A comparison of domain integral evaluation techniques for boundary element methods (original) (raw)
Related papers
Proceedings of the Indian National Science Academy, 2016
The proposed developments are based on a consistent implementation of the conventional, collocation boundary element method (BEM). A scheme is used to expand a generic (not problem-dependent) fundamental solution about hierarchical levels of source and field poles, which is particularly advantageous to make the technique seamlessly applicable to 2D and 3D problems of elasticity or potential, in terms of different types of curved elements for generally complicated geometry and topology. The proposed compact algorithm is more straightforward to lay out and seems to be more efficient than the ones available in the technical literature-particularly because the outermost loop refers to field nodes and geometry, in what may be called a reverse implementation. Some numerical results are shown for the conventional BEM, with validation and assessment for a few simple, but very large-scale, 2D potential problems with complicated geometry and topology for constant, linear and quadratic elements. Since iterative solvers are not required in this first step of numerical simulations, an isolated assessment of accuracy, computational effort and storage allocation of the proposed fast multipole technique becomes possible.
Recent Advances and Emerging Applications of the Boundary Element Method
Applied Mechanics Reviews, 2012
Workshop on the Emerging Applications and Future Directions of the Boundary Element Method," University of Akron, Ohio, September 1-3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Green's functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.
Boundary Elements and Other Mesh Reduction Methods XXXVIII, 2015
The proposed implementations are based on a consistent development of the conventional, collocation boundary element method (BEM)-with concepts taken from the variationally-based hybrid BEM-for large-scale 2D and 3D problems of potential and elasticity. The formulation is especially advantageous for problems of complicated topology or requiring complicated fundamental solutions. This paper, which is the sequel of a first paper presented at the PACAM 2014 Conference in Santiago, Chile, proposes a scheme for expansions of a generic fundamental solution about hierarchical levels of source and field poles. This makes the fast multipole technique directly applicable to different kinds of potential and elasticity problems with generally curved boundaries. The basic concept of the FMM, with the expansion of the fundamental solution about successive layers of source and field poles, is described in a compact algorithm that is more straightforward to lay out and seems to be more efficient than the ones available in the technical literature. The hierarchical tree of poles is built upon a topological concept of superelements inside superelements. The formulation is initially assessed and validated in terms of a simple 2D potential problem. Since iterative solvers are not required in this first step of numerical simulations, an isolated efficiency assessment of the implemented fast multipole technique is possible.
Communications in Numerical Methods in Engineering, 2007
In this paper, a sufficient condition, for transforming domain integrals into boundary integral is described. The transformation is accomplished by Green's and Gauss' theorems. It is shown that a wide range of domain integrals including some integrals in boundary element method (BEM) satisfy this sufficient condition and can be simply transformed to boundary. Although emphasis is made on potential and elastostatic problems, this method can also be used for many other applications. Using the present method, a wide range of 2D and 3D domain integrals over simply or multiply connected regions can be transformed exactly to the boundary. The resultant boundary integrals are numerically evaluated using an adaptive version of the Simpson integration method. Several examples are provided to show the efficiency and accuracy of the present method.
A convergence analysis of the performance of the DRM-MD boundary integral approach
International Journal for Numerical Methods in Engineering, 2007
In this article, we study the performance of the dual reciprocity multi-domains approach (DRM-MD) when the shape functions of the boundary elements, for both the approximation of the geometry and the surface variables of the governing equations, are quadratic functions. A series of tests are carried out to study the consistency of the proposed boundary integral technique. For this purpose a limiting process of the subdivision of the domain is performed, reporting a comparison of the computed solutions for every refining scheme. Furthermore, the DRM-MD is solved in its dual reciprocity approximation using two different radial basis interpolation functions, the conical function r plus a constant, i.e. (1 + r), and the augmented thin plate spline. Special attention is paid to the contrast between numerical results yielded by the DRM-MD approach using linear and quadratic boundary elements towards a full understanding of its convergence behaviour.
International Journal for Numerical Methods in Engineering, 1991
This paper presents a new boundary element approach to transform domain integrals into equivalent boundary integrals. The technique, called the Multiple Reciprocity Method, is applied to 2-D elasticity problems and operates on domain integrals resulting from different types of body forces such as gravitational and centrifugal forces, as well as loadings due to linear and quadratic temperature distributions. Numerical examples are presented to demonstrate the accuracy and efficiency of the method.
A robust and non-singular formulation of the boundary integral method for the potential problem
Engineering Analysis with Boundary Elements, 2014
A non-singular formulation of the boundary integral method (BIM) is presented for the Laplace equation whereby the well-known singularities that arise from the fundamental solution are eliminated analytically. A key advantage of this approach is that numerical errors that arise due to the proximity of nodes located on osculating boundaries are suppressed. This is particularly relevant in multi-scale problems where high accuracy is required without undue increase in computational cost when the spacing between boundaries become much smaller than their characteristic dimensions. The elimination of the singularities means that standard quadrature can be used to evaluate the surface integrals and this results in about 60% savings in coding effort. The new formulation also affords a numerically robust way to calculate the potential close to the boundaries. Detailed implementations of this approach are illustrated with problems involving osculating boundaries, 2D domains with corners and a wave drag problem in a 3D semi-infinite domain. The explicit formulation of problems with axial symmetry is also given.
The Boundary Element Method for Potential Problems
Fundamental Concepts and Models for the Direct Problem
This chapter presents the boundary element method applied to potential problems. The integral equation is obtained for the Laplace equation and discretized into boundary elements. Constant, linear, and quadratic boundary elements are considered. The method is applied to some numerical examples and results are compared to analytical solutions. A convergence study is carried out in order to access the behaviour of the method with mesh refinement.
Generalized boundary element method for galerkin boundary integrals
Engineering Analysis with Boundary Elements, 2005
A meshless approach to the Boundary Element Method in which only a scattered set of points is used to approximate the solution is presented. Moving Least Square approximations are used to build a Partition of Unity on the boundary and then used to construct, at low cost, trial and test functions for Galerkin approximations. A particular case in which the Partition of Unity is described by linear boundary element meshes, as in the Generalized Finite Element Method, is then presented. This approximation technique is then applied to Galerkin boundary element formulations. Finally, some numerical accuracy and convergence solutions for potential problems are presented for the singular, hypersingular and symmetric approaches.