Finite elements with penalties in nonlinear elasticity

Abstract

AbstractThis paper describes the use of a penalty method to enforce the constraint of incompressibility in nonlinear elasticity. As an example, a problem involving the use of the Newton–Raphson method in conjunction with incremental loading and a successive mesh refinement scheme is presented. It is shown that during the incremental loading phase and the Newton–Raphson refinement on a fixed mesh, all tangent stiffness matrices are positive definite for the chosen energy density and load increment. But when the mesh is refined and the solution is interpolated as a starting value on the new mesh, the tangent stiffness matrix is indefinite. A theoretical analysis of the associated mixed method and a new equivalence theorem are seen to lead to a way to retain positive definiteness. The key is the use of an equivalent tangent stiffness matrix which is the reduced Hessian matrix. The numerical example shows that both positive definiteness and the quadratic convergence rate of the Newton–Raphson method are obtained.