The solution to the elastodynamic equation in the exterior of a polyhedral domain or a screen exhibits singular behavior from the corners and edges. The detailed expansion of the singularities implies quasi-optimal estimates for piecewise polynomial approximations of the Dirichlet trace of the solution and the traction. The results are applied to hp and graded versions of the time domain boundary element method for the weakly singular and the hypersingular integral equations. Numerical examples confirm the theoretical results for the Dirichlet and Neumann problems for screens and for polygonal domains in 2d. They exhibit the expected quasi-optimal convergence rates and the singular behavior of the solutions.
翻译:摘要: 在多面体域或屏幕的外部求解弹性动力学方程会出现奇异行为。细致的奇异性展开意味着对解的Dirichlet迹和牵引力的分段多项式逼近具有准最优估计。将结果应用于弱奇异积分方程和超奇异积分方程的hp版本和分级版本的时间域边界元法。数值实验证实了二维屏幕问题和多边形域的Dirichlet和Neumann问题的理论结果。它们展示了预期的准最优收敛速率和解的奇异行为。