Hybrid Chebyshev Polynomial Scheme for the Numerical Solution of Partial Differential Equations

In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including the method of fundamental solution (MFS) and the equilibrated collocation Trefftz method. However, this scheme can be applied to solve PDEs with constant coefficients only. So, for solving a wide variety of PDEs, one-step hybrid Chebyshev polynomial scheme is proposed. This approach combines two matrix systems of two-step approach into a single matrix system. The solution is approximated by the sum of particular solution and homogeneous solution. The Laplacian or biharmonic operator is kept on the left hand side and all the other terms are moved to the right hand side and treated as the forcing term. Various boundary value problems governed by the Poisson equation in two and three dimensions are considered for the numerical experiments. HCPS is also applied to solve an inhomogeneous Cauchy-Navier equations of elasticity in two dimensions. Numerical results show that HCPS is direct, easy to implement, and highly accurate

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods

Fundamental solution based numerical methods for three dimensional problems: efficient treatments of inhomogeneous terms and hypersingular integrals

In recent years, fundamental solution based numerical methods including the meshless method of fundamental solutions (MFS), the boundary element method (BEM) and the hybrid fundamental solution based finite element method (HFS-FEM) have become popular for solving complex engineering problems. The application of such fundamental solutions is capable of reducing computation requirements by simplifying the domain integral to the boundary integral for the homogeneous partial differential equations. The resulting weak formulations, which are of lower dimensions, are often more computationally competitive than conventional domain-type numerical methods such as the finite element method (FEM) and the finite difference method (FDM). In the case of inhomogeneous partial differential equations arising from transient problems or problems involving body forces, the domain integral related to the inhomogeneous solutions term will need to be integrated over the interior domain, which risks losing the competitive edge over the FEM or FDM. To overcome this, a particular treatment to the inhomogeneous term is needed in the solution procedure so that the integral equation can be defined for the boundary. In practice, particular solutions in approximated form are usually applied rather than the closed form solutions, due to their robustness and readiness. Moreover, special numerical treatment may be required when evaluating stress directly on the domain surface which may give rise to hypersingular integral formulation. This thesis will discuss how the MFS and the BEM can be applied to the three-dimensional elastic problems subjected to body forces by introducing the compactly supported radial basis functions in addition to the efficient treatment of hypersingular surface integrals. The present meshless approach with the MFS and the compactly supported radial basis functions is later extended to solve transient and coupled problems for three-dimensional porous media simulation

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations

Finite element methods with local Trefftz trial functions

In the development of numerical methods for boundary value problems, the requirement of flexible mesh handling gains more and more importance. The available work deals with a new kind of conforming finite element methods on polygonal/polyhedral meshes. The idea is to use basis functions which are defined implicitly as local solutions of the underlying homogeneous problem with constant coefficients. They are referred to local Trefftz functions. These local problems are treated by means of boundary integral equations and are approximated by the use of the boundary element method in the numerics. The method is applied to the stationary diffusion equation, where lower as well as higher order basis functions are introduced in two space dimensions. The convergence is analysed with respect to the H^1- as well as the L_2-norm and rates of convergence are proven. In case of non-constant diffusion coefficients, a special approximation is proposed. Beside the uniform refinement, an adaptive strategy is given which makes use of the residual error estimator and an introduced refinement procedure. The reliability of the residual error estimate is proven on polygonal meshes. Finally, the generalization to arbitrary polyhedral meshes with polygonal faces is discussed. All theoretical results and considerations are confirmed by numerical experiments.In der Entwicklung numerischer Verfahren zur Approximation von Randwertaufgaben werden flexible Vernetzungen der zugrunde liegenden Gebiete immer wichtiger. Die vorliegende Arbeit beschäftigt sich mit neuartigen Finiten Element Methoden, die zu konformen Approximationen auf polygonalen und polyhedralen Gittern führen. Der Gedanke dieser Vorgehensweise liegt darin, die Ansatzfunktionen implizit als Lösungen von lokalen Randwertaufgaben zu definieren, wie dies auch schon E. Trefftz vorgeschlagen hat. Hierbei wird die Differentialgleichung des Ursprungsproblems mit konstanten Koeffizienten und homogener rechter Seite verwendet. Die lokalen Probleme werden mit Randintegralgleichungen und in der Realisierung mit Randelementmethoden behandelt. Das Verfahren wird auf die stationäre Diffusionsgleichung angewendet, wofür Ansatzfunktionen niedriger als auch höherer Ordnung eingeführt werden. Konvergenzraten bezüglich der H^1- sowie der L_2-Norm werden untersucht und bewiesen. Im Falle eines nicht konstanten Diffusionskoeffizienten wird eine spezielle Vorgehensweise vorgeschlagen. Neben der gleichmäßigen Verfeinerung der Netze wird ebenso eine adaptive Strategie angegeben, die von dem residualen Fehlerschätzer und einer eingeführten Verfeinerung Gebrauch macht. Die Zuverlässigkeit des Fehlerschätzers auf polygonalen Netzen wird bewiesen und schließlich wird das Verfahren erweitert, so dass es auf polyhedralen Gittern mit polygonalen Elementflächen angewendet werden kann. Alle theoretischen Resultate und Überlegungen werden durch numerische Experimente bestätigt

Enriched and Isogeometric Boundary Element Methods for Acoustic Wave Scattering

This thesis concerns numerical acoustic wave scattering analysis. Such problems have been solved with computational procedures for decades, with the boundary element method being established as a popular choice of approach. However, such problems become more computationally expensive as the wavelength of an incident wave decreases; this is because capturing the oscillatory nature of the incident wave and its scattered field requires increasing numbers of nodal variables. Authors from mathematical and engineering backgrounds have attempted to overcome this problem using a wide variety of procedures. One such approach, and the approach which is further developed in this thesis, is to include the fundamental character of wave propagation in the element formulation. This concept, known as the Partition of Unity Boundary Element Method (PU-BEM), has been shown to significantly reduce the computational burden of wave scattering problems. This thesis furthers this work by considering the different interpolation functions that are used in boundary elements. Initially, shape functions based on trigonomet- ric functions are developed to increase continuity between elements. Following that, non-uniform rational B-splines, ubiquitous in Computer Aided Design (CAD) soft- ware, are used in developing an isogeometric approach to wave scattering analysis of medium-wave problems. The enriched isogeometric approach is named the eXtended Isogeometric Boundary Element Method (XIBEM). In addition to the work above, a novel algorithm for finding a uniform placement of points on a unit sphere is presented. The algorithm allows an arbitrary number of points to be chosen; it also allows a fixed point or a bias towards a fixed point to be used. This algorithm is used for the three-dimensional acoustic analyses in this thesis. The new techniques developed within this thesis significantly reduce the number of degrees of freedom required to solve a problem to a certain accuracy—this reduc- tion is more than 70% in some cases. This reduces the number of equations that have to be solved and reduces the amount of integration required to evaluate these equations

Hybrid-Trefftz finite elements for elastostatic and elastodynamic problems in porous media

The displacement and stress models of the hybrid-Trefftz finite element formulation are applied to the elastostatic and elastodynamic analysis of two-dimensional saturated and unsaturated porous media problems. The formulation develops from the classical separation of variables in time and space, but it leads to two time integration strategies. The first is applied to periodic problems, which are discretized in time using Fourier analysis. A mixed finite element approach is used in the second strategy for discretization in time of non-periodic/transient problems. These strategies lead to a series of uncoupled problems in the space dimension, which is subsequently discretized using either the displacement or the stress model of the hybrid-Trefftz finite element formulation. The main distinction between the two models is in the way that the interelement continuity is enforced. The displacement model enforces the interelement compatibility, while the stress model enforces the interelement equilibrium. As is typical of Trefftz methods, for both models, the approximation bases are constrained to satisfy locally the homogeneous form of the domain (Navier) equations. The free-field solutions of these equations are derived in cylindrical coordinates and used to construct the domain approximations of the hybrid-Trefftz displacement and stress elements. If the original equations are non-homogeneous, the influence of the source terms is modelled using Trefftz-compliant solutions of the corresponding static problem. For saturated porous media, the finite element models are based on the Biot's theory. It assumes an elastic solid phase fully permeated by a compressible liquid phase obeying the Darcy's law. For the modelling of unsaturated porous media, the finite elements are formulated using the theory of mixtures with interfaces. The model is thermodynamically consistent and considers the full coupling between the solid, fluid and gas phases, including the effects of relative (seepage) accelerations. Small displacements and linear-elastic material behaviour are assumed for all models

Boundary integral equations of time harmonic wave scattering at complex structures

The first chapter will be a brief recapitulation of well known results concerning layer potentials in the context of wave propagation in harmonic regime. In Chapter 2, we give an overview of the Rumsey reaction principle that is the most popular boundary integral formulation for multi-subdomain scattering, and we present a new alternative integral formulation that seems to be the first boundary integral formulation of the second kind for multi-subdomain scattering in geometrical configurations involving junction points. Chapter 3 is dedicated to the multi-trace formalism which is a completely new approach to boundary integral formulation of multi-subdomain scattering: we briefly describe the local multi-trace formulation, and describe in detail the derivation of the global multi-trace formulation developed by us, as well as its sparsified counterpart that we dubbed quasi-local multi-trace formulation. In Chapter 4 we present a new functional framework adapted to well-posed boundary integral equations for scattering by particular types of object dubbed multi-screens as they take the form of arbitrary arrangements of thin panels of impenetrable material. In Chapter 5 we describe several works on asymptotic modelling in the context wave propagationin harmonic regime. The last chapter presents research perspectives