Numerical methods in classical and continuum mechanics

Dynamic inversion of a 3D, nonlinearly elastic cap.

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Overview

In classical mechanics one typically studies Lagrangian and Hamiltonian differential equations (both ordinary and partial), sometimes with additional external force terms or with dissipative terms. When studying these systems numerically, it is desirable to use numerical methods that accurately reflect underlying structure. In particular, it is desirable to have a scheme that

My research to date in this area has been focused on the design of conservative integration schemes for (unconstrained and constrained) Hamiltonian systems, and the detailed analysis of the stability and approximation properties of these and more general structure-preserving schemes.

Table of Contents

Design of conservative integration schemes

General Hamiltonian systems

First integrals or conservation laws for Hamiltonian systems with symmetry are typically lost under numerical integration in time. In some cases, failure to maintain certain conservation laws can lead to physically impossible solutions (due to numerical dissipation of energy and degradation of physically meaningful invariants) and in other cases to numerical instability.

The following example from the work of Simo & Tarnow (1992) shows plots of energy versus time for a tumbling elastic structure. After a brief loading period, the structure tumbles freely and the energy should be conserved; however, standard time integration schemes can produce growth or dissipation in the energy as shown in the energy versus time plots below.

For Hamiltonian systems with symmetry it is thus generally desirable that numerical time integration schemes preserve physically meaningful integrals from the underlying system. These types of integrators are usually referred to as conserving integrators and have been the subject of my attention for the past few years. A general formalism for these schemes is contained in the paper Time Integration and Discrete Hamiltonian Systems.

Constrained Hamiltonian systems

An extension of conserving integrators to mechanical systems subject to holonomic constraints, such as the following two-dumbbell problem,

can be found in the article Mechanical Systems Subject to Holonomic Constraints: Differential-Algebraic Formulations and Conservative Integration.

Infinite-dimensional Hamiltonian systems

Conserving schemes can also be formulated for infinite-dimensional systems, such as arise in the study of three-dimensional elastic bodies. Deformable bodies exhibiting elastic response have applications in many diverse areas of engineering and the physical sciences. For example, engineering materials such as natural rubbers are well-approximated as elastic continua and there is some interest in modeling large-scale atomic systems using elastic continuum approximations.

Depending on the nature of the external loads and boundary conditions, various constants of motion can be identified in the dynamics of an elastic continuum. These conservation laws, however, are typically lost under numerical approximation. For the past few years I've been developing numerical techniques which preserve underlying conservation laws for general hyperelastic continuum models, both compressible and incompressible. Conservative time-integration schemes, and how they couple with finite element spatial discretization, for compressible and incompressible hyperelastic continuum models can be found in the article Exact Energy-Momentum Conserving Algorithms for General Models in Nonlinear Elasticity .

In the above article you will see some numerical examples illustrating the dynamics of an elastic cap

and an incompressible elastic slab

Approximation properties of structure-preserving schemes

What can be said about the qualitative properties of a numerical solution produced by a structure-preserving scheme? One way to answer this question is to study the associated modified equations. Modified equations are a concept from backward error analysis that characterize discretization errors in a numerical method. In particular, if a method has order r on a given equation, then for any N it has order r+N on a certain perturbed or modified equation.

In an article with Des Higham and Andrew Stuart, namely Qualitative Properties of Modified Equations, I introduced a new technique to show that structure-preserving methods are characterized by structure-preserving modified equations. Knowing properties of these equations often provides valuable insight into the methods. For example, our techniques show that symplectic schemes on Hamiltonian equations possess Hamiltonian modified equations. Thus, results from Hamiltonian perturbation theory can be used to gain insight into the behavior of symplectic methods.

More generally, if there exist perturbation results for equations with a certain structure, then much can be said about the approximation properties of a structure-preserving scheme.

Stability properties of conserving schemes

Some standard, robust time-integration schemes such as the Implicit Midpoint Rule can exhibit an interesting instability when applied to mechanical systems that can undergo large rotations in space. Essentially, these schemes introduce an artificial coupling between internal vibratory motions in the system and overall spatial rotations of the system.

To see this, consider an elastic bar that is freely rotating in space. In the Implicit Midpoint Rule, forces in the bar are calculated from the strain in an interpolated configuration (n+1/2). However, as the diagram below shows, spatial rotations cause the interpolated configuration to be artificially strained.

This instability shows up in the ``simple'' n-particle problem

and in more complex systems. For example, consider the motion of a solid rubber body, which is "pinched" and then released to tumble freely in space.

The motion of the cylinder (colored by contours of a component of the nominal stress tensor) should look as in the following picture.

However, after some time the Implicit Midpoint Rule begins to become unstable. In contrast, a conserving scheme looks okay as can be seen in the following comparison

The conserving scheme has been specifically designed to avoid artificial coupling of rotations and internal motions.

A nice analysis of these types of numerical stability problems can be done within the context of the central force problem. This was done in the article On the Stability of Symplectic and Energy-Momentum Algorithms for Nonlinear Hamiltonian Systems with Symmetry.

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