Creating geometrically robust designs for highly sensitive problems using topology optimization

Abstract

Resonance and wave-propagation problems are known to be highly sensitive towards parameter variations. This paper discusses topology optimization formulations for creating designs that perform robustly under spatial variations for acoustic cavity problems. For several structural problems, robust topology optimization methods have already proven their worth. However, it is shown that direct application of such methods is not suitable for the acoustic problem under consideration. A new double filter approach is suggested which makes robust optimization for spatial variations possible. Its effect and limitations are discussed. In addition, a known explicit penalization approach is considered for comparison. For near-uniform spatial variations it is shown that highly robust designs can be obtained using the double filter approach. It is finally demonstrated that taking non-uniform variations into account further improves the robustness of the designs.

Appendix:

1.1 Derivation of sensitivities for the double filter

The sensitivities, $\frac {\text {d} {\Phi }}{\text {d} \xi _{i}}$, for the double filter may be derived as follows:

• 1. Apply the chain rule for calculating the sensitivities.

  $$ \frac{\text{d} {\Phi}}{\text{d} \xi_{i}} = \sum\limits_{j, k, l, h}\frac{\partial \tilde{\xi_{l}}}{\partial \xi_{i}} \frac{\partial \bar{\tilde{\xi_{h}}}}{\partial \tilde{\xi_{l}}} \frac{\partial \tilde{\bar{\tilde{\xi_{k}}}}}{\partial \bar{\tilde{\xi_{h}}}} \frac{\partial \bar{\tilde{\bar{\tilde{\xi_{j}}}}}}{\partial \tilde{\bar{\tilde{\xi_{k}}}}} \frac{\text{d} {\Phi}}{\text{d} \bar{\tilde{\bar{\tilde{\xi}}}}_{j}}. $$

  (32)

• 2. Eliminate two sums using the fact that $\frac {\partial \bar {\tilde {\xi _{h}}}}{\partial \tilde {\xi _{l}}} = 0 \ \forall \ l \neq h$ and that $\frac {\partial \bar {\tilde {\bar {\tilde {\xi _{j}}}}}}{\partial \tilde {\bar {\tilde {\xi _{k}}}}} = 0 \ \forall \ k \neq j$ due to the locality of (23).

  $$ \frac{\text{d} {\Phi}}{\text{d} \xi_{i}} = \sum\limits_{j} \sum\limits_{h} \frac{\partial \tilde{\xi_{h}}}{\partial \xi_{i}} \frac{\partial \bar{\tilde{\xi_{h}}}}{\partial \tilde{\xi_{h}}} \frac{\partial \tilde{\bar{\tilde{\xi_{j}}}}}{\partial \bar{\tilde{\xi_{h}}}} \frac{\partial \bar{\tilde{\bar{\tilde{\xi_{j}}}}}}{\partial \tilde{\bar{\tilde{\xi_{j}}}}} \frac{\text{d} {\Phi}}{\text{d} \bar{\tilde{\bar{\tilde{\xi}}}}_{j}} \ \ i \in \lbrace 1, 2, ..., N \rbrace. $$

  (33)

• 3. Utilize that $\tilde {\xi _{h}}$ only depends on the design variables ξ _i within the density filter radius reducing the sum over h significantly. The same argument applied to $\tilde {\bar {\tilde {\xi _{j}}}}$ and $\bar {\tilde {\xi _{h}}}$ reduces the sum over j. The set of indices for the dependent variables are denoted, $\mathcal {B}_{e, i}$ and $\mathcal {B}_{e, h}$ respectively. The sensitivities now take the form,

  $$ \frac{\text{d} {\Phi}}{\text{d} \xi_{i}} = \sum\limits_{j \in \mathcal{B}_{e, h}} \sum\limits_{h \in \mathcal{B}_{e, i}} \frac{\partial \tilde{\xi_{h}}}{\partial \xi_{i}} \frac{\partial \bar{\tilde{\xi_{h}}}}{\partial \tilde{\xi_{h}}} \frac{\partial \tilde{\bar{\tilde{\xi_{j}}}}}{\partial \bar{\tilde{\xi_{h}}}} \frac{\partial \bar{\tilde{\bar{\tilde{\xi_{j}}}}}}{\partial \tilde{\bar{\tilde{\xi_{j}}}}} \frac{\text{d} {\Phi}}{\text{d} \bar{\tilde{\bar{\tilde{\xi}}}}_{j}}. $$

  (34)

• 4. Rewriting the expression gives,

  $$ \frac{\text{d} {\Phi}}{\text{d} \xi_{i}} = \sum\limits_{h \in \mathcal{B}_{e, i}} \frac{\partial \tilde{\xi_{h}}}{\partial \xi_{i}} \frac{\partial \bar{\tilde{\xi_{h}}}}{\partial \tilde{\xi_{h}}} \left[ \sum\limits_{j \in \mathcal{B}_{e, h}} \frac{\partial \tilde{\bar{\tilde{\xi_{j}}}}}{\partial \bar{\tilde{\xi_{h}}}} \frac{\partial \bar{\tilde{\bar{\tilde{\xi_{j}}}}}}{\partial \tilde{\bar{\tilde{\xi_{j}}}}} \frac{\text{d} {\Phi}}{\text{d} \bar{\tilde{\bar{\tilde{\xi}}}}_{j}} \right]. $$

  (35)

• 5. For a given h the expression in the bracket in 35 only depends on j. Thus we may define,

  $$ {\Delta} {\Phi}_{h} = \sum\limits_{j \in \mathcal{B}_{e, h}} \frac{\partial \tilde{\bar{\tilde{\xi_{j}}}}}{\partial \bar{\tilde{\xi_{h}}}} \frac{\partial \bar{\tilde{\bar{\tilde{\xi_{j}}}}}}{\partial \tilde{\bar{\tilde{\xi_{j}}}}} \frac{\text{d} {\Phi}}{\text{d} \bar{\tilde{\bar{\tilde{\xi}}}}_{j}}, \ \ h \in \lbrace 1, 2, ..., N \rbrace. $$

  (36)

  This illustrates that the application of the double filter simply corresponds to applying the single filter twice. $\square$

1.2 Application of robust approach for frequency bands

Single frequency problems have been the focus of the paper due to the high sensitivity in the performance of the optimized designs under geometric variations. In this section we provide an example showing the method applied for a band of frequencies as well. As will be demonstrated, this problem is far less sensitive towards geometric variations in the design. A requirement for considering optimization for a band of frequencies for the cavity problem is that a small amount of damping is added to the model problem to avoid problems caused by resonances in the frequency band of interest. The need for damping has nothing to do with the double filter or the robust approach and must be added regardless of the optimization strategy. Mass proportional damping is introduced by adding the term “$\alpha _{\text {damp}} \ \text {i} \ \hat {\omega } \hat {p}$” to (7) where α _damp = 0.01 is the damping factor.

In the following we consider the model problem presented in Fig. 14 and seek to minimize the mean of the average sound pressure in Ω _OP over a 1/3 octave frequency band, f _b≈[61.85, 77.92] Hz, centered at, f _c = 62.5 Hz. The objective function may thus be stated as,

$$ {\Phi(\xi) = \frac{1}{N_{i}} {\sum}_{i = 1}^{N_{i}} \frac{1}{A_{\text{OP}}} \int | \hat{p}(\bar{\tilde{\bar{\tilde{\xi}}}}, f_{i}) |^{2} d\boldsymbol{\Omega}_{\text{OP}}, } $$

(37)

where f _i are the frequencies optimized for and N _i is the number of frequencies. An optimization is performed using the standard approach with the double filter and the second projection at η ₂ = 0.5. For comparison an optimization is performed using the robust approach with double filtering and five realizations of the second projection level, η ₂∈{0.3, 0.4, 0.5, 0.6, 0.7}. For both cases we use N _i = 20 and consider equidistant frequencies in f _b including both endpoints. For the PDE problem we use a pure FEM discretization with $\mathcal {N}_{x} = 216, \ \mathcal {N}_{y} = 108$ elements. A filter range of R ₁ = 5 elements is used. Figure 23 shows the resulting SPS and SPSP variables for the two optimizations.

Fig. 23

(i) SPS and (ii) PSPS design variables obtained using the a) standard and b) robust approach for the 1/3 octave frequency band optimization

Figure 24 shows the mean of the average sound pressure level in Ω _OP over the 1/3 octave frequency band, $\langle L_{\hat {p}} \rangle _{\boldsymbol {\Omega }_{\text {OP}}, f_{\text {b}}}$, scaled by the same quantity in Ω _OP for the empty cavity, $\langle L_{\hat {p}} \rangle _{\boldsymbol {\Omega }_{\text {OP}_{\mathbf {ED}}}, f_{\text {b}}} \approx 89.79$ dB, as a function of projection level η ₂, for both the standard and the robust approach. The mean over the frequency is calculated using 100 equidistant frequencies in f _b. This variation in projection level corresponds to a near-uniform erosion/dilation of 1 element or approximately 8 cm in the design. The presented results have been evaluated with the same amount of damping as the one used in the optimization.

Fig. 24

$\langle L_{\hat {p}} \rangle _{\boldsymbol {\Omega }_{\text {OP}}, f_{\text {b}}} / \langle L_{\hat {p}} \rangle _{\boldsymbol {\Omega }_{\text {OP}_{\mathbf {ED}}, f_{\text {b}}}}$ for the designs in Fig. 23 under near-uniform variations imposed by varying η ₂. The performance of the designs is seen to be almost constant under the prescribed uniform variations

From the figure it is clearly observed that both the standard and robust approach produce results which do not show any significant sensitivity towards uniform erosion or dilation of the design. It is noted that the robust approach produces a design with better performance. This is likely due to the additional restrictions on the optimization when using the robust approach which eliminates the local minimum trapping the optimization performed with the standard approach.

An investigation of the sensitivity of the performance under non-uniform geometric variations for the design optimized using the standard approach is also performed. Here it is shown that, just as for the uniform geometric perturbations, the sensitivity drops significantly when considering a band of frequencies compared to a single frequency. Twenty five non-uniform geometric variations are applied as described in Section 11 using A = 6, B ∈ {2, 3.5, 5, 6.5, 8}, C ∈ 2π⋅{1, 2, 3, 4, 5}, $\eta _{\min } = 0.3, \eta _{\max } = 0.7$. Figure 25i show the sensitivity of the performance under the twenty five non-uniform geometric variations for the average response over f _b while Fig. 25ii show the performance sensitivity under the same twenty five non-uniform geometric variations for the single frequency f _s = 70.15 Hz.

Fig. 25

Scaled performance for the design in Fig. 23. a) under non-uniform variations imposed through η ₂(x). The performance of the design is seen to be less sensitive for a frequency band (i) then for a single frequency (ii). The red line marks the performance of the nominal design while the black line with circles mark the performance for the perturbed design

From the figure it is clearly seen that the average response over f _b is far less sensitive to geometric perturbations than when only considering f _s .