The elliptic eigenvalue optimization problem can be extended to incorporate various types of objective functions and constraints, leading to a more diverse set of optimization scenarios. Some examples of additional objective functions and constraints include:
Multiple Eigenvalue Optimization: Instead of optimizing a single eigenvalue, the objective function could involve optimizing multiple eigenvalues simultaneously. This could lead to designs that optimize the system's behavior across multiple modes or frequencies.
Nonlinear Constraints: Introducing nonlinear constraints, such as material nonlinearity or geometric nonlinearity, can add complexity to the optimization problem. These constraints could represent physical limitations or design requirements that need to be satisfied.
Shape Constraints: Including constraints on the shape of the domain or the phase-field function can further tailor the optimization process. For example, constraints on symmetry, smoothness, or specific geometric properties can be incorporated.
To adapt the AFEM to handle these variations in the optimization problem, the algorithm would need to be modified to accommodate the specific form of the objective function and constraints. This may involve adjusting the error estimators, refining the mesh based on new criteria, and updating the optimization process to account for the additional complexities introduced by the new objectives and constraints.