Dual-variational-formulation-based A Posteriori Error Bounds For Finite Element Analysis

The finite element method has been widely used to make critical decisions in the design of engineering structures.In order to control the simulation quality,several a pos-teriori error estimators have been presented to evaluate the discretization error of finite element analysis.Among the existing estimators,the constitutive relation error(CRE),which is based on a pair of admissible solution fields and defined via a convex constitu-tive relation,can be guaranteed as a strict upper bound of the global discretization error.Practically,a posteriori error analysis is frequently focused on the goal-oriented error estimation assessing the discretization error in some specific quantities of interest.Two of the available goal-oriented error estimation techniques,the CRE-based method and the constrained optimization method with convex objective functions,have been claimed to provide strict upper and lower bounds of the quantity error.Several types of existing a posteriori error estimation techniques for finite element analysis are reviewed in the introduction.Then the equivalence of the two strictly bound-ing approaches of goal-oriented errors is addressed,both in formulation and in principle.It is also shown that the strict bounding property in global or goal-oriented error estima-tion is essentially a natural result of the dual variational formulation corresponding to the primal one that the finite element analysis is based on.In the context of linear problems,the CRE-based goal-oriented error estimation tech-nique are extended to two types of problems.One is about the elastic foundation system with double shear effect.Strict upper and lower bounds of errors in some quantities of interest are given,and a modification scheme to overcome the difficulty of shear lock-ing is proposed based on the error estimation technique as well.The other considers the cases with non-symmetric bilinear forms,in which the static response sensitivity analy-sis is discussed as a typical instance.Strict error bounds are provided for the quantities associated with response sensitivity derivatives,which is also exemplified by some mod-e1 problems,such as the Bernoulli-Euler beam model and the membrane settled on an elastic foundation.In the context of nonlinear problems,CREs for the two types of elliptic variational inequalities(EVIs)with quadratic minimizing variational forms are defined based on the corresponding dual variational formulations,and their forms in some specific EVI prob-lems are explicitly presented.The CREs are guaranteed to provide strict upper bounds of global energy-norm errors in the finite element solutions to these EVI problems,which is also assessed by some numerical results.As an extension and generalization of the CRE theory,at last,a generalized consti-tutive relation error(GCRE)is introduced in an analogous form of the Fenchel-Young inequality for the general class of convex problems in mechanics.The GCRE is exempli-fied by a hyperelastic problem and a frictional contact problem.The dissertation aims at further developing the CRE theory and duality-based error estimation methods in both theoretical bases and engineering applications.