The development of sequential explicit, convex approximation schemes has allowed for expansion of the size of optimization problems that can now be achieved. These approximation schemes use information from the original optimization statement to generate a series of approximate subproblems allowing for an efficient solution strategy. This thesis reviews established sequential explicit, convex approximations in the literature along with a brief overview of their associated solution schemes. A primary focus is placed on the theory and application of the Method of Moving Asymptotes (MMA) approximation due to its continued regard in the field of structural topology optimization. Numerical examples explore optimization problems solved by the MMA approximation in order to demonstrate the behavior of this method and impact of the prescribed empirical parameters. Other numerical examples study structural topology optimization problems in the 2D and 3D setting to compare with alternative, competitive update schemes such as the OC and to highlight the benefit of using the MMA in more complex settings.

The traditional Optimality Criteria (OC) update in topology optimization suffers from slow convergence, thereby requiring a large number of iterations to result in only a small improvement in the performance and design. To address this problem, we propose to use a novel fixed-point formulation of the OC update to accelerate the convergence. Such strategies can achieve higher convergence rates without overly complexifying the update process. In this thesis, we first provide some mathematical background on fixed-point iteration methods. Then, based on theoretical analysis and numerical experiments, we analyze these methods’ respective advantages and drawbacks in the context of topology optimization. The analysis focuses on the methods’ design update stability, effectiveness in reducing the design cycles, computational cost, and robustness. Through numerical studies, we found one of the methods, called Periodic Anderson Extrapolation (PAE), is the most stable, effective, economic, and robust approach to speed up OC’s convergence. The overall update is named Periodically Anderson Extrapolated Optimality Criteria (PAE-OC). Via several 2D and 3D benchmarks, we demonstrate that the PAE-OC can effectively reduce both the number of iterations and computation time. In addition, this scheme shows good robustness with respect to the change of boundary conditions, problem sizes, and parameters. Finally, we show the scalability of the PAE-OC through a 3D problem consisting of more than 3 million degrees of freedom.

Structural design methodologies were strongly influenced by the advent of computing. The advances in numerical analyses, such as the finite element method, and Computer Aided Design software have literally helped shape the engineering world as it is today. Structural optimization methods such as topology optimization aim to take the next step by letting the computer guide the design, in order to achieve new and more efficient designs. This approach has the potential to change the future of various industries, including aircraft, automobile, construction, etc. The introduction of stress constraints on traditional topology optimization allows for safer and more reliable solutions that will more closely resemble the final structure. The successful solution of this problem poses several conceptual and numerical difficulties. Thus this dissertation details the main issues of this problem and reviews the current techniques discussed in the literature including some critiques of their performance. The main contribution of this work is a novel technique based on the Augmented Lagrangian method that can efficiently handle a large number of constraints. In contrast to existing methods which are both problem- and mesh-dependent, the presented approach contains only a few parameters which need to be adjusted for each new case. In order to verify the technique's capabilities, a user friendly MATLAB code was developed that is both effective and robust. Several representative examples, including large-scale problems, are presented. Finally, the solutions obtained here, including some unexpected complications, are thoroughly discussed and suggestions for future work are also addressed.

The general shape of a shell structure can have a great impact on its structural performance. This thesis presents a numerical implementation that finds the funicular form of grid shell structures. Two methods are implemented for the form finding: the potential energy method (PEM) and the force density method (FDM). The PEM, inspired by the 3D hanging chain model, find the funicular form by minimization of the total potential energy. On the other hand, the FDM find the funicular form by solving a linear system, in which a geometric stiffness matrix is constructed with the force density and nodal connectivity information. The form finding process is nonlinear, because the nodal loads are calculated with the rationale of tributary area and evolve with the structural form. Apart from the form finding, a member sizing procedure is implemented for a preliminary estimation of the member cross sectional area. In the member sizing, the stress-ratio method is used to achieve a fully-stressed design. Finally, three numerical examples are examined to demonstrate the effectiveness of the current implementation.

The ground structure method is used to find an optimal solution for the layout optimization problem. The problem domain is discretized with a union of highly connected members, which is called a ground structure. The objective typically is to minimize the total volume of material while satisfying nodal equilibrium constraint and predefined stress limits (plastic formulation). However, such approach may lead to very slender members and unstable nodes that might cause instability issues. This thesis presents the implementation of the ground structure method involving instability consideration. The plastic formulation is implemented considering buckling constraint and nodal instability constraint either in isolation or in combination. The Euler buckling criteria is taken as the buckling constraint in the implementation with local instability consideration. The nominal lateral force method is used in the implementation involving nodal instability consideration. Moreover, the efficiency of the nonlinear programming is addressed. Several numerical examples are presented to illustrate the features of the implementation.

This thesis proposes an efficient gradient-based optimization algorithm to solve reliability-based topology optimization (RBTO) of structures under loading and material uncertainties. Topology optimization is a powerful design tool as it can provide the most efficient material layout for structural design problems under given conditions and limitations. However, most attempts are formulated in a deterministic manner, which may be impractical as this formulation ignores the inherent uncertainty and randomness in structural design problems. The objective of RBTO considered in this research is to identify the optimal topology of truss structures with minimum weight which also satisfy certain requirements on the reliability of the structures. As a subtopic of reliability-based design optimization (RBDO), RBTO problems are primarily performed with algorithms based on a first-order reliability method (FORM) which are well developed in the literature for RBDO. However, those algorithms may lead to deficient or even invalid results for RBTO problems since the gradient of probabilistic constraint, calculated by first order approximation, is not accurate enough for RBTO to converge correctly regardless of how accurate the failure probability is approximated. A segmental multi-point linearization (SML) method is proposed for a more accurate estimation of failure probability and its gradient. Numerical examples show that the RBTO algorithm based on the SML is more stable numerically and is able to converge to a solution that is closer to the true optimum than conventional FORM-based algorithms. The obtained optimal topology can serve as a starting point for engineers to make the design of structures both economic and reliable.

Nonlinear elastic materials are of great engineering interest, but challenging to model with standard finite elements. The challenges arise because nonlinear elastic materials are characterized by non-convex stored-energy functions as a result of their ability to undergo large reversible deformations, are incompressible or nearly incompressible, and often times possess complex microstructures. In this work, we propose and explore an alternative approach to model finite elasticity problems in two dimensions by using polygonal discretizations. We present both lower order displacement-based and mixed polygonal finite element approximations, the latter of which consist of a piecewise constant pressure field and a linearly-complete displacement field at the element level. Through numerical studies, the mixed polygonal finite elements are shown to be stable and convergent. For demonstration purposes, we deploy the proposed polygonal discretization to study the nonlinear elastic response of rubber filled with random and periodic distributions of rigid particles, as well as the development of cavitation instabilities in elastomers containing vacuous defects. These physically-based examples illustrate the potential of polygonal finite elements in studying and modeling nonlinear elastic materials with complex microstructures under finite deformations.

In this thesis, the generation of initial ground structures for generic domains in two and three dimensions are discussed. Two methods of discretization are compared: Voronoi-based discretizations and structured quadrilateral discretizations. In addition, a simple and effective member generation approach is proposed: the Macro-element approach; which can be implemented with both types of discretization. The features of the approach are discussed: efficient generation of initial ground structures; reduction in matrix bandwidth for global stiffness matrix; finer control of bar connectivity; and reduction of overlapped bars. Numerical examples are presented which display the features of the proposed approach, and highlight the comparison with literature results and traditional ground structure generation methods.

There is a constant demand from the industry to provide better, more situation dependent construction materials; materials which are able to satisfy strength requirements while also able to accommodate other design requirements such as ductility, fracture resistance, thermal resistance/insulation, etc. Functionally graded materials (FGMs) are one such material. This study investigates the fracture process of sustainable concrete, fiber reinforced concrete, and of functionally graded concrete slabs. Both two-dimensional and three-dimensional problems are analyzed.

The primary focus of the thesis is on sustainable, functionally graded concrete slabs, emphasizing the computational/mechanical aspects of fracture. A model of the slabs is developed; which incorporates a variety of cohesive zone models (CZMs) into an implicit, nonlinear finite element scheme. Intrinsic cohesive zone elements, with traction-separation relationships defined along the crack surface, are utilized to simulate mode I fracture of the slabs. Based on the load to crack mouth opening displacement (CMOD) relationships of the slab, one is able to optimize concrete properties and placement to reach predefined goals. A parametric study is conducted on the fracture parameters of the slab; the results of which show that the variations in the CZMs have a direct correlation with the overall behaviour of the slab. Additionally, in conducting the experiments for the slabs, a new fracture test for concrete is developed. The attractive feature of the test is that it uses a specimen geometry which is easily obtained from in-situ concrete in the field. Technology exists which allows us to extract cylindrical cores from concrete structures at relative ease. This study proposes a specimen geometry which can easily be developed from these cylindrical cores called the disk-shaped compact tension (DCT) specimen. A series of experiments are conducted on the specimen, and computational simulations are carried out. A parametric study is done; the results of which, show that the specimen geometry is able to predict the mode I fracture properties of concrete, with both virgin and recycled aggregates, with relative accuracy and ease.

Current work utilizes cohesive zone modeling to study the fracture properties of metals. This study proposes a hybrid technique experimental/numerical to extract cohesive fracture properties of elasto-plastic material using inverse analysis and digital image correlation. Two approaches are suggested - a shape optimization technique and a parameter optimization for the PPR potential-based cohesive zone model. In shape optimization approach, CZM is obtained by piecewise interpolation of the optimized interpolation points whereas in parameter optimization for the PPR potential-based CZM, the CZM is obtained by using the PPR model which utilized the parameters coming from an optimization scheme. Unconstrained, derivative free Nelder-Mead scheme is used for optimization purpose. The bulk material is modeled as plane-stress J2 plastic material. The proposed schemes are verified for various plausible cases, such as different displacement field data, various initial guess and noisy displacement field data. As a proof of concept, both schemes are applied to Polymethyl Methacrylate (PMMA) quasistatic crack growth experiment, which is modeled as elastic material, to substantiate its utility. Near tip displacement field is obtained experimentally using DIC and used as input to the optimization schemes. Computationally predicted global responses of the PMMA specimen, using the CZMs extracted from the inverse analysis, shows good agreement with the experimental global response.

A library of nonlinear solution schemes including load, displacement, arc-length, work, generalized displacement, and orthogonal residual control are cast into a unified framework for solving nonlinear finite element systems. Each of these solution schemes differs in the use of a constraint equation for the incremental-iterative procedure. The governing finite element equations and constraint equation for each solution scheme are combined into a single matrix equation, which characterizes the unified approach. This theoretical model leads naturally to an effective object-oriented implementation and potential for integration into a finite element analysis code. Using this framework, the strengths and weaknesses of the various solution schemes are examined through several numerical examples.

This work explores the use of manufacturing-type constraints, in particular pattern gradation and repetition, in the context of layout optimization. By placing constraints on the design domain in terms of number and size of repeating patterns along any direction, the conceptual design for buildings is facilitated. To substantiate the potential future applications of this work, examples within the context of high-rise building design are presented. Successful development of such ideas will lead to practical engineering solutions, especially during the building design process. Throughout this work, a continuous topology optimization formulation is utilized with compliance as the objective function and constraints on the pattern geometry. Examples are given to illustrate the ideas developed both in two-dimensional and three-dimensional building configurations.

The present work investigates the feasibility of nite element methods and topology optimization for unstructured meshes in massively parallel computer architectures, more specically on Graphics Processing Units or GPUs. Algorithms for every step in these methods are proposed and benchmarked with varied results. The ultimate goal of this work is to speed up the topology optimization process by means of parallel computing using o-the-shelf hardware. To further facilitate future application and deployment, a transparent massively parallel topology optimization code was written and tested. Examples are compared with both, a standard sequential version of the code, and a massively parallel version to better illustrate the advantages and disadvantages of this approach.

Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these conventional elements exhibit the well-known “checkerboard” pathology in the iterative solution of topology optimization problems. A feasible alternative to eliminate such long-standing problem consists of using hexagonal (honeycomb) elements with Wachspresstype shape functions. The features of the hexagonal mesh include two-node connections (i.e. two elements are either not connected or connected by two nodes), and three edge-based symmetry lines per element. In contrast, quads can display one-node connections, which can lead to checkerboard; and only have two edge-based symmetry lines. In addition, Wachspress rational shape functions satisfy the partition of unity condition and lead to conforming finite element approximations. We explore the Wachspress-type hexagonal elements and present their implementation using three approaches for topology optimization: elementbased, continuous approximation of material distribution, and minimum length-scale through projection functions. Examples are presented that demonstrate the advantages of the proposed element in achieving checkerboard-free solutions and avoiding spurious finescale patterns from the design optimization process.

The topology optimization problem is generally an ill-posed problem, where the solution is not unique. Because of that, implementation of topology optimization faces complications such as mesh-dependency of the solution, and numerical instabilities. Also, the results obtained by topology optimization are not usually fabrication friendly due to the fine and arbitrary patterns. An effective method that addresses both above mentioned issues consists of imposing a minimum length scale to the resulting structural members. Several approaches to achieve minimum length scale in topology optimization have been proposed in the literature. However, while some problems are solved, others are created or remain unsolved; and the search for better methods continues. In this thesis, we review several prominent approaches and propose a new approach to achieve minimum length scale. We discuss the potential of the new approach for obtaining other fabrication and design constraints, in addition to the minimum length scale constraint. Thus the new approach has the potential of improving the quality of topology optimization in various engineering applications where design constraints must be placed.

The size effect is the change of structural properties, especially nominal strength, due to scaling of geometrically similar structures. Due to the relatively large non-linear fracture process zone, the size effect on the nominal strength of a concrete structure is explained by non-linear fracture mechanics employing both an equivalent elastic crack model and a cohesive zone model (CZM) approach. The concept of equivalent elastic crack model provides the theoretical background for the size effect method (SEM) and the two-parameter fracture model (TPFM), which provide two size-independent fracture parameters. In addition, the CZM characterizes non-linear fracture process behavior through the bi-linear softening curve, which is determined by four experimental fracture parameters: tensile strength (ft), initial fracture energy (Gf ), total fracture energy (GF ) and critical crack tip opening displacement (CTODc). The location of the kink point in the bi-linear softening model has been estimated empirically in the literature. Thus a formal criterion to determine the kink point is proposed and discussed. The bi-linear softening curve in the CZM enables prediction of the load versus crack mouth opening displacement (CMOD) experimental curves as well as the size effect. Several examples and a sensitivity analysis are given to illustrate these points.

Dynamic stress intensity factors (DSIFs) are important fracture parameters in understanding and predicting dynamic behavior of a cracked body. To evaluate DSIFs for both homogeneous and nonhomogeneous materials, the interaction integral (conservation integral) originally proposed to evaluate SIFs for a static homogeneous medium is extended to incorporate dynamic eÞects and material nonhomogeneity, and is implemented in conjunction with the finite element method. In this study, a research code is developed and verified using benchmark problems. Then, various homogeneous and nonhomogeneous cracked bodies under dynamic loading are employed to investigate dynamic fracture behavior such as the variation of DSIFs for different material profiles, the relation between initiation time and the domain size (for integral evaluation), and the contribution of each distinct term in the interaction integral.

The dynamic failure of homogeneous and Functionally Graded Materials (FGMs) can be simulated by incorporating a Cohesive Zone Model (CZM) into the numerical scheme. The failure criterion is incorporated by the CZM using both a ßnite cohesive strength and work to fracture in the material description. In this study, ßrst the general dynamic behavior of FGMs (without initial crack) is investigated considering bulk material modeled with graded elements, i.e. elements possessing a spatially varying material property ßeld, such as Young's modulus, Poisson's ratio, mass density, etc. The results reveal some interesting features of dynamic behavior of FGMs compared to that of homogeneous materials. Next, two CZMs developed for FGMs are described and the numerical implementation scheme is discussed. Finally, the inàuence of material property variation on the crack propagation pattern for FGM structures under impact loading is investigated with a number of examples. The powerful features of CZMs in simulating branching and spontaneous crack initiation behaviors are also presented. The present ßnite element code is named I-CD (Illinois Cohesive Dynamic).