运动方程自适应步长求解的高性能Galerkin时程单元初探

AN INITIAL STUDY OF A NOVEL HIGH-PERFORMANCE GALERKIN FINITE ELEMENT AND APPLYING FOR THE ADAPTIVE TIME-STEP METOD IN SOLVING MOTION EQUATION

该文以一阶运动方程为例,利用其非自伴随性质,构建了新型的凝聚检验函数,进而提出了一套高性能Galerkin有限单元——凝聚单元。该单元为无条件稳定的单步法单元,对于次多项式单元,其端结点位移和速度均可达到O(h^(2m+2))阶的超高收敛性,比常规Galerkin单元的结点精度高2阶。采用此单元,该文进而实现了无需额外的结点修正技术的自适应步长的高效算法。该文对这一研究进展做一简介,并给出初步算例验证了该法的可行性和有效性。

Taking the first-order equations of motion as the model problem,a novel condensed test function is constructed by using its non-self-adjoint property and then a new type of high-performance Galerkin finite element,called condensed element,is proposed.The proposed element,being of one-step type and unconditionally stable,can produce O(h^(2m+2)) super-convergence for displacement and velocity at end-nodes of elements of degree,which is two orders higher than traditional elements.Further,an efficient adaptive time step-size algorithm is achieved without additional nodal displacement recovery technique being used.The paper gives a brief report of this initial and promising study with some preliminary numerical examples given to show the feasibility and effectiveness of the proposed approach.