A combined space–time extended finite element method

By J. Réthoré, Anthony Gravouil, Alain Combescure. 2005

75 citations recorded by the source corpus.

The Newmark method for the numerical integration of second order equations has been extensively used and studied along the past fifty years for structural dynamics and various fields of mechanical engineering. Easy implementation and nice properties of this method and its derivatives for linear problems are appreciated but the main drawback is the treatment of discontinuities. Zienkiewicz proposed an approach using finite element concept in time, which allows a new look at the Newmark method. The idea of this paper is to propose, thanks to this approach, the use of a time partition of the unity method denoted Time Extended Finite Element Method (TX‐FEM) for improved numerical simulations of time discontinuities. An enriched basis of shape functions in time is used to capture with a good accuracy the non‐polynomial part of the solution. This formulation allows a suitable form of the time‐stepping formulae to study stability and energy conservation. The case of an enrichment with the Heaviside function is developed and can be seen as an alternative approach to time discontinuous Galerkin method (T‐DGM), stability and accuracy properties of which can be derived from those of the TX‐FEM. Then Space and Time X‐FEM (STX‐FEM) are combined to obtain a unified space–time discretization. This combined STX‐FEM appears to be a suitable technique for space–time discontinuous problems like dynamic crack propagation or other applications involving moving discontinuities. Copyright © 2005 John Wiley & Sons, Ltd.

Subfields: Civil and Structural Engineering, Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/a34df45c7f9d153784d0c73e9b7b7a6684d80169

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Civil and Structural Engineering

In Computational Mechanics

In Mechanics of Materials

Citations in the corpus

X-FEM explicit dynamics for constant strain elements to alleviate mesh constraints on internal or external boundaries (2008, background): “Other authors purpose a generalization of X-FEM by introducing enrichment strategies for time-dependent problems [16].”
A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks (2008, unknown): “[18,19], de Borst et al.”
Dynamic crack propagation analysis of orthotropic media by the extended finite element method (2009, methodology): “Furthermore, Réthoré et al. (2005) proposed a combined space–time extended finite element method, based on the idea of the time extended finite element method (TXFEM), allowing a suitable form of the time stepping formulae to satisfy stability and energy conservation criteria.”
An explicit dynamics extended finite element method. Part 2: Element-by-element stable-explicit/explicit dynamic scheme (2009, background): “As in various references [3, 4, 17, 29, 33, 34], we now consider the case of stationary then moving crack.”