Design optimization and sensitivity analysis on time-domain sound radiation of laminated curved shell structures

Abstract

Compared with the frequency-domain sound radiation analysis, the time-domain analysis is more suitable for complicated engineering problems. However, the research on design optimization of time-domain sound radiation was rarely reported. To reduce the undesired time-domain noise radiated from laminated curved shells, the sensitivity formulation of transient sound pressure is obtained by directly differentiating response equations and the corresponding optimization procedure is presented. The Newmark integral method is applied to calculate the vibration response, and the results of which are input into the sound radiation analysis as boundary conditions. Combined with the time-domain boundary element method (BEM), the time-domain boundary integral equation is numerically discretized in both the spatial and time domains, and the transient sound pressure is obtained by solving an algebraic equation. To reduce the time-domain noise, ply thicknesses are taken as the design variables to minimize the square of sound pressure on a prescribed reference surface in the sound medium or the structural surface over a certain period of time. In addition, the constraint on the structural mass is considered. The calculation of time-domain sound radiation sensitivity is transformed into the following two processes: (a) the derivation of transient vibration response based on finite element method (FEM); (b) the derivation of transient sound pressure based on time-domain BEM. The optimal solution is obtained by using the method of moving asymptotes (MMA). Numerical examples verify the accuracy of the sensitivity formulae, and show that the time-domain sound radiation is significantly reduced within allowable constraints.

Appendix 1. Finite element equations for laminated curved shell elements

Figure 

Fig. 18

Curved shell element

18 shows an eight-node curved shell element. In this figure, $(x{ - }y{ - }z)$ is the global coordinate system; $(x^{\prime}{ - }y^{\prime}{ - }z^{\prime})$ is the local coordinate system; and $(\xi { - }\eta { - }\zeta )$ is the natural coordinate system with $- 1 \le \xi ,\zeta ,\eta \le 1$. Here, we assume that ${\mathbf{v}}_{3i}$ is the nodal normal unit vector perpendicular to the middle surface of the element and that ${\mathbf{v}}_{1i}$ and ${\mathbf{v}}_{2i}$ are nodal unit vectors that are perpendicular to ${\mathbf{v}}_{3i}$ and orthogonal to each other. The displacement vector of any point within the element can be expressed in the following interpolation form:

$$\left\{ {\begin{array}{*{20}c} u \\ v \\ w \\ \end{array} } \right\} = \sum\limits_{i = 1}^{8} {N_{i} (\xi ,\eta )\left\{ {\begin{array}{*{20}c} {u_{i} } \\ {v_{i} } \\ {w_{i} } \\ \end{array} } \right\}} + \sum\limits_{i = 1}^{8} {\frac{{T_{i} }}{2}\zeta N_{i} (\xi ,\eta )\left[ {\begin{array}{*{20}c} {l_{1i} } & { - l_{2i} } \\ {m_{1i} } & { - m_{2i} } \\ {n_{1i} } & { - n_{2i} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\alpha_{i} } \\ {\beta_{i} } \\ \end{array} } \right\}}$$

(A1)

where the subscript $i$ denotes the ith node; $\left( {u_{i} \, v_{i} \, w_{i} \, \alpha_{i} \, \beta_{i} } \right)^{{\text{T}}}$ is the generalized nodal displacement vector; $\alpha_{i}$ and $\beta_{i}$ are the rotation angles of ${\mathbf{v}}_{3i}$ around ${\mathbf{v}}_{2i}$ and ${\mathbf{v}}_{1i}$, respectively; $N_{i}$ (Zhai et al. 2017) and $T_{i}$ are the two-dimensional interpolation function and nodal thickness, respectively; and $(l_{ji} \, m_{ji} \, n_{ji} )^{{\text{T}}}$, with $j = 1, \, 2,{ 3}$, are the direction cosines of ${\mathbf{v}}_{1i}$, ${\mathbf{v}}_{2i}$ and ${\mathbf{v}}_{3i}$, respectively. The stiffness matrix of the element can be written as:

$${\mathbf{K}}^{e} = \int_{ - 1}^{1} {\int_{ - 1}^{1} {\int_{ - 1}^{1} {{\mathbf{B}}^{{\text{T}}} {\mathbf{DB}}\left| {\mathbf{J}} \right|} } } {\text{ d}}\xi {\text{d}}\eta {\text{d}}\zeta$$

(A2)

where ${\mathbf{B}}$ (Zhai et al. 2017) and ${\mathbf{D}}$ are the strain matrix and elastic matrix, respectively; and ${\mathbf{J}}$ is the Jacobian matrix:

$${\mathbf{J}} = \sum\limits_{i = 1}^{8} {\left[ {\begin{array}{*{20}c} {N_{i,\xi } \left( {x_{i} + \frac{{\zeta T_{i} l_{3i} }}{2}} \right)} & {N_{i,\xi } \left( {y_{i} + \frac{{\zeta T_{i} m_{3i} }}{2}} \right)} & {N_{i,\xi } \left( {z_{i} + \frac{{\zeta T_{i} n_{3i} }}{2}} \right)} \\ {N_{i,\eta } \left( {x_{i} + \frac{{\zeta T_{i} l_{3i} }}{2}} \right)} & {N_{i,\eta } \left( {y_{i} + \frac{{\zeta T_{i} m_{3i} }}{2}} \right)} & {N_{i,\eta } \left( {z_{i} + \frac{{\zeta T_{i} n_{3i} }}{2}} \right)} \\ {\frac{{N_{i} T_{i} l_{3i} }}{2}} & {\frac{{N_{i} T_{i} m_{3i} }}{2}} & {\frac{{N_{i} T_{i} n_{3i} }}{2}} \\ \end{array} } \right]}$$

(A3)

where the subscript $i$ denotes the ith node; $(x_{i} \, y_{i} \, z_{i} )$ is the global nodal coordinate; and $N_{i,j}$, with $j = \xi , \, \eta , \, \zeta$, represent the derivatives of $N_{i}$ with respect to the natural coordinates.

For an orthotropic shell, the material coordinate system is different from the global coordinate system due to the influence of the ply angle. Therefore, the elastic matrix ${\mathbf{D}}$ in the material coordinate system needs to be transformed into ${\overline{\mathbf{D}}}$ in the global coordinate system:

$${\overline{\mathbf{D}}} = {\mathbf{TDT}}^{{\text{T}}}$$

(A4)

with:

$${\mathbf{T}} = \left[ {\begin{array}{*{20}c} {\cos^{2} \theta } & {\sin^{2} \theta } & 0 & {2\sin \theta \cos \theta } & 0 & 0 \\ {\sin^{2} \theta } & {\cos^{2} \theta } & 0 & { - 2\sin \theta \cos \theta } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ { - \sin \theta \cos \theta } & {\sin \theta \cos \theta } & 0 & {\cos^{2} \theta - \sin^{2} \theta } & 0 & 0 \\ 0 & 0 & 0 & 0 & {\cos \theta } & { - \sin \theta } \\ 0 & 0 & 0 & 0 & {\sin \theta } & {\cos \theta } \\ \end{array} } \right]$$

(A5)

in which $\theta$ is the ply angle.

As shown in Fig. 

Fig. 19

Laminated curved shell element

19, a laminated curved shell element consists of a stack of curved shell elements with different material properties and ply parameters. For such a laminated curved shell element, the elastic matrix is not a continuous function of the thickness coordinate $\zeta$. Therefore, integration in the thickness direction is achieved by splitting the limits through each ply. In the calculation of the stiffness matrix of the kth ply, a new natural coordinate $\zeta^{*}$ with a value range of $\left[ {{ - }1, \, 1} \right]$ is introduced to replace the coordinate $\zeta$. Thus, the stiffness matrix of the laminated element can be expressed by using the following superposition formula:

$${\mathbf{K}}^{e} = \sum\limits_{k = 1}^{{\overline{k}}} {\int_{ - 1}^{1} {\int_{ - 1}^{1} {\int_{ - 1}^{1} {{\mathbf{B}}_{k}^{{\text{T}}} {\overline{\mathbf{D}}}_{k} {\mathbf{B}}_{k} } } } \left| {\mathbf{J}} \right|_{k} \left| {{\mathbf{J}}^{{\mathbf{*}}} } \right|_{k} {\text{d}}\xi {\text{d}}\eta {\text{d}}\zeta^{*} }$$

(A6)

where the subscript $k$ denotes the kth ply; and $\overline{k}$ is the number of plies.

Figure 

Fig. 20

Schematic diagram of coordinate transformation

20 shows the relationships between different coordinates of the kth ply. For a laminated curved shell element with a total thickness of $T_{0}$, we can obtain the following relationships:

$$\begin{gathered} \zeta_{k - 1} = \frac{{h_{k - 1} }}{{T_{0} }} \times 2 - 1 \\ \zeta_{k} = \frac{{h_{k} }}{{T_{0} }} \times 2 - 1 \\ \end{gathered}$$

(A7)

where $h_{k - 1}$ and $h_{k}$ are the bottom surface height and top surface height of the kth ply, respectively; $\zeta_{k - 1}$ and $\zeta_{k}$ represent the values of the two surfaces in the $\zeta$ coordinate. Therefore, the following relationship can be obtained:

$$\zeta = \frac{{\zeta_{k} + \zeta_{k - 1} }}{2} + \frac{{\zeta^{*} (\zeta_{k} - \zeta_{k - 1} )}}{2}$$

(A8)

Combining Equations (A7) and (A8) yields:

$$\zeta = \frac{{T_{k} }}{{T_{0} }}\zeta^{*} + \left( {\frac{{2h_{k - 1} + T_{k} }}{{T_{0} }} - 1} \right)$$

(A9)

where $T_{k} = h_{k} - h_{k - 1}$ represents the thickness of the kth ply.

Here, we let $\omega_{k} = T_{k} /T_{0}$ and $\varphi_{k} = \left( {2h_{k - 1} + T_{k} } \right)/T_{0} - 1$. Thus, the expressions for $\left| {{\mathbf{J}}^{{\mathbf{*}}} } \right|_{k}$, ${\mathbf{J}}_{k}$ and ${\mathbf{B}}_{k}$ in Equation (A6) are shown in Equations (A10) to (A13):

$$\left| {{\mathbf{J}}^{{\mathbf{*}}} } \right|_{k} = \frac{{{\text{d}}\zeta }}{{{\text{d}}\zeta^{*} }} = \frac{{T_{k} }}{{T_{0} }}$$

(A10)

$${\mathbf{J}}_{k} = \sum\limits_{i = 1}^{8} {\left[ {\begin{array}{*{20}c} {N_{i,\xi } \left[ {x_{i} + \frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} l_{3i} }}{2}} \right]} & {N_{i,\xi } \left[ {y_{i} + \frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} m_{3i} }}{2}} \right]} & {N_{i,\xi } \left[ {z_{i} + \frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} n_{3i} }}{2}} \right]} \\ {N_{i,\eta } \left[ {x_{i} + \frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} l_{3i} }}{2}} \right]} & {N_{i,\eta } \left[ {y_{i} + \frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} m_{3i} }}{2}} \right]} & {N_{i,\eta } \left[ {z_{i} + \frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} n_{3i} }}{2}} \right]} \\ {\frac{{N_{i} \left( {T_{k} } \right)_{i} l_{3i} }}{2}} & {\frac{{N_{i} \left( {T_{k} } \right)_{i} m_{3i} }}{2}} & {\frac{{N_{i} \left( {T_{k} } \right)_{i} n_{3i} }}{2}} \\ \end{array} } \right]}$$

(A11)

$${\mathbf{B}}_{k} = {{\varvec{\Gamma}}}\left[ {\begin{array}{*{20}c} {{\mathbf{J}}_{k}^{ - 1} } & 0 & 0 \\ 0 & {{\mathbf{J}}_{k}^{ - 1} } & 0 \\ 0 & 0 & {{\mathbf{J}}_{k}^{ - 1} } \\ \end{array} } \right]\sum\limits_{i = 1}^{8} {\left[ {\begin{array}{*{20}c} {N_{i,\xi } } & 0 & 0 & {\frac{{ - (\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\xi } l_{2i} }}{2}} & {\frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\xi } l_{1i} }}{2}} \\ {N_{i,\eta } } & 0 & 0 & {\frac{{ - (\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\eta } l_{2i} }}{2}} & {\frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\eta } l_{1i} }}{2}} \\ 0 & 0 & 0 & {\frac{{ - (T_{k} )_{i} N_{i} l_{2i} }}{2}} & {\frac{{(T_{k} )_{i} N_{i} l_{1i} }}{2}} \\ 0 & {N_{i,\xi } } & 0 & {\frac{{ - (\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\xi } m_{2i} }}{2}} & {\frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\xi } m_{1i} }}{2}} \\ 0 & {N_{i,\eta } } & 0 & {\frac{{ - (\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\eta } m_{2i} }}{2}} & {\frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\eta } m_{1i} }}{2}} \\ 0 & 0 & 0 & {\frac{{ - (T_{k} )_{i} N_{i} m_{2i} }}{2}} & {\frac{{(T_{k} )_{i} N_{i} m_{1i} }}{2}} \\ 0 & 0 & {N_{i,\xi } } & {\frac{{ - (\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\xi } n_{2i} }}{2}} & {\frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\xi } n_{1i} }}{2}} \\ 0 & 0 & {N_{i,\eta } } & {\frac{{ - (\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\eta } n_{2i} }}{2}} & {\frac{{(\omega_{k} + \varphi_{k} \zeta^{*} )(T_{k} )_{i} N_{i,\eta } n_{1i} }}{2}} \\ 0 & 0 & 0 & {\frac{{ - (T_{k} )_{i} N_{i} n_{2i} }}{2}} & {\frac{{(T_{k} )_{i} N_{i} n_{1i} }}{2}} \\ \end{array} } \right]}$$

(A12)

with

$${{\varvec{\Gamma}}} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ \end{array} } \right]$$

(A13)