Structure-preserving model reduction for mechanical systems

Dedicated to Alan Newell on the occasion of his 60th birthday
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Abstract

This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen–Loève expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

Introduction

The problem setting. The problem of constructing simple, yet predictive, models for complex physical systems operating on many length and time scales has a long and distinguished history, from finding finite dimensional Galerkin truncation models to inertial manifolds and to finding envelope equations, a particular love of Alan’s. Another approach, the focus of the present paper, is to use what is variously known as model reduction, the Karhunen–Loève expansion (KLE), empirical eigenfunction, or proper orthogonal decomposition method.

Systems are often modeled by nonlinear partial differential equations that contain phenomena on many scales, which can be both difficult to analyze mathematically and computationally expensive for simulation, design and control problems. The reasons for this difficulty are manifold. For example, the system itself may have complex geometry, such as flow through a jet engine or the dynamic motions of an automobile. Thus, if one attempts to model the fluid equations or those of elasticity in such a complex geometry, the amount of computation will of course be significant, even for simple flows or motions. However, there may be intrinsic difficulties in simple geometries as well, such as in turbulent flow with its characteristic feature of a cascade of energy to small scales. It is very important to understand to what extent one has to model the small scale dynamics to achieve accurate models of the large-scale motions. Recent work on large eddy simulation models and averaged fluid equations [10], [23], [25], [26] suggests that indeed one can do this with considerable savings in computational cost. In general, multiscale phenomena, both temporal and spatial, are of great importance as well as the source of many of the difficulties.

Fluid computations. The Karhunen–Loève method is perhaps best known in fluid computations, as described in [11]. The literature is huge in this area and we cite only the recent work of [34], [35] as examples. While fluids have a well-known variational structure (see, for instance, [10]), it is more subtle than that of solids because of the large particle relabeling symmetry group. Thus, in this paper we have focused on solid mechanical examples.

General goals of this paper. In this paper we focus on the problem of constructing low-dimensional models for mechanical systems. Our aim is to develop a general methodology which is applicable to a wide range of mechanical systems, including systems of jointed rigid bodies such as robotic systems, as well as fluid and elastic systems modeled through finite-element analysis.

Such systems are well studied, and a significant amount is known about their mathematical and geometric structure. This geometric structure has fundamental implications for our understanding of the behavior of many mechanical systems. It also leads to computational methods which take advantage of this structure, for example in ensuring that numerical integration methods conserve energy or momentum.

In this paper, our goal is to develop a model reduction procedure that is consistent with and indeed preserves the geometric structure underlying the mechanics, and that ties in with standard computational methods for analysis and simulation of both finite-dimensional and continuum mechanical systems. Our main focus is to develop a basic theory behind this area of mechanical model reduction, applicable to nonlinear high-dimensional systems, whose configuration spaces may have constraints; that is, be manifolds. Non-trivial configuration manifolds are, of course, often introduced in holonomic mechanical systems by imposing configuration space constraints, such as those encountered in articulated and robotic systems.

We first discuss our motivation behind the basic problem, explaining why reduced-order models of high-dimensional systems are of both mathematical and computational interest. We then give an overview of the basic theory we develop later in the paper for model reduction of mechanical systems.

The fundamental motivation behind model reduction is that low-dimensional systems should both be simpler to work with analytically, and be faster and more convenient to work with computationally. There is of course a great need for such computational savings in problems of both structural design and control design.

Structural design. Many high-dimensional or continuum mechanical systems exhibit behavior that it is perhaps not unreasonable to expect to be well modeled by appropriate low-dimensional nonlinear systems. An example is given by the dynamic motion of an aircraft wing in flight, where the typical motions of the wing are large-scale and often relatively simple bulk bending dynamics.

The computational advantages are multiplied when considering, for example, performance evaluation of an aircraft wing under dynamic loading. Here finite-element methods are typically used, and performance checked via Monte Carlo sampling. Since a large number of repeated simulation trials must be performed, any reduction in the computational costs per simulation can allow a greater exploration of Monte Carlo space.

Control design. Other applications include control design, where the design of a stabilizing feedback controller may be extremely difficult for a high-dimensional nonlinear system, but much simpler for the low-dimensional system. Model reduction should preserve the essential features of the system dynamics, and for some control applications this is all that is necessary; the controller does not depend on the highly uncertain fine-scale features of the dynamics. See [3], [12], [18], [30], [37].

Systems analysis. Analysis problems of interest include an understanding of the bifurcation structure of the system. A further benefit of model reduction is that often a low-dimensional model can provide qualitative understanding of the phenomena under consideration.

The most basic method of model reduction for linear systems is that of modal expansion, where the phase space of the system is decomposed into subspaces corresponding to an eigendecomposition of the generator of the differential equations. This is a technique of fundamental importance for many applications. However, it is important to realize that the modal decomposition alone does not provide enough information to decide upon a good reduced-order model. What is additionally necessary is a method of deciding which modes should be preserved in the model. Typically the low-frequency modes are kept, however, there are many applications where this is not the best choice; in particular, in control systems where the frequency at which an accurate model is necessary is at crossover, and this may not correspond to the low-frequency modes of the system.

One of the most widely used methods for model reduction of general dynamical systems is to apply a KLE to the state space, and use a Galerkin projection to construct the reduced system. The KLE was introduced by Pearson [32], and developed by several authors, including [14], [21]. The use of this method for analysis of turbulent flows was pioneered by Lumley [22].

For linear control systems, the method of balanced truncation due to [29] has proved to be extremely useful in practical applications, as it tends to preserve the dynamics which are important for control. A mathematically precise version of this statement is possible, using the results of [6], [9] on the errors obtained using balanced truncation.

The relationship between balancing and the KLE method was developed in the papers by Lall et al. [17], [18], where a method of using the KLE in order to construct the balanced truncation of a linear system of n first-order differential equations was constructed. In fact, the standard KLE methods applied to linear systems in first-order form is equivalent to the method known as input-balancing for controlled systems with a single-input.

For mechanical systems, many of the above relationships remain to be worked out. A possible approach is to use balancing methods as used in [18] in combination with the methods in this paper to construct reduced-order nonlinear models for a mechanical system.

The reduced-order systems we construct are integrated using globally-supported basis functions, known as Ritz functions, in contrast with the standard locally-supported shape function approach. Previous approaches deriving the Ritz functions from the linearization of the equations of motion were probably initiated by Nickell [31], who extended the use of modal superposition methods for systems with nonlinear dynamics. Wilson et al. [38] later introduced the so-called load-dependent vectors. A recent overview of these techniques was given by Leger and Dussault [19], and this method was also used in [15], [16].

Section snippets

An example: the docking device

We will illustrate the theory with an example of finite-element analysis of nonlinear three-dimensional elasticity, called the docking device, shown in Fig. 1. This model consists of two rectangular blocks of rubber, connected by a steel rod. One block is mounted on fixed supports. The other block is clamped to an unsupported rigid and massive steel frame.

The elasticity in this system is modeled by a nonlinear finite-element model with 9600 degrees of freedom. For this example, we construct a

Model reduction of mechanical systems

In this paper, our goal is to develop model reduction methods for mechanical systems. We now make precise our notion of mechanical system, following the modern approach to mechanics; see, for example, the treatment in [24]. Let the space of configurations of a mechanical system be a differentiable manifold Q. The Lagrangian is a function L:TQ→R, where TQ is the tangent bundle of Q, typically given by the difference between the kinetic and potential energy of the system: L(q,q̇)=T(q,q̇)−V(q).The

Finite-element models of elasticity

The model reduction described below is specialized to three-dimensional elastodynamics, but applies equally well to Lagrangian finite-element models of other types of solids independently of the constitutive equation used.

Improvements

It is clear that the procedures outlined here can be improved in many ways that need further exploration for future applications.

Symmetry and travelling bases. Experience has shown that if a system has symmetry, then using travelling Karhunen–Loève bases can dramatically cut down the dimension of the reduced system; see, for example, [8], [36] and references therein. It is likewise well understood that discrete symmetries are also important to take into account when building bases. Thus, for a

Acknowledgements

We thank Ronald Coifman, John Doyle, Darryl Holm, Yannis Kevrekidis, Clancy Rowley, and Peter Schröder for helpful comments and inspiration.

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    Research partially supported by AFOSR MURI grant F49620-96-1-0471 and NSF/DARPA OPAAL grant DMS-9874082.

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