Error computation and optimal mesh in elasticity and elastoplasticity

The proposed method is based on the computation of the error on the constitutive law associated with a finite element mesh. The computation of this error is made using a statically admissible stress field which is obtained explicitly from the finite element solution (displacement approach). From the knowledge of the error, on optimal mesh can be derived. This approach is different from the one developed by BABUSKA and RHEINBOLT which only leads to error indicators and, more recently from the one developed by ZIENKIEWICZ et al. This work is based on the method of P. LADEVEZE which has been previously applied to thermal problems. This method allows to construct a statically admissible stress field using the following procedure: 1) a force density is derived from the 'kinematic' finite element stress field, solving explicitly local problems associated with each vertix, 2) given the force density on the edges of elements, the 'static' stress field is then derived explicitly for each element of the mesh. This numerical procedure only needs a small amount of new calculations in addition to the finite element analysis. Since the two stress field are available, it is then possible to compute, element by element, the local contribution to the error on the constitutive law. This allows a local measure of the mesh quality and permits a means of obtaining, at any location, the new optimal mesh size which will lead to a uniform error of given value. The mesh size being available at any location, a semi-automatic or automatic mesh generation procedure can be applied. In plasticity, the definition of the error on the constitutive law has been extended in order to take into account the incremental and non-reversible nature of the stress-strain relationship. (orig.)