Preconditioned Krylov subspace methods for boundary element solution of the Helmholtz equation

Discretization of boundary integral equations leads, in general, to fully populated complex valued non-Hermitian systems of equations. In this paper we consider the efficient solution of these boundary element systems by preconditioned iterative methods of Krylov subspace type. We devise preconditioners based on the splitting of the boundary integral operators into smooth and non-smooth parts and show these to be extremely efficient. The methods are applied to the boundary element solution of the Burton and Miller formulation of the exterior Helmholtz problem which includes the derivative of the double layer Helmholtz potential—a hypersingular operator. © 1998 John Wiley & Sons, Ltd.     

Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H¹-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).     

A Galerkin least-squares finite element method for the two-dimensional Helmholtz equation

In this paper a Galerkin least-squares (GLS) finite element method, in which residuals in least-squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one-dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two-dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher-order quadratic interpolations is also presented.     

Exact finite element solutions to the Helmholtz equation

For one-, two- and three-dimensional co-ordinate systems finite element matrices for the wave or Helmholtz equation are used to produce a single difference equation holding at any point of a regular mesh. Under homogeneous Dirichlet or Neumann boundary conditions, these equations are solved exactly. The eigenfunctions are the discrete form of sine or cosine functions and the eigenvalues are shown to be in error by a term of + O(h²ⁿ) where n is the order of the polynomial approximation of the wave function. The solutions provide the means of testing computer programs against the exact solutions and allow comparison with other difference schemes.     

A comparative study on finite element methods for dynamic fracture

The performance of finite element methods for dynamic crack propagation in brittle materials is studied. Three methods are considered: the extended finite element method (XFEM), element deletion method and interelement crack method. The extended finite element method is a method for arbitrary crack propagation without remeshing. In element deletion methods, elements that meet a fracture criterion are deleted. In interelement crack methods, the crack is limited to element edges; the separation of these edges is governed by a cohesive law. We show that XFEM and interelement method show similar crack speeds and crack paths. However, both fail to predict a benchmark experiment without adjustment of the energy release rate. The element deletion method performs very poorly for the refinements studied, and is unable to predict crack branching.     

A residual‐based finite element method for the Helmholtz equation

A new residual‐based finite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter‐element boundaries. The inclusion of residuals on inter‐element boundaries distinguishes this method from the well‐known Galerkin least‐squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral finite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.     

A two-level finite element method and its application to the Helmholtz equation

A two-level finite element method is introduced and its application to the Helmholtz equation is considered. The method retains the desirable features of the Galerkin method enriched with residual-free bubbles, while it is not limited to discretizations using elements with simple geometry. The method can be applied to other equations and to irregular-shaped domains. © 1998 John Wiley & Sons, Ltd.     

Internal field error reduction in boundary element analysis for Helmholtz equation

A systematic analysis has been made to clarify the cause of very large errors in the boundary element solutions at internal points near the boundary. Following the results of the analysis, two new ideas have been developed to reduce the errors when solving the Helmholtz equation. The first idea, which is an extended application of the equipotential condition to Helmholtz type problem, is easily applied to any existing program. The second idea which gives us more accurate solutions is based on the analytic integration of boundary integrals by expanding the fundamental solutions into a sum of elementary functions. Test calculations demonstrate that the two present techniques are efficient to reduce the errors near the boundary.     

Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method

This paper presents a gradient field representation using an analytical regularization of a hypersingular boundary integral equation for a two-dimensional time harmonic wave equation called the Helmholtz equation. The regularization is based on cancelation of the hypersingularity by considering properties of hypersingular elements that are adjacent to a singular node. Advantages to this regularization include applicability to evaluate corner nodes, no limitation for element size, and reduced computational cost compared to other methods. To demonstrate capability and accuracy, regularization is estimated for a problem about plane wave propagation. As a result, it is found that even at a corner node the most significant error in the proposed method is due to truncation error of non-singular elements in discretization, and error from hypersingular elements is negligibly small.     

A multidomain boundary element method for two equation turbulence models

The paper deals with the multidomain Boundary Element Method (BEM) for modelling 2D complex turbulent flow using low Reynolds two equation turbulence models. While the BEM is widely accepted for laminar flow this is the first case, where this method is applied for a complex flow problems using k–ε turbulence model. The integral boundary domain equations are discretised using mixed boundary elements and a multidomain method also known as subdomain technique. The resulting system matrix is overdetermined, sparse, block banded and solved using fast iterative linear least squares solver. The simulation of turbulent flow over a backward step is in excellent agreement with the finite volume method using the same turbulent model.     

A hardware acceleration of the time domain boundary integral equation method for the wave equation in two dimensions

We aimed at reducing the runtime of the time domain boundary integral equation method (TDBIEM) for the two-dimensional scalar wave equation by the help of a special-purpose computer MDGRAPE-2, which is a high-performance vector-parallel computer dedicated to computing long-range forces in classical molecular dynamics simulations. To this end, we presented a formulation to make MDGRAPE-2 compute the layer potential that is the most time-consuming in a run of the TDBIEM, considering to minimise the amount of data-transfer between MDGRAPE-2 and the host computer. The newly implemented TDBIEM code using three MDGRAPE-2s boards was about 109 times faster than the conventional code in an analysis of 5400 boundary elements and 640 time steps. We also demonstrated that the present hardware acceleration helps to perform large-scale simulations such as light propagation in photonic crystals.     

Tetrahedral polynomial finite elements for the Helmholtz equation

It is shown that the Helmholtz equation in three dimension leads to finite element approximations on tetrahedral elements that closely resemble the corresponding two-dimensional treatment on triangle. For each polynomial order, there exist two numeric universal matrices independent to tetrahedron size and shape; the element matrices are always given as linear combinations of row and column permutations of these. Numeric matrices are given up to third-order, and the permutation schemes are shown in detail. Experimental computer programs using these elements have shown fast matrix assembly times; convergence rates are essentially similar to those obtained with the corresponding triangular elements.     

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.     

Dispersion and pollution of the FEM solution for the Helmholtz equation in one,two and three dimensions

For high wave numbers, the Helmholtz equation suffers the so‐called ‘pollution effect’. This effect is directly related to the dispersion. A method to measure the dispersion on any numerical method related to the classical Galerkin FEM is presented. This method does not require to compute the numerical solution of the problem and is extremely fast. Numerical results on the classical Galerkin FEM (p‐method) is compared to modified methods presented in the literature. A study of the influence of the topology triangles is also carried out. The efficiency of the different methods is compared. The numerical results in two of the mesh and for square elements show that the high order elements control the dispersion well. The most effective modified method is the QSFEM [1,2] but it is also very complicated in the general setting. The residual‐free bubble [3,4] is effective in one dimension but not in higher dimensions. The least‐square method [1,5] approach lowers the dispersion but relatively little. The results for triangular meshes show that the best topology is the ‘criss‐cross’ pattern. Copyright © 1999 John Wiley & Sons, Ltd.     

Coupling finite and boundary element methods for two-dimensional potential problems

A finite-element-boundary-element (FE-BE) coupling method based on a weighted residual variational method is presented for potential problems, governed by either the Laplace or the Poisson equations. In this method, a portion of the domain of interest is modelled by finite elements (FE) and the remainder of the region by boundary elements (BE). Because the BE fundamental solutions are valid for infinite domains, a procedure that limits the effect of the BE fundamental solution to a small region adjacent to the FE region, called the transition region (TR), is developed. This procedure involves a judicious choice of functions called the transition (T) functions that have unit values on the BE-TR interface and zero values on the FE-TR interface. The present FE-BE coupling algorithm is shown to be independent of the extent of the transition region and the choice of the transition functions. Therefore, transition regions that extend to only one layer of elements between FE and BE regions and the use of simple linear transition functions work well.     

A numerical integration scheme for special finite elements for the Helmholtz equation

The theory for integrating the element matrices for rectangular, triangular and quadrilateral finite elements for the solution of the Helmholtz equation for very short waves is presented. A numerical integration scheme is developed. Samples of Maple and Fortran code for the evaluation of integration abscissæ and weights are made available. The results are compared with those obtained using large numbers of Gauss–Legendre integration points for a range of testing wave problems. The results demonstrate that the method gives correct results, which gives confidence in the procedures, and show that large savings in computation time can be achieved. Copyright © 2002 John Wiley & Sons, Ltd.     

GPU‐accelerated boundary element method for Helmholtz' equation in three dimensions

Recently, the application of graphics processing units (GPUs) to scientific computations is attracting a great deal of attention, because GPUs are getting faster and more programmable. In particular, NVIDIA's GPUs called compute unified device architecture enable highly mutlithreaded parallel computing for non‐graphic applications. This paper proposes a novel way to accelerate the boundary element method (BEM) for three‐dimensional Helmholtz' equation using CUDA. Adopting the techniques for the data caching and the double–single precision floating‐point arithmetic, we implemented a GPU‐accelerated BEM program for GeForce 8‐series GPUs. The program performed 6–23 times faster than a normal BEM program, which was optimized for an Intel's quad‐core CPU, for a series of boundary value problems with 8000–128000 unknowns, and it sustained a performance of 167 Gflop/s for the largest problem (1 058 000 unknowns). The accuracy of our BEM program was almost the same as that of the regular BEM program using the double precision floating‐point arithmetic. In addition, our BEM was applicable to solve realistic problems. In conclusion, the present GPU‐accelerated BEM works rapidly and precisely for solving large‐scale boundary value problems for Helmholtz' equation. Copyright © 2009 John Wiley & Sons, Ltd.