Meshfree methods have become strong alternatives to conventional numerical methods used in solid mechanics after significant progress in recent years. Radial point interpolation method (RPIM) is a meshfree method based on Galerkin formulation and constructs shape functions which enable easy imposition of essential boundary conditions. This thesis analyses plane elasticity problems using RPIM. A computer code implementing RPIM for the solution of plane elasticity problems is developed. Selected problems are solved and the effect of shape parameters on the accuracy of RPIM with and without polynomial terms added in the interpolation is studied. The optimal shape parameters are determined for plane elasticity problems.

Thesis (M.S.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on March 19, 2008) Includes bibliographical references.

Thesis (Ph.D.)--University of Delaware, 2008. / Principal faculty advisor: Tobin A. Driscoll, Dept. of Mathematical Sciences. Includes bibliographical references.

Thesis (M.S.)--University of Alabama at Birmingham, 2009. / Description based on contents viewed June 2, 2009; title from PDF t.p. Includes bibliographical references (p. 71-76).

Thesis (MTech (Mechanical Engineering))--Cape Peninsula University of Technology, 2009. / Includes bibliographical references (leaves 94-98). Also available online.

Kernel-based meshless methods for approximating functions and solutions of partial differential equations have many applications in engineering fields. As only scattered data are used, meshless methods using radial basis functions can be extended to complicated geometry and high-dimensional problems. In this thesis, kernel-based least-squares methods will be used to solve several direct and inverse problems. In chapter 2, we consider discrete least-squares methods using radial basis functions. A general l^2-Tikhonov regularization with W_2^m-penalty is considered. We provide error estimates that are comparable to kernel-based interpolation in cases in which the function being approximated is within and is outside of the native space of the kernel. These results are extended to the case of noisy data. Numerical demonstrations are provided to verify the theoretical results. In chapter 3, we apply kernel-based collocation methods to elliptic problems with mixed boundary conditions. We propose some weighted least-squares formulations with different weights for the Dirichlet and Neumann boundary collocation terms. Besides fill distance of discrete sets, our weights also depend on three other factors: proportion of the measures of the Dirichlet and Neumann boundaries, dimensionless volume ratios of the boundary and domain, and kernel smoothness. We determine the dependencies of these terms in weights by different numerical tests. Our least-squares formulations can be proved to be convergent at the H^2 (Ω) norm. Numerical experiments in two and three dimensions show that we can obtain desired convergent results under different boundary conditions and different domain shapes. In chapter 4, we use a kernel-based least-squares method to solve ill-posed Cauchy problems for elliptic partial differential equations. We construct stable methods for these inverse problems. Numerical approximations to solutions of elliptic Cauchy problems are formulated as solutions of nonlinear least-squares problems with quadratic inequality constraints. A convergence analysis with respect to noise levels and fill distances of data points is provided, from which a Tikhonov regularization strategy is obtained. A nonlinear algorithm is proposed to obtain stable solutions of the resulting nonlinear problems. Numerical experiments are provided to verify our convergence results. In the final chapter, we apply meshless methods to the Gierer-Meinhardt activator-inhibitor model. Pattern transitions in irregular domains of the Gierer-Meinhardt model are shown. We propose various parameter settings for different patterns appearing in nature and test these settings on some irregular domains. To further simulate patterns in reality, we construct different kinds of domains and apply proposed parameter settings on different patches of domains found in nature.

abstract: Divergence-free vector field interpolants properties are explored on uniform and scattered nodes, and also their application to fluid flow problems. These interpolants may be applied to physical problems that require the approximant to have zero divergence, such as the velocity field in the incompressible Navier-Stokes equations and the magnetic and electric fields in the Maxwell's equations. In addition, the methods studied here are meshfree, and are suitable for problems defined on complex domains, where mesh generation is computationally expensive or inaccurate, or for problems where the data is only available at scattered locations. The contributions of this work include a detailed comparison between standard and divergence-free radial basis approximations, a study of the Lebesgue constants for divergence-free approximations and their dependence on node placement, and an investigation of the flat limit of divergence-free interpolants. Finally, numerical solvers for the incompressible Navier-Stokes equations in primitive variables are implemented using discretizations based on traditional and divergence-free kernels. The numerical results are compared to reference solutions obtained with a spectral method. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

Patient-specific biomechanical analysis is an important tool used to understand the complex processes that occur in the body due to physical stimulation. Patient-specific models are generated by processing medical images; once an object from the image is identified via segmentation, a point cloud representation of the object is extracted. Generating an analysis suitable representation from the point cloud has traditionally required generating a finite element mesh, which often requires a well defined surface to accomplish. Point clouds lack a well defined geometry, meaning that the surface definition is incomplete at best. Point clouds that have been generated from images have a fuzzy boundary, meaning that no direct sampling of the boundary is guaranteed to exist and any method for completing the geometry definition requires assumptions on the part of the modeler. The process of generating a finite element mesh of the point cloud is difficult and tedious often requiring manual editing to alleviate poorly constructed elements. An alternative to generating a finite element mesh is to use meshfree analysis to solve the boundary value problem. The geometric representation of meshfree analysis is a point cloud, thus making it a natural choice for patient-specific analysis. When using meshfree analysis, it is common to skip the geometric reconstruction and use the point cloud directly as an analysis suitable geometry. The lack of a well defined surface can be problematic for a variety of reasons, namely the visualization of results and solving contact driven problems. The goal of this dissertation is to exploit some characteristics of the meshfree analysis to generate a well defined geometry for point clouds. Meshfree methods are commonly used for the solution of boundary value problems; their lack of a well defined geometry representation is a hindrance that is often remedied by accompanying the meshfree particle distribution with a secondary geometry representation, such as a mesh. The present work outlines a framework that can be used to define and study meshfree geometry representations. A particular meshfree geometry representation called the Meshfree Correction Implicit Geometry is introduced and studied under the guidelines of the framework.

In the presented thesis work, meshfree method with distance fields was extended to obtain solution of nonlinear transient heat transfer problems. The thesis work involved development and implementation of numerical algorithms, data structure, and software. Numerical and computational properties of the meshfree method with distance fields were investigated. Convergence and accuracy of the methodology was validated by analytical solutions, and solutions produced by commercial FEM software (ANSYS 12.1). The research was focused on nonlinearities caused by temperature-dependent thermal conductivity. The behavior of the developed numerical algorithms was observed for both weak and strong temperature-dependency of thermal conductivity. Oseen and Newton-Kantorovich linearization techniques were applied to linearized the governing equation and boundary conditions. Results of the numerical experiments showed that the meshfree method with distance fields has the potential to produced fast accurate solutions. The method enables all prescribed boundary conditions to be satisfied exactly.

The recent advances in the Materials Engineering have led to the development of new materials with customized microstructure in which the properties of its constituents and their geometric distribution have a considerable effect on determination of the macroscopic properties of the substance. Direct inclusion of the material microstructure in the analysis on a macro level is challenging since spatial meshes created for the analysis should have enough resolution to be able to accurately capture the geometry of the microstructure. In most cases this leads to a huge finite element model which requires a substantial amount of computational resources. To circumvent this limitation a number of homogenization techniques were developed. By considering a small element of the material, referred to as Representative Volume Element (RVE), homogenization methods make it possible to include the effects of a material’s microstructure on the overall properties at the macro level. However, complexity of the microstructure geometry and the necessity of satisfying periodic boundary conditions introduce additional difficulties into the analysis procedure. In this dissertation we propose a hybrid homogenization method that combines Asymptotic homogenization with MeshFree Solution Structures Method (SSM). Our approach allows realistic inclusion of complex geometry of the microstructure that can be captured from micrographs or micro CT scans. In addition to unprecedented flexibility in handling complex geometries, this method also provides a completely automatic analysis procedure. Using meshfree solution structures simplifies meshing to creating a simple cartesian grid which only needs to contain the domain. This also eliminates manual modifications which usually needs to be performed on meshes created from image data. A computational platform is developed in C++ based on meshfree/asymptotic method. In this platform also a novel meshfree solution structure is designed to provide exact satisfaction of periodic boundary conditions for boundary value problems such as homogenization. Performance of the developed platform is tested over 2D and 3D domains against previously published data and/or conventional finite element methods. After getting satisfactory results, homogenized properties are used to compute localized stress and strain distributions over inhomogeneous structures. Furthermore, effects of geometric features of pores/inclusions on homogenized mechanical properties is investigated and it is demonstrated that the developed platform could provide an automated quantitative analysis tool for studying effects of different design parameters on homogenized properties.