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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 169
An Iterative Radial Simplex Method for Elastostatic and Elastodynamic Boundary Elements K. Davey and M.T. Alonso Rasgado
School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, United Kingdom Full Bibliographic Reference for this paper
K. Davey, M.T. Alonso Rasgado, "An Iterative Radial Simplex Method for Elastostatic and Elastodynamic Boundary Elements", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 169, 2006. doi:10.4203/ccp.83.169
Keywords: integration, simplexes, boundary elements, elastodynamics.
Summary
In this paper the iterative radial simplex method is introduced for the evaluation of singular
integrals in boundary elements. The method involves the careful employment of multiple
integration where the inner integral is performed along a radial direction. Evaluation of the
radial-inner integral on a simplex of dimension n provides n+1 integrals on simplex
domains of dimension n-1. On each simplex in the lower dimension the procedure is
repeated and so on until the final dimension is 0 which yields an analytical solution or the
singularity is sufficiently remote to facilitate numerical integration. Attention in the paper is
restricted to dimensions 3, 2 and 1 with integration performed on tetrahedrons, triangles and
closed intervals. In addition the super-singular integral equation is obtained
from the differentiation of the hyper-singular integral equation. The super-singular integral
equation has been developed for the purpose of analysing surface waves, which requires
continuity of inter-element stresses, which is not present with the standard forms of
boundary elements. Illustrated in the paper through detailed examples is the method's
ability to evaluate to high accuracy the severe singular integrals involved in elastostatics and
elastodynamics.
The boundary element method (BEM) offers distinct advantages over domain methods such as the finite element method (FEM), one being the avoidance of domain discretisation. The avoidance of domain integration is achieved with the use of Green's functions, which are solutions to the governing equations for the case of an instantaneous source applied in an infinite or semi-infinite domain. The embodiment of these analytical solutions in the governing integral equations provides the boundary element method with high stability and accuracy that is often lacking with other competing methods. The use of Green's function offers advantages as well as disadvantages in that they can be complex in form and are singular making their integration problematic. Discussed in this paper is a method utilising domain integration. Although this could be viewed as a retrograde step, the advantages can outweigh the disadvantages.
The background mathematics underpinning the method requires the concepts of solid angle
and spherical co-ordinates in an Euclidean space of dimensions 3, 2 and 1. A point
To demonstrate the application of the recursive integration scheme to the elastodynamics,
BEM integration over a 2-simplex is investigated. The simplex utilised is the triangular
element with nodal co-ordinates
Described in the paper is a recursive integration method that can be employed to evaluate, to high accuracy, singular boundary and domain integrals on simplex element domains. The following conclusions can be made:
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