Symmetric coupling of multi‐zone curved Galerkin boundary elements with finite elements in elasticity

The coupling of Finite Element Method (FEM) with a Boundary Element Method (BEM) is a desirable result that exploits the advantages of each. This paper examines the efficient symmetric coupling of a Symmetric Galerkin Multi‐zone Curved Boundary Element Analysis method with a Finite Element Method for 2‐D elastic problems. Existing collocation based multi‐zone boundary element methods are not symmetric. Thus, when they are coupled with FEM, it is very difficult to achieve symmetry, increasing the computational work to solve the problem. This paper uses a fully Symmetric curved Multi‐zone Galerkin Boundary Element Approach that is coupled to an FEM in a completely symmetric fashion. The symmetry is achieved by symmetrically converting the boundary zones into equivalent ‘macro finite elements’, that are symmetric, so that symmetry in the coupling is retained. This computationally efficient and fast approach can be used to solve a wide range of problems, although only 2‐D elastic problems are shown. Three elasticity problems, including one from the FEM‐BEM literature that explore the efficacy of the approach are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Symmetric Galerkin boundary integral formulation for interface and multi-zone problems

Domains containing an ‘internal boundary’, such as a bi-material interface, arise in many applications, e.g. composite materials and geophysical simulations. This paper presents a symmetric Galerkin boundary integral method for this important class of problems. In this situation, the physical quantities are known to satisfy continuity conditions across the interface, but no boundary conditions are specified. The algorithm described herein achieves a symmetric matrix of reduced size. Moreover, the symmetry can also be invoked to lessen the numerical work involved in constructing the system of equations, and thus the method is computationally very efficient. A prototype numerical example, with several variations in the boundary conditions and material properties, is employed to validate the formulation and corresponding numerical procedure. The boundary element results are compared with analytical solutions and with numerical results obtained with the finite element method. © 1997 John Wiley & Sons, Ltd.     

Symmetric Galerkin boundary integral fracture analysis for plane orthotropic elasticity

This paper discusses the formulation and implementation of the symmetric Galerkin boundary integral method for two dimensional linear elastic orthotropic fracture analysis. For the usual case of a traction-free crack, the symmetry of the coefficient matrix can be effectively exploited to significantly reduce the computational work required to construct the linear system. In addition, computation time is reduced by employing efficient analytic integration formulas for the analysis of the orthotropic singular and hypersingular integrals. Preliminary test calculations indicate that the method is both accurate and efficient.     

Symmetric Galerkin boundary element method in plasticity and gradient plasticity

A boundary integral equation (BIE) formulation and a boundary element (BE) method which confer symmetry to key operators are concisely described with reference to quasi-static plasticity. This formulation is based on the combined use of static and kinematic sources, on Galerkin weighted residual enforcement of integral equations for displacements and tractions along the boundary and for stresses in the potentially yielding domain and on space discretizations in terms of generalized variables in Prager's sense. Typical theoretical results of computational interest not available in conventional nonsymmetric BE methods are surveyed. The subjectivity (mesh-dependence) implied by material instability is illustrated by examples. As a remedy, a symmetric BIE-BE formulation for nonlocal, gradient-dependent plasticity is developed and discussed on the basis of its variationally consistent discretization.Extended version of a key-note lecture at the 4th International Conference on Computational Plasticity, Barcelona, April 3–6, 1995.Dedicated to J. C. Simo     

Symmetric Galerkin boundary element method for quasi-brittle-fracture and frictional contact problems

The analysis of elastic quasi-brittle structures containing cohesive cracks and contacts with friction is given a unitary formulation in the framework of incremental plasticity. Integral equations for displacements and tractions are enforced by a weighted-residual Galerkin approach so that symmetry is preserved in the key operators (in contrast to collocation BE approaches) and cracks (either internal or edge cracks) can be dealt with by a single-domain BE formulation. The space-discrete problem in rates is expressed as a linear complementarity problem centered on a symmetric matrix or, equivalently, as a quadratic programming problem in variables pertaining to the displacement discontinuity locus only. Criteria for overall instabilities and bifurcations are derived from this formulation. The BE approach proposed and implemented by a suitable time-stepping technique, is comparatively tested by numerical solutions of cohesive-crack propagation problems.     

Numerical transfer-method with boundary conditions for arbitrary curved beam elements

A novel numerical transfer-method is presented to solve a system of linear ordinary differential equations with boundary conditions. It is applied to determine the structural behaviour of the classical problem of an arbitrary curved beam element. The approach of this boundary value problem yields a unique system of differential equations. A Runge–Kutta scheme is chosen to obtain the incremental transfer expression. The use of a recurrence strategy in this equation permits to relate both ends in the domain where boundary conditions are defined. Semicircular arch, semicircular balcony and elliptic–helical beam examples are provided for validation.     

Apparent stiffness tensors for porous solids using symmetric Galerkin boundary elements

Representative volume elements (RVEs) from porous or cellular solids can often be too large for numerical or experimental determination of effective elastic constants. Volume elements which are smaller than the RVE can be useful in extracting apparent elastic stiffness tensors which provide bounds on the homogenized elastic stiffness tensor. Here, we make efficient use of boundary element analysis to compute the volume averages of stress and strain needed for such an analysis. For boundary conditions which satisfy the Hill criterion, we demonstrate the extraction of apparent elastic stiffness tensors using a symmetric Galerkin boundary element method. We apply the analysis method to two examples of a porous ceramic. Finally, we extract the eigenvalues of the fabric tensor for the example problem and provide predictions on the apparent elastic stiffnesses as a function of solid volume fraction.     

Symmetric BE method in two-dimensional elasticity: evaluation of double integrals for curved elements

A procedure for the evaluation of the double integrals involved in the symmetric BE method is proposed, which consists in a regularization via integration by parts and in a successive numerical evaluation of weakly singular integrals. The procedure is applicable to curved elements and higher order shape functions and lends itself to a fairly straightforward implementation. Continuity is required for displacement modelling, while tractions may be modelled as piece-wise continuous. Some example problems characterized by the presence of curved boundaries or of cracks are analyzed using the proposed technique to demonstrate its effectiveness. This study is part of a research project supported by CNR (Italian National Research Council)     

A symmetric Galerkin BEM implementation for 3D elastostatic problems with an extension to curved elements

 Like the finite element method (FEM), the symmetric Galerkin boundary element method (SGBEM) can produce symmetric system matrices. While widely developed for two dimensional problems, the 3D-applications of the SGBEM are very rare. This paper deals with the regularization of the singular integrals in the case of 3D elastostatic problems. It is shown that the integration formulas can be extended to curved elements. In contrast to other techniques, the Kelvin fundamental solutions are used with no need to introduce the new kernel functions. The accuracy of the developed integration formulas is verified on a problem with known analytical solution. Received 6 November 2000     

Symmetric Galerkin BEM for shear deformable plates

A symmetric Galerkin boundary element formulation is developed for shear deformable plates. A mixed strategy is used for the integration process, i.e. partial regularization using simple solutions followed by a singularity subtraction technique. For the shear equation, full regularization is achieved using new kernel relationships found through a constant shear mode of deformation. Some of the strong singular integrals are avoided altogether by using a modified traction obtained through a very simple variable change; appropriate boundary conditions are defined. Details of the implementation are given and several example problems solved to verify the accuracy of the proposed formulation. Copyright © 2003 John Wiley & Sons, Ltd.     

Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

The solution of three-dimensional elastostatic problems using the Symmetric Galerkin Boundary Element Method (SGBEM) gives rise to fully populated (albeit symmetric) matrix equations, entailing high solution times for large models. This paper is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for elastic solid with cracks. Arbitrary geometries and boundary conditions may be considered. Numerical results on test problems involving a cube, single or multiple cracks in an unbounded medium, and a cracked cylindrical solid are presented. BEM models involving up to 10⁶ BEM unknowns are considered, and the desirable predicted trends of the elastostatic FM-SGBEM, such as a O(N) complexity per iteration, are verified.     

High-order curved finite elements

Recent finite element analysis of adaptive refinement has focused attention on high-order approximation within elements. High-order approximation may be attained over curved elements with either rational basis functions expressed directly in the global co-ordinates or with a particular class of isoparametric basis functions. Theory for construction of high-order rational bases yields modifications to the blending method for a special class of elements in order to achieve higher order approximation within these elements.     

A symmetric Galerkin multi-zone boundary element formulation

The recent development of the symmetric Galerkin approach to boundary element analysis (BEA) has been demonstrated to be superior to the collocation method for medium to large problems. This fact has been shown in both heat conduction and elasticity. Accounts of collocation multi-zone analysis techniques have also been prevalent in the literature, dealing with multiple boundary integral relations associated with portions of overall objects. This technique results in overall system matrices with a blocked, sparse, but unsymmetric character. It has been shown that multi-zone techniques can produce smaller solution times than a single zone analysis for large problems. These techniques are useful for multi-material problems as well. This paper presents an approach for combining the benefits of both techniques resulting in a Galerkin multi-zone method, that is overall unsymmetric but contains a significant amount of block symmetry. A condensation technique in the multi-zone solver is shown to exploit the symmetry of the Galerkin formulation as well as the blocked sparsity of the multi-zone technique. This method is compared to collocation multi-zone on two elasticity problems from the literature. It is concluded that an appropriate implementation of the symmetric Galerkin multi-zone BEA indeed has the potential to be superior to the collocation based multi-zone BEA, for medium to large-scale elasticity problems. © 1997 John Wiley & Sons, Ltd.     

Infinite boundary elements

Special types of boundary elements are discussed which can be used for the modelling of surfaces which extend to infinity. The theoretical background and details of implementation are discussed. On test examples it is shown that the elements perform extremely well even for cases in which they are located close to the area of interest. A practical application of the use of the elements for the modelling of mining excavations is given.     

Spectral properties of Galerkin boundary element matrices

The motivation for the present article is to review some key features of the Symmetric Boundary Element Method from the point of view of the algebraic properties of the matrices arising from the Galerkin discretization of the displacement- and traction-Somigliana identities. The focus is on showing which features of these linear pseudo-differential operators are preserved and which are lost due to discretization.     

Forbidden nodes for curved finite elements

It is shown here that certain interpolating polynomials of degrees four and five may not always be uniquely defined on triangular-shaped elements which have one curved side. Conditions which indicate non-uniqueness are given, together with some geometrical interpretations concerning the location of the node on the curved side. A numerical example is given to demonstrate that there ar curves for which every point is unsuitable to be chosen as a node.     

Modified meshless local Petrov–Galerkin formulations for elastodynamics

This paper is concerned with the extension to the elastodynamic problems of the idea of analytical integrations in meshless local weak formulations. In this context, Taylor series expansions of the incognita fields are considered, and the related integrals of the meshless formulations are solved analytically, rendering a so‐called modified methodology. The moving least squares approximation is employed for the spatial variation of the displacement fields, and two variants of the meshless local Petrov–Galerkin (MLPG) formulation are discussed here, which are based on the use of Heaviside or Gaussian weight test functions. Once the spatial discretization is considered, the semi‐discretized ordinary differential equations for nodal unknowns that arise are treated in the time‐domain by the Houbolt's method. Considering the modified meshless methodology, it was found that the standard derivatives of the shape functions must also be modified to achieve numerically stable procedures, taking into account dynamic analyses. Thus, an efficient and easy‐to‐implement technique is developed to properly compute the shape function derivatives. As described in the paper, the proposed modified MLPG formulations are more effective than standard MLPG formulations, especially taking into account well refined large scale problems, providing much better computational efficiency, good accuracy and more robust convergence. Copyright © 2012 John Wiley & Sons, Ltd.