A parallel multipolar indirect boundary element method for the Neumann interior Stokes flow problem

A multipolar expansion technique is applied to the indirect formulation of the boundary element method in order to solve the two‐dimensional internal Stokes flow second kind boundary value problems. The algorithm is based on a multipolar expansion for the far field and numerical evaluation for the near field. Due to the nature of the algorithm, it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations. A parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvements. Copyright © 2000 John Wiley & Sons, Ltd.     

A boundary element numerical scheme for the two‐dimensional convection–diffusion equation

In this study, the theoretical and numerical fundamentals of BIEM techniques for the two‐dimensional convection–diffusion problem are presented. After an extended presentation of the basic integral formulation, the discretizing and iterative processes for its resolution are introduced. Interesting remarks on general expressions versus previously published particularized results are worth mentioning. A numerical solution scheme is provided, which has been completely developed and designed to the physical problem posed. A novel scheme based in the simultaneous solving of the potential and the gradient of the potential boundary integral equations is included. A diversity of problems is tested to prove the possibilities of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Compression of time‐generated matrices in two‐dimensional time‐domain elastodynamic BEM analysis

This paper describes a new scheme to improve the efficiency of time‐domain BEM algorithms. The discussion is focused on the two‐dimensional elastodynamic formulation, however, the ideas presented apply equally to any step‐by‐step convolution based algorithm whose kernels decay with time increase. The algorithm presented interpolates the time‐domain matrices generated along the time‐stepping process, for time‐steps sufficiently far from the current time. Two interpolation procedures are considered here (a large number of alternative approaches is possible): Chebyshev–Lagrange polynomials and linear. A criterion to indicate the discrete time at which interpolation should start is proposed. Two numerical examples and conclusions are presented at the end of the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

A 2‐D time‐domain boundary element method with damping

A new material damping model which is convenient for use in the time‐domain boundary element method (TDBEM) is presented and implemented in a proposed procedure. Since only fundamental solutions for linear elastic material are employed, the procedure has high efficiency and is easy to be integrated into current TDBEM codes. Analytical and numerical results for benchmark problems demonstrate that the accuracy of the proposed method is high. Copyright © 2001 John Wiley & Sons, Ltd.     

The extended finite element method for rigid particles in Stokes flow

A new method for the simulation of particulate flows, based on the extended finite element method (X‐FEM), is described. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment, allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh. We present a weak form of the equations of motion useful for the simulation of freely moving particles, and solve example problems for particles with prescribed and unknown velocities. Copyright © 2001 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.     

Singularities in anisotropic steady‐state heat conduction using a boundary element method

In many heat conduction problems, boundaries with sharp corners or abrupt changes in the boundary conditions give rise to singularities of various types which tend to slow down the rate of convergence with decreasing mesh size of any standard numerical method used for obtaining the solution. In this paper, it is shown how this difficulty may be overcome in the case of an anisotropic medium by a modified boundary element method. The standard boundary element method is modified to take account of the form of the singularity, without appreciably increasing the amount of computation involved. Two test examples, the first with a singularity caused by an abrupt change in a boundary condition and the second with a singularity caused by a sharp re‐entrant corner, are investigated and numerical results are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

Implementation of regularized isogeometric boundary element methods for gradient‐based shape optimization in two‐dimensional linear elasticity

The present work addresses shape sensitivity analysis and optimization in two‐dimensional elasticity with a regularized isogeometric boundary element method (IGABEM). Non‐uniform rational B‐splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations. With the advantage of tight integration of design and analysis, the application of IGABEM in shape optimization reduces the mesh generation/regeneration burden greatly. The work is distinct from the previous literatures in IGABEM shape optimization mainly in two aspects: (1) the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations, eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives, which were the main implementation difficulty in IGABEM, and (2) although based on the same Computer Aided Design (CAD) model, the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh, thus achieving a balance between less design variables for efficiency and refined mesh for accuracy. This technique was initially used in isogeometric finite element method and was incorporated into the present IGABEM implementation. Copyright © 2016 John Wiley & Sons, Ltd.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

A two‐scale approach for fluid flow in fractured porous media

A two‐scale numerical model is developed for fluid flow in fractured, deforming porous media. At the microscale the flow in the cavity of a fracture is modelled as a viscous fluid. From the micromechanics of the flow in the cavity, coupling equations are derived for the momentum and the mass couplings to the equations for a fluid‐saturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two‐scale approach and integrated over time. By exploiting the partition‐of‐unity property of the finite element shape functions, the position and direction of the fractures is independent from the underlying discretization. The resulting discrete equations are non‐linear due to the non‐linearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach, and show that faults in a deforming porous medium can have a significant effect on the local as well as on the overall flow and deformation patterns. Copyright © 2006 John Wiley & Sons, Ltd.     

Stokes flow around an asymmetric channel divider; a computational approach using MATLAB

While computational methods for solving Stokes-flow problems have existed for some time, these have depended on specialized codes developed specifically, for this type of problem. This work shows how to combine traditional applied mathematics and a modern over-the-counter software package Matlab to solve and study Stokes flow in a channel with a splitter plate. Specifically exact unidirectional flow solutions are used as a basis for choosing boundary conditions for Matlab to anticipate the boundary conditions of a Stokes flow. A method for selecting zeroth and first-order approximate boundary conditions is presented, along with a suggestion for finding a second-order approximation. It is also shown that small errors made in choosing the approximate boundary conditions do not grow as one moves away from the boundary into the interior of the flow. Finally several computational examples using this approach are presented.     

A sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals

A new transformation technique is introduced for evaluating the two‐dimensional nearly singular integrals, which arise in the solution of Laplace's equation in three dimensions, using the boundary element method, when the source point is very close to the element of integration. The integrals are evaluated using (in a product fashion) a transformation which has recently been used to evaluate one‐dimensional near singular integrals. This sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element. The method is straightforward to implement and, when it is compared with a number of existing techniques for evaluating two‐dimensional near singular integrals, it is found that the sinh method is superior to the existing methods considered, both for potential integrals across the full range of b values considered (0<b?10), and for flux integrals where b>0.01. For smaller values of b, the use of the Lmethod is recommended for flux integrals. Copyright © 2006 John Wiley & Sons, Ltd.     

Coupling of finite element and boundary integral methods for a capsule in a Stokes flow

We introduce a new numerical method to model the fluid–structure interaction between a microcapsule and an external flow. An explicit finite element method is used to model the large deformation of the capsule wall, which is treated as a bidimensional hyperelastic membrane. It is coupled with a boundary integral method to solve for the internal and external Stokes flows. Our results are compared with previous studies in two classical test cases: a capsule in a simple shear flow and in a planar hyperbolic flow. The method is found to be numerically stable, even when the membrane undergoes in‐plane compression, which had been shown to be a destabilizing factor for other methods. The results are in very good agreement with the literature. When the viscous forces are increased with respect to the membrane elastic forces, three regimes are found for both flow cases. Our method allows a precise characterization of the critical parameters governing the transitions. Copyright © 2010 John Wiley & Sons, Ltd.     

An adaptive domain decomposition coupled finite element–boundary element method for solving problems in elasto‐plasticity

The purpose of this paper is to present an adaptive finite element–boundary element method (FEM–BEM) coupling method that is valid for both two‐ and three‐dimensional elasto‐plastic analyses. The method takes care of the evolution of the elastic and plastic regions. It eliminates the cumbersome of a trial and error process in the identification of the FEM and BEM sub‐domains in the standard FEM–BEM coupling approaches. The method estimates the FEM and BEM sub‐domains and automatically generates/adapts the FEM and BEM meshes/sub‐domains, according to the state of computation. The results for two‐ and three‐dimensional applications in elasto‐plasticity show the practicality and the efficiency of the adaptive FEM–BEM coupling method. Copyright © 2009 John Wiley & Sons, Ltd.     

A fast hierarchical dual boundary element method for three‐dimensional elastodynamic crack problems

In this work a fast solver for large‐scale three‐dimensional elastodynamic crack problems is presented, implemented, and tested. The dual boundary element method in the Laplace transform domain is used for the accurate dynamic analysis of cracked bodies. The fast solution procedure is based on the use of hierarchical matrices for the representation of the collocation matrix for each computed value of the Laplace parameter. An ACA (adaptive cross approximation) algorithm is used for the population of the low rank blocks and its performance at varying Laplace parameters is investigated. A preconditioned GMRES is used for the solution of the resulting algebraic system of equations. The preconditioners are built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. An original strategy, based on the computation of some local preconditioners only, is presented and tested to further speed up the overall analysis. The reported numerical results demonstrate the effectiveness of the technique for both uncracked and cracked solids and show significant reductions in terms of both memory storage and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

Solving the steady‐state groundwater flow equation for finite linear aquifers using a generalized Fourier series approach in two‐dimensional domains

A method to solve steady linear groundwater flow problems using generalized Fourier Series is developed and particularized for multiple Fourier series in two‐dimensional domains. It leads to a linear vector equation whose solution provides a finite number of generalized Fourier coefficients approximating the hydraulic head field. Its implementation is shown and two relevant properties are found for the system matrix. It is always symmetric and, once computed, if additional Fourier terms are needed for a better approximation of the hydraulic head field, previously computed matrix elements remain invariant, i.e. only new rows and columns are added to the system matrix. The method is demonstrated in three simple cases with different geometries and transmissivity fields, where solutions are compared with analytical and finite element method results. Thus, the method is verified as an alternative to other flow solvers. Additionally, it provides a direct way to obtain the spectral form of the flow equation solution, given a spectral representation of transmissivity, and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions. Copyright © 2009 John Wiley & Sons, Ltd.     

Rayleigh wave correction for the BEM analysis of two‐dimensional elastodynamic problems in a half‐space

A simple, elegant approach is proposed to correct the error introduced by the truncation of the infinite boundary in the BEM modelling of two‐dimensional wave propagation problems in elastic half‐spaces. The proposed method exploits the knowledge of the far‐field asymptotic behaviour of the solution to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocal theorem of elastodynamics is used for a convenient computation of this contribution involving the same boundary integrals that form the original BEM system. The method is formulated for a two‐dimensional homogeneous, isotropic, linearly elastic half‐space and its implementation in a frequency domain boundary element scheme is discussed in some detail. The formulation is then validated for a free Rayleigh pulse travelling on a half‐space and successfully tested for a benchmark problem with a known approximation to the analytical solution. Copyright © 2004 John Wiley & Sons, Ltd.     

A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

This article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two‐dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low‐frequency FMM and the high‐frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low‐frequency FMM and the quadrature order for the high‐frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton–Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, Nlog N, where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite‐size phononic crystal was demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.