Parameter Optimization

To set up a parameter optimization in COMSOL Multiphysics^®, you only need to add a general-purpose Optimization study. An associated settings window prompts you to add an objective function, control variables and parameters, and constraints. The parameters that you use for parameter optimization can be the same that were used to set up the model in the first place, such as geometry dimensions, material properties, or boundary loads. Whereas a parametric sweep will give you an overview of the design parameter space, a parameter optimization will give you the optimal parameters and objective function values.

When running a parameter optimization with parameters that define geometry dimensions, remeshing is needed at each iteration — a process that is fully automatic with the Optimization Module. The optimal solution is always a true CAD part that can be immediately exported to industry-standard CAD formats. This requires the CAD Import Module, Design Module, or one of the LiveLink™ products for CAD.

Topology Optimization

Topology optimization represents even greater freedom in geometry variation than both parameter and shape optimization. This approach allows for material removal and addition during the optimization process, allowing for holes to be created in the geometry not originally present in the design. This method typically results in an organic-looking design, and is a popular method for light-weighting. A dedicated user interface and study are available for topology optimization.

The extreme design freedom associated with topology optimization can result in designs that cannot be manufactured with conventional methods. It is therefore common to introduce manufacturing constraints, because this enables the optimized design to be produced using extrusion or milling.

In addition, just like with shape optimization, remeshing is not required for topology optimization. The optimal and smoothed design is made available in the STL, 3MF, or PLY file formats for further use in another software or for verification analysis within COMSOL Multiphysics^®.

Gradient-Based Solvers

Gradient-based optimization methods are used when the derivatives can be computed efficiently using the adjoint method. This is possible for custom objectives or constraints, as long as these are differentiable. This is thanks to the core technology of COMSOL Multiphysics^® that features symbolic differentiation, which also provides the flexibility needed for solving custom multiphysics problems.

Gradient-based optimization can be used with thousands, even millions, of design variables. This is often the case for shape or topology optimization, where the design variables represent field quantities that are distributed throughout space and represented by different values in each mesh element.

The gradient-based methods simultaneously compute all analytic derivatives, whereas the derivative-free methods have to approximate each derivative and will take more time as the number of design variables increases.

The gradient-based methods included in the Optimization Module are:

• Method of moving asymptotes (both MMA and GCMMA)
• Interior point optimizer (IPOPT)
• Sparse nonlinear optimizer (SNOPT)
• Levenberg–Marquardt

Shape Optimization

As an alternative to varying a set of CAD parameters, you can allow for the geometry to deform more or less freely by using the built-in features for shape optimization. This approach allows for greater freedom and can sometimes produce even better results than parameter optimization. A set of dedicated user interfaces are available for easily defining the allowed deformations of boundaries in a 2D or 3D model. Additionally, a specialized shell shape optimization feature is available, as well as a shape optimization study type for controlling the solvers.

The tools used for shape optimization in solids are based on methods that deform the mesh in a controlled manner where remeshing is not required. The optimal geometry is available in a faceted surface mesh format, such as STL, 3MF, or PLY. This geometry can then be reused in a separate analysis within COMSOL Multiphysics^® or exported for use in another software.

Parameter Estimation

A model is only as accurate as the input, but it can be difficult to source accurate material parameters from suppliers. To account for nonlinearities, you might need to perform experiments. However, trying to design experiments that allow you to extract the desired parameters with analytical methods can be challenging.

A solution to these problems is to use the parameter estimation functionality of the Optimization Module to find the set of parameters of a model that minimizes the deviation between the physical and the simulated experiment. In addition to the interface for general parameter estimation, a specialized user interface for curve fitting is available for fitting a curve (represented by a model expression) to time-dependent data.

The parameter estimation method is based on least-squares fitting and can be used when the reference data is a function of time or a single parameter. In many cases, you will get an estimate of the variance and confidence of the estimated parameters.

To get started with parameter estimation, a ready-to-use app is available with the capabilities of using built-in tutorial samples or imported measurement data, as well as entering custom model expressions for the curve to be fitted.

Derivative-Free Solvers

Derivative-free optimization methods are used when the search directions needed by the optimization solver can only be computed indirectly. This is often the case for parameter optimization where the control variables represent geometry dimensions and remeshing is needed in each iteration step.

The derivative-free methods included in the Optimization Module are:

• Trust-region methods
  • Bound optimization by quadratic approximation (BOBYQA)
  • Constraint optimization by linear approximation (COBYLA)
• Direct search methods
  • Nelder–Mead (the Simplex method)
  • Coordinate search