The gradient of the finite element variational indicator with respect to nodal and applications in fracture mechanics and mesh optimization

Abstract

Abstract We derive a closed‐form expression for the change in the variational indicator of a finite element mesh with respect to perturbations in nodal point co‐ordinates. The expression is evaluated very effectively from standard finite element data obtained in one solution, and may be easily programmed as part of a general finite element code. We present the derivation for two‐ and three‐dimensional isoparametric elements used in linear and nonlincar elasticity. The expression has practical applications in the computation of stress intensity factors in fracture mechanics and in the determination of the ‘optimal’ mesh with a given element‐node connectivity. We demonstrate both applications by accurately determining the stress intensity factor of a Mode I crack using a finite element mesh which was improved using mesh optimization.