Stress Analysis using Boundary Elements

A finite element–boundary element method for advection–diffusion problems with variable advective fields and infinite domains

A coupled system of finite elements and boundary elements is presented for the modelling of sound propagation in a nonuniform mean flow. Finite elements are used to discretize the acoustic pressure in a bounded computational domain including the nonuniform flow field. The radiation of acoustic waves in the exterior uniform flow field is modelled by a boundary integral representation formula. The interface condition on the shared boundary yields an integral equation which is solved using boundary elements. The coupling of finite elements and boundary elements is described in detail for sound propagation in a stationary fluid where the acoustic pressure is governed by the Helmholtz equation. The coupled computational system is applied to estimate scattering and refractive effects on the acoustic pressure waves through a boundary layer surrounding an aircraft’s fuselage. The coupled mathematical formulation for three dimensional sound radiation around an engine in subsonic flow is discussed.

A direct traction BIE approach for three-dimensional crack problems

This paper reports on the development of a finite element — boundary element coupling procedure for the analysis of arbitrary shaped elastic bodies subjected to dynamic loads. The coupling is accomplished through equilibrium and compatibility considerations along the boundary element — finite element interface.

A 2D RVE formulation by the boundary element method considering phase debonding

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.

A Monte Carlo solution method for linear elasticity

Dynamic 2.5-D Green's function for a point load or a point fluid source in a layered poroelastic half-space

In the boundary element method (BEM) for stress analysis, it is well known that thermal loads give rise to an additional volume integral in the primary form of the boundary integral equation (BIE). This volume integral needs to be further transformed to surface ones in order to retain the characteristic of the BEM as a boundary solution technique. In this study of the BEM for 3D thermoelasticity in general anisotropy, the fundamental solutions are expressed as Fourier series with coefficients calculated using an explicit-form Green’s function. In the exact volume-to-surface integral transformation associated with the term for the thermal effects in the BIE, a new kernel function is constructed. All formulations are implemented in an existing BEM code for 3D elastostatic analysis. Some numerical examples are presented to demonstrate the veracity of the formulations and the implementation, where the numerical results are compared with those obtained using the finite element method (FEM).

Sensitivity of proximal femoral stiffness and areal bone mineral density to changes in bone geometry and density

Finite Element Method (FEM1) is pervasively used in most of 3D product design analysis, in which Computer Aided Design (CAD) models need to be converted in to mesh models first and then enriched with some material features and boundary conditions data, etc. The interaction between CAD models and FEM models is intensive. Boundary Element Method (BEM) has been expected to be advantageous in large-scale problems in recent years owing to its reduction of the dimensionality and its reduced complexity in mesh generation. However, the BEM application has so far been limited to relatively small problems due to the memory and computational complexity for matrix buildup are O(N2). The fast multipole BEM (FMBEM) combined with BEM and fast multipole method (FMM) can overcome the defect of the traditional BEM, and provides an effective method to solve the large-scale problem. Combining GPU parallel computing with FMBEM can further improve its efficiency significantly. Based on the three-dimensional elastic mechanics problems, the parallelisms of the multipole moment (ME), multipole moment to multipole moment (M2M) translation, multipole moment to local expansion (M2L) translation, local expansion to local expansion (L2L) translation and near-field direct calculation were analyzed respectively according to the characteristics of the FMM, and the parallel strategies under CUDA were presented in this paper. Three main major parts are included herein: (1) FMBEM theory in 3D elastostatics, (2) the parallel FMBEM algorithm using CUDA, and (3) comparison the GPU parallel FMBEM with BEM, FEM and FMBEM respectively by engineering examples. Numerical example results show the 3D elastostatics GPU FMBEM not only can speed up the boundary element calculation process, but also save memory which can be effective to solve the large-scale engineering problems.

8 - Open boundary problems

The use of modal methods to determine acoustic transient and harmonic radiation and scattering from fluid loaded shells of revolution will be reviewed. Using such methods the velocity field of a fluid loaded shell is first expressed as an invacuo modal vector expansion where the modes are determined either analytically for simple structures or via finite element methods for more complex structures. Boundary element or other methods are subsequently used to evaluate mode-dependent acoustic radiation impedances and impulse responses which account for the coupling of the fluid. In contrast to the harmonic problem which results in sets of algebraic equations for the coefficients of the modal expansion, the transient problem results in sets of coupled time-dependent convolution integral equations for the time-dependent coefficients. Our experience in solving both two- and three-dimensional radiation and scattering problems using modal methods will be reviewed. Numerical results for a wide variety of two- and three-dimensional radiation and scattering problems will be presented. In particular results will be presented and discussed for cylindrical, spherical, and spheroidal shells.

The boundary element method has evolved rapidly within the past decade and is now recognized as a reliable and efficient alternative to finite element and finite difference procedures, especially for problems encountered in Potential Theory and Elasto-statics. The technique consists in transforming the partial differential equations which govern the behavior in the domain over to a set of integral equations relating quantities associated with the boundary, and applying numerical procedures to generate approximate solutions for the boundary variables. This paper describes a general procedure, based on the weighted residual approach, for transforming the differential equations to integral expressions that form the basis for the Direct Boundary Element Method, and shows that boundary element and finite element methods can be interpreted as variants of the fundamental weak formulation for the problem.

In this study, a combined finite element (FE)–boundary element (BE) method is presented to investigate the dynamic characteristics of shell and plate structures in contact with fluid. The numerical procedure consists of two parts. In the first part, the dynamic characteristics of structures are obtained under in-vacuo condition by using the finite element method. Then, in the second part, fluid-structure interaction effects are computed in terms of generalized added mass coefficients by using the boundary element method. In analyses, surrounding fluid is assumed ideal, i.e. inviscid, incompressible and irrotational, in the context of linear hydroelasticity theory. In order to show the applicability of the proposed method, the dynamic characteristics of two different structures—a vertical rectangular plate in contact with fluid on one side and a horizontal cylindrical shell partially filled with fluid—are investigated and compared with the results obtained with a commercial software, ANSYS.