Level Set Method

In the level set method, the boundary of the design is implicitly represented as the isosurface (the zero level set) of a function $ϕ (x)$ defined on the finite element mesh, as shown in Figure 1. In the level set model, the domain is defined based on the value of the level set function:

{\begin{matrix} ϕ (x) > 0 : x \in Ω / \partial Ω \\ ϕ (x) = 0 : x \in \partial Ω \\ ϕ (x) < 0 : x \in D / Ω \end{matrix}

Where, $D$ specifies the design domain; $Ω$ represents the material region, $\partial Ω$ stands for the boundary and $D / Ω$ represents the region with no material. The dynamic motion of the boundary is governed by the so-called, level set equation:

\frac{\partial ϕ}{\partial t} = v_{n} | \nabla ϕ |

Where,

v_{n}

is the normal velocity and

| \nabla ϕ |

is the norm of the gradient of the level set function. The basic idea of the level set equation is to map the boundary evolution into an evolution of the level set function

ϕ (x)

level_set_rep — Figure 1. 2D Design and its Corresponding Level Set Representation

Level-set based topology optimization can be considered as advanced shape optimization. It works like conventional shape optimization, where the design is changed by moving the boundary, while at the same time topological changes, such as boundary emerging and splitting can be handled naturally.