On the duality of finite element discretization error control in computational Newtonian and Eshelbian mechanics

Abstract

In this paper, goal-oriented a posteriori error estimators of the averaging type are presented for the error obtained while approximately evaluating the J-integral in nonlinear elastic fracture mechanics. Since the value of the J-integral is one component of the material force acting on the crack tip of a pre-cracked elastic body, the appropriate mechanical framework to be chosen is the one named after Eshelby rather than classical Newtonian mechanics. However, in a finite element setting, the discretized Eshelby problem is generally not solved explicitly. Rather, its solution is approximated by the finite element solution of the corresponding discretized dual Newton problem. As a consequence, discrete material forces arise not only at the crack tip but also at other nodes of the current finite element mesh. It is the objective of this paper to establish goal-oriented a posteriori error estimators in both the framework of Eshelbian and Newtonian mechanics and to elaborate their dual relations. This allows to control the error of the J-integral while, at the same time, no further discrete material forces arise during the adaptive mesh refinement process which could lead to misleading mechanical interpretations of the results obtained by the finite Dedicated to the memory of the esteemed colleague Professor Karl Popp, University of Hannover, who unexpectedly passed away on April 24, 2005. M. Rüter (B)

Strain smoothing in FEM and XFEM

“…This is also the subject of recent work on derivative recovery [106] and a posteriori error estimation in XFEM [20,16,34]. Duality techniques such as those proposed in [81,80,78,79,69] are promising tools to devise goal-oriented error estimates where the stress intensity factor (SIF) is the quantity of interest.…”

Energy based methods applied in mechanics by using the extended Noether's formalism

“…In this manner, possible explanations arise for difficult concepts such as contact formulations [115]. We understand that the additional equations from the extended Noether formalism is necessary to fulfill in order obtain physically correct models in fracture mechanics [116,117]. These configurational forces are employed to compute the crack propagation in linear elasticity [118] and elasto-plasticity [119].…”

An estimation of the sensitivity of numerical error norm using adjoint model

“…The estimation of truncation error is obtained using the action of a high-order accuracy finite-difference stencil on a previously computed field. The resulting adjoint state provides the global sensitivity of the numerical error norm to the local truncation error, which constitutes the main feature of the present approach that distinguishes it from other contributions [5][6][7] related to this topic. This technique is applicable to parabolic and elliptic problems of rather general form restricted only by the requirement for existence of a continuous Gateaux differential.…”

“…From the perspective of estimation of the total solution quality, the methods for the estimation of error norms and their sensitivities using adjoint equations are of a significant interest. However, the adjoint equations are only seldom used for the estimation of the error norm and the authors are aware of only several works addressing this topic [5][6][7] and limited to some particular cases for elliptic equations. In the present work we consider the sensitivity of norms of the solution perturbation to a source-like disturbing term.…”