Finite element recovery techniques for local quantities of linear problems using fundamental solutions

Abstract

We show that local quantities of interest such as displacements or stresses of a FE-solution can be calculated with improved accuracy if fundamental solutions are employed. The approach is based on Betti's theorem and an integral representation of the local quantities via Green's function. The unknown Green's function is split into a regular part and a fundamental solution so that only the regular part must be approximated on the finite element ansatz space. Some numerical studies for linear elasticity will illustrate our approach. IntroductionIn finite element technology a posteriori recovery techniques for displacements or stresses are becoming increasingly popular. The best known technique in this regard is the method of Zienkiewicz and Zhu where discontinuous stresses are smoothed by an L 2 -projection, [16]. But often we are only interested in the behavior of the solution in certain regions of the problem domain. In such situations it would make no sense to recover the displacement or stress field from the raw FE-output in the whole domain but to minimize the error only in the region of interest. These so-called goal-oriented error estimates are based on duality techniques (Betti's theorem) where the improvement of local quantities of interest is achieved by an iterative local mesh refinement strategy, [3, 6-8, 11, 13, 14].A fundamental result of FE-technology is that any local quantity can be calculated by forming the L 2 -scalar product between the approximated Green's function corresponding to this quantity and the right-hand side of the governing differential equation. This insight is the starting point of our approach. It is based on an integral representation of the local quantity of interest via Green's function. The Green's function is split into an unknown regular part and a known fundamental solution of the adjoint differential operator. Since the exact singularity is used in the integral representation no regularization is necessary. Only the regular part of the Green's function must be approximated on the FE-ansatz space resulting in a higher accuracy of the local quantities. Also no mesh refinement is necessary to catch the singular part. Furthermore the new approach can easily be incorporated into existing FE-codes.The paper is organized as follows: In Sect. 2 we introduce a model problem, the plane stress state of 2-D linear elasticity. In Sect. 3 we briefly introduce the application of Green's function. In Sect. 4 and 5 we shall explain our strategy to improve the accuracy of finite element calculations by employing fundamental solutions. In Sect. 6 an upper error bound of the local quantities is presented and finally in Sect. 7 numerical results for 2-D problems are discussed. Model problemWe consider a problem of 2-D linear elasticity namely the plane stress state in a bounded plane body X & R 2 with a Lipschitz continuous boundary C ¼ C D þ C N . The boundary value problem consists of finding the solution u ¼ fu i g of the elliptic Lamé-equationwith body forces p i 2 L 2 ðXÞ and boundary ...

A posteriori error estimation techniques in practical finite element analysis

“…Indeed, in some numerical experiments the GreenÕs function decomposition method gave excellent results in that high convergence rates in the required solutions were observed (see Ref. [55], and also Section 9.3). But, as mentioned already, the method is rather restrictive in that the fundamental solution needs to be available (see (97) and (98)).…”

“…have meaningful point values) in two or three dimensions. Although an approximation of the GreenÕs function can still be obtained, this problem can be circumvented either by using regularized functionals, as proposed for example in [44] or [48], or, when available, using analytical functions (see Kelly et al [53,54]) or Grätsch and Hartmann [55]. The idea is to approximate the GreenÕs function by splitting it into a regular part and a wellknown fundamental solution, and for this reason the method can be referred to as the GreenÕs function decomposition method.…”

Pointwise error estimation and adaptivity for the finite element method using fundamental solutions

“…where the notationG ij 1;h ðy; xÞ indicates a similar decomposition as in (38), see [18] for details.…”

Influence functions and goal‐oriented error estimation for finite element analysis of shell structures

“…and F h (.) indicates that we are using the approximated geometry defined by (21) to compute the bilinear and linear forms. We denote with h the 3D approximate geometry, which in general is different from the actual reference domain .…”

“…A posteriori error estimation procedures have become an important tool for assessing the reliability and accuracy of finite element computations [7,17]. These techniques have been extended to the so-called goal-oriented error estimations where the error is estimated in some output data of engineering interest [7][8][9][18][19][20][21][22][23][24]. The essential tool for estimating the error in such a quantity of interest is the influence function which filters out the necessary information for an accurate error estimate.…”