All dissertations are copyrighted by the research group of G. H. Paulino or the authors.
Publications available here are for educational or academic use only. All rights of reproduction or distribution in any form are reserved.
Junho Chun. "ReliabilityBased Topology Optimization Frameworks for the Design of Structures Subjected to Random Excitations" PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2016. Link to Dissertation  Link to Dissertation Presentation  Link to Dissertation on Illinois IDEALS View/Hide Abstract
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 nonstructural 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. 
Daniel W. Spring. "Failure Processes in Soft and QuasiBrittle Materials with Nonhomogeneous Microstructures" PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2015. Link to Dissertation  Link to Dissertation Presentation  Link to Dissertation on Illinois IDEALS View/Hide Abstract
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 biocomposites and medical implants. In this thesis, we numerically investigate a wide spectrum of failure behavior; in soft and quasibrittle 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 ParkPaulinoRoesler tractionseparation relation. To account for friction, we present a new, coupled cohesivefriction 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 twodimensional dynamic cohesive fracture simulations of quasibrittle 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 randomlyseeded 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 elementsplitting 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 threedimensions. 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 quasibrittle 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. 
Sofia Leon. "Adaptive Finite Element Simulation of Fracture: From Plastic Deformation to Crack Propagation" PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2015. Link to Dissertation  Link to Dissertation Presentation  Link to Dissertation on Illinois IDEALS View/Hide Abstract
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: quasibrittle 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 Liegroup 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 threedimensional problems on CPU and GPU systems with the main goals of either improving computational efficiency or fidelity of the final solution. We investigate quasibrittle fracture by means of the interelement 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 interelement 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 threedimensions, the size of the problem increases so rapidly that mesh adaptivity is critical to enable the simulation of largescale 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 reducedscale 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.

Tomás Zegard. "Structural Optimization: From Continuum and Ground Structures to Additive Manufacturing" PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2014. Link to Dissertation  Link to Dissertation Presentation  Link to Dissertation on Illinois IDEALS View/Hide Abstract
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. Gradientbased 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 closedform solutions for common problems in structural engineering. These closedform 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 densitybased 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.

Arun L. Gain. "Polytopebased Topology Optimization Using a MimeticInspired Method" PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2013. Link to Dissertation  Link to Dissertation Presentation  Link to Dissertation on Illinois IDEALS View/Hide Abstract
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 neighborbased coordinates, metric coordinate method and maximum entropy shape functions. The extension of the shape functions to threedimensions, 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 nonpolynomial 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 threedimensional 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 firstorder 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 threedimensional 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 onenode connections, are present in such formulations. Constraints in the geometrical features of spatial discretization can also result in mesh dependent suboptimal 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 densitybased methods and differential equationdriven methods (further subclassified as levelset and phasefield methods). Before choosing densitybased methods for polyhedral topology optimization, a couple of differential equationdriven methods; which are representative of the literature, are exhaustively analyzed in twodimensions. Finally, we also investigate aesthetics in topology optimization designs. In this work, twodimensional 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 Differenceinspired approach (VEM) facilitates accurate numerical analysis on any non selfintersecting closed polygons such as tessellations.

Lauren Beghini. "Building Science Through Topology Optimization." PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2013. Link to Dissertation  Link to Dissertation Presentation  Link to Dissertation on Illinois IDEALS View/Hide Abstract
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 highrise 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, multiobjective optimization, including linearized buckling, has been studied in this context. To handle the large amounts of data associated with a highrise, this new framework has been written to take advantage of a topological data structure together with objectoriented 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 highrise 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.

Cameron Talischi. "Restriction Methods for Shape and Topology Optimization." PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2012. Link to Dissertation  Link to Dissertation Presentation (Video file size: 552 MB)  Link to Dissertation on Illinois IDEALS View/Hide Abstract
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
highrise buildings to the design of patienttailored craniofacial bone replacements.
Optimal shape problems and more generally PDEconstrained inverse problems, however,
pose several fundamental challenges. For example, these problems are often illposed 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 nonconvergent sequences of shapes that exhibit progressively
finer features. To address the illposedness, one either enlarges the admissible design space
allowing for generalized microperforated 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 wellposed 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 socalled variable
thickness problem, whose theoretical optimal solutions do not have a clearlydefined
boundary. Within the restriction setting, we also analyze and provide a justification for the
socalled 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 largescale nonconvex 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 twodimensional topology optimization problems.
We show that these elements, unlike their loworder Lagrangian counterparts, are not susceptible
to certain gridscale 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 loworder finite element spaces obtained from polygonal
discretizations satisfy the wellknown BabuskaBrezzi 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 ngons. 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 selfcontained discretization
and analysis Matlab code using polygonal elements, along with a general framework for
topology optimization.

Tam Nguyen. "System ReliabilityBased Design and Multiresolution Topology Optimization." PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2010. Link to Dissertation  Link to Dissertation Presentation  Link to Dissertation on Illinois IDEALS View/Hide Abstract
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 largescale 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
reliabilitybased design optimization (SRBDO) procedure; and (3) to investigate the
reliabilitybased topology optimization (RBTO) problem. First, it is noted that the
material distribution method often requires a large number of design variables, especially
in threedimensional applications, which makes topology optimization computationally
expensive. A multiresolution topology optimization (MTOP) scheme is thus developed to
obtain highresolution 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 reliabilitybased design optimization (RBDO) due to the complexity of
system reliability analysis. A new singleloop system RBDO approach is developed by
using the matrixbased 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 firstorder 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 secondorder reliability method (SORM), termed as
SORMbased RBDO, are proposed. These technical advances enable us to perform
RBTO of largescale 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 largescale
problems in engineering design practices.

Eshan Dave. "Asphalt Pavement Aging and Temperature Dependent Properties Using Functionally Graded Viscoelastic Model." PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2009. Link to Dissertation  Link to Dissertation Presentation View/Hide Abstract
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 nonhomogeneity. Current pavement analysis and simulation procedures dwell on the use of layered approach to account for these nonhomogeneities. The conventional finiteelement 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 nonhomogeneities. Specialized elements such as “graded elements” allow for nonhomogenous 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 inservice age. Other application areas include simulation of functionally graded fiberreinforced concrete, geotechnical materials, metal and metal composites at high temperatures, polymers, and several other naturally existing and engineered materials.

Kyoungsoo Park. "PotentialBased Fracture Mechanics Using Cohesive Zone and Virtual Internal Bond Modeling." PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2009. Link to Dissertation  Link to Dissertation Presentation View/Hide Abstract
The characterization of nonlinear constitutive relationships along fracture surfaces
is a fundamental issue in mixedmode cohesive fracture simulations. A generalized
potentialbased constitutive theory of mixedmode fracture is proposed in conjunction
with physical quantities such as fracture energy, cohesive strength and shape of
cohesive interactions. The potentialbased model is verified and validated by investigating
quasistatic fracture, dynamic fracture, branching and fragmentation. For
quasistatic 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 edgeswap 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 edgesplit
and vertexremoval operators to reduce computational cost. Computational results
demonstrate that the potentialbased 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 quasibrittle material fracture behavior. All the
computational models have been developed in conjunction with verification and/or
validation procedures.

Bin Shen. "Functionally Graded FiberReinforced Cementitious Composites – Manufacturing and Extraction of Cohesive Fracture Properties using Finite Elements and Digital Image Correlation." PhD Dissertation, Department of Civil and Environmental Engineering, UIUC, 2009. Link to Dissertation View/Hide Abstract
A novel fourlayer functionally graded fiberreinforced 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 fullfield digital image correlation (DIC) technique has been adopted to compute the twodimensional 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 NelderMead optimization method is used to solve the illposed 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.

Shun Wang. "Krylov Subspace Methods for Topology Optimization on Adaptive Meshes." PhD Thesis, Department of Computer Science, UIUC, 2007. Link to Dissertation View/Hide Abstract
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 illconditioned 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 illconditioning 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 shortterm 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 convectiondiffusion problems, it achieves a levelindependent 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 levelindependent 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.

Zhengyu Zhang "Extrinsic Cohesive Modeling of Dynamic Fracture and Microbranching Instability Using A Topological Data Structure." PhD Thesis, Department of Civil and Environmental Engineering, UIUC, 2007. Link to Dissertation View/Hide Abstract
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 topologybased 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 twodimensional and threedimensional problems are analyzed. A series of dynamic fracture phenomena, including spontaneous crack initiation, dynamic crack microbranching 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 ratedependent problems.

Seong Hyeok Song "Fracture of Asphalt Concrete: A Cohesive Zone Modeling Approach Considering Viscoelastic Effects." PhD Thesis, Department of Civil and Environmental Engineering, UIUC, 2006. Link to Dissertation View/Hide Abstract
Asphalt concrete is a quasibrittle 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 generalpurpose 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 rateindependent and ratedependent CZMs, e.g. a powerlaw 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 threedimensional mode I fracture tests, e.g. diskshaped compact tension (DC(T)), are performed considering viscoelastic effects. The ability to simulate mixedmode fracture and crack competition phenomenon is demonstrated in conjunction with singleedge notched beam (SE(B)) test simulation. The predicted mixedmode fracture behaviors are found to be in close agreement with experimental results. Fracture behavior of pavement under tire and temperature loadings is explored.

Alok Sutradhar "Galerkin boundary element modeling of threedimensional functionally graded material systems." PhD Thesis, Department of Civil and Environmental Engineering, UIUC, 2005. Link to Dissertation View/Hide Abstract
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 threedimensional 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.

Matthew Walters "Domainintegral methods for computation of fracturemechanics parameters in threedimensional functionallygraded solids." PhD Thesis, Department of Civil and Environmental Engineering, UIUC, 2005. Link to Dissertation View/Hide Abstract
A natural or engineered multiphase composite with macroscale 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 postprocessing techniques in a finiteelement framework to compute the J integral, mixedmode stress intensity factors and nonsingular T stresses along generallycurved, planar cracks in threedimensional 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 modeI thermomechanical loading. The proposed domain integral formulation accommodates both linearelastic and deformationplastic behavior in FGMs. Next discussed is the extension of interactionintegral procedures to compute directly mixedmode stress intensity factors and T stresses along planar, curved cracks in FGMs under linearelastic loading. The investigation addresses effects upon interaction integral procedures imposed by crackfront curvature, applied crackface tractions and material nonhomogeneity. Additional considerations for T stress evaluation include the influence of mode mixity and computation of the antiplane shear component of nonsingular stress, T_13 .

JeongHo Kim "Mixedmode crack propagation in functionally graded materials." PhD Thesis, Department of Civil and Environmental Engineering, UIUC, 2003. Link to Dissertation View/Hide Abstract
The fracture parameters describing the crack tip fields in functionally graded materials (FGMs) include stress intensity factors (SIFs) and Tstress (nonsingular stress). These two fracture parameters are important for determining the behavior of a crack under mixedmode loading conditions in brittle FGMs (e.g. ceramic/ceramic such as TiC/SiC). The mixedmode SIFs and Tstress 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. Mixedmode crack propagation in homogeneous and graded materials is performed by means of a remeshing algorithm of the finite element method considering general mixedmode and nonproportional loadings. Each step of crack growth simulation consists of calculation of mixedmode SIFs, determination of crack growth based on fracture criteria, and local automatic remeshing along the crack path. The present approach requires a userdefined 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.

YounSha Chan "Hypersingular integrodifferential equations and applications to fracture mechanics of homogeneous and functionally graded materials with straingradient effects." PhD Thesis, Graduate Group in Applied Mathematics (GGAM), Department of Civil and Environmental Engineering, University of California, Davis, 2001. Link to Dissertation View/Hide Abstract
The focus of this work is to solve crack problems in functionally graded materials (FGMs) with straingradient 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 wellknown 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 cracktip asymptotics; we have detailed the numerical approximation procedures; also, we have generalized the definition of stress intensity factors (SIFs) under straingradient theory and provided formulas for computing SIFs.
Different types of crack problems have been solved: Conventional classical linear elastic fracture mechanics (LEFM) vs. straingradient 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 caseMode III crack problems in homogeneous materials with the characteristic length [cursive l] ' responsible for surface straingradient term being zero.
