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

Abstract

SUMMARYIn this paper, we first present a consistent procedure to establish influence functions for the finite element analysis of shell structures, where the influence function can be for any linear quantity of engineering interest. We then design some goal-oriented error measures that take into account the cancellation effect of errors over the domain to overcome the issue of over-estimation. These error measures include the error due to the approximation in the geometry of the shell structure. In the calculation of the influence functions we also consider the asymptotic behaviour of shells as the thickness approaches zero. Although our procedures are general and can be applied to any shell formulation, we focus on MITC finite element shell discretizations. In our numerical results, influence functions are shown for some shell test problems, and the proposed goal-oriented error estimation procedure shows good effectivity indices.

A posteriori error estimation techniques in practical finite element analysis

“…In [49] it is found in some numerical studies that the error representation (90) leads to accurate error estimates and to effectivity indices close to 1.0 (since (90) takes into account the cancellation effect of the error over the domain). The error estimate includes the error due to the approximation in the geometry of the shell structure.…”

“…This approach was used by Grätsch and Bathe in shell analyses [49]. Using a continuum mechanics shell formulation [1,50], the error in a linear quantity of interest can be written as corresponding to a MITC9 shell finite element formulation such that…”

Goal-oriented error estimation in the analysis of fluid flows with structural interactions

“…as defined in (30). To actually obtain a computable quantity of the error in the quantity of interest, one useful strategy (and commonly accepted practice) is to replace the unknown solution Z by a solution obtained using a richer space than V 0;h : For instance, Grätsch and Bathe [20,21] used the same mesh of 4-node and 9-node elements in shell analyses. Hence, if in a 2D or shell analysis Z 9-node h denotes the 9-node element solution and Z h the 4-node element solution (h denotes the element size in each case), we obtain from (48)…”

“…Of course, the level of accuracy must be sufficiently high to obtain the structural response to the desired accuracy, and to ensure that this is the case, the solution error needs to be measured and controlled. In principle, a number of error measures but in particular goaloriented error measures have the potential to be very useful [11][12][13][14][15][16][17][18][19][20][21]. Indeed, the basic aim with goal-oriented error measures, to assess the error in certain solution quantities, is directly applicable.…”

On Reliable Finite Element Methods for Extreme Loading Conditions

“…It may not be necessary to solve for the fluid flow very accurately in the complete fluid domain in order to only predict the stresses, accurately, in the structure at a certain location. In these cases, the concept of goal-oriented error estimation can be very effective and has considerable potential for further developments 31,36,37 .…”