Adjoint-based error estimation and mesh adaptation for stabilized finite deformation elasticity

Adjoint-based error estimation provides the ability to approximate the discretization error for a functional quantity of interest, such as point-wise displacements or stresses. Mesh adaptation provides the ability to control the discretization error to obtain more accurate solutions while still remaining computationally feasible. In this paper, we develop an approach for adjoint-based error estimation and mesh adaptation for nonlinear finite deformation elasticity using a mixed stabilized finite element method. We apply our developed technique to a well known test case, the Cook's membrane problem, to validate and demonstrate its effectiveness. We then demonstrate the utility of adjoint-based error estimation and mesh adaptation for a three-dimensional example motivated by the study of a cell embedded in a matrix.