Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. No such class of radial basis functions that reflects these physical properties was known until 1994, when Narcowich and Ward introduced a family of matrix-valued radial basis functions that are divergence free. They also obtained error bounds and stability estimates for interpolation by means of these functions. These divergence-free functions are very smooth, and have unbounded support. In this thesis we introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. This leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We develop error bounds and stability estimates which hold for a broad class of functions. We conclude with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.

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

Thesis (M. Eng.(Mechanical Engineering))--University of Pretoria, 2004. / Includes bibliographical references.

Thesis (M.S.)--Florida State University, 2005. / Advisor: Dr. Anke Meyer Baese, Florida State University, College of Engineering, Dept. of Electrical and Computer Engineering. Title and description from dissertation home page (viewed June 7, 2005). Document formatted into pages; contains ix, 39 pages. Includes bibliographical references.

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

In this paper the obstacle description with Radial Basis Functions for contact problems in three dimensional elasticity will be done. A short Introduction of the idea of Radial Basis Functions will be followed by the usage of Radial Basis Functions for approximation of isosurfaces. Then these isosurfaces are used for the obstacle-description in three dimensional elasticity contact problems. In the last part some computational examples will be shown.

Radial basis function (RBF) methods for partial differential equations (PDEs), either in bulk domains, on surfaces, or in a combination of the formers, arise in a wide range of practical applications. This thesis proposes numerical approaches of RBF-based meshless techniques to solve these three kinds of PDEs on stationary and nonstationary surfaces and domains. In Chapter 1, we introduce the background of RBF methods, some basic concepts, and error estimates for RBF interpolation. We then provide some preliminaries for manifolds, restricted RBFs on manifolds, and some convergence properties of RBF interpolation. Finally, implicit-explicit time stepping schemes are briefly presented. In Chapter 2, we propose methods to implement meshless collocation approaches intrinsically to solve elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, that is, Kansa methods. Our main theoretical contribution is a robust convergence analysis under some standard smoothness assumptions for high-order convergence. We simulate reaction-diffusion equations to generate Turing patterns and solve shallow water problems on manifolds. In Chapter 3, we consider convective-diffusion problems that model surfactants or heat transport along moving surfaces. We propose two time-space algorithms by combining the methods of lines and kernel-based meshless collocation techniques intrinsic to surfaces. We use a low-order time discretization for fair comparison, and higher-order schemes in time are possible. The proposed methods can achieve second-order convergence. They use either analytic or approximated spatial discretization of the surface operators, which do not require regeneration of point clouds at each temporal iteration. Thus, they are alternatively applied to handle models on two types of evolving surfaces, which are defined as prescribed motions and governed by geometric evolution laws, respectively. We present numerical examples on various evolving surfaces for the performance of our algorithms and apply the approximated one to merging surfaces. In Chapter 4, a kernel-based meshless method is developed to solve coupled second-order elliptic PDEs in bulk domains and on surfaces, subject to Robin boundary conditions. It combines a least-squares kernel-based collocation method with a surface-type intrinsic approach. We can thus use each pair for discrete point sets, RBF kernels (globally and restrictedly), trial spaces, and some essential assumptions, to search for least-squares solutions in bulks and on surfaces, respectively. We first analyze error estimates for a domain-type Robin-boundary problem. Based on this analysis and the existing results for surface PDEs, we discuss the theoretical requirements for the Sobolev kernels used. We then select the orders of smoothness for the kernels in bulks and on surfaces. Finally, several numerical experiments are demonstrated to test the robustness of the coupled method in terms of accuracy and convergence rates under different settings.

Distributed modal filters based on piezoelectric polymer have especially become popular in the field of active vibration control to reduce the problem of spillover. While distributed modal filters for one-dimensional structures can be found analytically based on the orthogonality between the mode shapes, the design for two-dimensional structures is not straightforward. It requires a continuous gain variation in two dimensions, which is not realizable from the current manufacturing point of view. In this thesis, a structural optimization problem is considered to approximate distributed modal sensors for two-dimensional plate structures, where the thickness is constant but the polarization can switch between positive and negative. The problem is solved through an explicit parametric level set method. In this framework, the boundary of a domain is represented implicitly by the zero isoline of a level set function. This allows simultaneous shape and topology changes. The level set function is approximated by a linear combination of Gaussian radial basis functions. As a result, the structural optimization problem can be directly posed in terms of the parameters of the approximation. This allows to apply standard optimization methods and bypasses the numerical drawbacks, such as reinitialization, velocity extension and regularization, which are associated with the numerical solution of the Hamilton-Jacobi equation in conventional methods.Since the level set method based on the shape derivative formally only allows shape but not topology transformation, the optimization problem is firstly tackled with a derivative-free optimization algorithm. It is shown that the approach is able to find approximate modal sensor designs with only few design variables. However, this approach becomes unsuitable as soon as the number of optimization variables is growing. Therefore, a sensitivity-based optimization approach is being applied, based on the parametric shape derivative which is with respect to the parameters of the radial basis functions. Although the shape derivatives does not exist at points where the topology changes, it is demonstrated that an optimization routine based on a SQP solver is able to perform topological changes during the optimization and finds optimal designs even from poor initial designs. In order to include the sensors' distribution as design variable, the parametric level set approach is extended to multiple level sets. It turns out that, despite the increased design space, optimal solutions always converge to full-material polarization designs. Numerical examples are provided for a simply supported as well as a cantilever square plate. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

Current methods for solving thennoelasticity problems involve using finite element analysis, boundary element analysis, or other meshed-type methods to determine the deflections under an imposed temperature/stress field. This thesis will detail a new approach using meshless methods to solve these types of thermoelasticity problems in which the solution is independent of boundary and internal meshing. With the rapidly increasing availability and performance of computer workstations and clusters, the major time requirement for solving a thermoelasticity model is no longer the computation time, but rather the problem setup. Defining the required mesh for a complex geometry can be extremely complicated and time consuming, and new methods are desired that can reduce this model setup time. The proposed meshless methods completely eliminate the need for a mesh, and thus, eliminate the need for complicated meshing procedures. Although the savings gain due to eliminating the meshing process would be more than sufficient to warrant further study, the localized meshless method can also be comparable in computational speed to more traditional finite element solvers when analyzing complex problems. The reduction of both setup and computational time makes the meshless approach an ideal method of solving coupled thermoelasticity problems. Through the development of these methods it can be determined whether they are feasible as potential replacements for more traditional solution methods. More specifically, two methods will be covered in depth from the development to the implementation. The first method covered will be the global meshless method and the second will be the improved localized method. Although they both produce similar results in terms of accuracy, the localized method greatly improves upon the stability and computation time of the global method.

Meshless methods have recently received much attention but are yet to reach their full potential as the required problem setup (i.e. collocation point distribution) is still significant and far from automated. The distribution of points still closely resembles the nodes of finite volume-type meshes and the free parameter, c, of the radial-basis expansion functions (RBF) still must be tailored specifically to a problem. The localized meshless collocation method investigated requires a local influence region, or topology, used as the expansion medium to produce the required field derivatives. Tests have shown a regular cartesian point distribution produces optimal results, however, in order to maintain a locally cartesian point distribution a recursive quadtree scheme is herein proposed. The quadtree method allows modeling of irregular geometries and refinement of regions of interest and it lends itself for full automation, thus, reducing problem setup efforts. Furthermore, the construction of the localized expansion regions is closely tied up to the point distribution process and, hence, incorporated into the automated sequence. This also allows for the optimization of the RBF free parameter on a local basis to achieve a desired level of accuracy in the expansion. In addition, an optimized auto-segmentation process is adopted to distribute and balance the problem loads throughout a parallel computational environment while minimizing communication requirements.