ii Preface Radial Basis Function (RBF) methods have become the primary tool for interpolating multidimensional scattered data. RBF methods also have become important tools for solving Partial Differential Equations (PDEs) in complexly shaped domains. Classical methods for the numerical solution of PDEs (finite difference, finite element, finite volume, and pseudospectral methods) are based on polynomial interpolation. Local polynomial based methods (finite difference, finite element, and finite volume) are limited by their algebraic convergence rates. Numerical studies, such as the comparison of the MQ collocation method with the finite element method in [136], have been done that illustrate the superior accuracy of the MQ method when compared to local polynomial methods. Global polynomial methods, such as spectral methods, have exponential convergence rates but are limited by being tied to a fixed grid. RBF methods are not tied to a grid and in turn belong to a category of methods called meshless methods. The large number of recent books, on meshfree methods illustrates the popularity that the methods have recently enjoyed. The global, non-polynomial, RBF methods may be successfully applied to achieve exponential accuracy where traditional methods either have difficulties or fail. An example is in multidimensional problems in non-rectangular domains. RBF methods succeed in very general settings by composing a univariate function with the Euclidean norm which turns a multidimensional problem into one that is virtually one dimensional. RBF methods are a generalization of the Multiquadric (MQ) RBF method which utilizes one particular RBF. The MQ RBF method has a rich history of theoretical development and applications. The subject of this monograph is the MQ RBF approximation method with a particular emphasis on using the method to numerically solve partial differential equations. This monograph differs from other recent books [31, 63, 179, 209] on meshless methods in that it focuses only on the MQ RBF while others have focused on meshless methods in general. It is hoped that this refined focus will result in a clear and concise exposition of the area. Matlab code that illustrates key ideas about the implementation of the MQ method has been included in the text of the manuscript. The included code, as well as additional Matlab code used to produce many of the numerical examples, can be found on the web at iii A Matlab programs 147 B Additional Meshfree Method References 153 viii CONTENTS

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Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations

The imposition of Dirichlet boundary conditions in a non conformal mesh can be done with the use of Lagrange multipliers. But this leads to an over-constrained problem that produces an unstable solution if the finite element subspace for the Lagrange multipliers field does not satisfy the inf-sup condition. A stabilization technique is presented, with on a parameter depending on the mesh size. A benchmark is resolved to validate the stability of our method.

Imposing Dirichlet boundary conditions in the eXtended Finite Element Method

Cracking elements method for dynamic brittle fracture

An investigation of fatigue crack growth of interfacial cracks in bi-layered materials using the extended finite element method is presented. The bi-material consists of two layers of dissimilar materials. The bottom layer is made of aluminium alloy while the upper one is made of functionally graded material (FGM). The FGM layer consists of 100 % aluminium alloy on the left side and 100 % ceramic (alumina) on the right side. The gradation in material property of the FGM layer is assumed to be exponential from the alloy side to the ceramic side. The domain based interaction integral approach is extended to obtain the stress intensity factors for an interfacial crack under thermo-mechanical load. The edge and centre cracks are taken at the interface of bi-layered material. The fatigue life of the interface crack plate is obtained using the Paris law of fatigue crack growth under cyclic mode-I, mixed-mode and thermal loads. This study reveals that the crack propagates into the FGM layer under all types of loads.

Fatigue crack growth simulations of interfacial cracks in bi-layered FGMs using XFEM

3-D local mesh refinement XFEM with variable-node hexahedron elements for extraction of stress intensity factors of straight and curved planar cracks

Meshless element free Galerkin method for unsteady nonlinear heat transfer problems

Fatigue crack growth simulations of 3-D problems using XFEM

In this paper, bi-material interfacial cracks have been simulated using element free Galerkin method (EFGM) and extended finite element method (XFEM) under mode-I and mixed mode loading conditions. Few crack interaction problems of dissimilar layered materials are also simulated using extrinsic partition of unity enriched approach. Material discontinuity has been modeled by a signed distance function whereas strong discontinuity has been modeled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The results obtained by EFGM and XFEM for bi-material edge and center cracks are compared with those available in literature. In order to check the validity of simulations, the results have been obtained for two different ratio of Young’s modulus.

Akhilendra Singh

Papers

Meshless element free Galerkin method for unsteady nonlinear heat transfer problems

Fatigue crack growth simulations of 3-D problems using XFEM

Numerical simulation of bi-material interfacial cracks using EFGM and XFEM

New enrichments in XFEM to model dynamic crack response of 2-D elastic solids

Fatigue crack growth simulations of homogeneous and bi-material interfacial cracks using element free Galerkin method