Introduction

Engineering structures are often designed and optimized to meet multiple and possibly conflicting objectives such as minimum weight and maximum strength. The merit of multi-objective optimization is to obtain a wide range of structural design choices so that the best tradeoff among different objectives can be achieved. This paper presents a multi-objective structural–acoustic optimization study of engineering structures. Specifically, we take non-uniform beams as an example.

A non-uniform beam is a common structural element in many applications. Many studies have been conducted on vibration, stability, and fatigue of non-uniform beams [1,2,3]. Numerous methods for solving vibration problems have also been studied, such as the finite element method [4], transfer matrix [5], and differential transfer [6]. Optimization of structural–acoustic properties is an important study of non-uniform beams. Adali studied the Pareto optimization of beams with a moving boundary in the 1980s [7]. Eschenauer et al. applied both deterministic and stochastic optimization for weight and static deflection reduction of non-uniform beams [8]. Continuous non-uniform beam optimization was considered in [9]. Optimization of boundary conditions was studied in [10]. Vibro-acoustic optimization was studied in [4, 11].

Two general categories of algorithms for solving multi-objective optimization problems (MOPs) exist. One is deterministic, and the other is evolutionary; the latter dominates among these categories. Representative deterministic search algorithms include multi-objective steepest descent [12], direct search [13], tangent space continuation [14], set-oriented algorithms [15], and cell mapping [16]. The convergence and global coverage of the Pareto set by deterministic algorithms can be usually guaranteed [12, 17]. However, for high-dimensional MOPs, deterministic algorithms face the curse of dimensionality. In addition, deterministic algorithms also have the drawback of being trapped in the local minima.

Evolutionary algorithms are stochastic and obtain the Pareto set in the form of a collection of random points in the design space. Therefore, evolutionary algorithms are not limited by the dimension of the problem. The mainstream evolutionary algorithms for MOPs include genetic algorithms [18, 19], multi-objective particle swarm optimization (MOPSO) [20], and the strength Pareto evolutionary algorithm [21]. This paper aims to solve the MOP of non-uniform beams with a modified particle swarm optimization (PSO) technique.

Multi-objective Particle Swarm Optimization

An MOP usually involves more than one conflicting objective. Unlike single-objective optimization problems, the solution of MOP usually forms a set. The definition of Pareto optimality is based on the concept of dominance.

Consider an MOP as

$$\mathop {\hbox{min} }\limits_{{\varvec{k} \in Q}} \{ \varvec{F}(\varvec{k})\} ,\,$$
(1)

where \(\varvec{k} \in Q \subset {\mathbf{R}}^{q}\) is a \(q\)-dimensional vector of design parameters; \(\varvec{F}\) is the map that consists of the objective function \(f_{i} \;:\;Q \to {\mathbf{R}}^{1}\).

$$\varvec{F}\;:\;Q \to {\mathbf{R}}^{m} ,\,\;\varvec{F}(\varvec{k}) = [f_{1} (\varvec{k}), \ldots ,\,f_{m} (\varvec{k})].$$
(2)

The design space \(Q \subset {\mathbf{R}}^{q}\) can be expressed in terms of inequality and equality constraints

$$\begin{aligned} Q & : = \{ \varvec{k} \in {\mathbf{R}}^{q} \;|\;g_{j} (\varvec{k}) \le 0,\,\;j = 1, \ldots ,\,l,\, \\ & \;\;\;\;\;\;\;\text{and}\;g_{j} (\varvec{k}) = 0,\,\;j = l + 1, \ldots ,\,m\} . \\ \end{aligned}$$
(3)

(a) Let \(\varvec{v}\), \(\varvec{w} \in {\mathbf{R}}^{m}\). The vector \(\varvec{v}\) is said to be less than or equal to \(\varvec{w}\) denoted as \(\varvec{v} \le_{p} \varvec{w}\) if \(v_{i} \le w_{i}\) for all \(i \in \{ 1, \ldots ,\,m\}\).

(b) A vector \(\varvec{v} \in Q\) is said to be dominated by a vector \(\varvec{w} \in Q\) denoted as \(\varvec{w} \prec \varvec{v}\) with respect to MOP (1) if \(\varvec{F}(\varvec{w}) \le_{p} \varvec{F}(\varvec{v})\) and \(\varvec{F}(\varvec{w}) \ne \varvec{F}(\varvec{v})\). Otherwise, \(\varvec{v}\) is non-dominated by \(\varvec{w}\).

(c) A point \(\varvec{w} \in Q\) is called Pareto optimal or a Pareto point of MOP (1) if no \(\varvec{v} \in Q\) dominates \(\varvec{w}\).

(d) The set \(\varvec{P}\) of all Pareto optimal solutions is called the Pareto set. The image \(\varvec{F}(\varvec{P})\) of \(\varvec{P}\) is called the Pareto front.

MOPSO integrates the non-dominant sorting and adaptive meshing in the objective space to reach the convergence and spreading of an evolutionary algorithm [20, 22]. The basic procedure of MOPSO is similar to that of the traditional single-objective PSO where the movements of each particle are influenced by the local best position and the current global best position [23].

The velocity \(\nu_{i}^{m}\) of particle \(i\) in the mth generation is updated to \(v_{i}^{m + 1}\) in the next generation according to the following equation:

$$v_{i}^{m + 1} = \omega \,v_{i}^{m} + R_{1} (B_{i} - p_{i} ) + R_{2} (A_{h} - p_{i} ),\,$$
(4)

where \(p_{i}\) is the current position of particle \(i\) in the design space; \(B_{i}\) is the best position occupied by particle \(i\); \(\omega\) is the inertia factor; \(R_{1}\) and \(R_{2}\) are two uniformly distributed random numbers in \([0, 1]\); and \(A_{h}\) is the reference particle of the current generation selected according to the measurement of particle distribution in the objective space. In this study, we set \(\omega = 0.4\) as suggested in [24]. Equation (4) has the same form as that of the single-objective PSO. However, the local and global best positions are determined by non-dominant relationships.

Table 1 presents the procedure of MOPSO. An archive denoted as \(A\) is created to store non-dominant particles generated at each iteration. The archive is updated constantly to ensure its non-dominancy. As will be shown in the following section, feasible points occupy only a small portion of the volume of the design space.

Table 1 Main structure of the MOPSO algorithm
Full size table

Before Eq. (4) is applied, a fitness function is evaluated to determine \(A_{h}\). The procedure is listed in Table 2. The objective space is discretized into a collection of cells, and the number of points that fall in each cell is counted. A fitness value is assigned, and cells that contain more particles are penalized. In other words, a greater number of particles in one cell correspond to a lower fitness value assigned to that cell to maintain an even level of particle spreading. The normalized fitness value serves as the selection probability of a cell from the roulette wheel run. Let \(z\) denote the cell chosen from the roulette wheel selection. \(A_{h}\) is determined by selecting a random particle from cell \(z\).

Table 2 Pseudo code of fitness assignment in the objective space
Full size table

The dominance check is applied to update the archive when a new position of one particle is calculated. Let \(p_{i}\) denote the new position of particle \(i\) at a certain generation. The principle to update the archive is to keep all non-empty positions being occupied with non-dominant points in the archive. Therefore, if \(p_{i}\) dominates \(A_{j} \in A\), then \(A_{j}\) will be removed. On the other hand, if \(p_{i}\) is dominated by any element in the archive \(A\), then \(p_{i}\) will not be added to the archive. Hence, the current position of particle \(i\) is not a potential solution.

Note that the capacity of the archive is usually larger than the population size but is still finite. During the evolution, the archive might be full. Under this circumstance, a secondary criterion will be applied to determine whether to add the new particle position to the archive. The code to enforce this criterion is presented in Table 3.

Table 3 Pseudo code of archive update when the archive is full
Full size table

To avoid the population from being trapped in the local optimum, the mutation operator is introduced. The percentage of the mutated population decays exponentially with respect to the number of the current generation [20],

$$p_{r} = \exp \,\left( { - \frac{{15{gen}}}{{g_{\hbox{max} } }}} \right),$$
(5)

where \(p_{r}\) is the percentage of population being mutated; \({gen}\) is the current number of generations; and \(g_{{\text{max}}}\) is the maximum number of generations. The mutation occurs only in the first few generations where the landscape of the potential solutions is unknown. The pseudo code of the mutation operator is shown in Table 4.

Table 4 Pseudo code of the mutation process
Full size table

Structural–Acoustic Analysis

Before we formulate a multi-objective structural–acoustic optimization problem, we present the structural–acoustic analysis of non-uniform beams by applying the transfer matrix method [5].

Vibration Analysis

We use a Euler–Bernoulli beam to demonstrate the multi-objective structural–acoustic optimization problem in this study. The equation of motion is given by

$$D(x)\frac{{\partial^{4} w}}{{\partial x^{4} }} + c(x)\frac{\partial w}{\partial t} + \gamma (x)\frac{{\partial^{2} w}}{{\partial t^{2} }} = g(x,\,t),\, \, 0 \le x \le L ,$$
(6)

where \(D(x)\) is the beam rigidity; \(\gamma (x)\) is the mass per unit area; \(c(x)\) is the structural damping; \(g(x,\,t)\) is the external excitation; and \(w(x,\,t)\) is the deflection of the beam. We consider the Rayleigh damping here, namely \(c(x) = \alpha {\kern 1pt} \gamma (x) + \eta D(x)\). Furthermore, we assume that \(D(x)\), \(\gamma (x)\), and \(c(x)\) are given smooth functions of \(x\).

The transfer matrix method proposes to divide the beam into \(N\) uniform segments such that \(D(x)\) and \(\gamma (x)\) can be written as

$$D(x) = \left\{ {\begin{array}{*{20}l} {D_{1} ,\,{\text{ }}\beta _{1} \le x \le \beta _{2} } \hfill \\ \vdots \hfill \\ {D_{i} ,\,{\text{ }}\beta _{i} \le x \le \beta _{{i + 1}} } \hfill \\ \vdots \hfill \\ {D_{N} ,\,{\text{ }}\beta _{N} \le x \le \beta _{{N + 1}} } \hfill \\ \end{array} } \right.,\,\,\gamma (x) = \left\{ {\begin{array}{*{20}l} {\gamma _{1} ,\,{\text{ }}\beta _{1} \le x \le \beta _{2} } \hfill \\ \vdots \hfill \\ {\gamma _{i} ,\,{\text{ }}\beta _{i} \le x \le \beta _{{i + 1}} } \hfill \\ \vdots \hfill \\ {\gamma _{N} ,\,{\text{ }}\beta _{N} \le x \le \beta _{{N + 1}} } \hfill \\ \end{array} } \right. ,$$
(7)

where \(\beta_{i}\) and \(\beta_{i + 1}\) are the coordinates of both ends of the ith segment, and \(\beta_{1} = 0\) and \(\beta_{N + 1} = L\). The free undamped vibration of each segment satisfies the following equation:

$$D_{i} \frac{{\partial^{4} w}}{{\partial x^{4} }} + \gamma_{i} \frac{{\partial^{2} w}}{{\partial t^{2} }} = 0,\, \, \beta_{i} \le x \le \beta_{i + 1} .$$
(8)

Consider the harmonic response of the ith segment as \(w(x,\,t) = X_{i} (x)\,{\text{e}}^{{{\text{j}}\omega \,t}}\). We have

$$D_{i} X_{i}^{{^{\prime \prime \prime \prime } }} - \gamma_{i} \omega^{2} X_{i} = 0,\, \, \beta_{i} \le x \le \beta_{i + 1} .$$
(9)

The general solution of Eq. (9) is given by

$$X_{i} (x) = X_{i} (\beta_{i} )f_{1} (\varphi_{i} ) + \frac{{X_{i}^{{^{\prime } }} (\beta_{i} )}}{{k_{i} }}f_{2} (\varphi_{i} ) + \frac{{M_{i} (\beta_{i} )}}{{D_{i} k_{i}^{2} }}f_{3} (\varphi_{i} ) + \frac{{Q_{i} (\beta_{i} )}}{{D_{i} k_{i}^{3} }}f_{4} (\varphi_{i} )\,,\,$$
(10)

where \(k_{i}^{4} = \tfrac{{\gamma_{i} \,\omega^{\,2} }}{{D_{i} }}\), \(\varphi_{i} = k_{i} (x - \beta_{i} )\) and

$$M_{i} (x) = D_{i} \frac{{\partial^{2} X_{i} }}{{\partial x^{2} }},\,\;\,Q_{i} (x) = D_{i} \frac{{\partial^{3} X_{i} }}{{\partial x^{3} }},\,$$
(11)

and

$$\begin{array}{*{20}l} {f_{1} (\varphi_{i} ) = \tfrac{1}{2}\left[ {\cosh \,(\varphi_{i} ) + \cos \,(\varphi_{i} )} \right],} \hfill & {f_{2} (\varphi_{i} ) = \tfrac{1}{2}\left[ {\sinh \,(\varphi_{i} ) + \sin \,(\varphi_{i} )} \right],} \hfill \\ {f_{3} (\varphi_{i} ) = \tfrac{1}{2}\left[ {\cosh \,(\varphi_{i} ) - \cos \,(\varphi_{i} )} \right],} \hfill & {f_{4} (\varphi_{i} ) = \tfrac{1}{2}\left[ {\sinh \,(\varphi_{i} ) - \sin \,(\varphi_{i} )} \right].} \hfill \\ \end{array}$$
(12)

The functions in Eq. (12) satisfy the following relations:

$$\frac{{\partial f_{1} }}{\partial x} = k_{i} f_{4} ,\,\,\frac{{\partial f_{2} }}{\partial x} = k_{i} f_{1} ,\,\,\frac{{\partial f_{3} }}{\partial x} = k_{i} f_{2} ,\,\,\frac{{\partial f_{4} }}{\partial x} = k_{i} f_{3} .$$
(13)

Define a vector \(\varvec{u}_{i} = \left[ {X_{i} (\beta_{i} ),\,X_{i}^{{^{\prime } }} (\beta_{i} ),\,M_{i} (\beta_{i} ),\,Q_{i} (\beta_{i} )} \right]^{\text{T}}\). From Eqs. (10) and (11), we can evaluate \(\varvec{u}_{i + 1} = \left[ {X_{i} (\beta_{i + 1} ),\,X_{i}^{{^{\prime } }} (\beta_{i + 1} ),\,M_{i} (\beta_{i + 1} ),\,Q_{i} (\beta_{i + 1} )} \right]^{\text{T}}\), leading to the following transfer relationship:

$$\varvec{u}_{i + 1} = \varvec{A}_{i} \,\varvec{u}_{i} ,\,$$
(14)
$$\varvec{A}_{i} = \left[ {\begin{array}{*{20}c} {f_{1} } & {\tfrac{{f_{2} }}{{k_{i} }}} & {\tfrac{{f_{3} }}{{D_{i} k_{i}^{2} }}} & {\tfrac{{f_{4} }}{{D_{i} k_{i}^{3} }}} \\ {k_{i} f_{4} } & {f_{1} } & {\tfrac{{f_{2} }}{{D_{i} k_{i} }}} & {\tfrac{{f_{3} }}{{D_{i} k_{i}^{2} }}} \\ {D_{i} k_{i}^{2} f_{3} } & {D_{i} k_{i} f_{4} } & {f_{1} } & {\tfrac{{f_{1} }}{{k_{i} }}} \\ {D_{i} k_{i}^{3} f_{2} } & {D_{i} k_{i}^{2} f_{3} } & {k_{i} f_{4} } & {f_{1} } \\ \end{array} } \right]_{{x = \beta_{i + 1} }},$$
(15)

where \(\varvec{A}_{i}\) is known as the transfer matrix of one segment. By repeating this process over all segments, we obtain the transfer relationship from \(\varvec{u}_{1}\) to \(\varvec{u}_{N + 1}\) as

$$\varvec{u}_{N + 1} = \varvec{A}_{N} \varvec{A}_{N - 1} \cdots \varvec{A}_{2} \varvec{A}_{1} \varvec{u}_{1} \equiv \varvec{Bu}_{1} ,\,$$
(16)

where the matrix \(\varvec{B}\) is the transfer matrix of the beam from \(x = 0\) to \(x = L\). Note that \(\varvec{B}\) is a function of the frequency \(\omega\) and that \(\varvec{u}_{1}\) and \(\varvec{u}_{N + 1}\) must satisfy the boundary conditions.

After imposing the boundary conditions to \(\varvec{u}_{1}\) and \(\varvec{u}_{N + 1}\) in Eq. (16), we obtain the transcendental equation for determining the natural frequencies of the beam. Examples of transcendental equations for different boundary conditions are shown in Table 5.

Table 5 Transcendental equations for different boundary conditions of the non-uniform beam
Full size table

Let \(\omega_{n}\) be the nth natural frequency of the beam obtained from the transcendental equation and \(X_{ni} (x)\) be the corresponding mode function of the ith segment. The nth mode function of the beam can be written as

$$\varphi_{n} (x) = \left\{ {\begin{array}{*{20}l} {X_{n1} (x),\, \, \beta_{1} \le x \le \beta_{2} } \hfill \\ \vdots \hfill \\ {X_{ni} (x),\, \, \beta_{i} \le x \le \beta_{i + 1} } \hfill \\ \vdots \hfill \\ {X_{nN} (x),\, \, \beta_{N} \le x \le \beta_{N + 1} } \hfill \\ \end{array} } \right.\,$$
(17)

We assume that \(\varphi_{n} (x)\) is normalized such that

$$\int_{0}^{L} {\gamma (x)\,\varphi_{n}^{2} (x)\,{\text{d}}x} = \sum\limits_{i = 1}^{N} {\gamma_{i} \int_{{\beta_{i} }}^{{\beta_{i + 1} }} {X_{ni}^{2} (x)\,{\text{d}}x = 1.} }$$
(18)

The mode functions are orthogonal,

$$\int_{0}^{L} {\gamma (x)\varphi_{m} (x)\varphi_{n} (x){\text{d}}x = \sum\limits_{i = 1}^{N} {\gamma _{i}}{\int_{{\beta_{i} }}^{{\beta_{i + 1} }} {X_{mi} (x)X_{ni} (x){\text{d}}x = \delta_{mn} } ,} }$$
(19)
$$\int_{0}^{L} {\,D(x)\,\varphi_{m}^{{^{\prime \prime \prime \prime } }} (x)\,\varphi_{n} (x)\,{\text{d}}x} = \sum\limits_{i = 1}^{N} {D_{i} \int_{{\beta_{i} }}^{{\beta_{i + 1} }} {X_{mi}^{{^{\prime \prime \prime \prime } }} (x)X_{ni} (x){\text{d}}x = \omega_{n}^{2} \delta_{mn} .} }$$
(20)

Consider a harmonic external excitation \(g(x,\,t) = G(x)\,{\text{e}}^{{{\text{j}}\omega {\kern 1pt} t}}\) and a complex harmonic response of the beam \(w\,(x,\,t) = X(x)\,{\text{e}}^{{{\text{j}}\omega {\kern 1pt} {\text{t}}}}\) such that the spatial part of the function can be expanded in terms of the mode functions

$$X(x) = \mathop \sum \limits_{n = 1}^{\infty } \,W_{n} \,\varphi_{n} (x),\,\;G(x) = \mathop \sum \limits_{n = 1}^{\infty } G_{n} \,\gamma (x)\,\varphi_{n} (x),$$
(21)

where \(W_{n}\) is the modal expansion coefficient of the response, and \(G_{n}\) can be computed as

$$ G_{n} = \int_{0}^{L} {\gamma (x){\kern 1pt} G(x)\,\varphi_{n} (x){\text{d}}x.} \ $$
(22)

From Eq. (6) together with the orthogonality properties of the mode functions, we obtain

$$W_{n} = \frac{{G_{n} }}{{\omega_{n}^{2} - \omega^{2} + {\text{j}}\omega \,(\alpha + \eta \omega_{n}^{2} )}}.$$
(23)

The forced damped solution of Eq. (6) then reads

$$w(x,\,t) = \mathop \sum \limits_{n = 1}^{\infty } \,W_{n} \,\varphi_{n} (x)\,{\text{e}}^{{{\text{j}}\omega {\kern 1pt} t}} = \mathop \sum \limits_{n = 1}^{\infty } \frac{{G_{n} \cdot \varphi_{n} (x)}}{{\omega_{n}^{2} - \omega^{2} + {\text{j}}\omega {\kern 1pt} (\alpha + \eta \omega_{n}^{2} )}}{\text{e}}^{{{\text{j}}\omega {\kern 1pt} t}} .$$
(24)

Acoustic Analysis

The wavenumber transformation of a mode function is defined as [5]

$$\begin{aligned} \varPhi_{n} (k) & = \int_{ - \infty }^{\infty } {\varphi_{n} (x)\,{\text{e}}^{{{\text{j}}}kx} {\text{d}}x} = \int_{0}^{L} {\,\varphi_{n} (x)\,{\text{e}}^{{{\text{j}}}kx} {\text{d}}x} \\ & = \sum\limits_{i = 1}^{N} {\int_{{\beta_{i} }}^{{\beta_{i + 1} }} {X_{ni} (x)\,{\text{e}}^{{{\text{j}}}kx} {\text{d}}x.} } \\ \end{aligned}$$
(25)

The wavenumber transformation of the deflection velocity of the forced response of the beam is an interesting function that reveals the acoustic property [5].

The spatial distribution of the deflection velocity of the forced response of the beam can be derived from the closed-form solution in Eq. (24),

$$v(x) = \mathop \sum \limits_{n = 1}^{\infty } \,{\text{j}}\omega \,W_{n} \,\varphi_{n} (x).$$
(26)

The wavenumber transformation of the velocity is given by

$$V(k) = \int_{ - \infty }^{\infty } {{\kern 1pt} v(x)\,{\text{e}}^{{{\text{j}}kx}} {\text{d}}x} = \int_{0}^{L} {\,v(x)\,{\text{e}}^{{{\text{j}}kx}} {\text{d}}x} = \sum\limits_{n = 1}^{\infty } {{\text{j}}\omega \,W_{n} \,\varPhi_{n} (k)} .$$
(27)

The average acoustic power radiated by the vibrating beam per unit width over one period of vibrations can be computed as [5]

$$\bar{P} = \frac{{\omega {\kern 1pt} \rho_{0} }}{4\pi }\int_{ - k}^{k} {\frac{{V(\hat{k})\,V^{ * } (\hat{k})}}{{\sqrt {k^{2} - \hat{k}^{2} } }}{\text{d}}\hat{k},}$$
(28)

where \(k = \omega /c_{0}\) is the wavenumber of the air at the frequency \(\omega\); \(c_{0}\) is the speed of sound; and \(\rho_{0}\) is the air density. \(\text{(}\;\text{)}^{ * }\) denotes the complex conjugate.

Let \(\left\langle {v^{2} } \right\rangle\) be the spatial and temporal average of the deflection velocity field of the beam given by

$$\left\langle {v^{2} } \right\rangle = \frac{1}{L}\int_{0}^{L} {v(x)\,v^{ * } (x){\kern 1pt} {\text{d}}x.}$$
(29)

The radiation efficiency of the vibrating beam is defined as

$$\sigma = \frac{{\bar{P}}}{{\tfrac{1}{2}\rho_{0} {\kern 1pt} c_{0} L\left\langle {v^{2} } \right\rangle }}.$$
(30)

Some computational notes are in order. The computations of Eqs. (28) and (30) require numerical integrations. The analytical expression of the integration in Eq. (25) helps speed up these numerical integrations. The transfer matrix method for structural–acoustic analysis offers a fast and accurate approach for multi-objective optimization studies.

Structural–Acoustic Optimization

Design Variables

The geometric and material properties of the non-uniform beam are assumed to be continuous functions of the spatial coordinate \(x\). In the following discussions, we assume that only the thickness of the beam denoted as \(h\,(x)\) is non-uniform. We take a number of the sampled thickness \(h\,(x_{i} )\) along the beam as design variables and use them to construct the smooth function \(h\,(x)\) by means of spline interpolation.

Figure 1 shows an example of a cubic spline representation of the thickness profile \(h\,(x)\) with five sampled points. Figure 1b shows 10 sampled thicknesses, while Fig. 1c shows the 10-segment stepped beam based on the sampled thicknesses that are used with the transfer matrix method. The number of segments is a user-defined variable for the structural–acoustic analysis with the transfer matrix method. It affects the accuracy of the solution but does not increase the complexity of the optimization problem.

Fig. 1
figure 1

Spline interpolation for the thickness profile h(x) with the design variables h(x i ). a the spline curve with five sampled points, b the spline curve with 10 sampled points, c the stepped beam of ten segments used in the solution with the transfer matrix method

Full size image

Objective Functions

The primary goal of structural–acoustic design is to create a lightweight and quiet structure. The multi-objective optimization of non-uniform beams aims to seek a balance between weight reduction and sound isolation.

The total mass of the beam can be expressed as

$$m_{\text{tot}} = \mathop \sum \limits_{i = 1}^{N} \frac{{\rho {\kern 1pt} h_{i\,} L}}{N},\,$$
(31)

where \(\rho\) is the mass density of the beam; \(L\) is the length of the beam; \(h_{i}\) is the average thickness of the ith segment determined by the thickness profile; and \(N\) is the number of segments of the beam.

The second objective function is the integration of the radiated sound power in Eq. (28) over a range of frequencies

$$I = \int_{{\omega_{1} }}^{{\omega_{2} }} {\bar{P}\,{\text{d}}\omega ,\,}$$
(32)

where \(\omega_{1}\) and \(\omega_{2}\) define the lower and upper bounds of the frequencies of interest, respectively. These bounds are determined based on practical considerations in engineering applications. In this study, we assume that they are given.

Constraints

To ensure that the optimally designed structure meets engineering requirements, we impose three constraints for optimization.

  1. 1.

    The lower and upper bounds of the mass \(m_{\text{tot}}\):

    $$m_{\hbox{min} } < m_{\text{tot}} < m_{\hbox{max} } .$$
    (33)
  2. 2.

    The lower bound of the fundamental frequency \(\omega_{1}.\)

    $$\omega_{\hbox{min} } < \omega_{1} .$$
    (34)

    This constraint guarantees minimum static strength for structural integrity.

  3. 3.

    The smoothness of the beam

    $$\mathop {\mathop {\text{max}}\limits }\limits_{i = 1, \ldots ,\,N} \left| {h_{i + 1} - h_{i} } \right| < \Delta h.$$
    (35)

    Although the spline interpolation generates a smooth profile of the non-uniform beam, this constraint on the maximum height difference between two neighboring segments in the transfer matrix solution limits the sharp change in the thickness.

Numerical Results

Two case studies are presented in this section with different boundary conditions. For both cases, the number of interpolation coordinates, i.e., the number of design variables, is 10. The number of segments used by the transfer matrix method is 50. The material properties of the non-uniform beam are listed in Table 6. For all the optimization studies, we set the population size of the MOPSO method as 100, the number of generations as 80, and the archive size as 120. The mutation rate is 0.025. The MOPSO method is executed eight times for each scenario. The non-dominant solutions from all eight experiments are taken as the final result. The partition of the objective space for fitness value assignment is \(\text{30} \times 30\). All computations presented in this paper are conducted in a laptop with a 2.4 GHz i7-4700MQ CPU.

Table 6 Parameters of the non-uniform beam
Full size table

Clamped–Clamped Beam

The constraints defined in Eqs. (33)–(35) significantly reduce the feasible solutions in the design space. We discuss the feasible solutions in the design space first under the following constraints:

$$10\,{\text{kg}} < m_{\text{tot}} < 30\,{\text{kg}},\,\;\frac{{\omega_{1} }}{2\pi } > 10\,{\text{Hz}},\,\;\mathop {\mathop {\hbox{max} }\limits }\limits_{i = 1, \ldots ,\,N} \left| {h_{i + 1} - h_{i} } \right| < 1.75\,{\text{mm}} .$$
(36)

A uniform unit magnitude loading is applied to the beam in the interval \(x \in [1.0125,\,\,1.3125]\text m\). The frequency of the excitation sweep is in the range from 50 to 150 Hz. Hence, the excitation is equivalent to a bandlimited noise defined in this frequency range. This range represents the frequencies of interest. The bounds of the spline coordinates \(x_{i}\) are given by \(1 \le x_{i} \le 10\) mm. The beam length is 1.5 m. The first 10 modes of the beam are used for the structural–acoustic analysis.

Table 7 presents the results of random samplings of feasible solutions in the design space for the clamped–clamped beam. The number of random sampling ranges from 1000 to half a million. We repeat the experiments five times to compute the average ratio of the feasible solutions. For this example, only 0.3% of the 10-dimensional design space is occupied by the feasible points. Therefore, the search for the Pareto set will be constrained within the 0.3% volume of the design space.

Table 7 Statistical study of the volumetric fraction of feasible solutions in the design space
Full size table

Figure 2 shows the Pareto front for the clamped–clamped beam, which consists of 21 solutions. The average computational time is 4765.2 s in the laptop. Figure 3 presents the thickness profile of two extreme designs, namely, the minimum weight and minimum sound radiation. The spline coordinates of the two extreme solutions are listed in Table 8. Figures 4 and 6 show the sound radiation efficiency and radiated sound power of the two beams. Both quantities are reduced in the frequency range \([\text{50},\,150]\) Hz as compared with the baseline uniform beam of the same mass, as highlighted in Figs. 5 and 7. The average reduction of the radiated sound power over the frequency range is defined as

Fig. 2
figure 2

Pareto front of the clamped–clamped beam. The population size of the MOPSO method is 100 with 80 generations. Twenty-one Pareto solutions are found

Full size image
Fig. 3
figure 3

Optimized thickness profile for the clamped–clamped beam. The harmonic load is applied in the interval marked by the red lines. a beam No. 1 has the minimum weight, while b beam No. 2 has the minimum integrated sound power

Full size image
Table 8 Spline coordinates of the optimal clamped–clamped beam
Full size table
Fig. 4
figure 4

a Sound radiation efficiency and b radiated sound power of beam No. 1 in Fig. 3. Solid lines the optimized non-uniform beam. Dashed lines the uniform beam with the same mass. The targeted frequency range is from 50 to 150 Hz

Full size image
Fig. 5
figure 5

Zoomed view of Fig. 4 in the frequency range 50–150 Hz. Solid lines the optimal non-uniform beam. Dashed lines the uniform beam with the same mass. Radiated sound power reduction of the optimal design is 68.99 dB compared with the baseline beam

Full size image
Fig. 6
figure 6

a Sound radiation efficiency and b radiated sound power of beam No. 2 in Fig. 3. Solid lines the optimized non-uniform beam. Dashed lines the uniform beam with the same mass. The targeted frequency range is from 50 to 150 Hz

Full size image
$$\Delta \bar{P} = 10\lg \left( {\frac{{\Delta \bar{P}_{I} }}{{P_{0} }}} \right)({\text{dB}}),$$
(37)
$$\Delta \bar{P}_{I} = \frac{1}{{\omega_{2} - \omega_{1} }}\int_{{\omega_{1} }}^{{\omega_{2} }} {\left( {\bar{P}_{u} - \bar{P}_{n} } \right){\text{d}}\omega } ,$$
(38)

where \(\bar{P}_{u}\) and \(\bar{P}_{n}\) are the radiated sound power of the uniform and non-uniform beam, respectively; and \(P_{0} = 10^{ - 12}\) W is the sound power reference. The sound power reductions in the frequency range are 68.88 and 77.97 dB for the optimal designs shown in Figs. 5 and 7, respectively.

Fig. 7
figure 7

Zoomed view of Fig. 6 in the frequency range 50–150 Hz. Solid lines the optimized non-uniform beam. Dashed line the uniform beam with the same mass. Radiated sound power reduction of the optimal design is 77.97 dB compared with the baseline beam

Full size image

Simply Supported Beam

A uniform unit magnitude loading is applied to the beam in the interval \([0.4714{\kern 1pt},\,0.7071]\)m. The frequency sweeps in the range from 200 to 600 Hz. The bounds of the spline coordinates \(x_{i}\) are given by \(1 \le x_{i} \le 15\) mm. The beam length is 1.0 m. The first 10 modes of the beam are used for the structural–acoustic analysis. The lower bound of the minimum fundamental frequency is set as 8 Hz.

Figure 8 shows the Pareto front for the simply supported beam. Fifty-five solutions are found. The average computational time is 5042.7 s. Figure 9 presents the beam shape of two extreme designs. The corresponding spline coordinates are listed in Table 9. Figures 10 and 12 show the sound radiation efficiency and radiated sound power of the two beams. Both quantities are reduced in the frequency range \([200,\,{\kern 1pt} 600]\) Hz as compared with the baseline uniform beam of the same mass, as highlighted in Figs. 11 and 13 (Fig .12). The reductions of the radiated sound power are 35.13 and 39.19 dB in the frequency range of interest as shown in Figs. 11 and 13, respectively.

Fig. 8
figure 8

Pareto front of the simply supported beam. The population size of the MOPSO method is 100 with 80 generations. Fifty-five Pareto solutions are found

Full size image
Fig. 9
figure 9

Optimized thickness profile for the simply supported beam. The harmonic load is applied in the interval marked by the red lines. a beam No. 1 has the minimum mass, while b beam No. 2 has the minimum integrated sound power

Full size image
Table 9 Spline coordinates of the optimal simply supported beam
Full size table
Fig. 10
figure 10

a Sound radiation efficiency and b radiated sound power of beam No. 1 in Fig. 9. Solid lines the optimized non-uniform beam. Dashed lines the uniform beam with the same mass. The targeted frequency range is from 200 to 600 Hz

Full size image
Fig. 11
figure 11

Zoomed view of Fig. 10 in the frequency range 200–600 Hz. Solid lines the optimized non-uniform beam. Dashed lines the uniform beam with the same mass. Radiated sound power reduction of the optimal design is 35.13 dB compared with the baseline beam

Full size image
Fig. 12
figure 12

a Sound radiation efficiency and b radiated sound power of beam No. 2 in Fig. 9. Solid lines the optimized non-uniform beam. Dashed lines the uniform beam with the same mass. The targeted frequency range is from 200 to 600 Hz

Full size image
Fig. 13
figure 13

Zoomed view of Fig. 12 in the frequency range 200–600 Hz. Solid lines the optimized non-uniform beam. Dashed lines the uniform beam with the same mass. The radiated sound power reduction of the optimal design is 39.19 dB compared with the baseline beam

Full size image

Concluding Remarks

This paper studied the multi-objective optimal design of non-uniform beams for minimum sound radiation. The structural weight and radiated sound power are two objective functions. The MOPSO algorithm is used to search for the optimal solutions. The transfer matrix method is used to obtain solutions of vibration and sound radiation of non-uniform beams with high efficiency and accuracy that are required in optimization studies. We discovered that when practical constraints such as the smoothness and lower bound of the fundamental frequency of the structure are imposed, the feasible solutions can be found in only a very small subset of the design space. Numerical results of multi-objective optimal design of non-uniform beams with two different boundary conditions are presented. We have shown that the optimized non-uniform beam has a smaller radiated sound power and radiation efficiency in the targeted frequency range compared with the uniform beam of the same mass.