A robust preconditioning technique for the extended finite element method

By Alexander Menk, Stéphane P.A. Bordas. 2011

159 citations recorded by the source corpus.

The extended finite element method enhances the approximation properties of the finite element space by using additional enrichment functions. But the resulting stiffness matrices can become ill‐conditioned. In that case iterative solvers need a large number of iterations to obtain an acceptable solution. In this paper a procedure is described to obtain stiffness matrices whose condition number is close to the one of the finite element matrices without any enrichments. A domain decomposition is employed and the algorithm is very well suited for parallel computations. The method was tested in numerical experiments to show its effectiveness. The experiments have been conducted for structures containing cracks and material interfaces. We show that the corresponding enrichments can result in arbitrarily ill‐conditioned matrices. The method proposed here, however, provides well‐conditioned matrices and can be applied to any sort of enrichment. The complexity of this approach and its relation to the domain decomposition is discussed. Computation times have been measured for a structure containing multiple cracks. For this structure the computation times could be decreased by a factor of 2. Copyright © 2010 John Wiley & Sons, Ltd.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/2ad102d1bc4e606fd785a5c7ee24f912c7192793

How researchers see this paper

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

In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

A well‐conditioned and optimally convergent XFEM for 3D linear elastic fracture (2016, methodology): “Bordas SPA, Rabczuk T, Zi G. Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic.”
Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics (2015, background): “Several authors [16, 28, 29, 34, 47] have shown that the conditioning of the GFEM can deteriorate significantly when the crack surface is close to nodes.”
Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth (2017, background): “This flexibility simplifies local adaptive processes, as the regenerated mesh need not conform to the crack faces [19–21] does not come for free, as (1) numerical integration requires special attention [22–24]; (2) optimal convergence requires enriching a volume independent of the mesh size, thereby”
Extended finite element method with global enrichment (2015, methodology): “The addition of enriched elements however, leads to poor conditioning of the resulting system matrices which can be solved either by the use of special preconditioners [24,26] or by using the so called degree of freedom (dof) gathering [23] technique.”
Stable Generalized Finite Element Method (SGFEM) (2011, background): “Preconditioning of the stiffness matrix, based on domain decomposition, have been recently suggested in [34] to address the conditioning problem.”
Computation of limit and shakedown loads using a node‐based smoothed finite element method (2012, methodology): “In addition to the above discussions, the authors believe that the method presented herein can be very promising to: (1) improve the load factor solution of limit and shakedown problems with strong discontinuities [81–83] in fracture structures by coupling the NS-FEM to the extended FEM (XFEM) [84–8”
Inexact Schwarz‐algebraic multigrid preconditioners for crack problems modeled by extended finite element methods (2012, background): “Another interesting study that combines domain decomposition and XFEM has just been published by Menk and Bordas [25].”
An improved stable XFEM (Is-XFEM) with a novel enrichment function for the computational modeling of cohesive cracks (2015, methodology): “Another possibility is to apply special preconditioners during the solution phase, such as those based on the local Cholesky decomposition [3], the domain decomposition [39] and the Jacobi one combined with diagonal scaling [58].”