On the convergence of a gradient method for the minimization of functionals in finite deformation elasticity theory and for the minimization of barrier grid functionals

Gradient descent methods are examined for the minimization of barrier-type polyconvex functionals arising in finite-deformation elasticity theory and grid optimization. The minimum of a functional is sought in the class of continuous piecewise affine deformations that preserve orientation. Sufficient conditions are found for a sequence of iterative approximations to belong to the feasible set and for the norm of the gradient of the functional to converge to zero on this set. As the functional, one can use a measure of the deformation of a grid, for instance, a grid formed of triangles or tetrahedra.