Special finite elements with holes and internal cracks

Abstract

SUMMARYFor the numerical treatment of stress concentration problems in plane elasticity, special finite elements with circular and elliptic holes and internal cracks have been developed. Two different variational formulations have been used to construct elements, which may be combined with conventional displacement elements. Using complex functions and conformal mapping techniques the systematic construction of trial functions is shown which not only satisfy a priori the governing differential equations but also the boundary conditions on such influential boundary portions as hole or crack surfaces. For the evaluation of the stiffness matrices of the special elements, only boundary integral computations are necessary. The numerical results of various examples are very accurate for both functionals.

Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

“…This can be used to generate hybrid finite elements. Earlier studies employed boundary-type approximation associated with Trefftz to develop special type finite elements, for example, elements with holes/ voids [35,36], for plate analysis [37][38][39]. Recently, the idea of employing local solutions over arbitrary finite elements has been investigated in [5,[31][32][33].…”

FM-DBEM Simulation of 3D Microvoid and Microcrack Graphite Models

“…Even under such a condition, the accuracy is not that satisfactory because the assumed displacement functions do not satisfy the dominance conditions at the free boundary. In order to solve such problems, some modified methods [16,17] are applied, but these methods need to model the vicinity of microvoids and microcracks with a mass of elements; besides, special elements are also defined to tackle the problem [18,19].…”

Trefftz solutions for piezoelectricity by Lekhnitskii's formalism and boundary‐collocation method

“…can be accurately accounted for by using Trefftz functions that satisfy the pertinent homogeneous boundary conditions [2][3][4][9][10][11]. In this section, special solution sets are constructed for arbitrarily orientated impermeable elliptical voids, impermeable sharp cracks and permeable sharp cracks in plane piezoelectrics by enforcing the pertinent homogeneous boundary conditions.…”

“…The complex potential functions k ( k ) for the elliptical void problem can be chosen in the form of a complex Laurent series [2,14], i.e.…”