Numerical time integration of lumped parameter systems governed by implicit constitutive relations

Time-integration for lumped parameter systems obeying implicit Bingham-Kelvin constitutive models is studied. The governing system of equations describing the lumped parameter system is a non-linear differential-algebraic equation and needs to be solved numerically. The response of this system is non-smooth and the kinematic variables can not be written in terms of the dynamic variables, explicitly. To gain insight into numerical time-integration of this system, a new time-integration scheme based on the trapezoidal method is derived. This method relies on two independent parameters to adjust for damping and is stable. Numerical examples showcase the performance of the proposed time-integration method and compare it to a benchmark algorithm. Under this scheme, implicit-explicit integration of the governing equations is possible. Using this new method, the limitations of the trapezoidal time-integration methods when applied to a non-smooth differential-algebraic equation are highlighted.