The Constitutive Relation Error Method: A General Verification Tool

Abstract

This chapter reviews the Constitutive Relation Error method as a general verification tool which is very suitable to compute strict and effective error bounds for linear and more generally convex Structural Mechanics problems. The review is focused on the basic features of the method and the most recent developments.

Appendix: Construction of Equilibrated Stress Fields

The construction of a statically admissible field is a key point of error estimation methods based on CRE. It particularly enables to obtain guaranteed error bounds for a large set of mechanical problems. A general construction approach, based on a post-processing of the FE stress field $\sigma _h$, has been introduced in [4, 35]. This approach, recently named EET (Element Equilibration Technique), can be decomposed in two steps:

• 1. tractions $\hat{\mathbf {F}}_h$, equilibrated with the external loading, are built on element edges;

• 2. in each element E, a stress field $\hat{\sigma }_{h|E}$ that verifies equilibrium:

  $$\begin{aligned} \mathbf div \, \hat{\sigma }_h + \mathbf {f}_d=\mathbf {0} \; \text {in}\, E \quad ; \quad \hat{\sigma }_h \mathbf {n}= \eta _E \hat{\mathbf {F}}_h \; \text {on}\, \partial E \end{aligned}$$

  (90)

  is computed, with $\eta _E = \pm 1$ a scalar value ensuring the continuity of the stress vector. The associated local problem is in practice solved with a quasi-explicit technique and polynomial basis, or with a dual approach with p enrichment (shape functions of degree $p+k$).

The first step leans on the following prolongation (energy) condition:

$$\begin{aligned} \int _E(\hat{\sigma }_h-\sigma _h) \varvec{\nabla }\phi _i \text {d}E = 0 \quad \Longrightarrow \quad \int _{\partial E}\hat{\sigma }_h\cdot \mathbf {n}\phi _i \text {d} S= \int _E(\sigma _h\varvec{\nabla }\phi _i - \mathbf {f}_d \cdot \phi _i)\text {d}E \quad \forall i \end{aligned}$$

(91)

where $\phi _i$ is the FE shape function associated to node i. This condition, which ensures equilibration of $\hat{\mathbf {F}}_h$ over E (as $\sum _i \phi _{i|E}=1$), leads to the solution to a system of the form:

$$\begin{aligned} \sum _{r=1}^{R_n} \mathbf {b}_n^r(i) = \mathbf {Q}_{E_n}(i) \quad \forall n=1,\ldots ,N \end{aligned}$$

(92)

over the set of N elements connected to each node i. $R_n$ is the number of edges for element $E_n$ connected to node i, $\mathbf {Q}_{E_n}(i) = \int _{E_n}(\sigma _h \varvec{\nabla }\phi _i - \mathbf {f}_d \phi _i)\text {d}E$, and unknowns $\mathbf {b}_n^r(i)$ are projections of tractions defined as $\hat{\mathbf {b}}_n^r(i) = \int _{\varGamma ^r_{E_n}} \eta _{E_n} \hat{\mathbf {F}}_h \phi _i \text {d} S$. Existence of a solution for each system is ensured by the equilibrium property (in the FE sense) verified by $\sigma _h$, and uniqueness may be obtained minimizing a cost function.

In [26], a new hybrid method called EESPT (Element Equilibration + Star Patch Technique) was introduced for the construction of admissible stress fields. As an intermediary between EET and SPET (flux-free [55, 56]) methods, it enables a nice compromise between accuracy of the computed stress fields, computational cost, and practical implementation in engineering softwares. The EESPT method still has two steps and leans on the construction of equilibrated tractions $\hat{\mathbf {F}}_h$ on element edges. The main change is in the way the tractions are constructed, with an increasing flexibility brought by a Partition of Unity Method (PUM); this leads to patch problems solved in an automatic and non-intrusive manner, from classical FE tools. The computation of $\hat{\sigma }_h$, over each element and from tractions $\hat{\mathbf {F}}_h$, remains unchanged and can be parallelized.

A comparison between EET, SPET and EESPT methods was performed in [36] on several industrial applications, one of them being the structure presented in Fig. 2. It was observed that the SPET method is more accurate than EET and EESPT methods, but it requires higher computational cost. The EESPT method, which provides results comparable to those of the EET method, seems to be a nice compromise between accuracy, computational cost and implementation issues.

Fig. 12

Norm of the FE stress field (a), and admissible stress fields computed with EET (b), standard SPET (c), and enhanced SPET (d) methods

In [57], an improved version of the EESPT method was studied. It uses ideas developed in [58] by considering a weak prolongation condition applied to high degree shape functions (non-vertex nodes). This results in a local minimization of the complementary energy and leads to optimized tractions in selected regions, particularly those with distorted elements or high gradients. The improved version of the EESPT method having a higher computational cost, criteria were introduced to select zones in which this version should be employed to get a nice compromise between accuracy and cost. One example is that of a plate with a hole subjected to a unit traction force (see Fig. 12). EET, standard EESPT, and enhanced EESPT methods were used to compute, from $\sigma _h$, a SA stress field $\hat{\sigma }_h$ and derive the associated CRE error estimate. Two criteria were introduced to detect zones in which the enhanced SPET method should be used; the first criterion is based on the element shape (distorsion level) and thus relies on the local quality of the mesh, whereas the second criterion considers local error contributions.