The Research On Numerical Methods For Nearly Incompressible Elasticity Problems In Three Dimensions And Fast Solving Algorithms

The finite element (FEM) method is an important class of numerical methods forsolving the three-dimensional elastic problems. In practical calculation，such as rubberand plastic, which show nearly incompressibility material property, i.e., Poisson’sratio ν is close to0.5. The volume locking phenomenon will happen when thecommonly used finite elements (such as a linear element) are applied to the solutionof this nearly incompressibility problems. Thus, we need to use some special methods.In this paper, first of all, based on the ANSYS software, we make some systematicstudies for the efficiency and robustness of high-order FEM method, reducedintegration and mixed higher-order FEM method based on u/pscheme by usinghexahedral mesh for3D nearly incompressible problems with mixed boundaryconditions. The numerical results have been shown that the aforementioned methodscan effectively overcome the locking phenomenon of3D elastic materials in whichmixed high-order FE method is the most accurate, and the calculated displacementsstably converge to the theoretical solution with the decrease of the mesh size. But theoverall stiffness matrix is an half positive definite, and to get the matrix, thecomputing scale cost by using the mixed element method is as two times as cost byusing the displacement method, which would cause inconvenience when we have tochoose the solver. We hope to obtain a symmetric, positive definite matrix for nearlyincompressible discrete problem, which is easy to select a more efficient solver. Thesecond part of this paper, we carefully derive the computing format of penaltyfunction finite element method, specifically analysis the conditions for success of theresulting method and verify the effectiveness and robustness of this method inovercoming locking phenomenon by some numerical experiments for nearlyincompressible elasticity problems with mixed boundary conditions. The quality ofthe mesh used in three-dimensional finite element analysis has a great effect on theaccuracy and computational efficiency. If the isotropic grids can be used in thepractical calculations, the method will have better convergence. In this paper, finallybased on penalty function in the finite element analysis of large, sparse and highlypathological positive definite equations, the design of several preconditioningconjugate gradient (PCG) method, Several kinds of pre-conditionedconjugate-gradient(PCG) approach are designed likeM1-PCG,M2-PCG and RS-GAMG-PCG. TheM1-PCG andM2-PCG are based on Block diagonal inversepre-conditioner and the RS-GAMG-PCG is based on global matrix. Also, thenumerical computational efficiencies are tested and analyzed when those method areapplied to solve the Quadratic penalty function element equation of cantilever beamand Cook membrane in this paper. The results shows that if we can take use of theparts of Geometry and analysis information like the equation type and the node degreefreedom which can be obtained easier from model, then，more efficiency AMGmethod and PCG method can be derived when combining the Coarse grid technologyand the construct interpolation operator constructor method in classical AMG method.And by using these methods, the overall efficiency of the finite element analysis canbe improved greatly.