Numerical method for solving of ill-conditioned boundary value problems in nonlinear elasticity - Brigadnov, Igor A. (Acta mechanica, 2023)
The article discusses the boundary value problem (BVP) of nonlinear elasticity for mapping (deformation) in two weak variational forms: as the static equilibrium equation and as the minimization problem for energy. Some questions of a mathematical correctness of the nonlinear elasticity BVP are discussed. Two simple examples prove that for some nonlinear elastic models the appropriate BVP either can have not a solution, or can have a discontinuous solution with discontinuities of sliding type. Both effects correspond to ill-conditioned BVP. Using finite element approximation the initial variational problems are transformed into a nonlinear system of algebraic equations and a finite-dimensional minimization problem, respectively. These problems can be ill-conditioned. The phenomenon is observed in problems where deformation has a large gradient heterogeneity. For numerically solving the original decomposition method of adaptive block relaxation is proposed. The main idea of this method consists of the iterative improvement in zones with “proportional” deformation by special decomposition of domain (variables) and separate calculation in these zones (on these variables). The numerical results show that for finding the deformed configuration this technique has obvious benefits over the standard methods.