Phase-field approaches in Topology Optimization

In a Topology Optimization problem, we often have a design variable in [-1, 1], representing how much material do we place at each point. A formal presentation of the problem is as follows

Alternatively, we can create an evolution equation for , such that the evolution process is equivalent with the above optimization process. In phase-field modeling, we have an evolution equation (either Allen-Cahn or Cahn-Hilliard type), and the goal is to minimize free energy. To solve the current Topology Optimization with phase-field modeling strategy, we adjust the free energy by adding the cost function. The volume constraint is accounted via a penalty term inspired by the augmented Lagrangian. So the equivalent evolution equation can be written as

Here, the math seems a little bit overwhelming, but the evolution equation is simply a diffusion-reaction equation (also an Allen-Cahn equation). The source term is composed of the double-well function (commonly used in phase-field), the cost function, and the penalty of volume contraint. The coefficients are adjustable variables that control the driving force of each piece of the evolution equation.

Now we only miss the PDE constraint. This is accomplished by a sensitivity analysis (adjoint problem). The derivation of the adjoint problem is skipped here. A good reference is dofin-adjoint documentation.

algorithm

The full solving algorithm is presented as follows.

This algorithm is implemented in an open-source code FEniCS.

simulation examples

The first working example is a standard one: optimizing the compliance of a cantilever beam. Various tests were carried out for this simple case to validate the algorithm and clarify the coefficient parameters.

The solution in 3D is also possible. Finer mesh ends with geometry with more details.

We can also solve the thermo-mechanical problem shown below. We observe a significant geometry change as the temperature increase becomes larger. In other words, when thermal expansion is stronger, we need a thicker lower strut to maintain stiffness. This result is consistent with the result in literature.

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