Explicit mixed strain-displacement finite element for dynamic geometrically non-linear solid mechanics

Abstract

Aquesta és una còpia de la versió author's final draft d'un article publicat a la revista Computational mechanics. Although appealing for their simplicity, low-order finite elements face inherent limitations related to their poor convergence properties. Such difficulties typically manifest as mesh-dependent or excessively stiff behaviour when dealing with complex problems. A recent proposal to address such limitations is the adoption of mixed displacement-strain technologies which were shown to satisfactorily address both problems. Unfortunately, although appealing, the use of such element technology puts a large burden on the linear algebra, as the solution of larger linear systems is needed. In this paper, the use of an explicit time integration scheme for the solution of the mixed strain-displacement problem is explored as an alternative. An algorithm is devised to allow the effective time integration of the mixed problem. The developed method retains second order accuracy in time and is competitive in terms of computational cost with the standard irreducible formulation.

Formulación mixta estabilizada explícita de elementos finitos para sólidos compresibles y quasi-incompresibles

“…Formulación mixta estabilizada explícita Para definir la ecuación discreta y estabilizada de la ecuación de movimiento (13), ésta tiene que ser discretizada en el tiempo. En la referencia [23] se discuten las ventajas de la formulación mixta explícita en conjunto con un estudio numérico de estabilidad y convergencia. Naturalmente, este método mixto-explícito es condicionalmente estable, pero su precisión es superior a su contraparte irreducible.…”

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

“…Doing this via an explicit time advancing scheme is a well-known procedure for the irreducible formulation. Reference [35] discusses the relative merits of doing this for the mixed formulation, that is, incorporating the solution of the weak form of the geometric equation into the time marching scheme. The resulting explicit mixed method is conditionally stable, but, compared with its irreducible counterpart: (i) it shows enhanced strain and stress accuracy and (ii) it does not require a reduced critical time step.…”

“…In this case, the load F at the free end of the short cantilever is applied instantly at t = 0, and it remains constant in time. For the integration in time, the time step is selected so that conditional stability is warranted [35]. The OSS method is used for the stabilization of the mixed problem, with algorithmic constants c u = 1.0, c ε = 1.0 and L 0 = 50 mm.…”

“…The relative rate of convergence of the different formulations is shown in Figures 8 and 9, for the compressible and quasi-incompressible cases, respectively. Results are also shown for the MEX-FEM formulation without displacement subscale, u = 0, as originally presented in reference [35].…”

“…In reference [35], the authors proposed an explicit ε − u strain/displacement mixed finite element formulation (MEX-FEM in the following) to address dynamic geometrically nonlinear problems in solid mechanics. It was shown there that combining the mixed formulation with an explicit time integration scheme yields a completely explicit mixed approach that is very cost effective, because of two reasons.…”