Topology and shape optimization of dissipative and hybrid mufflers

Abstract

This article presents a topology optimization (TO) method developed for maximizing the acoustic attenuation of a perforated dissipative muffler in the targeted frequency range by optimally distributing the absorbent material within the chamber. The finite element method (FEM) is applied to the wave equation formulated in terms of acoustic pressure (chamber) and velocity potential (central duct, due to the existence of thermal gradients and mean flow) in order to evaluate the acoustic performance of the noise control device in terms of transmission loss (TL). Sound propagation through the chamber fibrous material is modeled considering complex equivalent acoustic properties, which vary spatially not only as a function of temperature but also as a function of the filling density, since non-homogeneous density distributions are considered. The acoustic coupling at the perforated duct is performed by introducing a coordinate-dependent equivalent impedance. The objective function to maximize is expressed as the mean TL in the targeted frequency range. The sensitivities of this function with respect to the filling density of each element in the chamber are evaluated following the standard adjoint method. The method of moving asymptotes (MMA) is used to update the design variables at each iteration of the TO process, keeping the weight of absorbent material equal or lower than a given value, while maximizing attenuation. Additionally, several particular designs inferred from the topology optimization results are analyzed. For example, the sizing optimization of a number of rings is carried out simultaneously with the aforementioned TO process (density layout). A reactive chamber is added in order to evaluate the TL of a hybrid muffler and its shape optimization is also carried out simultaneously with the aforementioned TO. Results show an increase in the muffler’s mean TL at target frequencies, for all cases under study, while the amount of absorbent material used is maintained or even reduced.

Appendix: Differentiation of the global matrix κ with respect to the design variables

In this appendix, the expressions of each of the components of κ are differentiated analytically with respect to each of the design variables considered in this study.

1.1 A.1 Calculation of ∂κ/∂ρ_b

Differentiating the global matrix κ in (54) with respect to the bulk density ρ_b assigned to the elements $e=1,...,N^{e}_{\rho }$, one obtains:

$$ \frac{\partial \boldsymbol{\kappa}}{\partial \rho_{b}} = \frac{\partial \mathbf{K}}{\partial \rho_{b}} +j\omega \frac{\partial \mathbf{C}}{\partial \rho_{b}} -\omega^{2} \frac{\partial \mathbf{M}}{\partial \rho_{b}} \ , $$

(61)

where, according to (24), (25) and (33–37), the terms considered in order to build the global matrices are:

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{K}_{m}}{\partial \rho_{b}} &=& \displaystyle\sum\limits_{e=1}^{N_{\rho}^{e}} {\int}_{\mathit{\Omega}_{\rho}^{e}} \frac{-\partial \rho_{m} / \partial \rho_{b}}{{\rho_{b}^{2}}} \left( \nabla \mathbf{N} \right)^{T} \left( \nabla \mathbf{N} \right) \text{ d} \mathit{\Omega} \ , \end{array} $$

(62)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{M}_{m}}{\partial \rho_{b}} &=& \displaystyle\sum\limits_{e=1}^{N_{\rho}^{e}} {\int}_{\mathit{\Omega}_{\rho}^{e}} \left( - \frac{\partial \rho_{m} / \partial \rho_{b} }{{\rho_{m}^{2}} {c_{m}^{2}}} - 2 \frac{\partial c_{m} / \partial \rho_{b} }{\rho_{m} {c_{m}^{3}}} \right) \mathbf{N}^{T} \mathbf{N} \text{ d} \mathit{\Omega} \end{array} $$

(63)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{K}_{aa}^{Z_{p}}}{\partial \rho_{b}} &=& \sum\limits_{e=1}^{N_{\rho}^{e}} {\int}_{\mathit{\Gamma}_{\rho}^{e} \cap \mathit{\Gamma}_{p}} -\frac{{\rho_{0}^{2}} U_{mf} \partial \tilde{Z}_{p} / \partial \rho_{b}}{\tilde{Z}_{p}^{2}} \mathbf{N}^{T} \frac{\partial \mathbf{N}}{\partial x} \text{ d} \mathit{\Gamma} \ , \end{array} $$

(64)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{K}_{am}^{Z_{p}}}{\partial \rho_{b}} &=& \sum\limits_{e=1}^{N_{\rho}^{e}} {\int}_{\mathit{\Gamma}_{\rho}^{e} \cap \mathit{\Gamma}_{p}} -\frac{\rho_{0} \partial \tilde{Z}_{p} / \partial \rho_{b}}{\tilde{Z}_{p}^{2}} \mathbf{N}^{T} \mathbf{N} \text{ d} \mathit{\Gamma} \ , \end{array} $$

(65)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{C}_{aa}^{Z_{p}}}{\partial \rho_{b}} &= &\sum\limits_{e=1}^{N_{\rho}^{e}} {\int}_{\mathit{\Gamma}_{\rho}^{e} \cap \mathit{\Gamma}_{p}} -\frac{{\rho_{0}^{2}} \partial \tilde{Z}_{p} / \partial \rho_{b}}{\tilde{Z}_{p}^{2}} \mathbf{N}^{T} \mathbf{N} \text{ d} \mathit{\Gamma} \ , \end{array} $$

(66)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{C}_{mm}^{Z_{p}}}{\partial \rho_{b}} &=& \sum\limits_{e=1}^{N_{\rho}^{e}} {\int}_{\mathit{\Gamma}_{\rho}^{e} \cap \mathit{\Gamma}_{p}}-\frac{ \partial \tilde{Z}_{p} / \partial \rho_{b} }{\tilde{Z}_{p}^{2}} \mathbf{N}^{T} \mathbf{N} \text{ d} \mathit{\Gamma} \ , \end{array} $$

(67)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{C}_{ma}^{Z_{p}}}{\partial \rho_{b}} &=& \sum\limits_{e=1}^{N_{\rho}^{e}} {\int}_{\mathit{\Gamma}_{\rho}^{e} \cap \mathit{\Gamma}_{p}} -\frac{\rho_{0} U_{mf} \partial \tilde{Z}_{p} / \partial \rho_{b} }{\tilde{Z}_{p}^{2}} \mathbf{N}^{T} \frac{\partial \mathbf{N}}{\partial x} \text{ d} \mathit{\Gamma} \ , \end{array} $$

(68)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{M}_{ma}^{Z_{p}}}{\partial \rho_{b}} &=& \sum\limits_{e=1}^{N_{\rho}^{e}} {\int}_{\mathit{\Gamma}_{\rho}^{e} \cap \mathit{\Gamma}_{p}} -\frac{\rho_{0} \partial \tilde{Z}_{p} / \partial \rho_{b} }{\tilde{Z}_{p}^{2}} \mathbf{N}^{T} \mathbf{N} \text{ d} \mathit{\Gamma} \ . \end{array} $$

(69)

Using the equivalent acoustic properties detailed in (5) and (6), the derivatives of ρ_m and c_m with respect to ρ_b can be obtained as:

$$ \begin{array}{@{}rcl@{}} \frac{\partial \rho_{m} }{\partial \rho_{b}} (\mathbf{x}) \!&=&\! \frac{1}{c_{m} (\mathbf{x})^{2}} \left( \frac{\partial Z_{m}}{\partial \rho_{b}} (\mathbf{x})c_{m} (\mathbf{x}) - Z_{m} (\mathbf{x})\frac{\partial c_{m}}{\partial \rho_{b}} (\mathbf{x})\right), \end{array} $$

(70)

$$ \begin{array}{@{}rcl@{}} \frac{\partial c_{m}}{\partial \rho_{b}} (\mathbf{x}) &=& -\frac{1}{k_{m} (\mathbf{x})^{2}}\frac{\partial k_{m}}{\partial \rho_{b}} (\mathbf{x}) \ , \end{array} $$

(71)

whereas according to (3) and (4), it can be obtained:

$$ \begin{array}{@{}rcl@{}} \frac{\partial Z_{m} }{\partial \rho_{b}} (\mathbf{x})&=&Z_{0} (\mathbf{x})\left( 1+a_{5} a_{6}\xi (\mathbf{x})^{a_{6}-1} - j a_{7} a_{8} \xi (\mathbf{x})^{a_{8}-1}\right) \frac{\partial \xi }{\partial \rho_{b}} (\mathbf{x}) \ ,\\ \end{array} $$

(72)

$$ \begin{array}{@{}rcl@{}} \frac{\partial k_{m} }{\partial \rho_{b}} (\mathbf{x})&=&k_{0} (\mathbf{x})\left( 1+a_{3} a_{4} \xi (\mathbf{x})^{a_{4}-1} - j a_{1} a_{2}\xi (\mathbf{x})^{a_{2}-1}\right) \frac{\partial \xi }{\partial \rho_{b}} (\mathbf{x}) \ .\\ \end{array} $$

(73)

Taking into account the thermal effects described in Christie’s power law recalled in (2), the derivative of the frequency parameter with respect to ρ_b is

$$ \frac{\partial \xi }{\partial \rho_{b}} (\mathbf{x}){}={}\frac{-A_{1} A_{2} \rho_{b} \left( \mathbf{x}\right)^{A_{2}-1}\rho_{0} (\mathbf{x})f}{R^{2} (\mathbf{x})} {}\left( \frac{T(\mathbf{x})+273.15}{T_{0}+273.15} \right)^{0.6}. $$

(74)

On the other hand, differentiating (40), $\partial \tilde {Z}_{p} / \partial \rho _{b}$ can be obtained as:

$$ \frac{\partial \tilde{Z}_{p} (x)}{\partial \rho_{b}}= Z_{0}(x) \frac{ j 0.425 k_{0} (x) d_{h} / \rho_{0} (x) F (\sigma)}{\sigma} \frac{\partial \rho_{m}}{\partial \rho_{b}} (\mathbf{x})\ . $$

(75)

1.2 A.2 Calculation of ∂κ/∂L_x

In Section 5.3, rings of absorbent material are defined as areas with constant ρ_b, and the dimensions of these are also subject to modification. Cartesian element grids are implemented within each ring. For the elements within a certain ring, $e=1,...,{N^{e}_{m}}$, and the elements within the corresponding duct zone underneath, $e=1,...,{N^{e}_{a}}$, the velocity field at the element integration points due to the modification of the ring length L_x must be taken into account during the computation of the terms listed below:

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{K}_{a}}{\partial L_{x}} &=& \sum\limits_{e=1}^{{N_{a}^{e}}} {\int}_{{\mathit{\Omega}_{a}^{e}}} \rho_{0} \left( \left( \frac{\partial\left( \nabla \mathbf{N} \right)^{T}}{\partial L_{x}} \mathbf{M} \left( \nabla \mathbf{N} \right) + \left( \nabla \mathbf{N} \right)^{T} \mathbf{M} \frac{\partial \left( \nabla \mathbf{N} \right)}{\partial L_{x}} \right) \text{d}\mathit{\Omega} \right. \\ & &\left. + \left( \nabla \mathbf{N} \right)^{T} \mathbf{M} \left( \nabla \mathbf{N} \right) \frac{\partial \left( \text{d}\mathit{\Omega} \right)}{\partial L_{x}} \right) \ , \end{array} $$

(76)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{M}_{a}}{\partial L_{x}} &=& \sum\limits_{e=1}^{{N_{a}^{e}}} {\int}_{{\mathit{\Omega}_{a}^{e}}} \frac{\rho_{0} \mathbf{N}^{T} \mathbf{N} }{{c_{0}^{2}}} \frac{\partial \left( \text{d}\mathit{\Omega} \right)}{\partial L_{x}} \ , \end{array} $$

(77)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{K}_{m}}{\partial L_{x}} &=& \displaystyle{\sum}_{e=1}^{{N_{m}^{e}}} {\int}_{{\mathit{\Omega}_{m}^{e}}} \frac{1}{\rho_{m}} \left( \left( \frac{\partial \left( \nabla \mathbf{N} \right)^{T}}{\partial L_{x}} \left( \nabla \mathbf{N} \right) + \left( \nabla \mathbf{N} \right)^{T} \frac{\partial\left( \nabla \mathbf{N} \right)}{\partial L_{x}} \right) \text{d}\mathit{\Omega} \right. \\ && \left. + \left( \nabla \mathbf{N} \right)^{T} \left( \nabla \mathbf{N} \right) \frac{\partial \left( \text{d}\mathit{\Omega} \right)}{\partial L_{x}} \right) \ , \end{array} $$

(78)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{M}_{m}}{\partial L_{x}} & =& \displaystyle{\sum}_{e=1}^{{N_{m}^{e}}} {\int}_{{\mathit{\Omega}_{m}^{e}}} \frac{\mathbf{N}^{T} \mathbf{N}}{\rho_{m} {c_{m}^{2}}} \frac{\partial\left( \text{d}\mathit{\Omega} \right)}{\partial L_{x}} \ , \end{array} $$

(79)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{K}_{aa}^{Z_{p}}}{\partial L_{x}} &=& {\sum}_{e=1}^{{N_{a}^{e}}} {\int}_{{\mathit{\Gamma}_{a}^{e}} \cap \mathit{\Gamma}_{p}} \frac{{\rho_{0}^{2}} U_{mf} \mathbf{N}^{T}}{\tilde{Z}_{p}} \left( \frac{\partial\left( \partial \mathbf{N}/\partial x \right)}{\partial L_{x}} \text{d} \mathit{\Gamma} + \frac{\partial \mathbf{N}}{\partial x} \frac{\partial\left( \text{d} \mathit{\Gamma} \right)}{\partial L_{x}} \right) \ ,\\ \end{array} $$

(80)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{K}_{am}^{Z_{p}}}{\partial L_{x}} &=& {\sum}_{e=1}^{{N_{a}^{e}}} {\int}_{{\mathit{\Gamma}_{a}^{e}} \cap \mathit{\Gamma}_{p}} \frac{\rho_{0}\mathbf{N}^{T} \mathbf{N} }{\tilde{Z}_{p}} \frac{\partial\left( \text{d} \mathit{\Gamma} \right)}{\partial L_{x}} \ , \end{array} $$

(81)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{C}_{aa}^{Z_{p}}}{\partial L_{x}} &=& {\sum}_{e=1}^{{N_{a}^{e}}} {\int}_{{\mathit{\Gamma}_{a}^{e}} \cap \mathit{\Gamma}_{p} } \frac{{\rho_{0}^{2}} \mathbf{N}^{T} \mathbf{N}}{\tilde{Z}_{p}} \frac{\partial\left( \text{d} \mathit{\Gamma} \right)}{\partial L_{x}} \ , \end{array} $$

(82)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{C}_{mm}^{Z_{p}}}{\partial L_{x}} &= &{\sum}_{e=1}^{{N_{m}^{e}}} {\int}_{{\mathit{\Gamma}_{m}^{e}} \cap \mathit{\Gamma}_{p} } \frac{\mathbf{N}^{T} \mathbf{N} }{\tilde{Z}_{p}} \frac{\partial\left( \text{d} \mathit{\Gamma} \right)}{\partial L_{x}} \ , \end{array} $$

(83)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{C}_{ma}^{Z_{p}}}{\partial L_{x}} &= &{\sum}_{e=1}^{{N_{m}^{e}}} {\int}_{{\mathit{\Gamma}_{m}^{e}} \cap \mathit{\Gamma}_{p}} \frac{\rho_{0} U_{mf} \mathbf{N}^{T}}{\tilde{Z}_{p}} \!\left( \frac{\partial \left( \partial \mathbf{N}/\partial x \right)}{\partial L_{x}} \text{d} \mathit{\Gamma} + \frac{\partial \mathbf{N}}{\partial x} \frac{\partial\left( \text{d} \mathit{\Gamma} \right)}{\partial L_{x}} \right) , \end{array} $$

(84)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{M}_{ma}^{Z_{p}}}{\partial L_{x}} &=& {\sum}_{e=1}^{{N_{m}^{e}}} {\int}_{{\mathit{\Gamma}_{m}^{e}} \cap \mathit{\Gamma}_{p}} \frac{\rho_{0} \mathbf{N}^{T} \mathbf{N}}{\tilde{Z}_{p}} \frac{\partial\left( \text{d} \mathit{\Gamma} \right) }{\partial L_{x}} \ . \end{array} $$

(85)

1.3 A.3 Calculation of ∂κ/∂L_r

In order to compute the derivative of the chamber terms with respect to the chamber dimension L_r = R_c − R_t (see Figs. 3 and 17), the velocity field generated by ∂L_r must be taken into account, but also the axisymmetric integration effect must be considered when computing $\partial \left (\text {d}\mathit {\Omega } \right ) / \partial L_{r}$. The terms are given by the expressions:

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{K}_{m}}{\partial L_{r}} &=& \displaystyle\sum\limits_{e=1}^{{N_{m}^{e}}} {\int}_{{\mathit{\Omega}_{m}^{e}}} \frac{1}{\rho_{m}} \left( \left( \frac{\partial \left( \nabla \mathbf{N} \right)^{T}}{\partial L_{r}} \left( \nabla \mathbf{N} \right) + \left( \nabla \mathbf{N} \right)^{T} \frac{\partial\left( \nabla \mathbf{N} \right)}{\partial L_{r}} \right) \text{d}\mathit{\Omega} \right.\\ && \left. + \left( \nabla \mathbf{N} \right)^{T} \left( \nabla \mathbf{N} \right) \frac{\partial \left( \text{d}\mathit{\Omega} \right)}{\partial L_{r}} \right) \ , \end{array} $$

(86)

$$ \begin{array}{@{}rcl@{}} \frac{\partial \mathbf{M}_{m}}{\partial L_{r}} & =& \displaystyle\sum\limits_{e=1}^{{N_{m}^{e}}} {\int}_{{\mathit{\Omega}_{m}^{e}}} \frac{\mathbf{N}^{T} \mathbf{N}}{\rho_{m} {c_{m}^{2}}} \frac{\partial\left( \text{d}\mathit{\Omega} \right)}{\partial L_{r}} \ . \end{array} $$

(87)