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

TL;DR

This paper introduces a new stable trapezoidal-based time-integration scheme for non-smooth, implicit Bingham-Kelvin systems modeled by differential-algebraic equations, enabling implicit-explicit integration and improved numerical stability.

Contribution

A novel trapezoidal-based time-integration method with adjustable damping parameters for non-smooth implicit systems is developed and analyzed.

Findings

The new method is stable and effective for non-smooth systems.

It allows implicit-explicit integration of complex differential-algebraic equations.

Performance comparisons show advantages over benchmark algorithms.

Abstract

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.