# A new conservative/dissipative time integration scheme for nonlinear mechanical systems

## TL;DR

This paper introduces a novel time integration scheme for nonlinear mechanical systems that ensures energy conservation or dissipation, improving accuracy and stability in simulations.

## Contribution

It develops a new algorithmic framework based on quadratic programs that guarantees strict energy properties while being easily integrable into existing models.

## Key findings

- The scheme guarantees energy conservation or dissipation as desired.
- It is applicable to reduced-order, finite element, and multibody models.
- Numerical examples demonstrate improved stability and accuracy.

## Abstract

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.