A posteriori error estimation for isogeometric analysis using the concept of Constitutive Relation Error

Abstract

The paper deals with the Isogeometric Analysis (IGA) technology, which has received much attention over the last decade due to its increased flexibility, accuracy, and robustness in many engineering simulations compared to classical Finite Element Analysis (FEA). In this context, we present a verification method, based on duality and the concept of Constitutive Relation Error (CRE), that enables to derive fully computable a posteriori error estimates on the numerical solution provided by IGA. Such estimates, which can be used for a wide class of structural mechanics models, thus constitute effective and practical tools to quantitatively control the numerical accuracy and drive adaptive algorithms. The focus is here on the construction of so-called admissible flux fields which is a key ingredient of the CRE concept, and which was until now almost exclusively addressed in the FEA framework alone. We show that this construction can be performed in a similar way for FEA and IGA, provided some technical issues (due to the use of B-Spline/NURBS basis functions instead of Lagrange polynomials) are carefully addressed. We also use the CRE concept along with adjoint techniques and local enrichments in order to derive accurate goal-oriented error estimates. Two- and three-dimensional numerical experiments are presented, for thermal, linear elasticity and nonlinear damage problems, to illustrate the capabilities and versatility of the proposed approach.