Computational Challenges for Multi-Physics Topology Optimization

Topology design involves working with multiple fields. Of primary interest is the density distribution of material that makes a certain objective function minimal while other objectives are satisfied as constraints in a mathematical programming statement. The evaluation of the objective and constraint functions will involve state variables that are fields that relate the design variables to physical behavior. The state fields are scalar or vector fields and multiple fields representing various physical responses may be involved; these fields will typically be coupled in multi-physics applications. Computational procedures for topology design (and for design optimization as a whole) thus encompass discretization schemes for design and state fields together with algorithms for optimization and for analysis. The prevailing computational approach to structural design and topology design in particular is to view the optimization procedure as a problem in the design variables only. This means that analysis is treated as a function call that provides information on function values and derivatives as a function of design. In optimization terms this is a nested format. An alternative to the nested format is to treat design and state fields on equal terms and formulate one unified optimization problem that involves also state equations as constraints. This is the typical approach in the areas of Mathematical Programs with Equilibrium Constraints (MPECs) and PDE-constrained optimization. Finally, in some cases it turns out to be advantageous to treat the design variables as functions of the states, for example where an explicit calculation of the optimal design for a fixed state is possible. This then leads to a variational statement for the optimal state field in itself.

Computational challenges are thus by the nature of the problem two-fold and successful implementations rely on both efficient analysis (and the associated sensitivity analysis) and on the efficiency of optimization algorithms. The discretization of the analysis and design fields play here a significant role for stability and for obtaining relevant results, and it is typical that the optimization will utilize a poor model to give results of little physical meaning. For the design field, the computational model will typically also include a relaxation of an integer valued field and suitable ways to handle this is a crucial issue especially when extending topology design from continuum structural applications to multi-physics settings.

The different approaches and the associated computational issues involved in their resolution will be illustrated by considering some recent work on design of multi-physics devices and the design of articulated mechanisms.