Optimization problem

An optimization problem is specified by an objective criterion to be minimized and a set of constraints criteria. More precisely, we focus on constrained optimization problems of the following form

\[\begin{split}\begin{equation} \begin{array}{rrclcl} \displaystyle \min_{x\in\mathcal{O}_{ad}} & {J(x)}\\ \textrm{s.t.} & g(x) = 0\\ &h(x)\geq 0 \\ \end{array} \end{equation}\end{split}\]

where \(\mathcal{O}_{ad}\) is the optimization set, \(x\) is the state variable, \(J\) is the objective, \(g\) and \(h\) are respectively the equality and inequality constraints.

A base class describing an optimization problem is the following OpenPisco.Optim.Problems.OptimProblemBase.

The auxiliary methods used to create an optimization problem are defined in the class OpenPisco.Optim.Problems.OptimProblemConcrete. The key methods are the following

  • OpenPisco.Optim.Problems.OptimProblemConcrete.SetObjectiveCriterion()

  • OpenPisco.Optim.Problems.OptimProblemConcrete.AddConstraint()

The following method updates the values of objective and constraints on the current optimization point

  • OpenPisco.Optim.Problems.OptimProblemConcrete.UpdateValues()

Note that, when defining an optimization problem, the criteria available in OpenPisco can be used indifferently as a constraint or objective.

Topology optimization problem

In the topology optimization context the space \(\mathcal{O}_{ad}\) is a set of admissible shapes. Thus, the current state of a topology optimization problem is a OpenPisco.LevelSetBase class.

See the optimization criteria documentation for a complete list of optimization criteria provided by OpenPisco.

Design space and implicit zones

The search for an optimal design is casted in a given design space. The default design space is the region described by the attribute OpenPisco.LevelSetBase.support.

It is possible to modify the default behaviour by indicating the following regions

  • OffZone : subregion excluded from the design space

  • OnZone : subregion included in the design space

  • Dirichlet : for physic-based optimization problems, subregion corresponding to the Dirichlet boundary condition location

  • Neumann : for physic-based optimization problems, subregion corresponding to the Neumann boundary condition location

The ImplicitGeometry module of the library BasicTools 1 is used to define specific zones.

Advance

The method OpenPisco.Optim.Problems.OptimTopoProblemBase.Advance() is responsible for updating the state, i.e. the level set, and checking if the new optimization point is acceptable. The routine tests the following geometric conditions:

  • the relative volume change is not too large : avoid drastic shape variations within one iteration

  • no deconnection between dirichlet and Neumann zones : avoid meaningless physical problem

  • neumann zone not in the void phase : avoid meaningless physical problem

1

https://gitlab.com/drti/basic-tools