Monolithic solution and null-space condensation of internal variables in finite viscoelasticity

Abstract

Finite-strain viscoelasticity is commonly discussed from a constitutive perspective, whereas the nonlinear solution architecture induced by time and space discretization is studied less systematically. In this work, we consider a representative viscoelastic model with a strain-like internal variable and focus on the fully discrete coupled problem in the deformation and the internal state. Starting from the underlying energy-dissipation structure, we derive the discrete weak forms and obtain a monolithic Newton system with a naturally non-symmetric block tangent. The increment of the internal variable is eliminated consistently at the level of the linearized system by a Schur complement, and the same reduction is shown to admit a geometric interpretation in terms of a null-space basis of the tangent space of the internal constraint manifold. This yields a genuinely monolithic counterpart to classical nested Gauss-point condensation, in which local constitutive equations are solved separately before the global equilibrium step. Numerical results for two- and three-dimensional Cook's membrane benchmarks show that the proposed strategy retains essentially the same outer Newton behavior as the classical approach while substantially reducing computational cost by avoiding repeated local Newton solves, it also remains convergent for load increments for which the nested scheme fails. Moreover, the numerical study suggests that suitably chosen approximation spaces for the internal variable can yield additional savings in computational effort without significant loss of accuracy. Although presented for finite viscoelasticity, the construction extends naturally to broader classes of thermodynamically consistent internal-variable models.