# Topology Optimization

It is by no mean our objective to be exhaustive regarding shape and topology optimization in this modest introduction. For detailed information regarding the various aspects inherent to such a vast field we refer for instance to 1 and 2 and the reference therein.

## Topology optimization in a nutshell

Topology optimization is devoted to the optimal design of structures. It aims at finding the best material distribution inside a working domain while fulfilling mechanical, geometrical and manufacturing specifications.

In the literature and the industry, it is mainly motivated by the following aspects:

Economical/ecological context: increase in the cost of raw materials and the necessity to decrease energy consumption

Design acceleration: alleviate workload at early stages of design to focus afterward on a physical-informed relevant initial design guess

Structure optimization automatization: domain specific intuition/skills and experience allow to manually optimize a structure through numerous trials and errors but slow, tedious and not within anyone’s grasp

The generic optimization problem associated can be described as followed

where

\(\Omega\) is the shape, or the design variable;

\(J(\Omega)\) is an objective function to be minimized;

\(C(\Omega)\) is a constraint set;

\(\mathcal{O}_{ad}\) is a set of admissible shapes;

Remark: While it is mainly used in structural mechanics in this library, it can also be used indifferently for CFD, design of cooling devices, biology, electromagnetism. This a quite a general framework.

Classically, the resolution of such a problem is done iteratively; each iteration aims to determine a “better” shape than the current solution. Through this iterative process, a descent direction is computed which requires one or several evaluations of the criteria involved in this optimization problem.

One notable issue in this field is that, in general, there is no global minima for this optimization problem. It means that, while you could improve the design by simply deforming the shape without altering the topology, doing so may lead to a falling into a “bad” local minima. To account for topology changes during the optimization, techniques abound in industry and engineering. Here, level set based methods, briefly described thereafter, are used.

## Principles of topology optimization with level set

Let \(\phi: D\rightarrow \mathbb{R}\) be a scalar function and \(\Omega\subset\mathbb{R}^3\), a bounded domain sctrictly included in \(D\). Let \(x\) be a point of \(D\), \(\phi\) is

Strictly negative if \(x\) is part of \(\Omega\)

0 if \(x\) belong to the boundary of \(\Omega\)

Stricly positive, else.

Typically, we consider the signed distance function for \(\phi\) , i.e. the euclidian distance from \(x\) to the boundary of \(\Omega\).

They are in fact 2 main types of level set method arising from this framework and proposed in this platform.

### First approach: Shape capturing with fixed mesh

In the first one, the whole set of admissible shapes \(\mathcal{O}_{ad}\) is assumed to lie within the computational domain \(D\) (e.g. a box) equipped with a fixed mesh. A new shape arising from the iterative optimization process is described by the value of the level set function at the vertices of this very mesh.

As the evaluation of the optimization criteria requires the computation of the PDEs solution by the physical solver, a computational mesh of the domain \(\Omega\) is required (no mesh of the shape is available) To handle this, the original problems are approximated by PDEs on the whole domain \(D\) as a whole instead of \(\Omega\)

### Second approach: Shape tracking with body-fitted meshes

In the second one, the level set method is coupled with a remeshing routine for the reconstruction of a body-fitted mesh.

It means that:

The shape is exactly meshed at each step of the underlying optimization process

The computational mesh of the domain \(\Omega\) is available (no approximation required, unlike the first approach)

In fact, the domain evolution strategy is based on two complementary representations:

The level set description: allows easily arbitrary topology changes (merging/splitting the interfaces)

An explicit mesh discretization: boundary fits smoothly the interface by an explicit discretization of the zero isovalue of the implicit domain

Since the structural interface is known explicitely at each step of the iterative procedure, the body-fitted approach simplifies the evaluation of the mechanical quantities of interest and provide a clear definition of the shape. It means in particular that the mechanical evaluation is not hindered by the underlying topology optimization method; it is in fact completly independant.

Obviously, in comparison with the first approach, a remeshing operation is required to recover a body-fitted mesh.

- 1
Allaire, C. Dapogny, F. Jouve, Shape and topology optimization, in Geometric partial differential equations, part II, A. Bonito and R. Nochetto eds., pp.1-132, Handbook of Numerical Analysis, vol. 22, Elsevier (2021).

- 2
Allaire, Conception optimale de structures, vol. 58 of Mathematiques & Applications (Berlin) [Mathematics & Applications], Springer (2007).