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

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.