The collocation Trefftz method for biharmonic equations with crack singularities
The purpose of this paper is to extend the boundary approximation method proposed by Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], i.e. the collocation Trefftz method called in this paper, for biharmonic equations with singularities. First, this paper derives the Green formulas for biharmonic equations on bounded domains with a non-smooth boundary, and corner terms are developed. The Green formulas are important to provide all the exterior and interior boundary conditions which will be used in the collocation Trefftz method. Second, this paper proposes three crack models (called Models I, II and III), and the collocation Trefftz method provides their most accurate solutions. In fact, Models I and II resemble Motz's problem in Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], and Model III with all the clamped boundary conditions originated from Schiff et al. [The mathematics of finite elements and applications III, 1979]. Moreover, effects on d1 of different boundary conditions are investigated, and a brief analysis of error bounds for the collocation Trefftz method is made. Since accuracy of the solutions obtained in this paper is very high, they can be used as the typical models in testing numerical methods. The computed results show that as the singularity models, Models I and II are superior to Model III, because more accurate solutions can be obtained by the collocation Trefftz method.
collocation trefftz methods biharmonic equation boundary approximation method boundary condition corner condition green function singular problems