Arisen from the geometric arrangements of panels and creases, unique mechanical properties such as foldability endow origami with promise for developing novel tunable and functional structural systems. To promote engineering applications of origami, a simplified but effective approach for investigation of the nonlinear mechanical behavior of non-rigid origami structures is essential. We propose a fully nonlinear, displacement-based formulation, for constructing quasi-static finite element analyses of origami structures based on a previously established bar-and-hinge simplification. The formulation leads to an efficient and robust numerical approach for predicting large displacements and large local deformations of origami structures. Comparison between actual paper-made models and numerical simulations hints the ability of the proposed approach in capturing key features of origami deformation. Thus the current work builds up a connection between theory and practice of origami structures, which has the potential to impact design, education, and applications of origami.

We propose a composite panel that can be assembled in a planar state and can be transformed with a one-DOF kinematic mechanism after the assembly. The proposed structure is comprised of generalized rigid-foldable tubes. The tubes are assembled such that they are non-trivially compatible with one another, but still share a desired single-curved surface. Because of the nontrivial assembly, the structure is expected to be flexible only in the desired one-DOF motion, deploying from a flat state to a 3D state, while it is significantly stiffer against other motions. The geometric construction follows the following procedure; (1) obtain an equivalent origami structure from a generating surface, (2) attach compatible tubular assemblies on both sides of the surface. This method produces a wide range of rigidly-foldable composite structures including corrugated surfaces with a flat-foldable compact state.

A new method is proposed to incorporate the first passage probability into stochastic topology optimization using sequential compounding method (Kang and Song 2010). Parameter sensitivities of the first passage probability in the probabilistic constraint are derived to facilitate the use of gradient-based optimizer for efficient topology optimization. The proposed method is applied to building structures subjected to stochastic ground motion to find optimal bracing systems which can resist future realization of stochastic excitations while achieving a desired level of reliability.

This paper presents an efficient numerical method for approximating the parameter sensitivity of the failure probability with respect to design parameters. The method is computationally inexpensive and the obtained approximations are more accurate than the approximations based on first order reliability method (FORM). The method is particularly suitable for applications in reliability-based design optimization (RBDO), including reliability-based topology optimization (RBTO).

Origami, an ancient paper folding art, has inspired many solutions to modern engineering challenges. The demand for actual engineering applications motivates further investigation in this field. Although rooted from the historic art form, many applications of origami are based on newly designed origami patterns to match the specific requirenments of an engineering problem. The application of origami to structural design problems ranges from micro-structure of materials to large scale deployable shells. For instance, some origami-inspired designs have unique properties such as negative Poisson ratio and flat foldability. However, origami structures are typically constrained by strict mathematical geometric relationships, which in reality, can be easily violated, due to, for example, random imperfections introduced during manufacturing, or non-uniform deformations under working conditions (e.g. due to non-uniform thermal effects). Therefore, the effects of uncertainties in origami-like structures need to be studied in further detail in order to provide a practical guide for scalable origami-inspired engineering designs. Through reliability and probabilistic analysis, we investigate the effect of randomness in origami structures on their mechanical properties. Dislocations of vertices of an origami structure have different impacts on different mechanical properties, and different origami designs could have different sensitivities to imperfections. Thus we aim to provide a preliminary understanding of the structural behavior of some common scalable origami structures subject to randomness in their geometric configurations in order to help transition the technology toward practical applications of origami engineering.

Origami systems inspired from the Miura-ori pattern are rigid and flat foldable meaning that they can fold completely by deforming only about prescribed fold lines. We investigate origami tubes and assemblages constructed from Miura-ori inspired sheets and use eigenvalue analyses to study their stiffness characteristics. A simplified bar model is used to model the stretching and shear of the flat panel segments and rotational hinges are used to simulate the bending stiffness of the panels and prescribed fold lines. We discuss the small to large deformation bending of thin sheets and show an improved method to estimate stiffness when modeling origami structures. The tube assemblages show interesting behaviors that make them suitable for applications in science and engineering.

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 nonconvex 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 study, 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.

When a particle inclusion is embedded in a polymer matrix, the polymer tends to adsorb on the surface of the inclusion. The effect of this may result in an interphasial zone between the particle and the polymer often referred to as “bound” rubber. The extent and composition of this zone depends on a number of factors, including the surface area and surface treatment of the particle, as well as the level of mixing and age of the composite [1]. Studies on the failure of particle reinforced polymers have indicated that, at large strains, cracks/debonding (occurring at the microscale) can have a significant influence on the macroscopic response of these composites [1, 2, 3]. There has been much work done on the debonding process of particle reinforced composite materials under small deformations, but in recent years the interest in finite deformation debonding has increased. In this study we present a fully three-dimensional model, using cohesive zone elements (CZEs) to account for the nonlinear debonding process between the particles and the interphase, to simulate the behavior of these composites under finite deformations. The nonlinear relation used for the cohesive model is the consistent, potential-based PPR model [4]. Our numerical model uses the concept of reduced volume elements (RVEs) with periodic boundary conditions to represent isotropic materials.

References
[1] J. L. Leblanc. Rubber-filled interactions and rheological properties in filled compounds. Progress in Polymer Science, 27:627-687, 2002
[2] J. Ramier. Comportement mécanique d’élastomères chargés, Influence de l’adhésion charge – polymère, Influence de la morphologie. PhD thesis, L’Institut National des Sciences Appliquées de Lyon, 2004.
[3] J. Ramier, L. Chazeau, and C. Gauthier. Influence of silica and its different surface treatments on the vulcanization process of silica filled SBR. Rubber Chemistry and Technology, 80:183-193, 2007.
[4] K. Park, G. H. Paulino, and J. R. Roesler. A unified potential-base cohesive model for mixed-mode fracture. Journal of the Mechanics and Physics of Solids, 57:891-908, 2009.

Origami patterns have been applied in spatial structures to make stiff shell structures as well as flexible transformable systems. Folding a planar sheet into a 3D configuration changes the stiffness and the behavior of the sheet. In this paper we discuss a scalable analytical model for simulating origami structures, and we use eigenvalue band-gaps to optimize both the flexibility and stiffness of the system. We focus our study on rigid, flat foldable tubes and investigate the influence that different parameters have on the stiffness characteristics.

Structural optimization has a long history of applications with buildings. Nonetheless, an optimal structure is a somewhat vague statement if no information is provided as to how was the scoring made (objective function). This work explores the geometrical aspects of lateral bracing systems commonly used in providing lateral support to buildings. A variety of for- mulations are described and used to attain the optimal bracing point location for a building system modeled as a truss. Results using the different formulations in two and three-dimensions are discussed and compared in order to arrive to a few useful conclusions for the practicing engineer. The analysis and results are then extended to include vertical loads, multiple stories and bays. Real life examples of lateral bracing systems in buildings are given and discussed on the framework of these findings.

Structural optimization has been frequently applied to problems of member sizing and/or shape finding, using discrete or continuum elements. Structures that combine both, discrete and continuum elements, has however received less attention. Reinforced concrete, beam-wall connections and slabs are a few examples of structures that can be modeled with a combination of discrete and continuum elements. This work presents a simple embedding formulation, that can be used to optimize the layout of a truss connected to a continuum. The formulation gives smooth gradient fields, that allow the use of gradient-based optimization algorithms. The embedding is based on a convolution (with an arbitrary degree of smoothness) of the degrees of freedom at the truss nodes. The coupling makes no use of the continuum topology, and treats the continuum as a cloud of points for which the solution is known. To efficiently search the linking nodes, a tree data structure is used, and because the continuum topology is fixed, the information from the tree is re-used for all iterations within the optimization loop. Advantages and shortcomings of the formulation are discussed, and compared to existing formulations. The solution for a simple problem for which an analytical solution can be obtained is used to validate and benchmark the proposed method. Further examples of varied complexity and dimensional space are also analyzed in order to provide better understanding of the method.

We investigate truss topology design with special attention to the pre-processing phase of ground structure generation together with its implications on the optimization process. Specifically we use a Voronoi-based ground structure generation approach at different levels (level 1, 2, etc.) with operators to treat overlapping bars, to create desirable connecting bars, etc. We also establish a metric to relate the effectiveness of the ground structure with respect to the properties of the underlining topology optimization problem. We address several computational aspects such as convergence of the method and type of optimality criteria (e.g. standard and modified versions) adopted. We present several examples for two-dimensional and three-dimensional problems and comment on the features of each problem addressed.

Topology optimization is becoming increasingly popular in the eld of civil engineering; however, research in the development of this technology for the design of high-rise buildings demands further attention. Thus, this work contributes to improve this application by describing an integrated topology optimization approach involving the concurrent optimization of both continuum (polygonal) and discrete (beam/truss) nite elements to design structural systems in high-rise buildings. By incorporating both types of ele- ments, the overall design process is simplied and improved. For instance, after the locations of the outer skin or shell of a high-rise building and its columns are determined, topology optimization can be used to design the internal structural system, while concurrently sizing the members. Several practical examples are given to show the importance and relevance of this work to the structural design industry for a variety of objective functions, including compliance, buckling and combinations thereof.

One of the fundamental considerations in structural engineering is to design structural system to withstand various types of stochastic excitations such as earthquake ground excitations, wind loads and ocean waves. Due to the randomness of stochastic excitations, random vibration analysis is required to characterize probabilistic structure responses, such that the incorporation of the probabilistic prediction leads to designing of the structures that can reliably withstand such hazardous events over prolonged periods of time. As the time histories do not provide an accurate prediction of future realization of a random process, probabilistic prediction of structural responses therefore needs to be based on random vibration analysis instead. Optimization of structural systems has been heavily implemented in structural engineering to achieve optimal performance that also satisfies constraints on the dynamic responses as demonstrated through mathematical programming based topology optimization. Despite its advances in fundamental theories, numerical algorithms and technical methodologies, there are computational challenges that inhibit the actual incorporation of stochastic response structures into topology optimization. Previously, the authors proposed a novel approach to overcome such technical difficulties in topology optimization under stochastic excitation, which allows computation of the instantaneous failure probability of a structure using a closed-form solution during the topology optimization by characterizing the input stochastic excitation by a discrete representation method and using structural reliability theory. In this paper, the proposed method is further developed to handle ‘system’ failure events, i.e. Boolean functions of multiple limit-states defined in terms of different locations, failure modes and time points. A system reliability-based topology optimization (SRBTO) approach is adopted for this purpose when considering structures under stochastic excitations. This approach employs the Matrix-based System Reliability (MSR) method to overcome challenges in evaluation of probability of system failure event of statistically dependent limit-states, and improve the accuracy of reliability analysis. The proposed method is demonstrated by numerical examples of optimization of structures under stochastic ground excitations.

One of the main objectives in the field of structural optimization is to achieve a structural design that produces the best performance while satisfying given design constraints. Among various applications of structural optimization, mathematical-programming-based topology optimization has gained recognition in the research community as well as industrial practice. This is primarily in response to the increasing number of labor hours and financial resources invested by structural engineers to control the responses 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 significance in industrial applications. Despite rapid technological advances, however, incorporating stochastic response of structures into topology optimization has not received attention as a new emerging field of research until recently, mainly due to computational challenges. This paper addresses technical difficulties in topology optimization under stochastic excitations by using a discrete representation method for stochastic processes (Chun et al. 2012). Using the characteristic representation of the uncertainty based on the discrete representation method and structural reliability theory, the failure probability is readily obtained by a closed-form solution for the linear system. In addition, system reliability-based topology optimization (SRBTO; Nguyen et al. 2011) is considered to account for the statistical dependence between multiple limit-states defined at different location, failure modes and time points. Using the Matrix-based System Reliability (MSR) method (Song and Kang 2009), the proposed SRBTO method overcomes difficulty of evaluating the probability of system failure events and improves the accuracy of reliability analysis. The proposed method is demonstrated by numerical examples of structures under stochastic ground motion excitations.

Reference
• Chun, J., J. Song, and G.H. Paulino (2012), Topology optimization of structures under stochastic excitations. 2012 Joint Conference of the Engineering Mechanics Institute and the 11th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, June 17-20, Notre Dame, IN
• Nguyen, T.H., J. Song, and G.H. Paulino (2011). Single-loop system reliability-based topology optimization considering statistical dependence between limit states. Structural and Multidisciplinary Optimization, Vol. 44(5), 593-611.
• Song, J., and W.-H. Kang (2009). System reliability and sensitivity under statistical dependence by matrix-based system reliability method. Structural Safety, Vol. 31(2), 148-156.

One of the main goals in structural optimization is to achieve a structural design with 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 the research community and structural engineering practice. This recent interest reflects the significance of time and financial resources dedicated by structural engineers 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 particular interest and is meaningful in industrial applications. Despite rapid technological advancesin this field however, computational challenges have prevented further development of its application, namely the actual incorporation of stochastic response of structures into topology optimization. In order to overcome such technical challenges in this field, this paper introduces a new method for incorporating random vibration theories into topology optimization in order to satisfy probabilistic constraints. This method uses a discrete representation method for stochastic processesto describe the stochastic response of a system subjected to random seismic excitations. Furthermore, a new formulation is developed for sensitivity of dynamic responses in order to use gradient-based optimization algorithms for the proposed topology optimization employing a discrete representation method. The proposed method is demonstrated by numerical examples of structures excited by random ground motion excitations.

The present research combines versatile polygonal elements with a multiresolution scheme to achieve computationally efficient and high resolution designs for structural dynamics problems. For typical quadrilateral and triangular elements, multiresolution methods have previously been developed that can use a coarse mesh for the displacement nodes, and finer meshes for design and density variables. This technique allows for a higher resolution of the solution, for only a slight increase in computational time. To implement the multiresolution approach for polygonal elements, ongoing work is focused on introducing conforming and nonconforming sub-discretizations within each polygon in order to obtain finer design and density variable meshes. The conforming approach uses the existing nodes and element centroid to divide the area of the polygonal element, while the non-conforming approach uses a mesh embedding approach to sub-discretize each of the larger convex polygonal finite elements. Both approaches use the centroids of the density variables as integration points for the finite element shape functions, as this has shown to provide accurate results for typical elements. The research work will focus on exploring optimization of eigenfrequency problems and also maximization of dynamic compliance problems. These problems often require more computational time within the system analysis as compared to the design optimization, and thus the multiresolution scheme is expected to provide faster computational time and improved quality solutions. We employ irregular domains to present sample solutions for structural dynamics problems. These include eigenfrequency and/or band gap (space between eigenfrequencies) maximization. Furthermore, we also investigate forced vibration problems where the displacement response at a point of the structure is minimized for a specified sinusoidal input.

In the past few decades, topology optimization methods have been applied to a wide range of practical applications. In the literature, typically a uniform grid of linear quads/bricks is used for topology optimization problems. Numerical anomalies, such as checkerboard pattern and onenode connections arise out of such formulations. Constraints in the geometrical features of spatial discretization can result in mesh dependent designs. Polygonal elements, which do not suffer from such numerical anomalies, have been investigated in the past in two-dimensional topology optimization. In the current work, we propose the use of polyhedral meshes to address the geometric features of the domain discretization. Polyhedral meshes provide a greater flexibility in discretizing complex domains. Moreover, techniques such as mesh refinement and coarsening produce elements which are inherently polyhedral. Typically, in order to solve the state equation on polyhedral meshes, the computation of global stiffness matrix would require conducting numerical integration in physical coordinates and dealing with each polyhedral element individually. In order to achieve numerical accuracy, a very high order quadrature is required which is computationally expensive. In the current work, we demonstrate the effectiveness of our Virtual Element Method (VEM) based approach for three-dimensional linear elastic topology optimization. The VEM is considered as the next evolutionary stage of the Mimetic Finite Difference (MFD) methods. In VEM approach, the stiffness matrix computation reduces to the evaluation of matrices which involve only surface integral terms, in contrast to the volume integrals encountered in conventional FEM, thus reducing the computational cost. The features of the current approach are demonstrated using various numerical examples for compliance minimization problem.

We investigate truss topology design with special attention to the pre-processing phase of ground structure generation together with its implications on the optimization process. Specifically we use a Voronoi-based ground structure generation approach at different levels (level 1, 2, etc.) with operators to treat overlapping bars, to create desirable connecting bars, etc. We also establish a metric to relate the effectiveness of the ground structure with respect to the properties of the underlining topology optimization problem. We address several computational aspects such as convergence of the method and type of optimality criteria (e.g. standard and modified versions) adopted. We present several examples for two-dimensional and threedimensional problems and comment on the features of each problem addressed.

We present a general topology optimization framework in Matlab, named PolyTop, using unstructured polygonal finite element meshes. The topology optimization code is structured in a modular fashion to separate the analysis routine from the particular formulation used. Therefore, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. In addition, we also present a robust Matlab implementation for polygonal mesh generation, named PolyMesher, that relies on an implicit description of the domain geometry. This work is based on the concept of Voronoi diagrams, which offer an effective scheme to discretize twodimensional geometries with complex domains. We remark that polygonal finite elements outperform linear triangles and quads in the field of topology optimization because they are not susceptible to numerical instabilities such as checkerboard patterns. Representative examples are provided to illustrate the capabilities of the framework composed by PolyTop and PolyMesher.

A prevalent problem in the field of topology optimization has been instabilities such as the appearance of checkerboard patterns when using low-order triangles and quads. It will be shown that discretizations based on polygonal finite elements naturally provide stable solutions. The better performance of polygonal discretizations is attributed to their enhanced approximation characteristics, which also alleviate shear locking in elasticity and lead to a stable low-order mixed variational formulation of incompressible Stokes flow. A simple but robust algorithm is provided, which utilizes centroidal Voronoi tessellations (CVTs) to generate convex polygonal meshes that possess enhanced regularity and isotropy. We will assess the performance of polygonal discretizations in elasticity and Stokes flow and discuss their applications to topology optimization problems in both solids and fluids.

Most papers in the literature which deal with topology optimization of trusses using the ground structure approach are constrained to linear behavior. Here we address the problem considering nonlinear behavior. More specifically, we concentrate on hyperelastic models, namely Ogden, Hencky, Saint-Venant and Neo-Hookean. In the optimization process, we consider different objective functions such as end compliance, strain energy and total potential energy. In the linear case, they are all equivalent; but in the nonlinear case, there are interesting peculiarities associated to each one. In addition, we discuss ground structure generation techniques and their relation to the underlining optimization problem. Some representative examples are given to demonstrate the features of each model. We conclude by exploring the role of nonlinearities in the overall topology design problem.

Structural optimization aims at achieving the best performance from a structural design while satisfying the given constraints. Among various applications of structural optimization, topology optimization based on mathematical programming and finite element analysis has recently gained great attention in the research community and practice. Many theories, methods and algorithms have been developed for topology optimization under various conditions [1, 2].

In practice, structural engineers invest significant time and financial resources 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 dynamic response constraints is of great importance and meaningful in industrial applications. Despite rapid technological advances, incorporating stochastic response of structures into topology optimization is a relatively new field of research due to computational challenges. One of the most widely used approaches to account for dynamic effects in topology optimization is to maximize the fundamental frequency [1, 2]. An approach to minimize the dynamical response of a structure for a given dominant frequency of dynamic loading was also developed. However, such methods are not able to handle the structural behavior under general random vibrations and thus have limits in practical applications.

This research introduces an alternative method incorporating random vibration theories into topology optimization in order to satisfy probabilistic constraints described in terms of inter-storey drift ratios, maximum displacements, crossing rate, etc. Using a discrete representation method of stochastic process [3], the stochastic response of a system subjected to random seismic excitations is described in a standard normal space. This is to compute the probability of failure in the constraint of the topology optimization effectively during the topology optimization. In addition, sensitivity formulation is developed in order to use gradient-based optimization algorithms (e.g. method of moving asymptotes) for the proposed topology optimization based on the discrete representation method. The proposed method is demonstrated by numerical examples of structures accelerated by random ground motions.

Reference
[1] M.P. BendsØe, O. Sigmund (2003). Topology Optimization Theory, Methods and Applications. Springer Verlag, Berlin Heidelberg.
[2] W.M. Rubio, G.H. Paulino, E.C.N. Silva (2011). Tailoring vibration mode shapes using topology optimization and functionally graded material concepts, Smart Materials and Structures, 20(2): 025009 (9pp).
[3] Der Kiureghian A. (2000). The geometry of random vibrations and solutions by FORM and SORM, Probabilistic Engineering Mechanics,15:81–90.

Though topology optimization has been applied to many fields, ranging from mechanical to aerospace engineering, more work must be done to tailor it to the needs of the structural engineer, especially in regards to the design of high-rise buildings. Thus, this work aims to improve its application to structural engineering by describing an integrated topology optimization approach involving continuum and discrete finite elements to design the lateral systems in structural braced frames for high-rise buildings. The approach is implemented using concurrent continuum finite elements and discrete beam/truss elements to simplify and improve the overall design process by creating optimal geometries for a given volume of material. For example, after an engineer develops a structural frame consisting of beams and columns sized for gravity loads, topology optimization on the continuum (e.g. quadrilateral) elements is used to create a conceptual design for the braces of the lateral system resulting in highly efficient structures. Several practical examples are demonstrated to show the importance and relevance of this work to the structural design industry.

In this work, a cohesive zone model of fracture is employed to study debonding in plastically deforming Al/epoxy T-peel joints. In order to model the adhesion between the bonded metal strips, the Park-Paulino-Roesler (PPR) potential based cohesive model (J Mech Phys Solids, 2009;57:891-908) is employed, and interface elements are implemented in a finite element commercial code. A study on the influence of the cohesive properties (i.e. cohesive strength, fracture energy, shape parameter and slope indicator) on the predicted peel-force versus displacement plots reveals that the numerical results are mostly sensitive to cohesive strength and fracture energy. In turn, these parameters are tuned until a match between experimental and simulated load displacement curves is achieved.

Progressive failure of engineering materials can be analyzed by means of cohesive zone models (CZM) of fracture. In such models, the fracture process is described by cohesive sur- faces whose behavior is characterized by a traction-separation relation. In order to deter- mine such a relation, the cohesive parameters, e.g. cohesive strength and the fracture energy, need to be specied. Owing to the diculties associated with the direct measurement of the aforementioned quantities, very often they are obtained by comparing the experimental load- displacement curve with nite element predictions based on idealized traction-separation rela- tions. However, it has been noted that obtaining such a local property from a global response is not always reliable [1]. In order to overcome such limitation, an inverse procedure for the identication of a cohesive model is employed in the present work. It is based on the full-eld measurement of surface displacements recorded during mechanical testing of ad-hoc fracture specimens (e.g. Double Cantilever Beam). To this aim, a traveling microscope is employed in order to track the fracture process at the microscale and the full eld displacement across the process using a multiscale Digital Image Correlation technique.

In summary, a numerical-experimental optimization process is developed in which the pa- rameters of a cohesive model are tuned until a regularized error function, dened by subtracting experimental and numerical displacements, is minimized. The cohesive model is implemented in the nite element framework and the inverse procedure is assessed on Double Cantilever Beam Aluminum samples bonded with epoxy adhesive. Encouraging preliminary examples are reported as well as directions for further research.

References [1] B. Shen and G. H. Paulino. Direct extraction of cohesive fracture properties from digital image correlation: A hybrid inverse technique. Experimental Mechanics, in press, 2010.

A nonlinear functionally graded beam model formulation, developed considering a tailored corotational formulation based on Rodrigues’ angles, is employed to evaluate geometric stability of risers under operating conditions. The finite element(FE) model implementation is based on computational abstractions of both mathematical and physical concepts associated to geometrical nonlinearities, with special emphasis on finite rotations. By means of the Rodrigues vector formula, employed in the evaluation of riser cross-section rotations in space, a consistent incremental formulation is derived considering geometric nonlinearities, involving large displacements and rotations but small strains. The element is a two node Timoshenko’s beam finite element formulation with active axial, bending and torsion displacements, all interpolated along its length using Hermite’s cubic functions. The formulation has been also extended to study risers under dynamic conditions in a stepby-step time integration, in its unconditionally stable form. The FE model allows for self-weight, buoyance, current, soil contact, buoys and, prescribed force and moment loading responses including path dependent buckling stability. The consistency of the formulation is evaluated through representative examples discussed in comparison with other alternatives. Several aspects regarding structural behavior will be presented such as the interplay of symmetric configurations with respect to geometry and material gradation, and the implications of material gradation distributions on riser stability.

This work aims to provide a numerical framework for the dynamic behavior representation of riser structures, considering the use of functionally graded materials (FGM). In this respect, a new corotational finite element formulation for the numerical representation of such risers is considered, including the effects of geometric presented to show the numerical model capabilities on representing the important kinematics of a riser structure in dynamics.

The estimation of concrete fracture properties is essential for an accurate cracking prediction of concrete pavement systems. The single-edge notched beam test has been used to characterize fracture parameters of concrete materials in the laboratory, but obtaining a field specimen with this geometry is not always practical. Currently, a standard exists, ASTM D7313, for the measurement of fracture energy in asphalt concrete using the disk-shaped compact tension (DCT) test. The benefit of this specimen geometry for both concrete and asphalt is that it can easily be fabricated in the laboratory or cored from the field. The total fracture energy (GF) of the material is estimated by using the concept of the work-of-fracture. Additional properties, such as the initial fracture energy (Gf) and the critical crack tip opening displacement (CTODC), can be extracted from the same test through employing compliance measurements and the concept of an equivalent elastic crack model. In this pilot study, the DCT specimen is adopted for concrete materials with small changes to the hole and notch geometry and loading rate of the specimen relative to ASTM D7313. The initial DCT experimental results for concrete containing virgin limestone aggregate and recycled concrete aggregate have been consistent and repeatable. A finite element model (FEM) of the specimen was developed to check the published KIC equation for this geometry and to derive the CTODC correction factor. A cohesive zone model was also successfully implemented to simulate the DCT specimen, which verified the validity of the calculated fracture properties from the DCT experiments.

Optimal structural topologies set a new frontier in modern architecture and provide a benchmark to evaluate the performance of existing and future structures. Several methodologies for the optimization of structural shapes and systems have been explored by engineers and architects at SOM (Skidmore, Owings & Merrill, LLP) in collaboration with universities and nearby academic institutions. These new technologies have consequently been integrated into the design process. The optimization is conducted with a combination of commercially available codes and custom written programs that interface with the commercial codes via the API (Advanced Programmer Interface). This paper highlights some of the optimization techniques and their applications to the conceptual design of high-rise projects.

Scaled-up microfluidics (as opposed to nanofluidics) are especially important in areas such as embryonic development, and thus consists on the focus of the present work. We propose a two-dimensional approach for micro- and millifluidics to create three-dimensional chemical profiles including pattern inversions using single-layer microfluidics modules. We will determine the response of the fluid pattern in the millifluidic systems with objects partially obstructing the main channel, which is critical when probing larger scale systems such as organized multicellular systems. We will consider both Newtonian and non-Newtonian flows, including viscous fluids and mixing of different fluids within a range of Reynolds numbers (from low to moderate). The topology optimation includes a multi-objective cost function (in terms of the desired flow characteristics) where polygonal finite elements are employed for the analysis and the method of moving asymptotes for the optimizer. Preliminary examples will be given to motivate the idea of the present multidisciplinary research. This research has potential applications in the fields of microfluidics, millifluidics, embryonic development, cellular stimulation, mixing layers, and chemical fabrication approaches.

A GPU-based computational framework is presented to deal with dynamic failure events simulated by means of cohesive zone elements. The work is divided into two parts. In the first part, we run computational experiments decoupled from mechanics and verify the effectiveness of dynamic insertion of cohesive elements in large meshes. In the second part, we present an explicit dynamics code that implements an extrinsic cohesive zone formulation where the elements are inserted on-the-fly, when needed and where needed. Examples demonstrate that our GPU-based system presents expressive gain in performance when compared to the corresponding serial CPU-based code.

The main challenge for implementing a GPU-based computational framework using extrinsic cohesive zone formulation resides on being able to dynamically adapt the mesh in a consistent way, inserting cohesive elements on fractures facets. In order to handle that, we extend the conventional data structure used in finite element code (based on element incidence) and store, for each element, references to the adjacent elements. This additional information suffices to consistently insert cohesive elements, duplicating nodes when needed. Currently, our data structure is specialized for triangular meshes, but an extent to tetrahedral meshes is straightforward.

To avoid concurrency on accessing shared entities, we employ the conventional strategy of graph coloring. In a pre-processing phase, each node of the dual graph (bulk element of the mesh) isassigned a color different to the colors assigned to adjacent nodes. In that way, elements of a same color can be processed in parallel without concurrency. All the procedures needed for the insertion of cohesive elements along fracture facets and for computing element and node properties are performed by threads assigned to elements, invoking one kernel per color. Computations on existing cohesive elements are also performed based on adjacent bulk elements.

A large portion of total fatigue crack life is spent in the microstructurally small fatigue crack (MSFC) phase. During the MSFC stages of incubation, nucleation, and microstructurally small propagation in AA7075-T651, several interfaces are mobilized including grain boundaries and particle-matrix bonds. Simulating MSFC life in three-dimensional polycrystals with the inclusion of these interfacial decohesion processes has been challenging due to the high degrees of nonlinearity and mode-mixity in the analyses. With the inception of a robust cohesive zone model (CZM), the Park-Paulino-Roesler (PPR) potential-based CZM [1], modeling these processes with massively parallel finite element analyses is now tractable. Two studies will investigate decohesion in AA7075-T651.

The first study will examine grain boundary decohesion in idealized cubical grain and irregularly-shaped grain polycrystals. The second study will investigate the incubation and nucleation of cracking within and from second-phase particles. Observations will be made on debonding processes with the PPR CZM and compared to those made in previous MSFC studies (such as [2]) in which all particle-grain interfaces were assumed perfectly bonded. The inclusion of the PPR CZM in these studies will offer new insights into MSFC life in a polycrystalline material. Computationally, these studies will serve as indications as to whether this methodology is viable for more complicated models. The overarching impetus for this work is to add to the body of knowledge of MSFC stages of total life in the development of a micromechanical model that provides a quantitative description of MSFC growth.

References
[1] Park, K., Paulino, G.H., Roesler, J. R. A unified potential-based cohesive model of mixedmode fracture. Journal of the Mechanics and Physics of Solids, 57, 2009.
[2] Bozek, J.E., Hochhalter, J.D., Veilleux, M.G., Liu, M., Heber, G., Sintay, S.D., Rollett, A.D., Littlewood, D.J., Maniatty, A.M., Weiland, H.,Christ R.J. Jr., Payne, J., Welsh, G., Harlow, D,G., Wawrzynek, P.A., Ingraffea, A.R. A Geometric Approach to Modeling Microstructurally Small Fatigue Crack Formation- Part I: Probabilistic Simulation of Constituent Particle Cracking in AA7075-T651. Modelling and Simulation in Materials Science and Engineering, 16, 2008.

Superposition principle is used in finding topological derivatives for elliptic partial di?erential equations. The superposition principle is applied to decompose both the solutions and boundary conditions (BCs). This leads to the investigation of the well-posedness issues in a more general setting of solving elliptic partial di?erential equations. Two type of cost functions have been investigated here. By using superposition principle we can handle Dirichlet, Neumann, and Robin BCs, and we also gain more insight of ?nding topological derivatives.

Structural optimization is concerned with achieving the best performance from a structural design while satisfying the given constraints. Among various applications of structural optimization, topology optimization has recently become a great interest in the research community and practice. Many theories, methods and algorithms have been developed for topology optimization under various conditions [1,2].

In practice, structural engineers invest significant time and financial resources 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 dynamic response constraints is of great importance and meaningful in industrial applications. Despite rapid technological advances, incorporating stochastic response of structures into topology optimization is a relatively new field of research due to computational challenges. One of the most widely used approaches to account for dynamic effects in topology optimization is to maximize the fundamental frequency [1,2]. An approach to minimize the dynamical response of a structure for a given dominant frequency of dynamic loading was also developed. However, such methods are not able to handle the structural behavior under general random vibrations and thus have limits in practical applications.

This research introduces an alternative method employing a random vibration theory that predicts the dynamic response of a structure under random vibrations in a stochastic manner in order to satisfy probabilistic constraints given in terms of inter-storey drift ratios, maximum displacements, etc. Using the power spectral density function of random seismic excitation and the frequency response function derived from the topology, spectral moments and the bandwidth of the process are computed. These are used to define mean crossing rates, first-passage probabilities, etc. that appear in the constraints of the structure under random excitation. The proposed method is demonstrated by a numerical example of a two-dimensional ground structure accelerated by random ground motion.

Keywords Topology optimization, dynamic response constraints, random process, means crossing rates.

References
[1] M.P. BendsØe, O. Sigmund, Topology Optimization Theory, Methods and Applications Springer Verlag, Berlin Heidelberg, 2003 [2] W. M. Rubio, G. H. Paulino, E. C. N. Silva, Tailoring Vibration Mode Shapes Using Topology Optimization and Functionally Graded Material Concepts, Smart Materials and Structures, Vol 20, No. 2, 9pp, 2011

Thermally induced cracking in asphalt pavements remains to be one of the prominent distress mechanisms in regions with cooler climates. At present, the AASHTO Mechanistic-Empirical Pavement Design Guide (MEPDG) is the most widely deployed pavement analysis and design procedure. For thermal cracking predictions, MEPDG utilizes a simplified one-dimensional stress evaluation model with a simple Paris-law (i.e. linear elastic fracture mechanics) based crack propagation procedure. The user-friendly graphical interface for MEPDG makes it an attractive design procedure of choice, however, the over simplicity of the model and lack of a physicsbased representation to accurately capture the nonlinear fracture behavior of ratedependent asphalt concrete reduce(s) the reliability of predictions. This study presents an interactive thermal cracking prediction model that utilizes a nonlinear finite element based thermal cracking analysis engine which can be easily employed using a user-friendly graphical interface. The analysis engine is comprised of (1) the cohesive zone fracture model for accurate simulation of crack initiation and propagation due to thermal loading and (2) the viscoelastic material model for time and temperature dependent bulk material behavior. The graphical user interface (GUI) is designed to be highly interactive and user-friendly in nature, and features screen layouts similar to those used in the AASHTO MEPDG, thus minimizing transition time for the user. This paper describes the individual components of the low temperature cracking prediction software (called LTC Model) including details on the graphical user interface, viscoelastic finite element analysis, cohesive zone fracture model, and integration of various software components for thermal cracking predictions.

Investigation of time dependent behavior combined with functionally graded property gradation can be accomplished by means of the non-homogeneous viscoelastic analysis procedure. This presentation describes the development of a generalized isoparametric finite element formulation to capture property gradients within elements, and an incremental-recursive formulation for solution of hereditary integral equations. The VFGM finite elements are implemented in commercial software ABAQUS using user defined material subroutine. The formulation and implementation is verified by comparison with analytical and numerical solutions.

Thin bonded overlay (TBO) systems have become popular options for pavement rehabilitation. The use of spray paver technology for construction of TBO leads to continuously varying asphalt binder content, up to approximately one-third of the layer thickness. Thus, TBOs behave in functionally graded viscoelastic manner. The formulations described in this presentation are applied for evaluation of cracking resistance in TBOs.

Topology optimization methods have been developed and applied to a wide range of practical applications in the past few decades. Recently a new set of methods have emerged for topology optimization, such as level set method and phase field method, where the outlines of the structures are represented using implicit mathematical functions. Current work makes use of phase field method for topology optimization [2]. The simplicity of the method along with no requirement of operations such as reinitialization makes it attractive. In this method, the structural shapes are represented using phase field functions and the shapes are evolved using a time-dependent reaction diffusion equation. In the literature, generally a uniform grid of linear quads is used for topology optimization problems. Numerical anomalies, such as checkerboard pattern and one-node connections arise out of such formulations. Such constraints in the geometrical features of spatial discretization can result in mesh dependent designs. In the current work, we propose to use polygonal meshes constructed using Voronoi tessellations [1] to implement the phase field method. The use of such unstructured meshes not only removes mesh bias but also provides a greater flexibility in discretizating complex domains. The features of current approach are demonstrated using various numerical examples.

References:
[1] Talischi C, Paulino GH, Pereira A, Menezes IFM (2010) Polygonal finite elements for topology optimization: A unifying paradigm. International Journal for Numerical Methods in Engineering 82: 671 – 698
[2] Takezawa A, Nishiwaki S, Kitamura M (2010) Shape and topology optimization based on the phase field method and sensitivity analysis. Journal of Computational Physics 229: 2697 – 2718

Filament-wound composite pressure vessels have been widely used in commercial and aerospace applications due to its high strength and light weight properties. Previous studies related to the optimization of composite pressure vessels have considered the optimization of the fiber angles (winding angle) of the reinforcement composite layers and their thickness, generally using analytical procedures, genetic algorithms, or optimality criterion methods. In this work, we present a methodology using the topology optimization method for static design of composite pressure vessels. We consider the optimization of the fiber angles of the orthotropic material layers while optimizing the material distribution of these layers, and keeping the thickness of the layers constant during the optimization process. The degenerated laminated continuum shell [1] element based on the kinematics of the first-order shell theory is used, which is attractive because it allows the analysis of plates and shell structures with any geometry, thick or thin, and accounts for the transverse shear deformation. The topology optimization formulation is formulated by combing two material models: the first is the SIMP model [2] (Solid Isotropic Material with Penalization), where the design variables describe the amount of material at each finite element; and the second is based on the Discrete Material Optimization [3] (DMO), where orientation variables determine the optimal fiber angle at each element of the domain. The optimization problem is formulated by minimizing the volume of the reinforcement composite layers subjected to a stress constraint, based on the Tsai-Wu failure criterion, with constant internal pressure. Examples of optimized composite pressure vessels are presented to illustrate the methodology.

References
[1] Reddy, J. N., “Mechanics of Laminated Composite Plates and Shells: Theory and Analysis”, Second Edition, CRC Press, Oxford, USA, 2004. [2] Bendsøe, M.P.; Sigmund, O., “Material interpolation schemes in topology optimization”, Archives on Applied Mechanics, Vol. 69, 1999, pp. 635-654. [3] Lund, E., “Buckling topology optimization of laminated multi-material composite shell structures,” Composite Structures, Vol. 91, No. 2, 2009, pp. 158–167

Nonlinear problems are prevalent in structural and continuum mechanics, and there is high demand for computation tools to solve these problems. Despite efforts to develop efficient and effective algorithms, no single algorithm is capable of solving any and all nonlinear problems; depending on the system and the degree of nonlinearity, one solution scheme may be preferred over another. 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 presentation focuses on the development of the theoretical model and its object-oriented implementation, the potential for integration into a finite element analysis code, and the strengths and weaknesses of the various solution schemes through numerical examples.

Many engineering applications of topology optimization cannot be defined on a rectangular domain or solved on a structured square mesh. The description and discretization of the design domain geometry, specification of the boundary conditions for the governing state equation, and accurate computation of the design response may require the use of unstructured meshes. In this talk we present a self-contained analysis tool in Matlab and show how the topology optimization code should be structured so as to separate the analysis routine fromthe particular formulation used. With this alternative code structure, the finite element and sensitivity analysis routines contain no information related to the formulation and thus can be extended, developed and modified independently. We focus on polygonal discretizations in this educational effort since the concept of Voronoi diagrams offers a simple way to discretize two-dimensional geometries with convex polygons. Also polygonal finite elements outperform linear triangles and quads in topology optimization as they are not susceptible to numerical instabilities such as checkerboard patterns. The isoparametric formulation for polygonal finite elements can be viewed as extension of the common linear triangles and bilinear quads to all convex n-gons. As a special case, these codes can generate and analyze structured triangular and quadrilateral meshes. Benchmark numerical examples are presented to illustrate the capabilities of the code.

This study presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation [1]. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method [2], which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The study provides numerical examples of two- and threedimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.

References
[1] Nguyen TH, Song J, Paulino GH (2010) Single-loop system reliability-based design optimization using matrix-based system reliability method: theory and applications. J Mech Des 132(1): 0110051~11 [2] Nguyen TH, Paulino GH, Song J, Le CH (2010) A computational paradigm for multiresolution topology optimization (MTOP). Struct Multidisc Optim 41(4): 525-539

Fatigue damage is a major failure phenomenon observed in infrastructure. Most fatigue failure investigations have been performed on the basis of linear elastic fracture mechanics using the well-known Paris relation [1] or related expressions. In order to capture nonlinear crack tip behavior, cohesive zone based fatigue crack models have been developed. However, previous cohesive zone based fatigue crack models possess several limitations. For example, previous models are unable to introduce fatigue damage if the cohesive traction does not reach the cohesive strength. Additionally, the model input should be free from the number of cycles because a real structure may experience arbitrary loading amplitude and frequency. In order to overcome such limitations, in this study, a novel cohesive zone based fatigue crack growth model is presented. The model clearly defines four stages during arbitrary cyclic loading: softening, unloading, reloading, and contact. The cohesive traction-separation relationship employs the PPR potential [2], in which the fatigue damage is accumulated by introducing two physically based damage measures. One damage measure is associated with the rate of separation while the other one is related to the rate of cohesive traction. Additionally, two modelconstants are introduced to account for crack closure/opening effects, which can be associated with crack face roughness, oxidation of fracture surface, etc. As a verification exercise, computational simulations show that the cohesive zone based fatigue crack growth model is able to represent stable fatigue crack growth, which corresponds to the Paris-type relations.

References
[1] P. Paris, and F. Erdogan, A critical analysis of crack propagation laws, Journal of Basic Engineering, v. 85, p.528-534, 1963. [2] K. Park, G.H. Paulino, and J.Roesler, A unified potential-based cohesive model of mixedmode fracture, Journal of the Mechanics and Physics of Solids, v. 57, p. 891-908, 2009.

Adaptive mesh refinement and coarsening schemes are presented for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge-split operators, whereas the adaptive mesh coarsening is based on a sequence of vertexremoval (or edge-collapse) operators [1]. Nodal perturbation and edge-swap operators are also employed around the crack tip region to improve the representation of crack geometry [2]. Cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS) [3]. The so-called PPR potential-based cohesive model [4] is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed-mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro-crack and local micro-cracks.

References
[1] K. Park, G.H. Paulino, W. Celes, and R. Espinha, Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture, International Journal for Numerical Methods in Engineering, 2011 (in press). [2] G.H. Paulino, K. Park, W. Celes, and R. Espinha, Adaptive dynamic cohesive fracture simulation using edge-swap and nodal perturbation operators, International Journal for Numerical Methods in Engineering, v. 84, p. 1303-1343, 2010. [3] W. Celes, G.H. Paulino and R. Espinha, A compact adjacency-based topological data structure for finite element mesh representation, International Journal for Numerical Methods in Engineering, v. 64, p. 1529-1556, 2005. [4] K. Park, G.H. Paulino, and J.Roesler, A unified potential-based cohesive model of mixedmode fracture, Journal of the Mechanics and Physics of Solids, v. 57, p. 891-908, 2009.

This paper presents an effective MATLAB implementation of a general topology optimization method for compliant mechanism synthesis of statically loaded structures. Our implementation is based on the educational framework PolyTop (Talischi et al., 2011b), which is easily extended to handle compliant mechanism design. The main features of PolyTop are preserved, including a general finite element module using polygons, which are superior to conventional linear triangles and quads in topology optimization as they are not susceptible to checkerboard patterns. The MATLAB code is explained in detail and benchmark numerical examples are presented to illustrate the capabilities of the code. Examples of mechanism synthesis are presented. Moreover, PolyTop offers room for further exploration of finite elements and topology optimization formulations both for research and for practical engineering applications.

We present a simple and robust Matlab code for polygonal mesh generation that relies on an implicit description of the domain geometry. The mesh generator can provide, among other things, the input needed for finite element and optimization codes that use linear convex polygons. In topology optimization, polygonal discretizations have been shown not to be susceptible to numerical instabilities such as checkerboard patterns in contrast to lower order triangular and quadrilaterial meshes. Also, the use of polygonal elements makes possible meshing of complicated geometries with a self-contained Matlab code. The main ingredients of the present mesh generator are the implicit description of the domain and the centroidal Voronoi diagrams used for its discretization. The signed distance function provides all the essential information about the domain geometry and others great flexibility to construct a large class of domains via algebraic expressions. Examples are provided to illustrate the capabilities of the code, which is compact and has fewer than 150 lines.

The field of topology optimization has neither been synchronized nor linked with the needs of the civil engineering profession in regard to the design of high-rise buildings. Thus, this work contributes to improve this linkage by describing an integrated topology optimization approach involving continuum and discrete finite elements to design the lateral systems in structural braced frames for high-rise buildings. As proof-of-concept, the approach is implemented using concurrent discrete beam/truss elements and continuum finite elements. Thus, the overall design process is simplified and improved. For instance, after an engineer develops a structural frame consisting of beams and columns sized for gravity loads, topology optimization on the continuum (e.g. quadrilateral) elements is used to create a conceptual design for the braces of the lateral system. Several practical examples are given to show the importance and relevance of this work to the structural design industry.

Craniofacial reconstruction surgery to restore functionality and appearance after massive bone loss is complicated. Currently there is no quantitative surgical pre-planning for these complex procedures. They are limited to adhoc bone reshaping by the surgeon during the surgery. As a first step to address this problem we proposed to use topology optimization into routine surgical planning [1]. This technique showed the potential to guide and clarify in which places skeletal materials are necessary to withstand the expected loads (e.g., for mastication) and support soft tissue structures, specialized organs (e.g., orbital contents), and prosthetic devices. We used a three-dimensional multi-resolution topology optimization method to design bone replacements. New improved simulation results for the bone replacements will be presented with varying load cases and boundary conditions. The method has the potential to improve current clinical methods and provide essential enabling technology to translate generic bone tissue engineering methods into patient-specific solutions.

References
[1] A. Sutradhar, G. H. Paulino, M. J. Miller, T. H. Nguyen. Topological optimization for designing large craniofacial segmental bone replacement. Proceedings of the National Academy of Sciences (PNAS),107(30):13222-13227, 2010.

Recently there has been great interest in level set methods for solving PDE-constrained optimization problems such as optimal shape and topology design but often the illposedness of the continuum problem is neglected in the construction of the algorithms. The main premise of this talk is that the ill-posedness has certain implications for the numerical solution schemes and a closer examination can shed light on the appropriate algorithmic choices. We begin with a clear statement of the optimal shape problem, a brief examination of the existence issue and the key elements of a well-posed restriction formulation. This is followed by a discussion on the consistency of the approximation and discretization schemes for the optimal shape problem, as well as a closer look at the existing level set based optimization algorithms. Finally, we present a new optimality criteria method for solving the optimization problem that is obtained from a consistent finite element discretization.

Graphical processing units or GPUs are massively parallel computer architectures that can be employed to speedup a varied type of numerical computations. Due to the architecture of these, structured problems achieve impressive performance due to the regular and ordered layout of the input and outputs. The present work investigates the feasibility of finite element methods and topology optimization for unstructured meshes in massively parallel computer architectures, more speciffically on GPUs. Algorithms and codes for each step in the method are proposed and benchmarked with varied results. 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. Results show that topology optimization on the GPU for unstructured meshes is feasible, and could potentially reach production level after certain improvements.

The checkerboard layout of material distribution is one of a number of serious numerical anomalies encountered in the solution of topology optimization problems. Regularization schemes such as filtering can be used to suppress the numerical instabilities, but these measures often involve heuristic parameters that can augment the optimization problem. Polygonal elements can be very useful in this aspect since they naturally exclude checkerboard layouts and provide flexibility in discretizing complex domains. Examples considering compliance minimization and compliant mechanism are presented that demonstrate the advantages of the proposed elements in achieving checkerboard-free solutions and avoiding one-node connections from the design optimization process. Potential extensions and impact of this work will also be discussed.

Severely graded properties are exhibited by asphalt concrete pavements through their thickness due to oxidative aging effects and thermal gradients. Most of the work to date has focused on use of layered-elastic models for the consideration of age stiffening. In the current work a graded viscoelastic model has been implemented within a numerical framework for the simulation of asphalt pavement responses. The graded approach eliminates the need for layered models, which are inaccurate due to the material mismatch present between layers and subsequent stress discontinuities at these interfaces. In areas of severe property gradations such as near the surface of asphalt concrete pavements, the layered approach requires a very detailed discretization. With the graded approach, even severe property gradations can be modeled without the need for extremely fine meshes.

The finite-element implementation incorporates generalized iso-parametric formulation for viscoelastic graded elements. Numerical time-stepping scheme has been implemented in the stand-alone analyses code for solving pavement responses under thermal and mechanical loading conditions. A functionally graded generalized Maxwell model is used as a constitutive model for asphalt concrete considering aging and temperature gradients. Finite element (2D) simulation results for typical asphalt pavement sections under tire and thermal loading will be presented along with details on the development and implementation of the approach.

An attractive approach for simulation of fracture, branching and fragmentation phenomena consists of using cohesive zone elements at the scale of interest. Such models require high levels of mesh refinement at the crack tip region so that nonlinear behavior can be captured and accurate results obtained. This imposes the use of large meshes that usually result in computational and memory costs prohibitively expensive for a single traditional workstation. If an extrinsic cohesive model is to be used, support for dynamic insertion of cohesive elements is also required. We present a Parallel Topological data Structure, called ParTopS, for supporting parallel adaptive fragmentation simulations that provides operations for dynamic insertion of cohesive elements, in a uniform way, for both two- and three-dimensional unstructured meshes. Those elements are truly represented and are treated like any other regular element. The framework is built as an extension of a compact adjacency-based serial topological data structure (TopS), which can natively handle the representation of cohesive elements. Symmetrical modifications of duplicated entities are used to reduce the communication of topological changes among mesh partitions. The correctness and efficiency of the proposed framework are demonstrated by a series of arbitrary insertions of cohesive elements into some sample meshes.

Titanium composites are promising materials for manufacturing aircraft components for supersonic applications. Fracture properties need to be studied and determined for an efficient design. Our current investigation utilizes Cohesive Zone Modeling to study the fracture behavior of commercially pure Ti grade 1 and grade 2. We employ the Inverse Analysis scheme proposed by Shen [1] and Digital Image Correlation (DIC) to extract the cohesive fracture properties. The traction separation relation resulting from the inverse analysis is verified and validated by means of experimental observations.

This presentation focuses on recently proposed enhancements of the existing thermal cracking model (TCMODEL) to introduce and capture fracture properties of hot-mix asphalt (HMA). The analysis is based on fracture mechanics concepts, including cracking evolution in space and time. For instance, cracking phenomenon is captured by means of a tailored cohesive zone model including physical parameters associated to strength and fracture energy, which can be obtained by means of practical experiments (e.g. indirect tensile testing, disk-shaped compact tension specimen). A graphical user interface (GUI) is presently under development, which allows an effective use of the model, including various user-defined features associated with project information, pavement materials and structure, and analyses parameters. In practical terms, the layout of the present GUI is similar to the mechanistic empirical design guide (MEPDG), however, it only requires relevant input associated to thermal cracking. In summary, the present GUI links all the tools/modules necessary for thermal cracking in a seamless fashion. The main features of the GUI will be illustrated, including data pre-processing for input into the thermal cracking model, analysis execution, and postprocessing.

This study proposes a single-loop performance measure approach system reliability-based topology optimization (SRBTO) using the matrix-based system relia- bility (MSR) method. The single-loop performance measure approach (PMA) is em- ployed to eliminate the inner loop of SRBTO that evaluates probabilistic constraints. The MSR method utilizes efficient matrix calculation to evaluate the system failure probability and its parameter sensitivities. Generic topology optimization problems which minimize the volume and satisfy the probabilistic constraints are investigated. Formulations of deterministic topology optimization (DTO), component reliability- based topology optimization (CRBTO), and system reliability-base topology optimiza- tion (SRBTO) are introduced. The implementation of the SRBTO/MSR procedure is explained. Numerical examples demonstrate the proposed SRBTO/MSR procedure for both two and three dimensional topology optimizations. Monte Carlo simulation is performed to verify the accuracy of the proposed approach.

This paper proposes a single-loop system reliability based design optimization (SRBDO) approach using the recently developed matrix-based system reliability (MSR) method. A single-loop method was employed to eliminate the inner loop of SRBDO that evaluates probabilistic constraints. The MSR method computes the system failure probability and its parameter sensitivities efficiently and accurately through efficient matrix calculations. The SRBDO/MSR approach proposed in this paper is uniformly applicable to general systems including series, parallel, cut-set and link-set system events. Two numerical examples demonstrate the proposed approach. In the first example, the cross-sectional areas of the members of a statistically indeterminate truss structure are determined for minimum total weight with a constraint on the system failure probability satisfied. The second example demonstrates the application of the proposed approach to topology optimization. The influences of the statistical correlation and the types of constraints, i.e. deterministic, probabilistic (component) and probabilistic (system) on the optimal topology are investigated.

In dynamic cohesive fracture simulation, cohesive surface elements are adaptively inserted whenever and wherever they are needed. An extrinsic cohesive zone modeling approach which requires access to adjacency information and management of a consistent data structure is employed. The topological data structure TopS [1], based on topological entities (node, element, vertex, edge and facet), is utilized in order to maintain data structure in time proportional to the number of retrieved entities. TopS provides adaptive topological operators such as nodal perturbation, edge-swap, adaptive mesh refinement, and adaptive mesh de-refinement. These adaptive topological operators are employed so that one reduces mesh bias in 4k structured meshes, improves crack patterns, and decreases computational cost. The constitutive relationship of fractured surfaces is also essential to simulate dynamic fracture phenomena. The potential-based cohesive zone model, called the PPR model [2], is presented and implemented in conjunction with the cohesive surface element approach. The PPR model represents different fracture energies and cohesive strengths, and describes various material softening behavior.

References
[1] Celes W., Paulino G.H., Espinha R., 2005, “A compact adjacency-based topological data structure for finite element mesh representation,” International Journal for Numerical Methods in Engineering, 64(11), pp. 1529–1556.
[2] Park K., Paulino G.H., Roesler J.R., 2009, “A unified potential-based cohesive model of mixed-mode fracture,” Journal of the Mechanics and Physics of Solids, In press, doi:10.1016/j.jmps.2008.10.003

This presentation makes use of inverse techniques used to extract the cohesive zone model (CZM). The focus is on 2-D Mode-I fracture. Full-field displacement data of plastics and micro-fiber reinforced cement composite with experimental noise are used as input to the inverse problem. Two inverse techniques are implemented and tested: the finite element displacement update and the virtual field methods. A solution scheme using the Nelder-Mead method and the Levenberg-Marquardt method obtains a compromise between robustness and efficiency. In the Nelder-Mead method, the physical constraint of the model parameters are satisfied by introducing barrier functions. It is found that, for smaller noise level, both inverse techniques yield satisfactory estimation of the CZM; however, when the noise level is higher, the finite element displacement update method yields better estimation.

Manufacturing constraints in topology optimization have had relevant applications in the mechanical and aerospace engineering industries. Further development of this field is needed in order to transition the technology toward practical industrial applications. Within this scope, this work aims at exploring the features of manufacturing constraints, especially pattern repetition, in the context of high-rise building design. The successful development of such ideas will lead to practical engineering solutions, especially during the conceptual phase of the building design process. The present work emphasizes a continuous topology optimization formulation in which the geometrical gradation of the patterns is investigated as an additional feature of the overall optimization process. Examples are given to illustrate the ideas developed both in 2D and 3D problems.

The separation of finite element and topology optimization discretizations can offer several advantages in obtaining high-fidelity solutions, especially when large-scale problems are considered. Thus this work proposes the use of a multilevel mesh representation involving finite element and topology optimization variables. This representation is based on a compact topological data structure named TopS, which has been extended to handle multilevel descriptions. A mapping-based framework provides a general approach to solve either two-dimensional or three-dimensional problems.

Achieving high-fidelity results from topology optimization simulations has been a common goal in the technical literature. To that effect, several techniques have been proposed with various degrees of success. By addressing the geometric features of the spatial discretization, we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions. More specifically, we consider polygonal meshes constructed from Voronoi tessellations, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities. These points are illustrated by means of several numerical examples, which are compared with results obtained by means of previous techniques.

In this paper, the Finite Particle Method (FPM) is introduced to analyze the large deflection of planar solids. Different from traditional variational principle based methods, the FPM is rooted in vector mechanics and physical thoughts [1]. It models the analyzed domain composed of finite particles. Newton’s second law is adopted to describe the motions of all particles. Instead of imposing a global equilibrium of the entire continuous system, the FPM holds a very strong dynamic equilibrium on particles. The basic procedures include the structure discretization, internal force evaluation and time integration. A triangular element is proposed to model the continuums. In the internal nodal force calculation, a simple manner to separate the rigid body motion and the deformation displacement is developed. To reduce the redundant degree of freedoms of nodal deformation components and obtain deformation distribution functions that satisfy continuity, a set of deformation coordinates for the element analysis is assumed. The explicit time integration is also included in this algorithm. Several numerical examples are presented to demonstrate the performance and applicability of the proposed method on the large deflection analysis of planar solids.

The Indirect Tension Test (IDT) is frequently used for evaluation of asphalt material viscoelastic creep properties. With the increased use of finer aggregate gradations and polymer modified asphalt binders in asphalt materials, crushing can occur under the narrow loading heads. The new specimen configuration proposed has a trimmed area under the loading heads, creating a “flattened-IDT.” This integrated modeling and testing study shows that the flattened IDT reduces the crushing observed in the regular IDT. This study shows that the flattened configuration alters creep properties approximately 10-16% within typical experimental variability.

The Indirect Tension Test (IDT) is frequently used for evaluation of asphalt material viscoelastic creep properties. With the increased use of finer aggregate gradations and polymer modified asphalt binders in asphalt materials, crushing can occur under the narrow loading heads. The new specimen configuration proposed has a trimmed area under the loading heads, creating a “flattened-IDT.” This integrated modeling and testing study shows that the flattened IDT reduces the crushing observed in the regular IDT. This study shows that the flattened configuration alters creep properties approximately 10-16% within typical experimental variability.

Low temperature cracking remains one of the major pavement distresses in asphalt concrete pavements in cold regions. An integrated laboratory testing, field performance data, and numerical simulation approach was used to study thermal cracking as part of a US National Pooled Fund Study on Low-Temperature Cracking. This paper focuses on testing, analysis, and field data from five controlled test sections at the Minnesota Road Research Program facility (MnROAD). Low temperature viscoelastic relaxation modulus master curves and tensile strength were obtained from indirect tension testing conducted at three temperatures. Fracture energy of field sampleswere obtained using the disc-shaped compact tension (DC[T]) test.Temperature-dependent thermal coefficient data was collected by one of the research partners (the University ofWisconsin) for each of the five field mixtures. Abi-linear cohesive zone modelwas used in the simulation of thermal cracking in five MnROAD pavement sections. Four custom-designed user subroutines were employed in the commercial finite element program ABAQUS, including: a bi-linear cohesive zone fracture model, a temperature shift factor routine, a time- and depth-dependent temperature profile algorithm, and a bi-linear thermal coefficient routine. The temperature boundary conditions were generated using the Enhanced Integrated Climatic Model (EICM) available in theAASHTO Mechanistic-Empirical Pavement Design Guide (MEPDG) using air temperatures obtained from National Weather Service databases. Detailed field performance crack maps were used to compare actual field cracking against numerical simulation results. This paper describes how this comprehensive, integrated testing and modeling program provided new insights towards the mechanisms of thermal cracking in asphalt pavements.

Asphalt concrete pavements are inherently graded viscoelastic structures. Oxidative aging of asphalt binder and temperature cycling due to climatic conditions are the major cause of such graded non-homogeneity. Current pavement analysis and simulation procedures either ignore or use a layered approach to account for non-homogeneities. For instance, the recently developed Mechanistic-Empirical Design Guide (MEPDG) [1], which was recently approved by the American Association of State Highway and Transportation Officials (AASHTO), employs a layered analysis approach to simulate the effects of material aging gradients through the depth of the pavement as a function of pavement age. In the current work, a graded viscoelastic model has been implemented within a numerical framework for the simulation of asphalt pavement responses under various loading conditions. A functionally graded generalized Maxwell model has been used in the development of a constitutive model for asphalt concrete to account for aging and temperature induced property gradients. The associated finite element implementation of the constitutive model incorporates the generalized iso-parametric formulation (GIF) proposed by Kim and Paulino [2], which leads to the graded viscoelastic elements proposed in this work. A solution, based on the correspondence principle, has been implemented in conjunction with the collocation method, which leads to an efficient inverse numerical transform procedure. This work is the first of a two-part paper and focuses on the development, implementation and verification of the aforementioned analysis approach for functionally graded viscoelastic systems. The follow-up paper focuses on the application of this approach.

This is the second article in a series of two papers describing simulation of functionally graded viscoelastic properties in asphalt concrete pavements. The techniques developed are applicable to other viscoelastic material systems with continuous, spatial grading of material properties. A full-depth asphalt concrete pavement has been simulated to demonstrate the applicability and importance of the graded viscoelastic analysis method. Based on the graded finite elements developed by Kim and Paulino[1], Buttlar et al. [2] used graded finite elements to determine typical responses to tire loading for an aged asphalt concrete pavement. In the current study, a similar pavement section is studied using the viscoelastic graded analysis (rather than elastic). Graded, layered and homogeneous material variations were used for a series of simulations, and the results from different approaches were compared.

Superelements offer several advantages for high-fidelity solutions of topology optimization problems. Thus this work proposes the use of a two-level mesh representation, involving finite element and topology optimization variables. The proposed mapping–based framework provides a general approach to solve either two-dimensional or three-dimensional problems considering either conventional or non-conventional finite elements.

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 elements with Wachspress-type shape functions. The features of the hexagonal mesh include twonode 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 1-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 Wachspresstype hexagonal elements and present their implementation using three approaches for topology optimization: element-based, 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 fine-scale patterns from the design optimization process.

Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the ¯eld of topology optimization. In general, ¯nite element meshes with these elements exhibit the well-known checkerboard pathology in the iterative solution of topology optimization problems. Voronoi and Wachspress-type ¯nite elements are less susceptible to such anomalies. Moreover, these elements provide more flexibility in mesh generation and are suitable for applications involving signi¯cant changes in the topology of the material domain. In particular, hexagonal Wachspress meshes 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 favor checkerboard con¯gurations; and only have two edge-based symmetry lines. Thus checkerboard-free solutions are obtained without any further restrictions on the local variation of material density or filtering techniques (e.g. filter of sensitivities). We explore general Voronoi-type elements and present their implementation using a couple of approaches for topology optimization: e.g. element-based, and minimum length-scale control through projection functions. Examples are presented that demonstrate the advantages of the proposed elements in achieving checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process. Potential extensions and impact of this work will also be discussed.

This paper deals with the application of Cohesive Zone Model (CZM) concepts to study mode I fracture in adhesive bonded joints. In particular, an intrinsic piece-wise linear cohesive surface relation is used in order to model fracture in a pre-cracked bonded Double Cantilever Beam (DCB) specimen. Finite element implementation of the CZM is accomplished by means of the user element (UEL) feature available in the FE commercial code ABAQUS. The sensitivity of the cohesive zone parameters (i.e. fracture strength and critical energy release rate) in predicting the overall mechanical response is first examined; subsequently, cohesive parameters are tuned comparing numerical simulations of the load-displacement curve with experimental results retrieved from literature.

In this paper Cohesive Zone Model (CZM) concepts are applied in order to study mode I fracture in a pre-cracked bonded Double Cantilever Beam (DCB) specimen. A cohesive surface element is implemented in a Finite Element commercial code using intrinsic cohesive zone models: exponential, bilinear and trapezoidal traction-separation laws. The sensitivity of cohesive zone parameters in predicting the overall mechanical response is examined, then the load displacement curves obtained with the different CZMs are compared and some interesting features concerning the prediction of damage onset in adhesive joints are illustrated. Finally, cohesive parameters are identified comparing numerically simulated load-displacement curves with experimental data retrieved from literature.

Functionally Graded Materials (FGMs) possess continuous variation of material properties and are characterized by spatially varying microstructures. Recently, the FGM concept has been explored in piezoelectric materials to improve properties and to increase the lifetime of piezoelectric actuators. Elastic, piezoelectric, and dielectric properties are graded along the thickness of a piezoceramic FGM. Thus, the gradation of piezoceramic properties can influence the performance of piezoactuators, and an optimum gradation can be sought through optimization techniques. However, the design of these FGM piezoceramics are usually limited to simple shapes. An interesting approach to be investigated is the design of FGM piezoelectric mechanisms which essentially can be defined as a FGM structure with complex topology made of piezoelectric and non-piezoelectric material that must generate output displacement and force at a certain specified point of the domain and direction. This can be achieved by using topology optimization method. Thus, in this work, a topology optimization formulation that allows the simultaneous distribution of void and FGM piezoelectric material (made of piezoelectric and non-piezoelectric material) in the design domain, to achieve certain specified actuation movements, will be presented. The method is implemented based on the SIMP material model where fictitious densities are interpolated in each finite element, providing a continuum material distribution in the domain. The optimization algorithm employed is based on sequential linear programming (SLP) and the finite element method is based on the graded finite element concept where the properties change smoothly inside the element. This approach provides a continuum approximation of material distribution, which is appropriate to model FGMs. Some FGM piezoelectric mechanisms were designed to demonstrate the usefulness of the proposed method. Examples are limited to two-dimensional models, due to FGM manufacturing constraints and the fact that most of the applications for such FGM piezoelectric mechanisms are planar devices. An one-dimensional constraint of the material gradation is imposed to provide more realistic designs.

The Indirect Tension Test (IDT) is frequently used in civil engineering because of its benefits over direct tension testing. During the Strategic Highway Research Program (SHRP), in the mid-1990’s, an IDT protocol was developed for evaluating tensile strength of Hot Mix Asphalt (HMA) mixtures. However, with the increased use of finer aggregate gradations and polymer modified asphalt binders in HMA mixtures, the IDT results can be misleading because of crushing failure under the narrow loading heads. For such mixtures the 150-mm diameter, 50-mm thick, cylindrical specimens tends to fail in crushing beneath the loading heads versus the desired indirect tension at the center of the specimen. Therefore, a new specimen configuration is proposed for strength testing of HMA. In place of the loading heads at the top and bottom, the specimen is trimmed to produce flat planes with parallel faces, creating a “flattened-IDT.” A viscoelastic finite element analysis of the flattened configuration was performed to evaluate the optimal trimming width. In addition, the numerically determined geometry was verified by means of laboratory testing of 3 different HMA mixtures. This integrated modeling and testing study shows that for the HMA mixtures with fine aggregate gradations and compliant asphalt binders used in this study, the flattened IDT eliminates the severe crushing observed in the regular IDT. It is recommended that further testing and analysis be performed on the flattened IDT arrangement, leading to a revision of the current AASHTO standard for IDT testing as asphalt mixtures.

Although asphalt concrete overlay systems represent a rapid and economical alternative for the repair of deteriorated pavements, reflective cracking continues to be major cause of premature deterioration of these systems. A better understanding of the complex mechanisms behind reflective cracking in asphalt overlays must first be obtained before significant advances in reflective crack prevention and mechanics-based overlay design can be fully realized. Traditional modeling approaches have not provided a direct means for the study of crack initiation and propagation in pavements. The cohesive zone fracture modeling approach provides a rational means for modeling cracking in structural systems consisting of quasi-brittle materials, as a finite length scale associated with the fracturing process is considered. A bi-linear cohesive zone model (Song et al., 2006) was used in the simulation of cracking in three field pavement sections studied in a recent NSF GOALI project. Detailed field performance data, especially crack maps from visual surveys were obtained and compared to the numerical simulation results. The temperature boundary conditions were generated using the Enhanced Integrated Climatic Model developed by Dempsey et al. (1990) based upon air temperatures obtained from National Weather Service databases. Viscoelastic bulk and cohesive fracture material properties for these pavement sections were obtained by laboratory testing of specimens fabricated from 150-mm field cores, in accordance with a new, efficient testing suite (Wagoner et al., 2006). A series of numerical simulations were performed using finite element models, which provided new insights towards the mechanisms of cracking in asphalt concrete overlays under thermal and mechanical loads. A series of finite element analyses were performed with hypothetical overlay configurations in an effort to demonstrate the concept of a “simulation-guided” interlayer/overlay design process, which allows the direct consideration of initiating and propagating cracks in one or more overlay layers.

Cohesive law is the key to FEM fracture simulation of quasi-brittle materials, yet it is normally empirically determined. A more convincing way to obtain cohesive law is to measure crack separation and crack surface traction. Recent development in experimental mechanics, e.g. photoelasticity and digital image correlation (DIC) enables accurate measurement of full field surface displacement. However, the cohesive stress at crack surface is impossible to measure directly. But the cohesive stress distribution is believed to determine the nearby displacement field uniquely. An inverse problem thereby is formulated in order to extract the cohesive law by fully utilizing the measured displacement field. The main focus is on how to solve the problem effectively and robustly. First by assuming the cohesive law with a few governing parameters, a forward problem is solved to obtain the complete displacement field at a certain loading level. This displacement field is then assumed known, while the cohesive law is to be solved in the inverse problem. The inverse problem is formulated in two ways: one to be solved using traditional Newton-Raphson or Newton-like Methods, and the other to be solved using optimization technique. Both methods can obtain the correct results, yet both methods depend on a good initial guess, which might not necessarily be a nearby point, of the cohesive law parameters. The nature of the problem is also explored and analyzed. This work can be generalized to compute mode II cohesive properties and other internal or boundary stress using full-field displacement field in a FEM frame.

Functionally Graded Materials (FGMs) possess continuous variation of material properties and are characterized by spatially varying microstructures. Recently, the FGM concept has been explored in piezoelectric materials to improve properties and to increase the lifetime of bimorph piezoelectric actuators. Elastic, piezoelectric, and dielectric properties are graded along the thickness of a piezoceramic FGM. Thus, the gradation of piezoceramic properties can influence the performance of piezoactuators. In this work, topology optimization is applied to find the optimum gradation variation in piezoceramics in order to improve actuator performance measured in terms of output displacements. A bimorph type actuator design is investigated. The corresponding optimization problem is posed as finding the optimized gradation of piezoelectric properties that maximizes output displacement or output force at the tip of the bimorph actuator. The optimization algorithm combines the finite element method with sequential linear programming. The finite element method is based on the graded finite element concept where the properties change smoothly inside the element. This approach provides a continuum approximation of material distribution, which is appropriate to model FGMs. The present results consider gradation between two different piezoceramic properties and two-dimensional models with plane stress assumption.

The micro-tools considered in this work consist essentially of multi-flexible structures actuated by two or more piezoceramic devices that must generate different output displacements and forces at different specified points of the domain and on different directions. The multiflexible structure acts as a mechanical transformer by amplifying and changing the direction of the piezoceramics output displacements. Micro-tools offer significant promise in a wide range of applications such as cell manipulation, microsurgery, and micro/nanotechnology processes. Although the design of these micro-tools is complicated due to the coupling among movements generated by various piezoceramics, it can be realized by means of topology optimization concepts. Recently, the concept of functionally graded materials (FGMs) has been explored in piezoelectric materials to improve performance and increase lifetime of piezoelectric actuators. Usually for an FGM piezoceramic, elastic, piezoelectric, and dielectric properties are graded along the thickness. Thus, the objective of this work is to study the influence of piezoceramic property gradation in the design of the multiflexible structures of piezoelectric micro-tools using topology optimization. The optimization problem is posed as the design of a flexible structure that maximizes different output displacements or output forces in different specified directions and points of the domain, in response to different excited piezoceramic portions: while minimizing the effects of movement coupling. The method is implemented based on the solid isotropic material with penalization (SIMP) model where fictitious densities are interpolated in each finite element, providing a continuum material distribution in the domain. As examples, designs of a single piezoactuator and an XY nano-positioner actuated by two FGM piezoceramics are considered. The resulting designs are compared with designs considering homogeneous piezoceramics. The present examples are limited to two-dimensional models because most of the applications for such micro-tools are planar devices.

Functionally Graded Materials (FGMs) possess continuously graded material properties and are characterized by spatially varying microstructures. The smooth variation of properties may offer advantages such as reduction of stress concentration and increased bonding strength. Recently, this concept has been explored in piezoelectric materials to improve properties and to increase the lifetime of bimorph piezoelectric actuators. Usually, elastic, piezoelectric, and dielectric properties are graded along the thickness of a piezoceramic PGM. Thus the gradation law of piezoceramic properties can influence the performance of piezoactuators. In this work, topology optimization has been applied to find the optimum gradation variation in piezoceramic FGMs to improve actuator performance measured in terms of output displacements. A bimorph type actuator design is considered. Accordingly, the optimization problem is posed as finding the optimized gradation variation of piezoelectric properties that maximizes output displacement or output force in the tip of bimorph actuator. The optimization algorithm combines the finite element method with sequential linear programming (SLP). The finite element method applied is based on the graded finite element concept where the properties change smoothly inside the element. This approach provides a continuum approximation of material distribution (CAMD), which is appropriate to model FGMs. The alternative FGM modelling using traditional FEM formulation and discretizing the FGM into layers gives a discontinuous stress distribution, which is not compatible with FGM behavior. The present results consider gradation between two different piezoceramic properties and consider two-dimensional models with plane stress assumption.

Co-extrasion involves extrasion of multiple layers at the same time. Functionally graded materials comprising various layers with different properties can be produced through co-extrusion. Rheological control is vital for successful co-extrusion of layered cement-based materials. The paste flow in the barrel and the die land in a ram extruder should be plug-like, while the paste should be sheared and uniformly elongated in the die entry region. In this work, the rheology of the layered pastes was adjusted by changing the water content in each layer. An economical method using thinwall tubes was effective to prepare layered feedrod of cementitious paste. The functionally graded cellular structures of cement-based materials were successfully fabricated by co-extrusion at a low extrudate velocity. Inspections showed that the roundness perpendicular to the extrusion direction, and the straightness parallel to the extrusion direction between layers were well preserved after extrusion. The transition between layers was gradual, which is critical for functionally graded materials.

Asphalt concrete pavements exhibit severely graded properties through their thickness due to oxidative aging effects, which are most pronounced at the surface of the pavement and decrease rapidly with depth from the surface. Most of the literature to date has focused on use of layered-elastic models for the consideration of age stiffening. In the current work, a graded viscoelastic model has been implemented within a numerical framework for the simulation of asphalt pavement responses under thermal and mechanical loading. The graded viscoelastic work is extension of the previous work by Paulino and Jin [1], Mukherjee and Paulino [2], and Buttlar et al. [3]. A functionally graded generalized Maxwell model has been used in the development of a constitutive model for asphalt concrete considering aging and temperature gradients. The aging gradient data from laboratory test results reported by Apeagyei [4] is used for obtaining material properties for the graded viscoelastic model. Finite element implementation of the constitutive model incorporates the generalized iso-parametric formulation (GIF) proposed by Kim and Paulino [5], which leads to the graded viscoelastic elements used in this work.

Large-scale topology optimization problems demand the solution of a large number of linear systems arising in the finite element analysis. These systems can be solved efficiently by special iterative solvers. Because the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems, and by using preconditioning and scaling techniques. We also provide a new implementation of the 8-node brick (B8) element for the continuous approximation of material distribution (CAMD) approach to improve designs of functionally graded materials. Specifically, we develop a B8/B8 implementation in which the element shape functions are used for the approximation of both displacements and material density at nodal locations. Finally, we evaluate the effectiveness of several solver and preconditioning strategies, and we investigate large-scale examples, including functionally graded materials, which are solved with a special version of the SIMP (solid isotropic material with penalization) model. The effectiveness of the solver is demonstrated by means of a topology optimization problem in a functionally graded material with 1.6 million unknowns on a fast PC.

Probabilistic fracture analyses are performed for investigating uncertain fracture response of Functionally Graded Material (FGM) structures. The First-Order-Reliability-Method (FORM) is implemented into an existing Finite Element code for FGM (FE-FGM), which was previously developed at the University of Illinois at Urbana-Champaign [2]. The computational simulation will be used in order to estimate the probability of crack initiation with uncertainties in the material properties only. The two-step probability analysis method proposed in the companion paper (Part I, [1]) is illustrated by a numerical example of a composite strip with an edge crack. First, the reliability index of a crack initiation event is estimated as we vary the mean and standard deviation of the slope and the location of the inflection point of the spatial profile of Young's modulus. Secondly, the reliability index is estimated as we vary the standard deviation and the correlation length of the random field that characterize the random spatial fluctuation of Young's modulus. Also investigated is the relative importance of the uncertainties in the toughness compared to those in Young's modulus.

Concrete fracture behavior is predicted by one of multi-scaling methods, called the virtual internal bond (VIB) model. The VIB model describes the microscopic interactions between the cement pastes and aggregates using the concept of homogenization. The microscopic behavior is connected to macroscopic behavior by the Cauchy-Born rule, which results in the strain energy function. From the macroscopic strain energy function, the VIB model represents both elastic and fracture behavior within the framework of continuum mechanics. In this study, a modified Morse functional potential is introduced for material particles interactions so that the potential is independent of the length scale lattice parameter. The other parameters in the potential function are determined on the basis of macroscopic fracture parameters, i.e. the fracture energy and the cohesive strength. Moreover, the fracture energy is evaluated in conjunction with the J-integral. Finally, the VIB model with the modified Morse potential is verified by the double cantilever beam test and validated by three-point bending tests.

This study develops a micromechanical damage model for two-phase functionally graded materials considering the interfacial debonding of particles and pair-wise interactions between particles. Given an applied mechanical loading, in the particle-matrix zones, the interactions from all other particles over the representative volume element are integrated to calculate the homogenized elastic fields. The progressive damage process is dependent on the applied loading and is represented by the debonding angles which are obtained from the relation between the particle stress and the interfacial strength. In terms of the elastic equivalency, the debonded, isotropic particles are replaced by the perfectly bonded, orthotropic particles. The effective elasticity distribution in the gradation direction is correspondingly solved. Numerical simulations are implemented to illustrate the capability of the proposed model.

This paper presents a computational framework appropriate for dynamic crack branching and fragmentation processes investigation. The ßnite element method incorporates special interface elements based on a cohesive zone model (CZM) to characterize the fracture process. A novel topology-based data structure is employed to facilitate fast and robust manipulation of evolving mesh information when extrinsic cohesive elements are inserted adaptively. To illustrate the application of the method, a set of Õquasi-steady-stateÔ crack propagation experiments exhibiting micro-branching phenomena in Polymethylmethacrylate (PMMA) are numerically simulated. The simulation results compare reasonably well with experimental observations both globally and locally, and demonstrate certain advantageous features of the extrinsic CZM with respect to the intrinsic CZM.

In concrete pavements, a single concrete mixture design is selected to resist mechanical loading without attempting to adversely affect the concrete pavement shrinkage, ride quality, or noise attenuation. An alternative approach is to design distinct layers within the concrete pavement surface which have specific functions thus achieving higher performance at a lower cost. The objective of this research was to address the structural benefits of functionally graded concrete materials (FGCM) for rigid pavements by testing and modeling the fracture behavior of different combinations of layered plain and synthetic fiber-reinforced concrete materials. Fracture parameters and the post-peak softening behavior were obtained for each FGCM beam configuration by the three point bending beam test. The peak loads and initial fracture energy between the plain, fiber-reinforced, and FGCM signified similar crack initiation. The total fracture energy indicated improvements in fracture behavior of FGCM relative to fulldepth plain concrete. The fracture behavior of FGCM depended on the position of the fiberreinforced layer relative to the starter notch. The fracture parameters of both fiber-reinforced and plain concrete were embedded into a finite element-based cohesive zone model. The model successfully captured the experimental behavior of the FGCMs and predicted the fracture behavior of proposed FGCM configurations and structures. This integrated approach (testing and modeling) demonstrates the viability of FGCM for designing layered concrete pavements system.

A functionally graded (FG) material system is employed to make fiber use more efficient in a fiber reinforced cement composite (FRCC). This preliminary study demonstrates beam elements that were functionally graded fiber reinforced cement composite (FGFRCC) with four layers, each with a different fiber volume ratio. Fiber volume ratio was graded in accordance with its potential contribution to the mechanical load-bearing capacity so as to reduce the overall fiber volume ratio while preserving the flexural strength and ductility of the beam. Extrusion was used to produce single homogeneous layers of constant fiber volume ratio. The FRCC layers with different fiber volume ratios were stacked according to a desired configuration and then pressed to make an integrated FGFRCC. Flexural tests were carried out to characterize the mechanical behavior, and the results were analyzed to evaluate the effectiveness of the designed fiber distribution. Compared with homogeneous FRCC with the same overall fiber volume fraction, the FGFRCC exhibited about 50% higher strength and comparable ductility.

Natural fibers are promising for engineering applications due to their low cost. They are abundantly available in tropical and subtropical regions of the world, and they can be employed as construction materials. Among natural fibers, bamboo has been widely used for housing construction around the world. Bamboo is an optimized composite material which exploits the concept of Functionally Graded Material (FGM). Biological structures, such as bamboo, are composite materials that have complicated shapes and material distribution inside their domain, and thus the use of numerical methods such as the finite element method and multiscale methods such as homogenization, can help to fiirther understanding of the mechanical behavior of these materials. The objective of this work is to explore techniques such as the finite element method and homogenization to investigate the structural behavior of bamboo. The finite element formulation uses graded finite elements to capture the varying material distribution through the bamboo wall. To observe bamboo behavior under applied loads, simulations are conducted considering a spatially-varying Young's modulus, an averaged Young's modulus, and orthotropic constitutive properties obtained from homogenization theory. The homogenization procedure uses effective, axisymmetric properties estimated from the spatially-varying bamboo composite. Three-dimensional models of bamboo cells were built and simulated under tension, torsion, and bending load cases.

A cohesive zone model (CZM) has been effective in exploring fracture behavior in various materials. In general, the cohesive parameters associated with material strength and cohesive fracture energy are considered more important than a CZM softening shape. However, the influence of the CZM softening shape becomes significant as the relative size of the fracture process zone compared to the structure size increases, which is relevant for asphalt concrete and other quasi-brittle materials. In this study, the power-law CZM is employed to investigate the influence of the CZM softening shape on asphalt concrete fracture. Three dimensional diskshaped compact tension (DC(T)) test simulation is performed considering bulk (background) material viscoelasticity.

Dynamic stress intensity factor (DSIF) is an important fracture parameter in understanding and predicting dynamic fracture behavior of a cracked body. To evaluate DSIFs for functionally graded materials (FGMs), the interaction integral originally proposed to evaluate SIFs for a static homogeneous medium is extended to incorporate dynamic effects and material nonhomogeneity, and is implemented in conjunction with the finite element method (FEM). To verify numerical implementations and to explore various dynamic fracture behaviors, both homogeneous and nonhomogeneous cracked bodies under dynamic loading are employed.

Probabilistic fracture analysis is performed for predicting uncertain fracture responses of Functionally Graded Material (FGM) structures. The uncertainties in material properties including Young's modulus and fracture toughness are considered. The limit state function for a crack initiation event is defined in terms of the J-integral for FGMs. The First-Order-Reliability-Method (FORM) is used in conjunction with a finite element code that computes the J-integral with high accuracy. A two-step probabilistic analysis procedure is proposed to investigate the effects of the uncertainties in the spatial distribution of Young's modulus on the probability of crack initiation in FGMs. First, we investigate the effects of the uncertainties in the shape of the spatial distribution by considering the slope and the location of the inflection point of a spatial distribution profile as random quantities. Second, we investigate the effects of the spatial fluctuations of Young's modulus by making use of a discretized random field. The companion paper (Part II) implements this method into a finite element fracture analysis code and presents numerical examples.

This presentation describes a topology optimization framework to design the material distribution of functionally graded structures with a tailored Von Mises stress field. The problem of interest consists in obtaining smooth continuous material fraction distribution that produces an admissible stress field. This work explores the topology optimization method for minimizing volume fraction of one of the phases considering stress constraints. Existence of inherent material microstructure requires consideration of the micro level stress field, which is computed through a mechanical concentration factor based on the local stress in each phase of the material. Thus, p-norm of the Von Mises stress in the microstructure is considered as a global constraint. To illustrate the method and discuss its essential features, we present engineering examples of axisymmetric FGM structures subjected to body forces.

Traditionally, standard Lagrangian-type finite elements, such as quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these elements exhibit the well-known "checkerboard" pathology in the solution of topology optimization problems. A feasible alternative to eliminate this long-standing problem consists of using hexagonal elements with Wachspress-type shape functions. The features of the hexagonal mesh include 2-node connections (i.e. 2 elements are either not connected or connected by 2 nodes), and 3 edge-based symmetry lines per element. In contrast, quads can display 1-node connection, which can lead to checkerboard; and only have 2 edge-based symmetry lines. We explore the Wachspress-type hexagonal elements and show their advantages in solving topology optimization problems. We also discuss extensions of the work to account for material gradient effects.

The present paper aims to develop a micromechanics-based effective elastic model of functionally graded composites. At the macroscopic scale, effective material properties of the composites are uniform in the same graded layer while gradually changing along the grading direction. Microstructurally, infinite particles are randomly dispersed in the matrix with gradual transitions. Particles are assumed to be spherical and nonintersecting. They are perfectly bonded with the matrix. A micromechanical framework is proposed to investigate effective mechanical properties along the grading direction. Within the context of the representative volume element (RVE), the effect of pair-wise interactions between particles is taken into account for the local stress and strain fields by using the modified Green's function method. Homogenization of the local field renders relations between the averaged strain, strain gradient and external loading. The effective elastic modulus tensor of the functionally graded composites is further constructed by numerical integration. The model prediction is compared with available experimental data.

This work employs the self-consistent method to investigate the effective thermal conductivity distribution in functionally graded materials (FGMs) considering the Kapitza interfacial thermal resistance. A heat conduction solution is first derived for one spherical particle embedded in a graded matrix with a prefect interface. The interfacial thermal resistance of a nanoparticle is simulated by a new particle with a lower thermal conductivity. A novel selfconsistent formulation is developed to derive the averaged heat flux field of the particle phase. Then the temperature gradient can be obtained in the gradation direction. From the relation between the effective flux and temperature gradient in the gradation direction, the effective thermal conductivity distribution is solved. If the gradient of the volume fraction distribution is zero, the PGM is reduced to a composite containing uniformly dispersed nanoparticles and a explicit solution of the effective thermal conductivity is provided. Disregarding the interfacial thermal resistance, the proposed model recovers the conventional self-consistent model. Mathematically, effective thermal conductivity is a quantity exactly analogous to effective electric conductivity, dielectric permittivity, magnetic permeability and water permeability in a linear static state, so this method can be extended to those problems for graded materials.

This paper presents a Cohesive Zone Model (CZM) approach for investigating dynamic crack propagation in homogeneous and Functionally Graded Materials (FGMs). The failure criterion is incorporated in the CZM using both a finite cohesive strength and work to fracture in the material description. A novel CZM for FGMs is explored and incorporated into a finite element framework. The material gradation is approximated at the element level using a graded element formulation. A numerical example is provided to demonstrate the efficacy of the CZM approach, in which the influence of the material gradation on the crack growth pattern is studied.

This paper presents a Cohesive Zone Model (CZM) approach for investigating dynamic crack propagation in homogeneous and Functionally Graded Materials (FGMs). The failure criterion is incorporated in the CZM using both a finite cohesive strength and work to fracture in the material description. A novel CZM for FGMs is explored and incorporated into a finite element framework. The material gradation is approximated at the element level using a graded element formulation. A numerical example is provided to demonstrate the efficiancy of the CZM approach, in which the influence of the material gradation on the crack branching pattern is studied.

In recent years the transportation materials research community has focused a great deal of attention on the development of testing and analysis methods to shed light on fracture development in asphalt pavements. Recently it has been shown that crack initiation and propagation in asphalt materials can be realistically modeled with cutting-edge computational fracture mechanics tools. However, much more progress is needed toward the development of practical laboratory fracture tests to support these new modeling approaches. The goal of this paper is twofold: (a) to present a disk-shaped compact tension [DC(T)] test, which appears to be a practical method for determining low-temperature fracture properties of cylindrically shaped asphalt concrete test specimens, and (b) to illustrate how the DC(T) test can be used to obtain fracture properties of asphalt concrete specimens obtained from field cores following dynamic modulus and creep compliance tests performed on the same specimens. Testing four mixtures with varied composition demonstrated that the DC(T) test could detect the transition from quasi-brittle to brittle fracture by testing at several low temperatures selected to span across the glass transition temperatures of the asphalt binder used. The tendency toward brittle fracture with increasing loading rate was also detected. Finally, the DC(T) test was used in a forensic study to investigate premature reflective cracking of an isolated portion of pavement in Rochester, New York. One benefit of the DC(T) test demonstrated during testing of field samples was the ability to obtain mixture fracture properties as part of an efficient suite of tests performed on cylindrical specimens.