Example Problems

Inverse Optimization

The following pages provide examples on how to use the optimization module to solve inverse optimization problems for force and material inversion. The MOOSE optimization module relies heavily on MultiApps to solve PDE constrained optimization problems. The main-app in the Multiapps system contains the optimization executioner which controls the execution and transfer of data in the sub-apps containing either the "forward" or "adjoint" numerical problem. The "forward" model is the numerical model of the system of interest and is used for computing the objective function. The "adjoint" model is the adjoint of the forward model and computes the adjoint variable used to compute the derivatives needed for optimization. Examples on the below pages follow the derivations given on the Theory page.

• Force Inversion Examples

• Material Inversion Examples

• Constrained Optimization Examples

• Debugging Help

In this section, we explore optimization techniques with a particular focus on two main types: inverse optimization and design optimization. Inverse optimization primarily involves minimizing the discrepancy between simulated and experimental data, often parameterizing aspects like force loads or material properties. This is elaborated upon in this section from the optimization theory page and in the theory section of the heat conduction MOOSE module. On the other hand, design optimization involves strategies such as shape and topology optimization, where the objective is to minimize a general function like strain energy or maximum stress, which is demonstrated in the Constrained Optimization Example: Shape Optimization

Regardless of which type of inversion problem you are trying to solve, it is recommended that you start with the Example 1: Point Loads. This example contains the best documentation on all of the MOOSE objects needed to solve an optimization and most importantly, this is the simplest optimization problem. When setting up your own optimization problems, you will likely run into problems and the Debugging Help page will help with the Tao executioner options and output that can help locate the problem.

Topology Optimization

Solid Isotropic Material Penalization (SIMP) examples typically require mechanical or thermal physics to meaningfully leverage the optimization module's capabilities. Some SIMP examples, including those that combine multiple loads, multiple materials, and multiple physics are available in the MOOSE combined module.

• Topology Optimization Examples