COMSOL 6.4 - Optimization Module Overview

The Optimization Module can be used throughout the COMSOL product family — it provides a general interface for calculating optimal solutions to engineering problems. Any model inputs, be it geometric dimensions, part shapes, material properties, or material distribution, can be treated as control variables, and any model output can be an objective function.

Simulation is a powerful tool in science and engineering for predicting the behavior of physical systems, particularly those governed by partial differential equations. In many cases a single or a few simulations are not enough to provide sufficient understanding of a system. Two important classes of problems whose resolution relies on a more systematic exploratory process are:

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Design problems, where the problem is to find the values of control variables or design variables that yield the best performance of a model, quantified by means of an objective function. Problems of this kind arise, for example, in structural optimization, antenna design, and process optimization.

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Inverse problems, and in particular parameter estimation in multiphysics models, where the problem is to reliably determine the values of a set of parameters that provide simulated data that best matches measured data. Such problems arise in, for example, geophysical imaging, nondestructive testing, and biomedical imaging. Curve fitting also belongs to this category.

Problems of the above types can often be formulated more generally as optimization problems. The Optimization interface and Optimization study step in COMSOL Multiphysics® are useful for solving design problems as well as inverse problems and parameter estimation.

Optimization Algorithms

There are four optimization algorithms for gradient-based optimization available in the Optimization Module.

The first algorithm is the IPOPT solver. When using IPOPT, the objective function can have any form and any constraints can be applied. The algorithm uses a gradient-based optimization technique to find optimal designs and when the underlying PDE is stationary, frequency dependent, or time dependent, analytic sensitivities of the objective function with respect to the control variables can be used. The main authors are Andreas Waechter and Carl Laird from IBM, but the code is still developed as an open-source project with many contributors.

The second and third algorithms are the MMA and GCMMA solvers, which are based on the method of moving asymptotes by Krister Svanberg of KTH (Royal Institute of Technology, Stockholm). The MMA solver can handle objective functions and constraints of the same very general form as IPOPT. It can solve min-max optimization problems, and it is well suited to handle problems with a large number of control variables, such as topology optimization. The globally convergent version of the solver is often referred to as GCMMA. Both MMA and GCMMA have support for limiting the step length via move limits.

The fourth algorithm is a Levenberg–Marquardt solver. When this solver is used, the objective function must be of least-squares type. Also, constraints are not supported. Since the Levenberg–Marquardt method is designed to solve problems of least-squares type, it typically converges faster than IPOPT for such problems.

In addition, the Optimization Module provides a number of gradient-free (derivative-free) optimization algorithms. Currently Nelder–Mead, BOBYQA, COBYLA, EGO, and a coordinate search are supported. These methods can optimize a model with respect to design parameters (model parameters) such as parameters which control the geometry sequence that defines the model’s geometry.

All optimization solvers are accessible from the same General Optimization study step, which contains the ordinary solver sequence over which the optimization method iterates. The gradient-free methods can contain any other study sequence, while the gradient-based methods are limited to optimizing over a single study step of a type supporting computation of analytic sensitivities: currently Stationary, Time Dependent, Frequency Domain, and Eigenfrequency studies.


	The Physics Interfaces and Building a COMSOL Multiphysics Model in the COMSOL Multiphysics Reference Manual