Projection Finite Element Method Research Articles

On Computational Issues in Large Deformation Analysis of Rubber Bushings*

ABSTRACT Accurate bushing analysis requires a locking free finite element formulation, an appropriate selection of the strain energy density function, and an adequate use of bulk modulus to assure numerical stability and accuracy. In this paper, the pressure projection finite element method is employed. The method projects displacement-calculated pressure onto a lower order pressure field, based on the Babuska-Brezzi condition, to avoid volumetric locking and pressure oscillation. Mooney-Rivlin and Cubic strain energy density functions are used to study the material effect on the predicted rubber behavior in tension-compression and shear deformation modes, and the need to use a higher order strain energy density function for bushing analysis is identified. The effect of bulk modulus on bonded rubber behavior in bushings with respect to bushing shape factor is studied, and the minimum allowable bulk modulus to impose incompressibility in bushing analysis is characterized. The load-deflection response of annular bushings subjected to axial, torsional, and radial deformations are analyzed and results are compared to linear approximations. An effort is made to demonstrate how a Mooney-Rivlin model cannot capture load-displacement nonlinearities in bushing axial and torsional deformations. Two- and three-dimensional results are compared and the applicability of two-dimensional analysis is discussed.

On the Implementation of Mixed Methods as Nonconforming Methods for Second- Order Elliptic Problems

In this paper we show that mixed finite element methods for a fairly general second-order elliptic problem with variable coefficients can be given a nonmixed formulation. (Lower-order terms are treated, so our results apply also to parabolic equations.) We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method's finite element space M h satisfies three conditions, then the two approximation methods are equivalent. These three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. We then construct appropriate nonconforming spaces M h for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in R 2 and R 3 , on simplices, or on prisms, is then implemented as a nonconforming method modified in a simple and computationally trivial manner. This new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution. A rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal-order multigrid methods for the solution of the linear system.

On the implementation of mixed methods as nonconforming methods for second-order elliptic problems

In this paper we show that mixed finite element methods for a fairly general second-order elliptic problem with variable coefficients can be given a nonmixed formulation. (Lower-order terms are treated, so our results apply also to parabolic equations.) We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method’s finite element space M h {M_h} satisfies three conditions, then the two approximation methods are equivalent. These three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. We then construct appropriate nonconforming spaces M h {M_h} for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in R 2 {\mathbb {R}^2} and R 3 {\mathbb {R}^3} , on simplices, or on prisms, is then implemented as a nonconforming method modified in a simple and computationally trivial manner. This new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution. A rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal-order multigrid methods for the solution of the linear system.