Comparison of boundary and finite element methods for moving-boundary problems governed by a potential

Three formulations of the boundary element method (BEM) and one of the Galerkin finite element method (FEM) are compared according to accuracy and efficiency for the spatial discretization of two-dimensional, moving-boundary problems based on Laplace's equation. The same Euler-predictor, trapezoid-corrector scheme for time integration is used for all four methods. The model problems are on either a bounded or a semi-infinite strip and are formulated so that closed-form solutions are known. Infinite elements are used with both the BEM and FEM techniques for the unbounded domain. For problems with the bounded region, the BEM using the free-space Green's function and piecewise quadratic interpolating functions (QBEM) is more accurate and efficient than the BEM with linear interpolation. However, the FEM with biquadratic basis functions is more efficient for a given accuracy requirement than the QBEM, except when very high accuracy is demanded. For the unbounded domain, the preferred method is the BEM based on a Green's function that satisfies the lateral symmetry conditions and which leads to discretization of the potential only along the moving surface. This last formulation is the only one that reliably satisfies the far-field boundary condition.     

An accurate scalar potential finite element method for linear,two-dimensional magnetostatics problems

We have assessed the accuracy of a commercially available computer software package for finite element method calculations of magnetostatic fields. The computer program, MSC/NASTRAN,

1 Available from the MacNeal-Schwendler Corporation, Los Angeles, CA 90041, U.S.A.

is well known for its wide applicability in structural analysis and heat transfer problems. We exploit the fact that the differential equations of magnetostatics are identical to those for heat transfer if the magnetic field problem is formulated with the reduced scalar potential.¹ Consequently, the powerful, optimized numerical routines of NASTRAN can immediately be applied to two- and three-dimensional linear magneto-statics problems. Application of the NASTRAN reduced scalar potential approach to a ‘worst case’ two-dimensional problem for which an analytic solution is available has yielded much better accuracy than was recently reported² for a reduced scalar potential calculation using a different finite element program. Furthermore, our method exhibits completely satisfactory performance with regard to computational expense and accuracy for a linear electromagnet with an air gap. Our analysis opens the way for large three-dimensional magnetostatics calculations at far greater economy than is possible with the more commonly used vector potential and boundary integral methods.     

Isoparametric finite element methods for two-dimensional transport calculations

Finite element methods for solving the two-dimensional x–y transport equation are considered. Numerical schemes are defined and tested on some simple practical problems.     

Galerkin finite element methods for wave problems

We compare here the accuracy, stability and wave propagation properties of a few Galerkin methods. The basic Galerkin methods with piecewise linear basis functions (called G1FEM here) and quadratic basis functions (called G2FEM) have been compared with the streamwise-upwind Petrov Galerkin (SUPG) method for their ability to solve wave problems. It is shown here that when the piecewise linear basis functions are replaced by quadratic polynomials, the stencils become much larger (involving five overlapping elements), with only a very small increase in spectral accuracy. It is also shown that all the three Galerkin methods have restricted ranges of wave numbers and circular frequencies over which the numerical dispersion relation matches with the physical dispersion relation — a central requirement for wave problems. The model one-dimensional convection equation is solved with a very fine uniform grid to show the above properties. With the help of discontinuous initial condition, we also investigate the Gibbs’ phenomenon for these methods.     

Hermitian cubic boundary elements for two-dimensional potential problems

A hermite interpolation based formulation is presented for the boundary element analysis of two-dimensional potential problems. Two three-noded Hermitian Cubic Elements (HCE) are introduced for the modelling of corners or points with non-unique tangents on the boundary. These elements, along with the usual two-noded HCE, are used in numerical examples. The results obtained show that faster convergence can be achieved using HCE compared with using Lagrange interpolation type Quadratic Elements (QE), for about the same amount of computing resources.     

A boundary element approach for non-homogeneous potential problems

This paper reports an implementation of a Boundary Element Method dealing with two-dimensional inhomogeneous potential problems. This method avoids the tedious calculation of the domain integral contributions to the boundary integral equations. This is achieved by applying approximate particular solutions which are obtained by expressing the source distribution in terms of a linear combination of radial basis functions. Numerical examples show that the method is efficient and can produce accurate results.     

A boundary element alternating method for two-dimensional mixed-mode fracture problems

A boundary element alternating method (BEAM) is presented for two dimensional fracture problems. An analytical solution for arbitrary polynomial normal and tangential pressure distributions applied to the crack faces of an embedded crack in an infinite plate is used as the fundamental solution in the alternating method. For the numerical part of the method the boundary element method is used. For problems of edge cracks a technique of utilizing finite elements with BEAM is presented to overcome the inherent singularity in boundary element stress calculation near the boundaries. Several computational aspects that make the algorithm efficient are presented. Finally the BEAM is applied to a variety of two-dimensional crack problems with different configurations and loadings to assess the validity of the method. The method gave accurate stress-intensity factors with minimal computing effort.     

A conforming finite element with internal equilibrium for two-dimensional problems

A method is presented for the derivation of displacement fields which satisfy compatibility of normal slopes at inter-element boundaries and the condition of internal equilibrium. The displacement field is obtained by integrating the governing partial differential equation over the element, using the initial value approach. The method is applied to calculate the stiffness matrix of a triangular bending element with nine degrees of freedom. Comparison with published solutions based on various element models shows a high degree of accuracy and convergence of the method. The advantage of internal equilibrium is illustrated by an example involving slab-column interaction. The solution correlates satisfactorily with existing experimental results.     

New complex Fourier shape functions for the analysis of two-dimensional potential problems using boundary element method

In this paper, boundary element analysis for two-dimensional potential problems is investigated. In this study, the boundary element method (BEM) is reconsidered by proposing new shape functions to approximate the potentials and fluxes. These new shape functions, called complex Fourier shape function, are derived from complex Fourier radial basis function (RBF) in the form of exp(iωr). The proposed shape functions may easily satisfy various functions such as trigonometric, exponential, and polynomial functions. In order to illustrate the validity and accuracy of the present study, several numerical examples are examined and compared to the results of analytical and with those obtained by classic real Lagrange shape functions. Compared to the classic real Lagrange shape functions, the proposed complex Fourier shape functions show much more accurate results.     

An infinite boundary element formulation for three-dimensional potential problems

An infinite boundary element (IBE) is presented for the analysis of three-dimensional potential problems in an unbounded medium. The IBE formulations are done to allow their coupling with the finite element (FE) matrices for finite domains and to obtain the overall matrices without destroying the banded structure of the FE matrices. The infinite body is divided into a number of zones whose contributions are expressed in terms of the nodal quantities at FE nodes by employing suitable decay functions and performing mainly analytical integrations of the boundary element kernels. The continuity and compatibility conditions for the potential and the flux at the FE-IBE interface are developed. The relationships for the contributions of the IBE flux vectors to the FE load vectors are given. The final equations for the IBE are obtained in the usual FE stiffness-load vector form and are easily assembled with the FE matrices for the finite object. A series of numerical examples in heat transfer and electromagnetics were solved and compared with alternative solutions to demonstrate the validity of the present formulations.     

Space adaptive finite element methods for dynamic Signorini problems

Space adaptive techniques for dynamic Signorini problems are discussed. For discretisation, the Newmark method in time and low order finite elements in space are used. For the global discretisation error in space, an a posteriori error estimate is derived on the basis of the semi-discrete problem in mixed form. This approach relies on an auxiliary problem, which takes the form of a variational equation. An adaptive method based on the estimate is applied to improve the finite element approximation. Numerical results illustrate the performance of the presented method. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

An application of the finite element technique to boundary value problems of potential flow

This report summarizes an experimental application of the finite element approach to two-dimensional inviscid fluid flow. The method results in a matrix equation relating the vector of velocities at the nodal points with the vector of singularities. The singularities, which are concentrated at the nodes, consist of a source and a vortex. On solution of this equation (defining the flow in the region), boundary conditions must te stipulated. This usually involves setting the internal singularities to zero and making certain modifications ro those lying on the boundary of the region. Some results of the application to a particular problem are included in the paper.     

A coupled cubic hermite finite element/boundary element procedure for electrocardiographic problems

The problem considered is that of trying to determine the potential distribution inside a human torso as a result of the heart's electrical activity. We describe here a high order (cubic Hermite) coupled finite element/boundary element procedure for solving such electrocardiographic potential problems inside an anatomically accurate human torso. Details of the cubic Hermite boundary element procedure and its coupling to the finite element method are described. We then present two and three dimensional test results showing the success, efficiency and accuracy of this high order coupled technique. Some initial results on an anatomically accurate torso are also given.     

Exact integration in the boundary element method for two-dimensional elastostatic problems

This paper derives the exact integrations for the integrals in the boundary element analysis of two-dimensional elastostatics. For facilitation, the derivation is based on the simple forms of the fundamental functions by taking constant, discontinuous linear and discontinuous quadratic elements as examples. The efficiency and accuracy of the derived exact integrations are verified against five benchmark problems; the results indicate that the derived exact integrations significantly reduces the CPU time for forming the matrices of the boundary element analysis and solving the internal displacements.     

A fast multipole boundary element method for solving two-dimensional thermoelasticity problems

A fast multipole boundary element method (BEM) for solving general uncoupled steady-state thermoelasticity problems in two dimensions is presented in this paper. The fast multipole BEM is developed to handle the thermal term in the thermoelasticity boundary integral equation involving temperature and heat flux distributions on the boundary of the problem domain. Fast multipole expansions, local expansions and related translations for the thermal term are derived using complex variables. Several numerical examples are presented to show the accuracy and effectiveness of the developed fast multipole BEM in calculating the displacement and stress fields for 2-D elastic bodies under various thermal loads, including thin structure domains that are difficult to mesh using the finite element method (FEM). The BEM results using constant elements are found to be accurate compared with the analytical solutions, and the accuracy of the BEM results is found to be comparable to that of the FEM with linear elements. In addition, the BEM offers the ease of use in generating the mesh for a thin structure domain or a domain with complicated geometry, such as a perforated plate with randomly distributed holes for which the FEM fails to provide an adequate mesh. These results clearly demonstrate the potential of the developed fast multipole BEM for solving 2-D thermoelasticity problems.     

An inverse finite element method for directly formulated free boundary problems

In free boundary problems, in addition to the primary field variables, velocity, pressure and temperature, unknown geometric parameters need to be evaluated. A characteristic feature of the problems is a demarcation line which separates two domains with different material properties. The problems are highly non-linear and difficult to handle computationally. Here an accurate and efficient method is presented which can appropriately handle the complexities associated with these problems. The method is based on a tesselation that is constructed by isolines as characteristic co-ordinate lines. Thus opposite sides of finite elements lie on isolines. The method allows the simultaneous determination of the location of the isolines with the primary unknowns. Its accuracy and efficiency is demonstrated in a number of one and two dimensional steady and unsteady model problems.     

Adaptive hp-versions of boundary element methods for elastic contact problems

The variational formulation of elastic contact problems leads to variational inequalities on convex subsets. These variational inequalities are solved with the boundary element method (BEM) by making use of the Poincaré–Steklov operator. This operator can be represented in its discretized form by the Schur-complement of the dense Galerkin-matrices for the single layer potential operator, the double layer potential operator and the hypersingular integral operator. Due to the difficulties in discretizing the convex subsets involved, traditionally only the h-version is used for discretization. Recently, p- and hp-versions have been introduced for Signorini contact problems in Maischak and Stephan (Appl Numer Math, 2005) . In this paper we show convergence for the quasi-uniform hp-version of BEM for elastic contact problems, and derive a-posteriori error estimates together with error indicators for adaptive hp-algorithms. We present corresponding numerical experiments.     

Interpolation functions in the immersed boundary and finite element methods

In this paper, we review the existing interpolation functions and introduce a finite element interpolation function to be used in the immersed boundary and finite element methods. This straightforward finite element interpolation function for unstructured grids enables us to obtain a sharper interface that yields more accurate interfacial solutions. The solution accuracy is compared with the existing interpolation functions such as the discretized Dirac delta function and the reproducing kernel interpolation function. The finite element shape function is easy to implement and it naturally satisfies the reproducing condition. They are interpolated through only one element layer instead of smearing to several elements. A pressure jump is clearly captured at the fluid–solid interface. Two example problems are studied and results are compared with other numerical methods. A convergence test is thoroughly conducted for the independent fluid and solid meshes in a fluid–structure interaction system. The required mesh size ratio between the fluid and solid domains is obtained.