Error representations for goal-oriented <i>a posteriori</i> error estimation in elasto-plasticity with applications to mesh adaptivity

Purpose The purpose of this paper is to develop new methods of error representation to improve the accuracy and numerical efficiency of a posteriori and goal-oriented adaptive framework of elastoplasticity with Prandtl–Reuss type material laws. Design/methodology/approach To obtain new methods of error representation for a posteriori and goal-oriented error estimators, weak forms of primal and dual problems are investigated starting with the initial boundary value problem (IBVP). Then, we approximate both problems using temporal discretization. Additionally, we introduce a secant form considering the nonlinearity of elasto-plastic constitutive equations, which is approximated by a tangent form. Finally, we obtain numerical primal and dual solutions and their corresponding error approximations of discretized primal and dual problems, allowing to build several goal-oriented a posteriori error estimators on temporal and spatial adaptive refinement by inserting primal solutions, dual solutions and their error approximations as arguments in residuals of both weak forms as well as in the secant form of the bilinear residual. Findings An elasto-plastic material is investigated in a framework of goal-oriented error estimator by using separately several methods of error representation to deal with either temporal or spatial adaptive refinement, as well as with both refinements leading to an effective reduction of computational effort. Specifically, new error representations based on goal-oriented error estimators are presented and obtained from primal and dual residuals, which use only primal solutions or only dual solutions or a combination of primal and dual solutions as arguments. Error representations obtained from primal residuals and evaluated using only primal arguments do not require the formulation of a dual problem. Research limitations/implications The effectiveness of the different proposed methods is illustrated by an example of a perforated sheet for adaptive spatial refinement where new mesh adaptation methods of error representation are compared against existing mesh adaptation methods such as uniform mesh refinement, mesh refinement based on gradient indicators and adjoint-based methods in literature. The framework generates a balanced mesh consisting of fine, medium and coarse elements for accurate results, avoiding a numerically costly simulation with only fine elements. Originality/value All new proposed methods of error representation successfully estimate actual errors during mesh adaptivity. Furthermore, the proposed methods of error representation allow us to obtain significant reduction and equidistribution of spatial error at the end of the mesh adaptivity process. Their application to a framework of goal-oriented error estimation due to time and mesh adaptivity remains an open issue.