A quadratic time finite element method for nonlinear elastodynamics within the context of hybrid finite elements

Abstract

Standard displacement-based finite elements are known to display overstiff behavior known as locking phenomenon for a wide variety of problems. To overcome this problem, hybrid elements have been proposed in the literature. They are based on a two-field variational principle, and involve an interpolation for the stress field that is chosen independently of that of the displacement field. Due to this independent choice of the stress interpolation, hybrid elements are less susceptible to the locking phenomenon, and thus provide better coarse mesh accuracy as compared to the displacement-based formulation. Hybrid elements are versatile elements with which one can model shell structures, almost incompressible materials as well as ‘chunky’ geometries. In this work, we use hybrid elements for the space discretization along with a higher-order momentum conserving variant of the time finite element method. The time finite element method provides a general variational framework for developing arbitrary-order time-stepping strategies for transient problems. It has been shown in the literature recently that for discrete chaotic systems the quadratic time finite element scheme is computationally more efficient and robust when compared to the linear time finite element strategy. In this work, we present a modified form of the quadratic time finite element method for the solution of nonlinear elastodynamics problems that conserves linear and angular momenta exactly, and energy in an approximate sense in the fully discrete setting. We present numerous examples to demonstrate the efficacy of hybrid elements over conventional elements within the context of the modified quadratic time finite element method. From our examples, we observe that for chaotic systems, the proposed strategy is significantly more computationally efficient compared to the linear transient strategy; however in other problems although the performance is good, it does not offer any significant computational advantage.