In this paper, the quadtree data structure and conforming polygonal interpolants are used to develop an h-adaptive finite element method. Quadtree is a hierarchical data structure that is computationally attractive for adaptive numerical simulations. Mesh generation and adaptive refinement of quadtree meshes is straight-forward. However, finite elements are non-conforming on quadtree meshes due to level-mismatches between adjacent elements, which results in the presence of so-called hanging nodes. In this study, we use meshfree (natural-neighbor, nn) basis functions on a reference element combined with an affine map to construct conforming approximations on quadtree meshes. Numerical examples are presented to demonstrate the accuracy and performance of the proposed h-adaptive finite element method. (c) 2005 Elsevier B.V. All rights reserved.

Quadtree is a hierarchical data structure that is well-suited for h-adaptive mesh refinement. Due to the presence of hanging nodes, classical shape functions are non-conforming on quadtree meshes. In this paper, we use natural neighbor basis functions to construct conforming interpolants on quadtree meshes. To this end, the recently proposed construction of polygonal basis functions is adapted to quadtree elements. A fast technique for calculating stiffness matrix on quadtree meshes is introduced. Residual-based error estimators and material force technique are used to estimate the error on quadtree meshes. The performance of the adaptive technique is demonstrated through the solution of linear and nonlinear boundary-value problems. (C) 2007 Elsevier B.V. All rights reserved.

In this paper, we present mesh-independent modeling of discontinuous fields on polygonal and quadtree finite element meshes. This approach falls within the class of extended and generalized finite element methods, where the partition of unity framework is used to introduce additional (enrichment) functions within the classical displacement-based finite element approximation. For crack modeling, a discontinuous function and the two-dimensional asymptotic crack-tip fields are used as enrichment functions. Linearly complete partition of unity approximations are adopted on polygonal (convex and nonconvex elements) and quadtree meshes. Excellent agreement with reference solution results is obtained for mixed-mode stress intensity factors on benchmark crack problems, and crack growth simulations without remeshing are conducted on polygonal and quadtree meshes to reveal the potential of the proposed techniques in computational failure mechanics. (c) 2007 Elsevier B.V. All rights reserved.

The Virtual Element Method (VEM) is a recently introduced extension of the Finite ElementMethod (FEM) to general polygonal and polyhedral (polytopal) meshes. By using a set of virtual canonical basis functions, the method provides flexibility for meshing complex geometries with convex and nonconvex polygonal elements, as well as providing a simple approach to handling non-matching meshes and fracturing. Polynomial projection operators are introduced to provide approximation accuracy and to preserve polynomial consistency. However, one limitation of the VEM, is the need to devise a problem dependent stabilization term to retain coercivity. The choice of stabilization adds complexity in formulating new problems and an incorrect choice can adversely affect the solution accuracy. In this dissertation, we develop virtual element methods that do not rely on a stabilization term for problems in planar linear elasticity. We first present strain-based approaches, which use higher order polynomials to enhance the strain polynomial approximation. In these methods, the polynomials are only chosen to preserve the stability of the system. We give theoretical arguments for the stability, well-posedness and prove convergence estimates for the first-order case. These approaches are numerically tested on benchmark elasticity problems, and the results show that the methods attain optimal convergence rates and provide a viable alternative to the standard VEM for compressible materials. However, for thin structures or nearly-incompressible materials, we find that the strain-based approaches and the standard VEM are overly stiff and suffer from locking phenomena. To alleviate locking, we appeal to stress-based approaches that rely on the Hellinger–Reissner variational formulation. These methods use selectively chosen higher order divergence-free polynomials that preserve the stability as well as alleviate locking. Starting with quadrilateral elements, we use a five-parameter expansion of the stress field to construct a method that is free of volumetric and shear locking. For six-noded triangular elements, we find that a fifteen-term divergence-free stress expansion resulted in a method that does not require stabilization and shows immunity to locking. An alternative to using divergence-free polynomials is also explored. This approach uses a penalty term to weakly enforce the equilibrium equations. Numerical results reveal that the stress-based approaches provide optimal convergence, and robustness for compressible and nearly-incompressible problems.

An irregular lattice model is proposed for simulating quasistatic fracture in softening materials. Lattice elements are defined on the edges of a Delaundy tessellation of the medium. The dual (Voronoi) tessellation is used to scale the elemental stiffness terms in a manner that renders the lattice elastically homogeneous. This property enables the accurate modeling of heterogeneity, as demonstrated through the elastic stress analyses of fiber composites. A cohesive description of fracture is used to model crack initiation and propagation. Numerical simulations, which demonstrate energy-conserving and grid-insensitive descriptions of cracking, are presented. The model provides a framework for the failure analysis of quasibrittle materials and fiber-reinforced brittle-matrix composites.

In this paper, an overview of the construction of meshfree basis functions is presented, with particular emphasis on moving least-squares approximants, natural neighbour-based polygonal interpolants, and entropy approximants. The use of information -theoretic variational principles to derive approximation schemes is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints. The maximization (minimization) of the Shannon-Jaynes entropy functional (relative entropy functional) is used to unity the construction of globally and locally Supported convex approximation schemes. A JAVA applet is used to visualize the meshfree basis functions, and comparisons and links between different meshfree approximation schemes are presented. Copyright (c) 2006 John Wiley & Sons, Ltd.

In this paper, we prove the continuity of maximum-entropy basis functions using variational analysis techniques. The use of information-theoretic variational principles to derive basis functions is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution, and the polynomial reproducing conditions acting as the linear constraints. For a set of distinct nodes {x(i)}(i=1)(n) in R-d, the convex approximation of a function u(x) is u(h)(x) = Sigma(n)(i=1) p(i)(x)u(i), where {p(i)}(i=1)(n) are nonnegative basis functions, and u(h)(x) must reproduce a. ne functions Sigma(n)(i=1) p(i)(x) = 1, Sigma(n)(i=1) p(i)(x) x(i) = x. Given these constraints, we compute p(i)(x) by minimizing the relative entropy functional (Kullback-Leibler distance), D(p parallel to m) = Sigma(n)(i=1) p(i)(x) ln(p(i)(x)/m(i)(x)), where m(i)(x) is a known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epiconvergence.