A Symmetric Galerkin Boundary Element Method for Linear Poroelasticity

A p-extension of the symmetric boundary element method

In this paper, the Symmetric Galerkin Boundary Element Method for Linear Elastic Fracture Mechanics is extended to non-linear cohesive cracks propagating through homogeneous linear elastic isotropic media. The cohesive model adopted is based on the concept of free energy density per unit undeformed area. The corresponding constitutive cohesive equations present a softening branch which induces a potential instability. Thus, a suitable solution algorithm capable of following the growth of the cohesive zone is needed, and in the present work the numerical simulation is controlled by an arc-length method combined with a Newton-Raphson algorithm for the iterative solution of nonlinear equations. The Boundary ElementMethod is very attractive for modeling cohesive crack problems as all nonlinearities are located on the boundaries of linear elastic domains. Moreover a Galerkin approximation scheme, applied to a suitable symmetric boundary integral equation formulation, ensures an easy and efficient treatment of cracks in homogeneous media and an excellent convergence behavior of the numerical solution. The cohesive zone model is applied to simulate a pure mode I crack propagation in concrete. Numerical results for three-point bending test are used to check the numerical results for mode I and are compared with some numerical results obtained by FEM analysis found in the literature.

A fast multilevel multipole algorithm (FMM) is derived for the Helmholtz equation and adopted to the symmetric Galerkin boundary element method (BEM) for acoustics.The FMM allows to evaluate a matrix-vector product of the BEM with a computational cost of O(N log 2 N ), thus leading to a significant reduction of computation time and memory requirements compared to standard BEM formulations.This allows the simulation of large scale acoustic models.A coupling algorithm based on Lagrange multipliers is proposed for the simulation of structure-acoustic field interaction.Finite plate elements are coupled to a Galerkin boundary element formulation of the acoustic domain.The interface pressure is interpolated as a Lagrange multiplier, thus, allowing the coupling of non-matching grids.The resulting saddle point problem is solved by an approximate Uzawa-type scheme in which the matrix-vector products of the boundary element operators are evaluated efficiently by the fast multipole boundary element method.The algorithm is demonstrated on the example of a cavity backed elastic panel.

Isogemetric analysis and symmetric Galerkin BEM: A 2D numerical study

A multipole Galerkin boundary element method for acoustics

Lower bound limit analysis by the symmetric Galerkin boundary element method and the Complex method

Dynamic crack analysis in piezoelectric solids under time-harmonic loadings with a symmetric Galerkin boundary element method

This paper describes the application of the symmetric Galerkin boundary element method (SGBEM) for the analysis of a two-dimensional linear elastic crack problem. The direct SGBEM for the multiple-crack problem is derived using the Betti reciprocal theorem, resulting in a completely symmetric matrix. The auxiliary fictitious state is used to obtain the equations. This approach can deal with any number of traction-free cracks within a single domain. Two-stage interpolation method called the ‘quasi-higher-order element method’ (QHOEM) is then proposed to solve the double integrals. In the initial stage, it uses higher-order elements to interpolate the field variables and, for the numerical integration involved, it further uses interpolation functions to decompose the higher-order elements into lower-order elements so that the existing analytical integration can be applied. Thus, it can be embedded into the present codes with ease and also reduces the computational cost. A finite rectangular plate containing a slanting crack and a kinked crack has been analysed to check the accuracy of the proposed method. A load-carrying fillet welded joint containing several cracks is then analysed, and the stress intensity factors at the crack tips are found to be in complete agreement with the dual-boundary-element method results.

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.

Boundary element methods for lower bound limit and shakedown analysis

P301 SGBEM/FEM for Non-Planar Evolution of Hydraulic Fractures in 3D Elastic Media 1 JAROON RUNGAMORNRAT MARY F. WHEELER MARK E. MEAR Center for Subsurface Modeling Institute of Computational Engineering and Sciences The University of Texas at Austin Texas USA A computational technique has been developed for simulating non-planar evolution of hydraulic fractures in three dimensional elastic media. The technique utilizes the finite element method (FEM) to model the fluid flow within the fracture while employing a symmetric Galerkin boundary element method (SGBEM) to treat the elasticity problem associated with a crack in an unbounded domain. The SGBEM is based upon

With reference to three-dimensional linear elastic solids susceptible to fracture processes, a symmetric Galerkin boundary element method is developed, based on the regularized version of the weak-form displacement and traction integral equations, and thus involving only kernels O(1/r). When the singularity is active, the numerical evaluation of the double surface integrals is carried out by using special integration schemes which exploit regularizing coordinate transformations. The performance of the method is assessed by solving some example problems involving cracks in unbounded domains and edge cracks in finite bodies.

Finite element alternating method has been suggested and used effectively to obtain the fracture parameters in assessing the integrity of cracked structures. The method obtains the solution from alternating independently between the FEM solution for an uncracked body and the crack solution in an infinite body. In the paper, the finite element alternating method is extended in order to obtain the elastic-plastic stress fields of a three dimensional inner crack. The three dimensional crack solutions for an infinite body were obtained using symmetric Galerkin boundary element method. As an example of a three dimensional inner crack, a penny-shaped crack in a finite body was analyzed and the obtained elastc-plastic stress fields were compared with the solution obtained from the finite element analysis with fine mesh. It is noted that in the region ahead of the crack front the stress values from FEAM are close to the values from FEM. But large discrepancy between two values is observed near the crack surfaces.

A numerical procedure is proposed to compute the T-stress for two-dimensional cracks in general anisotropic elastic media. T-stress is determined from the sum of crack-face displacements which are computed via an integral equation of the boundary data. To smooth out the data in order to perform accurately numerical differentiation, the sum of crack-face displacement is established in a weak-form integral equation in which the integration domain is simply the crack-tip element. This weak-form integral equation is then solved numerically using standard Galerkin approximation to obtain the nodal values of the sum of crack-face displacements. The procedure is incorporated in a weakly-singular symmetric Galerkin boundary element method in which all integral equations for the traction and displacement on the boundary of the domain and on the crack faces include (at most) weakly-singular kernels. To examine the accuracy and efficiency of the developed method, various numerical examples for cracks in infinite and finite domains are treated. It is shown that highly accurate results are obtained using relatively coarse meshes.

Weakly singular integral equations for Darcy's flow in anisotropic porous media