Shape optimization with the level-set-method using local volume constraints

Abstract

This paper presents a method to locally constrain multiple material volume domains for structural optimization with the Level Set Method (LSM). Two different Lagrangian formulations and multiplier update methods are used, for both the global and local problem. The local volume domains can be constrained by both equality and inequality constraints. The optimization objective is compliance minimization for well-posed statically loaded structures. For validation, several example problems are established and solved using the proposed method. Results show that the volume ratios for user established sub-domains can be controlled successfully. The local constraint values are met accurately in the case of equality constraints and remain in their feasible domain in the case of inequality constraints. Optimization results are not significantly hindered by the introduction of local volume constraints for comparable problems.

Appendix: Derivatives

The Lagrangian $\mathcal {L}$ is derived in order to transform the inequality constrained optimization problem into an unconstrained problem:

$$ \mathcal{L}(\phi(\mathbf{x})) = c(\phi(\mathbf{x})) + \lambda \left( g(\phi(\mathbf{x})) + s^{2} \right) $$

(42)

where s is a slack variable which converts the inequality into an equality constraint. The optimum is defined by meeting the KKT optimality conditions.

Let us take the compliance equal to the total strain energy:

$$ c = \mathbf{u}^{T} \cdot \tilde{\mathbf{K}} \cdot \mathbf{u} $$

(43)

The stiffness matrix $\tilde {\mathbf {K}}(\mathbf {\rho })$ is determined as follows:

$$ \tilde{\mathbf{K}}(\mathbf{\rho}) = \bigcup_{e=1}^{N_{e}} \rho_{e}(\phi) \mathbf{K}_{e} $$

(44)

where $\bigcup$ denotes the assembly of element components, N _e is the total number of elements, K _e is the element stiffness matrix and ρ _e is the element density determined by the LSF values. The strain energy density can be determined as follows:

$$ \begin{array}{ll} c &= \sum\limits_{e=1}^{N_{e}} \rho_{e}(\phi) \mathbf{u}^{T}_{e} \cdot \mathbf{K}_{e} \cdot \mathbf{u}_{e} \\[0.3cm] &= \sum\limits_{e=1}^{N_{e}} \tilde{H}\left( \phi(\mathbf{x})\right) \mathbf{u}^{T}_{e} \cdot \mathbf{K}_{e} \cdot \mathbf{u}_{e} \end{array} $$

(45)

The shape derivative of the Lagrangian $\mathcal {L}$ is derived as the Fréchet derivative with respect to ϕ as follows:

$$ \frac{\text{d} \mathcal{L}}{\text{d} {\Omega}} = \left\langle \frac{\text{d} \mathcal{L}}{\text{d} \phi}, \psi \right\rangle $$

(46)

where ψ is the variation of the level set function such that ψ ∈ Ψ. Combining (42) and (46), we get:

$$ \frac{\text{d} \mathcal{L}}{\text{d} {\Omega}} = \frac{\text{d} c}{\text{d} {\Omega}} + \lambda \frac{\text{d} g}{\text{d} {\Omega}} $$

(47)

1.1 A.1 Shape derivative of the compliance

Now let us define the shape derivative of the strain energy density as the Fréchet derivative with respect to ϕ:

$$ \frac{\text{d} c}{\text{d} {\Omega}} = \left\langle \frac{\text{d} c}{\text{d} \phi}, \psi \right\rangle $$

(48)

Now by rule of total derivative, (48) can be rewritten as follows:

$$ \begin{array}{ll} \left\langle \frac{\text{d} c}{\text{d} \phi}, \psi \right\rangle &= \left\langle \frac{\partial c}{\partial \phi}, \psi \right\rangle + \left\langle \frac{\partial c}{\partial \mathbf{u}} \frac{\text{d} \mathbf{u}}{\text{d} \phi}, \psi \right\rangle \\[0.5cm] & = \left\langle \frac{\partial c}{\partial \phi}, \psi \right\rangle + \left\langle \frac{\partial c}{\partial \mathbf{u}}, \mathbf{w} \right\rangle \end{array} $$

(49)

The partial derivatives from (49) are derived as follows:

$$ \begin{array}{ll} \frac{\partial c}{\partial \phi} &= {\sum}_{e=1}^{N_{e}} \tilde{\delta}\left( \phi(\mathbf{x})\right) \mathbf{u}^{T}_{e} \cdot \mathbf{K}_{e} \cdot \mathbf{u}_{e} \\[0.5cm] \frac{\partial c}{\partial \mathbf{u}} &= {\sum}_{e=1}^{N_{e}} \tilde{H}\left( \phi(\mathbf{x})\right) 2 \mathbf{K}_{e} \cdot \mathbf{u}_{e} \end{array} $$

(50)

The Fréchet derivatives for the condition $\tilde {\mathbf {K}} \mathbf {u} - \mathbf {f}^{\text {ext}} = 0$ are as follows:

$$ \begin{array}{ll} \frac{\text{d} \mathbf{f}^{\text{ext}}}{\text{d} {\Omega}} &= 0 \\[0.5cm] \frac{\text{d} \mathbf{a}}{\text{d} {\Omega}} &= \left\langle \frac{\text{d} \mathbf{a}}{\text{d} \phi}, \psi \right\rangle = \left\langle \frac{\partial \mathbf{a}}{\partial \phi}, \psi \right\rangle + \left\langle \frac{\partial \mathbf{a}}{\partial \mathbf{u}}, \mathbf{w} \right\rangle \end{array} $$

(51)

where $\mathbf {a} = \tilde {\mathbf {K}} \mathbf {u}$. The partial derivatives from (51) are derived as follows:

$$ \begin{array}{ll} \frac{\partial \mathbf{a}}{\partial \phi} &= {\sum}_{e=1}^{N_{e}} \tilde{\delta}\left( \phi(\mathbf{x})\right) \mathbf{K}_{e} \cdot \mathbf{u}_{e} \\[0.5cm] \frac{\partial \mathbf{a}}{\partial \mathbf{u}} &= {\sum}_{e=1}^{N_{e}} \tilde{H}\left( \phi(\mathbf{x})\right) \mathbf{K}_{e} \end{array} $$

(52)

From combining (51) and (52) follows:

$$\begin{array}{@{}rcl@{}} \begin{array}{l} \left\langle \frac{\partial \mathbf{a}}{\partial \mathbf{u}}, \mathbf{w} \right\rangle = - \left\langle \frac{\partial \mathbf{a}}{\partial \phi}, \psi \right\rangle {\sum}_{e=1}^{N_{e}} \tilde{H}\left( \phi(\mathbf{x})\right) \mathbf{K}_{e} \cdot \mathbf{w} {\ldots} \\ \hspace{2.6pc} = - {\sum}_{e=1}^{N_{e}} \tilde{\delta}\left( \phi(\mathbf{x})\right) \mathbf{K}_{e} \cdot \mathbf{u}_{e} \psi \end{array} \\ \end{array} $$

(53)

Now combining (49) and (50) and substituting (53), we get the following:

$$\begin{array}{@{}rcl@{}} \begin{array}{rl} \frac{\text{d} c}{\text{d} {\Omega}} =& {\sum}_{e=1}^{N_{e}} \tilde{\delta}\left( \phi(\mathbf{x})\right) \mathbf{u}^{T}_{e} \cdot \mathbf{K}_{e} \cdot \mathbf{u}_{e} \psi \\ &+ \ldots {\sum}_{e=1}^{N_{e}} \tilde{H}\left( \phi(\mathbf{x})\right) 2 \mathbf{K}_{e} \cdot \mathbf{u}_{e} \cdot \mathbf{w} \\ = & {\sum}_{e=1}^{N_{e}} \tilde{\delta}\left( \phi(\mathbf{x})\right) \mathbf{u}^{T}_{e} \cdot \mathbf{K}_{e} \cdot \mathbf{u}_{e} \psi \\&- \ldots {\sum}_{e=1}^{N_{e}} 2 \tilde{\delta}\left( \phi(\mathbf{x})\right) \mathbf{u}^{T}_{e} \cdot \mathbf{K}_{e} \cdot \mathbf{u}_{e} \psi \end{array} \\ \end{array} $$

(54)

This leads to the following relation for the shape derivative of the strain energy density:

$$ \frac{\text{d} c}{\text{d} {\Omega}} = - \sum\limits_{e=1}^{N_{e}} \tilde{\delta}\left( \phi(\mathbf{x})\right) \mathbf{u}^{T}_{e} \cdot \mathbf{K}_{e} \cdot \mathbf{u}_{e} \psi $$

(55)

1.2 A.2 Shape derivative of the global volume constraint

The derivation of the shape derivative of the Volume constraint follows the same procedure as with the compliance. The volume V is expressed as follows:

$$ V(\phi(\mathbf{x})) = \sum\limits_{e=1}^{N_{e}} \tilde{H}\left( \phi(\mathbf{x})\right) $$

(56)

The shape derivative of the constraint g(ϕ(x)) is derived as follows:

$$ \frac{\text{d} g}{\text{d} {\Omega}} = \left\langle \frac{\text{d} g}{\text{d} \phi}, \psi \right\rangle = \left\langle \frac{\partial g}{\partial \phi}, \psi \right\rangle $$

(57)

This leads to the following definition of the constraint shape derivative:

$$ \frac{\text{d} g}{\text{d} {\Omega}} = \sum\limits_{e=1}^{N_{e}} \tilde{\delta}\left( \phi(\mathbf{x})\right) \psi $$

(58)

1.3 A.3 Shape derivative of the Lagrangian

The shape derivative of the Lagrangian can now be defined by substituting the results from (55) and (58) into (47) and taking the derivative with respect to the LSF values:

$$ \frac{\text{d} \mathcal{L}}{\text{d} \phi} = \sum\limits_{e=1}^{N_{e}} \tilde{\delta}\left( \phi(\mathbf{x})\right) \left[ - \mathbf{u}^{T}_{e} \cdot \mathbf{K}_{e} \cdot \mathbf{u}_{e} + \lambda \right] $$

(59)

The boundary normal velocity V _N (x, t) can now be defined as:

$$ V_{N}(\mathbf{x},t) = \frac{\text{d} \mathcal{L}}{\text{d} \phi} $$

(60)

The Lagrangian formulation of the optimization problem contains a slack variable s to account for the inequality constraint. The switching condition from the KKT conditions can be satisfied in two ways:

• λ = 0: This implies that the inequality condition is inactive, meaning that the suggested optimum features a lower volume fraction than $V_{\max }$. However, for problems with fixed boundary conditions and fixed loads, not considering body forces, the compliance is minimized when the design domain is completely filled with material. This fact makes this case physically irrelevant.

• s = 0: Zero slack implies an active inequality constraint, g(ϕ) = 0, indicating that $V(\phi ) = V_{\max }$ for the optimum solution.

These cases show that the optimum will always lie at $V(\phi ) = V_{\max }$. As a result of this, one could define the volume constraint in (21) as an equality constraint. The slack variable s is now redundant and omitted.