Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D

We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Green's third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary integrals can then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions

Planewave density interpolation methods for 3D Helmholtz boundary integral equations

This paper introduces planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green's third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nystr\"om and boundary element discretization of boundary integral equation, can then be numerically evaluated by standard quadrature rules that are irrespective of the kernel singularity. Closed-form and purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nystr\"om and Galerkin boundary element methods. A variety of numerical examples---including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique

Accurate computation of Galerkin double surface integrals in the 3-D boundary element method

Many boundary element integral equation kernels are based on the Green's functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, which may be accelerated via the fast multipole method. There are advantages to Galerkin formulations for such integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires each entry in the system matrix to be created via the computation of a double surface integral over one or more pairs of triangles. There are a number of semi-analytical methods to treat these integrals, which all have some issues, and are discussed in this paper. We present novel methods to compute all the integrals that arise in Galerkin formulations involving kernels based on the Laplace and Helmholtz Green's functions to any specified accuracy. Integrals involving completely geometrically separated triangles are non-singular and are computed using a technique based on spherical harmonics and multipole expansions and translations, which results in the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. Example results are presented, and the developed software is available as open source

Enrichment of the Boundary Element Method through the Partition of Unity Method for Fracture Analysis using Local and Global Formulations

The present thesis proposes an innovative technique of applying enrichment to the Boundary Element Method to allow accurate analysis of 2D crack problems. An overview of fracture mechanics is given, with particular emphasis given to numerical methods and the techniques used to extract the highly important stress intensity factors - a measure of the singularity of a crack tip. The Boundary Element Method framework is described and later, the implementation of the new technique of enrichment is defined in detail. Finally, several crack problems are used to verify the accuracy of the method where the results are shown to compare very favourably with other well-established numerical methods

General-purpose kernel regularization of boundary integral equations via density interpolation

This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calder\'on calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same Green's function used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known Green's function. For the sake of definiteness, we focus here on Nystr\"om methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers are demonstrated by means of a variety of large-scale three-dimensional numerical examples

A high-performance boundary element method and its applications in engineering

As a semi-numerical and semi-analytical method, owing to the inherent advantage, of boundary-only discretisation, the boundary element method (BEM) has been widely applied to problems with complicated geometries, stress concentration problems, infinite domain problems, and many others. However, domain integrals and non-symmetrical and dense matrix systems are two obstacles for BEM which have hindered the its further development and application. This thesis is aimed at proposing a high-performance BEM to tackle the above two drawbacks and broaden the application scope of BEM. In this thesis, a detailed introduction to the traditional BEM is given and several popular algorithms are introduced or proposed to enhance the performance of BEM. Numerical examples in heat conduction analysis, thermoelastic analysis and thermoelastic fracture problems are performed to assess the efficiency and correction of the algorithms. In addition, necessary theoretical derivations are embraced for establishing novel boundary integral equations (BIEs) for specific engineering problems. The following three parts are the main content of this thesis. (1) The first part (Part II consisting of two chapters) is aimed at heat conduction analysis by BEM. The coefficient matrix of equations formed by BEM in solving problems is fully-populated which occupy large computer memory. To deal with that, the fast multipole method (FMM) is introduced to energize the line integration boundary element method (LIBEM) to performs better in efficiency. In addition, to compute domain integrals with known or unknown integrand functions which are caused by heat sources or heterogeneity, a novel BEM, the adaptive orthogonal interpolation moving least squares (AOIMLS) method enhanced LIBEM, which also inherits the advantage of boundary-only discretisation, is proposed. Unlike LIBEM, which is an accurate and stable method for computing domain integrals, but only works when the mathematical expression of integral function in domain integrals is known, the AOIMLS enhanced LIBEM can compute domain integrals with known or unknown integral functions, which ensures all the nonlinear and nonhomogeneous problems can be solved without domain discretisation. In addition, the AOIMLS can adaptively avoid singular or ill-conditioned moment matrices, thus ensuring the stability of the calculation results. (2) In the second part (Part III consisting of four chapters), the thermoelastic problems and fracture problems are the main objectives. Due to considering thermal loads, domain integrals appear in the BIEs of the thermoelastic problems, and the expression of integrand functions is known or not depending on the temperature distribution given or not, the AOIMLS enhanced LIBEM is introduced to conduct thermoelasticity analysis thereby. Besides, a series of novel unified boundary integral equations based on BEM and DDM are derived for solving fracture problems and thermoelastic fracture problems in finite and infinite domains. Two sets of unified BIEs are derived for fracture problems in finite and infinite domains based on the direct BEM and DDM respectively, which can provide accurate and stable results. Another two sets of BIEs are addressed by employing indirect BEM and DDM, which cannot ensure a stable result, thereby a modified indirect BEM is proposed which performs much more stable. Moreover, a set of novel BIEs based on the direct BEM and DDM for cracked domains under thermal stress is proposed. (3) In the third part (Part IV consisting of one chapter), a high-efficiency combined BEM and discrete element method (DEM) is proposed to compute the inner stress distribution and particle breakage of particle assemblies based on the solution mapping scheme. For the stress field computation of particles with similar geometry, a template particle is used as the representative particle, so that only the related coefficient matrices of one template particle in the local coordinate system are needed to be calculated, while the coefficient matrices of the other particles, can be obtained by mapping between the local and global coordinate systems. Thus, the combined BEM and DEM is much more effective when modelling a large-scale particle system with a small number of distinct possible particle shapes. Furthermore, with the help of the Hoek-Brown criterion, the possible cracks or breakage paths of a particle can be obtained