparse approximate inverse preconditioner will be faster for (sufficiently) large problems. This is important as we are more interested in scalable methods.

Realistic numerical analysis of dynamic failure process has long been a challenge in the ßeld of computational mechanics. The challenge consists of two aspects: a realistic representation of fracture criteria, and its eácient incorporation into a viable numerical scheme. This study investigates the dynamic failure process in a variety of materials by incorporating a Cohesive Zone Model (CZM) into the ßnite element scheme. The CZM failure criterion uses both a ßnite cohesive strength and work to fracture in the material description. Based on crack initiation criteria, CZMs can be categorized into two groups, i.e., intrinsic and extrinsic. This study focuses on extrinsic CZMs, which eliminates many of the inherent drawbacks present in the intrinsic CZMs. The extrinsic CZM approach allows spontaneous and adaptive insertion of arbitrary cracks in space and time, i.e., where needed and when needed. To that eÞect, a novel topology-based data structure is employed in the study, which provides both versatility and robustness, and allows adaptive insertion of cohesive elements as required by simulation. Both two-dimensional and three-dimensional problems are analyzed. A series of dynamic fracture phenomena, including spontaneous crack initiation, dynamic crack micro-branching and crack competition, are successfully captured by the CZM simulations. To better analyze mesh size dependence of the numerical scheme, an investigation of cohesive zone size is also presented, which indicates limitations of conventional cohesive zone size estimates in dynamic and rate-dependent problems.

Asphalt concrete is a quasi-brittle material that exhibits time and temperature dependent fracture behavior. Softening of the material can be associated to interlocking and sliding between aggregates, while the asphalt mastic displays cohesion and viscoelastic properties. To properly account for both progressive softening and viscoelastic effects occurring in a relatively large fracture process zone, a cohesive zone model (CZM) is employed. Finite element implementation of the CZM is accomplished via user subroutines that can be used in conjunction with general-purpose software. The bulk properties (e.g. relaxation modulus) and fracture parameters (e.g. cohesive fracture energy) are obtained from experiments. In this study, artificial compliance and numerical convergence (which are associated with the intrinsic CZM and the implicit finite element scheme, respectively) are addressed in detail. New rate-independent and rate-dependent CZMs, e.g. a power-law CZM, tailored for fracture of asphalt concrete are proposed. A new operational definition of crack tip opening displacement (CTOD), called [delta] 25 , is employed to considerably minimize the contribution of bulk material in measuring fracture energy. Predicted numerical results match well with experimental results without calibration. Simulations of various two- and three-dimensional mode I fracture tests, e.g. disk-shaped compact tension (DC(T)), are performed considering viscoelastic effects. The ability to simulate mixed-mode fracture and crack competition phenomenon is demonstrated in conjunction with single-edge notched beam (SE(B)) test simulation. The predicted mixed-mode fracture behaviors are found to be in close agreement with experimental results. Fracture behavior of pavement under tire and temperature loadings is explored.

Recent advances in material processing technology have enabled the design and manufacture of new functionally graded material systems that can withstand Very high temperature and large thermal gradient. Galerkin boundary element method is a powerful numerical method with good efficiency and accuracy which uses C 0 elements for hypersingular integrals which are essential for solving fracture problem. Novel Galerkin boundary element method formulations for steady state and transient heat conduction, and fracture problems involving multiple interacting cracks in three-dimensional graded material systems are developed. In the boundary element formulation, treatment of the singular and hypersingular integrals is one of the main challenges. A direct treatment of the hypersingular integral using a hybrid analytical/numerical approach is presented. Symmetric Galerkin formulation for exponentially graded material using the Green's function approach is developed. In the Green's function approach, each material variation requires different fundamental solution to be derived and consequently, new computer codes to be developed. In order to alleviate this constraint a "simple" Galerkin boundary element method is proposed where the nonhomogeneous problems can be transformed to known homogeneous problems for a class of variations (quadratic, exponential and trigonometric) of thermal conductivity. The material property can have a functional variation in one, two and three dimensions. Recycling existing codes for homogeneous media, the problems in nonhomogeneous media can be solved maintaining a pure boundary only formulation. This method can be used for any problem governed by potential theory. The transient heat conduction is carried out using a Laplace transform Galerkin formulation whereas the crack problem is formulated using the dual boundary element method approach. The implementations of all the techniques involved in this work are discussed and several numerical examples are presented to demonstrate the accuracy and efficiency of the methods. Finally, new techniques of scientific visualization, which is an integral part of computational science research, are explored in the context of boundary element method. This investigation includes developing new modules for viewing the boundary and the domain data using modern visualization tools, developing virtual reality based visualization and concluding with web based interactive visualization.

A natural or engineered multiphase composite with macro-scale spatial variation of material properties may be referred to as a functionally graded material, or FGM. FGMs can enhance structural performance by optimizing stiffness, improving heat, corrosion or impact resistance, or by reducing susceptibility to fracture. One promising application of FGMs is to thermal barrier coatings, in which a ceramic coating with high heat and corrosion resistance transitions smoothly to a tough metallic substrate. The absence of a discrete interface between the two materials reduces the occurrence of delamination and spallation caused by growth of interface and surface cracks. Fracture remains an important failure mechanism in FGMs, however, and the ability to predict critical flaw sizes is necessary for the engineering application of these materials. This presentation describes the development of numerical methods used to compute fracture parameters necessary for the evaluation of flaws in elastic continua. The current investigation employs post-processing techniques in a finite-element framework to compute the J -integral, mixed-mode stress intensity factors and non-singular T -stresses along generally-curved, planar cracks in three-dimensional FGM structures. Domain and interaction integrals developed over the past thirty years to compute these fracture parameters have proved to be robust and accurate because they employ field quantities remote from the crack. The recent emergence of promising engineering applications of FGMs motivates the extension of these numerical methods to this new class of material. This work first develops and applies a domain integral method to compute J -integral and stress intensity factor values along crack fronts in FGM configurations under mode-I thermo-mechanical loading. The proposed domain integral formulation accommodates both linear-elastic and deformation-plastic behavior in FGMs. Next discussed is the extension of interaction-integral procedures to compute directly mixed-mode stress intensity factors and T -stresses along planar, curved cracks in FGMs under linear-elastic loading. The investigation addresses effects upon interaction integral procedures imposed by crackfront curvature, applied crack-face tractions and material nonhomogeneity. Additional considerations for T -stress evaluation include the influence of mode mixity and computation of the anti-plane shear component of non-singular stress, T_13 .

The fracture parameters describing the crack tip fields in functionally graded materials (FGMs) include stress intensity factors (SIFs) and T-stress (non-singular stress). These two fracture parameters are important for determining the behavior of a crack under mixed-mode loading conditions in brittle FGMs (e.g. ceramic/ceramic such as TiC/SiC). The mixed-mode SIFs and T-stress in isotropic and orthotropic FGMs are evaluated by means of the interaction integral method, in the form of an equivalent domain integral, in combination with the finite element method, and are compared with available reference solutions. Mixed-mode crack propagation in homogeneous and graded materials is performed by means of a remeshing algorithm of the finite element method considering general mixed-mode and non-proportional loadings. Each step of crack growth simulation consists of calculation of mixed-mode SIFs, determination of crack growth based on fracture criteria, and local automatic remeshing along the crack path. The present approach requires a user-defined crack increment at the beginning of simulation. Crack trajectories obtained by the present numerical simulation are compared with available experimental results. Other numerical results such as load and SIF history versus crack extension are also provided for an improved understanding of fracture behavior of FGMs.

The focus of this work is to solve crack problems in functionally graded materials (FGMs) with strain-gradient effect. The method used and developed is called hypersingular integral equation method in which the integral is interpreted as a finite part integral, and it can be considered as a generalization of the well-known singular integral equation method. In developing the method, we have derived the exact formulas for evaluating the hypersingular integrals and used Mellin transform to study the crack-tip asymptotics; we have detailed the numerical approximation procedures; also, we have generalized the definition of stress intensity factors (SIFs) under strain-gradient theory and provided formulas for computing SIFs. Different types of crack problems have been solved: Conventional classical linear elastic fracture mechanics (LEFM) vs. strain-gradient theory; scalar problems (Mode III fracture) vs. vector ones (Mode I fracture); homogeneous materials vs. FGMs; different geometric setting of crack location and material gradation. In particular, we obtain a closed form solution for the crack profile in one simple case--Mode III crack problems in homogeneous materials with the characteristic length [cursive l] ' responsible for surface strain-gradient term being zero.