The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints.
In many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain.
Topology optimization is, in addition, concerned with the number of connected components/boundaries belonging to the domain.
Mathematically, shape optimization can be posed as the problem of finding a bounded set
, minimizing a functional possibly subject to a constraint of the form Usually we are interested in sets
which are Lipschitz or C1 boundary and consist of finitely many components, which is a way of saying that we would like to find a rather pleasing shape as a solution, not some jumble of rough bits and pieces.
Sometimes additional constraints need to be imposed to that end to ensure well-posedness of the problem and uniqueness of the solution.
with Shape optimization problems are usually solved numerically, by using iterative methods.
The idea of using a function to represent the shape is at the basis of the level-set method.
is evolved in the direction of negative shape gradient in order to reduce the value of the cost functional.
Higher order derivatives can be similarly defined, leading to Newtonlike methods.
Typically, gradient descent is preferred, even if requires a large number of iterations, because, it can be hard to compute the second-order derivative (that is, the Hessian) of the objective functional
is present, one has to find ways to convert the constrained problem into an unconstrained one.
Sometimes ideas based on Lagrange multipliers, like the adjoint state method, can work.
A convenient approach, suitable for a wide class of problems, consists in the parametrization of the CAD model coupled with a full automation of all the process required for function evaluation (meshing, solving and result processing).
Mesh morphing is a valid choice for complex problems that resolves typical issues associated with re-meshing such as discontinuities in the computed objective and constraint functions.
There are several algorithms available for mesh morphing (deforming volumes, pseudosolids, radial basis functions).
The approach of using a penalty function is an effective technique which could be used in the first stage of optimization.
The GA real-coded technique is applied in the present optimization problem.