基于EEP法的三维有限元超收敛计算初探  

EXPLORATION ON SUPER-CONVERGENT SOLUTIONS OF 3D FEM BASED ON EEP METHOD

作　　者：袁驷[1] 吴越[1] 徐俊杰[2] 邢沁妍[1] 

机构地区：[1]清华大学土木工程系,土木工程安全与耐久教育部重点实验室,北京100084 [2]中国地震局工程力学研究所,中国地震局地震工程与工程振动重点实验室,哈尔滨150080

出　　处：《工程力学》2016年第9期15-20,共6页Engineering Mechanics

基　　金：国家自然科学基金项目(51378293,51078199)

摘　　要：二维有限元法(FEM)的超收敛计算,借助有限元线法(FEMOL)作为桥梁,分两步采用单元能量投影(EEP)法导出超收敛公式,初步形成"逐维离散、逐维恢复"的方案。然而这一思路直接应用于三维问题却遇到了困扰:一维问题的EEP解(位移和导数)均可达到相同的超收敛阶,而二维问题却难以做到。研究发现,为了得到三维问题的EEP超收敛位移,只需提供二维问题最低阶的超收敛位移即可。该文按此思路推导了非规则网格下三维六面体单元的EEP超收敛位移公式,给出了一个实施方案,并通过数值算例验证了此方案的有效性。In the super-convergence computation of a 2D Finite Element Method（FEM）, the ＂discretization and recovery by dimension＂ scheme has basically formed by taking the Finite Element Method of Lines（FEMOL） as a bridge and iteratively by adopting the super-convergent formulas derived from the Element Energy Projection（EEP） method. However, when applying this idea to 3D problems, it occurs a puzzle that the EEP solutions（including displacements and derivatives） of 1D problems all share the same super-convergence order whereas those of 2D problems can hardly do. Recent studies show that in order to obtain the EEP super-convergent solution of a 3D problem, the displacement of a 2D problem with the least super-convergence order is merely necessary. Following this idea, this paper derives EEP super-convergent formulas for 3D hexahedron elements on irregular meshes, proposes an implementation scheme, and verifies its effectiveness with numerical examples.

关 键 词：有限元法 有限元线法 超收敛 三维问题 单元能量投影 六面体单元 

分 类 号：TU311.4[建筑科学—结构工程]