Shape and topology optimization based on the convected level set method

Abstract

The aim of this research is to construct a shape optimization method based on the convected level set method, in which the level set function is defined as a truncated smooth function obtained by using a sinus filter based on a hyperbolic tangent function. The local property of the hyperbolic tangent function dramatically reduces the generation of red the error between the specified profile of the hyperbolic tangent function and the level set function that is updated using a time evolution equation. In addition, the small size of the error facilitates the use of convective reinitialization, whose basic idea is that the reinitialization is embedded in the time evolution equation, whereas such treatment is typically conducted in a separate calculation in conventional level set methods. The convected level set method can completely avoid the need for additional calculations when performing reinitialization. The validity and effectiveness of our presented method are tested with a mean compliance minimization problem and a problem for the design of a compliant mechanism.

Appendix A Update scheme for the level set function based on the method of characteristics

Here, we discuss the basic idea of the method of characteristics and its application to solve the time evolution equation in (20).

For this brief explanation, we first deal with the simple advection equation (15). Consider a vector field X that satisfies the following ordinary differential equation,

$$ \frac{\partial\mathbf{X}(t)}{\partial t}=\mathbf{V}(\mathbf{X}(t),t)\ $$

(A.1)

The time derivative of the composite function ψ∘X(t) = ψ(X(t),t) is obtained as follows:

$$\begin{array}{@{}rcl@{}} \frac{\text{d}\psi(\mathbf{X}(t),t)}{\text{d} t}&=&\frac{\partial\psi(\mathbf{X}(t),t)}{\partial t}+\mathbf{V}\cdot\nabla\psi(\mathbf{X}(t),t) \\ &=&\frac{\text{D}\psi(\mathbf{X}(t),t)}{\text{D} t}, \end{array} $$

(A.2)

where Dψ/Dt represents the material derivative of ψ and corresponds to the left-hand side of (15). Based on the backward Euler scheme and using a time step Δt, this can be discretized as

$$\begin{array}{@{}rcl@{}} \frac{\text{D}\psi(\mathbf{X}(t),t)}{\text{D} t}&=&\frac{\psi(\mathbf{X}(t),t)-\psi(\mathbf{X}(t-\Delta t),t-\Delta t)}{\Delta t}\\ &&+O(\Delta t). \end{array} $$

(A.3)

We now assume that the discrete time is defined as 0<t ₁<⋯<t _n <⋯<t _N with t _n = nΔt, and consider the following initial value problem:

$$ \left\{\begin{array}{l} \cfrac{\partial\mathbf{X}(t)}{\partial t}=\mathbf{V}(\mathbf{X}(t),t)\ \text{ for }t\in(t_{n-1},t_{n}) \\ \mathbf{X}(t_{n})=\mathbf{x}. \end{array}\right. $$

(A.4)

Using an approximate solution, X ⁿ(x): = x−V(x,t _n )Δt, we obtain the following relation:

$$ \mathbf{X}(t_{n-1})-\mathbf{X}^{n}(\mathbf{x})=O(\Delta t^{2}). $$

(A.5)

Consequently, (A.3) can be reformulated as

$$ \frac{\text{D}\psi(\mathbf{x},t_{n})}{\text{D} t}=\frac{\psi(\mathbf{x},t_{n})-\psi(\mathbf{X}^{n}(\mathbf{x}),t_{n-1})}{\Delta t}+O(\Delta t). $$

(A.6)

Using a test function $\tilde {\psi }$, the notation ψ ⁿ: = ψ(⋅,t _n ), and the scalar product (⋅,⋅) in L ²(D), the material derivative of ψ ⁿ can be approximated as follows:

$$ \left( \frac{\text{D}\psi^{n}}{\text{D} t},\tilde{\psi} \right)\approx \left( \frac{\psi^{n}-\psi^{n-1}\circ\mathbf{X}^{n}}{\Delta t},\tilde{\psi}\right), $$

(A.7)

where ψ and $\tilde {\psi }$ are chosen from a finite element space. Due to the definition of V = v n with n=∇ψ/|∇ψ|, it should be noted that we need to assume that V(x,t _n )≈V(ψ ⁿ⁻¹), to avoid the nonlinearity in (A.7).

Equation (A.7) can be only used for solving (15), so it must be expanded in order to solve (20) in which the convection velocity is defined as (V−λsign(ψ)n) instead of only V as it is in (15). Considering the source term λsign(ψ)G(ψ), (20) can be approximated as

$$\begin{array}{@{}rcl@{}} &&\left( \frac{\text{D}\psi^{n}}{\text{D} t}-\lambda\text{sign}(\psi^{n})G(\psi^{n}),\tilde{\psi} \right)\\ &&\quad\approx \left( \frac{\psi^{n}-\psi^{n-1}\circ\mathbf{X}^{n}}{\Delta t}-\lambda\text{sign}(\psi^{n-1})G(\psi^{n-1}),\tilde{\psi}\right),\\ \end{array} $$

(A.8)

where X ⁿ = x−(V(x,t _n )−λsign(ψ ⁿ⁻¹)n)Δt. Note that the use of sign(ψ ⁿ⁻¹) and G(ψ ⁿ⁻¹) in the right-hand side of (A.8) is necessary to avoid the nonlinearity.