INFINITE SIMILAR BOUNDARY ELEMENT METHOD FOR DYNAMIC FRACTURE MECHANICS

In the conventional boundary element method, in order toobtain the last algebraic equation system a large number of integralsmust be calculated numerically on all boundary elements and internalcells so that a large amount of computing time is needed. If we canform similar boundary elements on the boundary and obtain the relationof the matrices on the similar boundary elements, it is not needed toobtain the matrices on all elements by numerical integrals, then agreat amount of numerical integrals will be decreased. In this paper,similar boundary element method (SBEM) for elastodynamic problems isdiscussed in detail. Similar boundary elements are classified and theirproperties are discussed. The interpolation method to obtain thematrices on the similar boundary elements is presented and the formulaeof the method are obtained. In similar boundary element method, theboundary is represented with some sub-domains on which the boundaryelements are similar. Then on a sub-domain of the boundary we only needto compute the matrices on a few boundary elements by numericalintegrals, and the ones on all other boundary elements can be obtainedby the interpolation method. Then superimposing the matrices on allboundary elements the coefficient matrix of the last algebraic equationsystem can be obtained. Comparing with the conventional boundaryelement method that the matrices on all boundary elements are obtainedindependently by numerical integrals, similar boundary element methodcan decrease the computing time to a great extent, and the solution isin total agreement with the one from the conventional boundary elementmethod. To obtain the singular stress at the tip of a dynamic crack,infinite similar boundary element method (ISBEM) is presented. In themethod the similar boundary element sub-domain at the tip of the crackcontains infinite similar boundary elements. From infinite similarboundary element method, the singular stress at the tip of a crack canbe obtained directly, but the singular boundary element is not neededand the degree of singular stress is not assumed. For some materialsthat we do not know the degree of singular stress at the tip of a crack,infinite similar boundary element method can be applied better than theconventional boundary element method does. In this method the numbersof boundary elements and nodes are infinite, so an infinite orderlinear algebraic equation system is formed, and then the numericalmethod for this infinite order system is discussed. For a problem withan irregular domain, we can use the curvilinear coordinate system onthe boundary of the domain, and then similar boundary element methodand infinite similar boundary element method presented in this papercan be applied too. Similar boundary element method can be applied toother problems which can be solved with the conventional boundaryelement method, and infinite similar boundary element method can beapplied to other crack problems.