Accuracy of desingularized boundary integral equations for plane exterior potential problems

In this article, computational results from boundary integral equations and their normal derivatives for the same test cases are compared. Both kinds of formulations are desingularized on their real boundary. The test cases are chosen as a uniform flow past a circular cylinder for both the Dirichlet and Neumann problems. The results indicate that the desingularized method for the standard boundary integral equation has a much larger convergence speed than the desingularized method for the hypersingular boundary integral equation. When uniform nodes are distributed on a circle, for the standard boundary integral formulation the accuracies in the test cases reach the computer limit of 10⁻¹⁵ in the Neumann problems; and O(N⁻³) in the Dirichlet problems. However, for the desingularized hypersingular boundary integral formulation, the convergence speeds drop to only O(N⁻¹) in both the Neumann and Dirichlet problems.     

Computability of series representations for Green's functions in a rectangle

Different series forms of Green's functions are analyzed for various boundary value problems stated for Laplace and Klein–Gordon equation in a rectangular region. The classical double-series representation for the Dirichlet problem for Laplace equation is converted into a single-series form, and a computational experiment is conducted to compare practical convergence of the two forms. By a partial summation of the single-series representation, the singular component of the Green's function is expressed in analytic form radically accelerating convergence of the remaining series for the regular component. Readily computable series forms are obtained for Green's functions of some mixed boundary value problems.     

A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems

We present a novel multigrid (MG) procedure for the efficient solution of the large non‐symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi‐layered elastic medium. The governing equations involve a highly non‐linear coupled system of integro‐partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N³). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green's functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub‐sets of the finest grid are used to treat mixed variable problems associated with ‘pinch points.’ Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss–Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

Embedded kinematic boundary conditions for thin plate bending by Nitsche's approach

A stabilized variational formulation, based on Nitsche's method for enforcing boundary constraints, leads to an efficient procedure for embedding kinematic boundary conditions in thin plate bending. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the C¹‐continuity requirements typical of these problems. Work‐conjugate pairs weakly enforce kinematic boundary conditions. The pointwise enforcement of corner deflections is key to good performance in the presence of corners. Stabilization parameters are determined from local generalized eigenvalue problems, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic C² B‐splines, exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Convergence analysis of the Q1-finite element method for elliptic problems with non-boundary-fitted meshes

The aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain's boundary. This problematic of the last century (e.g. the finite difference methodology) returns to topical studies with the huge development of domain embedding, fictitious domain or Cartesian-grid methods. These methods use regular structured meshes (most often Cartesian) for non-aligned domains. Although non-boundary-fitted approaches become more and more applied, very few studies are devoted to theoretical error estimates. In this paper, the convergence of a Q₁-non-conforming finite element method is analyzed for second-order elliptic problems with Dirichlet, Robin or Neumann boundary conditions. The finite element method uses standard Q₁-rectangular finite elements. As the finite element approximate space is not contained in the original solution space, this method is referred to as non-conforming. A stair-step boundary defined from the Cartesian mesh approximates the original domain's boundary. The convergence analysis of the finite element method for such a kind of non-boundary-fitted stair-stepped approximation is not treated in the literature. The study of Dirichlet problems is based on similar techniques as those classically used with boundary-fitted linear triangular finite elements. The estimates obtained for Robin problems are novel and use some more technical arguments. The rate of convergence is proved to be in 𝒪(h^1/2) for the H¹-norm for all general boundary conditions, and classical duality arguments allow one to obtain an 𝒪(h) error estimate in the L²-norm for Dirichlet problems. Numerical results obtained with fictitious domain techniques, which impose original boundary conditions on a non-boundary-fitted approximate immersed interface, are presented. These results confirm the theoretical rates of convergence. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical analysis of dissolution processes in cementitious materials using discontinuous and continuous Galerkin time integration schemes

The present paper is concerned with the numerical integration of non‐linear reaction–diffusion problems by means of discontinuous and continuous Galerkin methods. The first‐order semidiscrete initial value problem of calcium leaching of cementitious materials, based on a phenomenological dissolution model, an electrolyte diffusion model and the spatial p‐finite element discretization, is used as a highly non‐linear model problem. A p‐finite element method is used for the spatial discretization. In the context of discontinuous Galerkin methods the semidiscrete mass balance and the continuity of the primary variables are weakly formulated within time steps and between time steps, respectively. Continuous Galerkin methods are obtained by the strong enforcement of the continuity condition as special cases. The introduction of a natural time co‐ordinate allows for the application of standard higher order temporal shape functions of the p‐Lagrange type and the well‐known Gauss–Legendre quadrature of associated time integrals. It is shown, that arbitrary order accurate integration schemes can be developed within the framework of the proposed temporal p‐Galerkin methods. Selected benchmark analyses of calcium dissolution demonstrate the robustness of these methods with respect to pronounced changes of the reaction term and non‐smooth changes of Dirichlet boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

A new boundary element technique without domain integrals for elastoplastic solids

A simple idea is proposed to solve boundary value problems for elastoplastic solids via boundary elements, namely, to use the Green's functions corresponding to both the loading and unloading branches of the tangent constitutive operator to solve for plastic and elastic regions, respectively. In this way, domain integrals are completely avoided in the boundary integral equations. Though a discretization of the region where plastic flow occurs still remains necessary to account for the inhomogeneity of plastic deformation, the elastoplastic analysis reduces, in essence, to a straightforward adaptation of techniques valid for anisotropic linear elastic constitutive equations (the loading branch of the elastoplastic constitutive operator may be viewed formally as a type of anisotropic elastic law). Numerical examples, using J₂‐flow theory with linear hardening, demonstrate that the proposed method retains all the advantages related to boundary element formulations, is stable and performs well. The method presented is for simplicity developed for the associative flow rule; however, a full derivation of Green's function and boundary integral equations is also given for the general case of non‐associative flow rule. It is shown that in the non‐associative case, a domain integral unavoidably arises in the formulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Stokes flow around cylinders in a bounded two‐dimensional domain using multipole‐accelerated boundary element methods

The multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N²) (iterative solvers) or O(N³) (gaussian elimination) to O(N log N) or O(N), where N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes flow results in Fredholm equations of the second kind, which generally produce a well‐posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the first kind, which, upon discretization, produce ill‐conditioned systems of equations. The model system here is a two‐dimensional wide‐gap couette viscometer, where particles are suspended in the fluid between the cylinders. This is a typical system that is efficiently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is sufficiently well‐conditioned to allow the use of fast multipole methods. The direct approach results in severe ill‐conditioning, to a point where application of the multipole method leads to non‐convergence of the solution iteration. Copyright © 1999 John Wiley & Sons, Ltd.     

Dynamic solution of unbounded domains using finite element method: discrete Green's functions in frequency domain

In this paper we present a new approach for finite element solution of time‐harmonic wave problems on unbounded domains. As representatives of the wave problems, discrete Green's functions are evaluated in finite element sense. The finite element mesh is considered to be of repeatable pattern (cell) constructed in rectangular co‐ordinates. The system of FE equations is therefore reduced to a set of well‐known dispersion equations by using a spectral solution approach. The spectral wave bases are constructed directly from the FE dispersion equations. Radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level. Dirichlet/Neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem. A new discrete transformation method, recently proposed by the authors, is used to satisfy the boundary conditions. Comprehensive studies are made for showing the validity, accuracy and convergence of the solutions. The results of the benchmark problems indicate that the proposed method can be used to evaluate discrete Green's functions whose analytical forms are not available. Copyright © 2006 John Wiley & Sons, Ltd.     

The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes

The finite cell method (FCM) is an immersed domain finite element method that combines higher‐order non‐boundary‐fitted meshes, weak enforcement of Dirichlet boundary conditions, and adaptive quadrature based on recursive subdivision. Because of its ability to improve the geometric resolution of intersected elements, it can be characterized as an immersogeometric method. In this paper, we extend the FCM, so far only used with Cartesian hexahedral elements, to higher‐order non‐boundary‐fitted tetrahedral meshes, based on a reformulation of the octree‐based subdivision algorithm for tetrahedral elements. We show that the resulting TetFCM scheme is fully accurate in an immersogeometric sense, that is, the solution fields achieve optimal and exponential rates of convergence for h‐refinement and p‐refinement, if the immersed geometry is resolved with sufficient accuracy. TetFCM can leverage the natural ability of tetrahedral elements for local mesh refinement in three dimensions. Its suitability for problems with sharp gradients and highly localized features is illustrated by the immersogeometric phase‐field fracture analysis of a human femur bone. Copyright © 2016 John Wiley & Sons, Ltd.     

Galerkin boundary integral analysis for the axisymmetric Laplace equation

The boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric‐Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non‐singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post‐processing of the surface gradient. Published in 2005 by John Wiley & Sons, Ltd.     

On the computation of nearly singular integrals in 3D BEM collocation

In this paper, we propose an efficient strategy to compute nearly singular integrals over planar triangles in R ³ arising in boundary element method collocation. The strategy is based on a proper use of various non‐linear transformations, which smooth or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1/r, 1/r² and 1/r³, r=∥ x ? y ∥ being the distance between a fixed near observation point x and a generic point y of a triangular element. Extensive numerical tests and comparisons with some already existing methods show that the approach proposed here is highly efficient and competitive. Copyright © 2007 John Wiley & Sons, Ltd.     

A fast multi‐level convolution boundary element method for transient diffusion problems

A new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods (BEM) for transient diffusion. This approach, which is based upon the multi‐level multi‐integration concepts of Brandt and Lubrecht, provides a fast, accurate and memory efficient time domain method for this entire class of problems. Conventional BEM approaches result in operation counts of order O(N²) for the discrete time convolution over N time steps. Here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order O(N^3/2) for three two‐dimensional model problems using the multi‐level convolution BEM. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach. Copyright © 2005 John Wiley & Sons, Ltd.     

Singular‐ended spline interpolation for two‐dimensional boundary element analysis

A method of interpolation of the boundary variables that uses spline functions associated with singular elements is presented. This method can be used in boundary element method analysis of 2‐D problems that have points where the boundary variables present singular behaviour. Singular‐ended splines based on cubic splines and Overhauser splines are developed. The former provides C²‐continuity and the latter C¹‐continuity across element edges. The potentialities of the methodology are demonstrated analysing the dynamic response of a 2‐D rigid footing interacting with a half‐space. It is shown that, for a given number of elements at the soil–foundation interface, the singular‐ended spline interpolation increases substantially the displacement convergence rate and delivers smoother traction distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements

A key challenge while employing non‐interpolatory basis functions in finite‐element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B‐spline basis functions, with application to both second‐ and fourth‐order problems. This is achieved using concepts borrowed from Nitsche's method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence. Copyright © 2010 John Wiley & Sons, Ltd.     

Convergence analysis of a hierarchical enrichment of Dirichlet boundary conditions in a mesh‐free method

Implementation of Dirichlet boundary conditions in mesh‐free methods is problematic. In Wagner and Liu (International Journal for Numerical Methods in Engineering 2001; 50 :507), a hierarchical enrichment technique is introduced that allows a simple implementation of the Dirichlet boundary conditions. In this paper, we provide some error analysis for the hierarchical enrichment mesh‐free technique. We derive optimal order error estimates for the hierarchical enrichment mesh‐free interpolants. For one‐dimensional elliptic boundary value problems, we can directly apply the interpolation error estimates to obtain error estimates for the mesh‐free solutions. For higher‐dimensional problems, derivation of error estimates for the mesh‐free solutions depends on the availability of an inverse inequality. Numerical examples in 1D and 2D are included showing the convergence behaviour of mesh‐free interpolants and mesh‐free solutions when the hierarchical enrichment mesh‐free technique is employed. Copyright © 2001 John Wiley & Sons, Ltd.     

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

In this paper, high‐order systems are reformulated as first‐order systems, which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretization with 1D‐integrated radial basis function networks (1D‐IRBFN) (Numer. Meth. Partial Differential Equations 2007; 23 :1192–1210). The present method is enhanced by a new boundary interpolation technique based on 1D‐IRBFN, which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well‐known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

Cell‐based volume integration for boundary integral analysis

The evaluation of volume integrals that arise in boundary integral formulations for non‐homogeneous problems was considered. Using the “Galerkin vector” to represent the Green's function, the volume integral was decomposed into a boundary integral, together with a volume integral wherein the source function was everywhere zero on the boundary. This new volume integral can be evaluated using a regular grid of cells covering the domain, with all cell integrals, including partial cells at the boundary, evaluated by simple linear interpolation of vertex values. For grid vertices that lie close to the boundary, the near‐singular integrals were handled by partial analytic integration. The method employed a Galerkin approximation and was presented in terms of the three‐dimensional Poisson problem. An axisymmetric formulation was also presented, and in this setting, the solution of a nonlinear problem was considered. Copyright © 2012 John Wiley & Sons, Ltd.     

A meshless BEM for isotropic heat conduction problems with heat generation and spatially varying conductivity

In this paper, a new and simple boundary‐domain integral equation is presented for heat conduction problems with heat generation and non‐homogeneous thermal conductivity. Since a normalized temperature is introduced to formulate the integral equation, temperature gradients are not involved in the domain integrals. The Green's function for the Laplace equation is used and, therefore, the derived integral equation has a unified form for different heat generations and thermal conductivities. The arising domain integrals are converted into equivalent boundary integrals using the radial integration method (RIM) by expressing the normalized temperature using a series of basis functions and polynomials in global co‐ordinates. Numerical examples are given to demonstrate the robustness of the presented method. Copyright © 2005 John Wiley & Sons, Ltd.