The Optimisation Module is an add-on package that you can use in conjunction with any existing COMSOL Multiphysics Product. Once you have created a COMSOL Multiphysics model of your product or process, you always want to improve on your design. This involves four steps. First, you define your objective function – a figure of merit that describes your system. Second, you define a set of design variables – the inputs to the model that you would like to change. Third, you define a set of constraints, bounds on your design variables, or operating conditions that need to be satisfied. Last, you use the Optimisation Module to improve your design by changing the design variables, while satisfying your constraints. The Optimisation Module is a general interface for defining objective functions, specifying design variables, and setting up these constraints. Any model input, whether it be geometric dimensions, part shapes, material properties, or material distribution, can be treated as a design variable, and any model output can be used to define the objective function. It can be used throughout the COMSOL Multiphysics product family and can be combined with the LiveLink™ add-on products to optimise a geometric dimension in a third-party CAD program.

The Optimisation Module includes two different optimisation techniques: Derivative-Free and Gradient-Based optimisation. Derivative-Free optimisation is useful when your objective functions and constraints may be discontinuous and do not have analytic derivatives. For example, you may want to minimise the peak stress in your part by changing the dimensions. However, as the dimensions change, the location of the peak stresses can shift from one point to another. Such an objective function is non-analytic and requires a derivative-free approach. Five such methods are available in the Optimisation Module: the Bound Optimisation by Quadratic Approximation (BOBYQA) method, the Constraint Optimisation by Linear Approximation (COBYLA) method, the Nelder-Mead method, the coordinate search method, and the Monte Carlo method.

The Optimisation Module will compute an approximate gradient to evolve the design variables towards an improved design. You may want to minimise the total mass of your part which can also be done using this approach. The mass of the part is usually directly differentiable with respect to the part dimensions, permitting use of a gradient-based approach. The Optimisation Module will compute the exact analytic derivative of your objective and constraint functions using the adjoint method of the SNOPT optimiser, developed by Philip E. Gill of the University of California San Diego and Walter Murray and Michael A. Saunders of Stanford University, to improve the design variables. A second gradient-based algorithm is a Levenberg-Marquardt solver. You can use this solver when the objective function is of a least-squares type, typically for parameter estimation and curve fitting applications. A third method, the Method of Moving Asymptotes (MMA), is a gradient-based optimisation solver written by Professor K. Svanberg at the Royal Institute of Technology in Stockholm, Sweden. It is designed with topology optimisation in mind. The method is called GCMMA in the literature and is available in the Optimisation Module under the name MMA.

The advantage of the gradient-based method is that it can address problems involving hundreds, or even thousands, of design variables with very low increase in computational cost as the number of design variables increases. The Adjoint method simultaneously computes all analytic derivatives, whereas the derivative-free method has to approximate each derivative, and will take more time as the number of design variables increases. The gradient-based methods can also include more complex constraint functions.

The advantage of the derivative-free method is in its simplicity. It does not require finding a differentiable objective function, and it requires less user interaction to set up. However, due to the computational costs, derivative-free methods are most attractive when the number of design variables is around 10 or less. In practice, this covers a wide range of practical engineering optimisation problems.

Parameter Optimisation involves optimising any scalar input to the model, such as flow rates and load magnitudes. This is usually the easiest kind of optimisation and can be addressed with any technique.

Parameter Estimation is more complex, and involves correlating a COMSOL model to experimental data. Typically, the objective is to use a model to estimate material properties used.

Optimisation methods can be further classified by the types of variables being optimised. Dimension, Shape, and Topology optimisation are all addressed within the Optimisation Module, and each has its own place in the design process.

Dimensional optimisation involves defining design variables that can be directly translated to manufacturing. Typical design variables may be hole sizes, or length, width, and height of structural members. Dimensional optimisation is usually used as the last step in the design process and is performed once the design is more or less fixed in terms of the overall shape. The derivative-free method is usually used here.