The Concept of Topological Derivative for Eigenvalue Optimization Problem for Plane Structures

Abstract

This paper presents the topological derivative of the first eigenvalue for the free vibration model of plane structures. We conduct a topological asymptotic analysis to account for perturbations in the domain caused by inserting a small inclusion. The paper includes a rigorous derivation of the topological derivative for the eigenvalue problem along with a proof of its existence. Additionally, we provide numerical examples that illustrate the application of the proposed methodology for maximizing the first eigenvalue in plane structures. The results demonstrate that multiple eigenvalues were not encountered.

1. Introduction

Some considerations such as dynamic response and loads are important for understanding the behavior and design of structures. Structures under dynamic analysis are governed by linear differential equations, which involve solving an eigenvalue problem. The eigenvalue problem is crucial in many real-world applications, particularly in structural optimization, as it significantly contributes to structural integrity. When the excitation frequency matches one of the natural frequencies of vibration, the structure becomes highly responsive, which can lead to damage or collapse. Structural behavior is affected when variations in system parameters lead to changes in eigenvalues; eigenvectors; and, consequently, the final response characteristics of the system. The magnitude of these variations is reflected in the derivatives of the system’s eigenvalues and eigenvectors. One of the main concerns of sensitivity analysis is the presence of multiple eigenvalues. During the topology optimization process, only simple eigenvalues are present in the initial steps. However, as the iterative process progresses and the geometry changes, multiple eigenvalues may appear. The presence of multiple eigenvalues leads to convergence issues in optimization algorithms because they are not differentiable in the common sense [1,2]. Since then, several methods have been proposed to avoid the presence of multiple eigenvalues [3,4,5,6]. The conventional notion of derivative is naturally extended to functionals through the concept of the topological derivative ($D_T$), where the variable is a geometric domain subject to singular topology changes.

Concerning structural topology optimization, the $D_T$ provides the exact sensitivity of the associated objective functionals due to perturbations such as the insertion of infinitesimal voids, inclusions, or even cracks. In particular, for dynamic problems, the $D_T$ for simple eigenvalues of the Laplacian was considered by [7], and for multiple eigenvalues in elasticity problems by [8]. A similar work concerning the augmented Lagrangian method based on the concept of the topological derivative was also addressed by [9], where numerical results considering compliance and eigenfrequency constraints were introduced by the authors. More recently, the $D_T$ of $L^2$ and energy norms associated with the solution to plate bending, considering both Kirchhoff and Reissner–Mindlin theories, was introduced by [10]. In addition to the maximization of the first eigenvalue, the sensitivities obtained were adapted to the context of plate topology optimization for plates under elastic support and free vibration conditions. In [11], the $D_T$ was derived for membrane eigenvalue maximization concerning the nucleation of inclusions endowed with different material properties from the background. According to the numerical results, the presence of multiple eigenvalues was not observed as the iterative process evolved, even as the geometry became complex.

The field of eigenvalue topology optimization remains significant and requires the development of new methodologies. These should incorporate classical techniques such as sensitivity analysis, level set methods, and the bubble method, among others, to effectively address various challenges. Ref. [12] introduced a phase field-based structural optimization method that simplifies computation and improves efficiency compared to traditional level-set methods. This method uses a phase field function and solves a time-dependent reaction–diffusion equation but relies on the initial shape and requires careful management of perimeter control effects. More recently, ref. [13] presented an enhanced phase field method with multi-level correction, which improves accuracy and reduces computational costs for eigenfrequency topology optimization, as demonstrated by numerical examples in 2D and 3D.

As mentioned earlier, eigenfrequency optimization is a prominent research topic. The present work revisits the subject discussed in [11] to incorporate the problem of plane structures. The focus is on deriving the $D_T$ for maximizing the first eigenvalue. A closed formula for the $D_T$ of the first eigenvalue for plane structures is obtained. The resulting formula is used together with a level-set domain representation method to develop a topology design algorithm, as implemented in [14]. The optimal topology for the plane structure under prescribed boundary conditions is achieved by maximizing the first eigenvalue while minimizing the volume. Several proposed numerical examples demonstrate the feasibility of the present methodology in addressing topology optimizations. It is worth mentioning that, to the authors’ knowledge, the concept of $D_T$ for the eigenvalue problem for plane structures, considering mathematical rigor, has not been addressed elsewhere. Additionally, several numerical examples are explored. This paper is organized as follows. The topological derivative for plane structure problems is introduced in Section 2. In Section 3, some numerical examples are presented, showing that the $D_T$ derived here successfully solves eigenvalue problems for plane structures. Finally, the conclusions are summarized in Section 4.

2. Topological Derivative

Consider, for instance, an open and bounded domain $\Omega \subset R^2$ such that it is subject to a nonsmooth perturbation confined in a small ball $B_{\epsilon }\left(\widehat{x}\right)$ of radius $\epsilon$ and center at $\widehat{x}\in \Omega$,

$$\psi \left({\chi }_{\epsilon }\left(\widehat{x}\right)\right)=\psi \left(\chi \right)+f\left(\epsilon \right)D_T\psi \left(\widehat{x}\right)+o\left(f\left(\epsilon \right)\right),$$

(1)

where $\psi \left(\chi \right)$ is the shape functional associated with the unperturbed domain, $f\left(\epsilon \right)$ is a positive first-order correction function of $\psi$, and $o\left(f\right(\epsilon \left)\right)$ is the remainder, such that $o\left(f\right(\epsilon \left)\right)/f\left(\epsilon \right)\to 0$ as $\epsilon \to 0$. The function $\chi$ is the characteristic function associated with the unperturbed domain, and ${\chi }_{\epsilon }$ is the characteristic function associated with the perturbed domain. The function $\widehat{x}\mapsto D_T\psi \left(\widehat{x}\right)$ is termed the topological derivative of $\psi$ at $\widehat{x}$. The domain $\Omega$ is then divided into two subdomains, $\omega \subset \Omega$ and its complement $\Omega \setminus \omega$. Finally, a set of piecewise constant functions $\alpha$, $\rho$, and $\beta$ (which are the contrasts in the material properties) is considered and introduced according to

Table 1. Values of $\alpha$, $\rho$, and $\beta$.

	$\mathit{\alpha }$	$\mathit{\rho }$	$\mathit{\beta }$
$\Omega \setminus \omega$	${\alpha }_0$	${\rho }_0$	${\beta }_0$
$\omega$	${\alpha }_1$	${\rho }_1$	${\beta }_1$

.

The topological perturbation results from the nucleation of a small circular inclusion of the form ${\omega }_{\epsilon }\left(\widehat{x}\right):=B_{\epsilon }\left(\widehat{x}\right)=|x-\widehat{x}|<\epsilon$ for $\widehat{x}\in \Omega$. In this specific case, the perturbation is governed by a set of piecewise constant functions ${\alpha }_{\epsilon }$, ${\rho }_{\epsilon }$, and ${\beta }_{\epsilon }$, as introduced in

Table 2. Values of ${\alpha }_{\epsilon }$, ${\rho }_{\epsilon }$, and ${\beta }_{\epsilon }$.

	${\mathit{\alpha }}_{\mathit{\epsilon }}$	${\mathit{\rho }}_{\mathit{\epsilon }}$	${\mathit{\beta }}_{\mathit{\epsilon }}$
$\Omega \setminus B_{\epsilon }$	$\alpha$	$\rho$	$\beta$
$B_{\epsilon }$	${\gamma }_{\alpha }\alpha$	${\gamma }_{\rho }\rho$	${\gamma }_{\beta }\beta$

and

Table 3. Values of ${\gamma }_{\alpha }$, ${\gamma }_{\rho }$, and ${\gamma }_{\beta }$.

	${\mathit{\gamma }}_{\mathit{\alpha }}$	${\mathit{\gamma }}_{\mathit{\rho }}$	${\mathit{\gamma }}_{\mathit{\beta }}$
$\Omega \setminus \omega$	${\alpha }_1/{\alpha }_0$	${\rho }_1/{\rho }_0$	${\beta }_1/{\beta }_0$
$\omega$	${\alpha }_0/{\alpha }_1$	${\rho }_0/{\rho }_1$	${\beta }_0/{\beta }_1$

.

2.1. Plane Structures

This section introduces the mathematical model for the plane structure problem and the shape functionals previously introduced herein in relation to the eigenvalue problem, for the sake of completeness. It also covers both the perturbed and unperturbed problems and demonstrates the existence of the associated topological derivative.

The original unperturbed problem can be stated as follows: Find $u\in \mathcal{V}(\Omega )$, such that

$${\int }_{\Omega }\alpha \sigma \left(\mathit{u}\right)·{\nabla }^s\mathit{v}+{\int }_{\Omega }\rho k\mathit{u}·\mathit{v}={\int }_{\Omega }\beta \mathit{f}·\mathit{v}\forall \mathit{v}\in \mathcal{V}(\Omega ),$$

(2)

where $\mathcal{V}(\Omega )=H_0^1(\Omega ;R^2)$. The coefficients $\alpha$, $\rho$, and $\beta$$\in {\mathbb{R}}^{+}$, are given in Table 1. In addition, $\sigma \left(\mathit{u}\right)=\mathbb{C}{\nabla }^s\mathit{u}$, $\mathit{u}:\Omega \mapsto R^2$ is the displacement field and k is a positive function. The symmetric part of the displacement gradient tensor ${\nabla }^s\mathit{u}$ and constitutive tensor $\mathbb{C}$ are given by

$${\nabla }^s\mathit{u}={\displaystyle \frac{\nabla \mathit{u}+{(\nabla \mathit{u})}^T}{2}}\mathrm{and}\mathbb{C}={\displaystyle \frac{E}{1+\nu }}\left(\mathbb{I}+{\displaystyle \frac{\nu }{1-\nu }}\mathbf{I}\otimes \mathbf{I}\right),$$

(3)

where the symbols $\mathrm{I}$ and $\mathbb{I}$ represent the second- and fourth-order identity tensors, respectively. It is worth mentioning that only the homogeneous Dirichlet condition on the boundary $\partial \Omega$ was considered. However, this could be replaced by any other boundary condition, provided the problem of interest remains well-posed.

Additionally, $\nu$ is the Poisson ratio and E is the Young’s modulus. The $L^2$ and energy norm shape functionals we are dealing with are defined as follows:

$$\mathcal{G}\left(u\right)={\int }_{\Omega }\rho k{\parallel \mathit{u}\parallel }^2\mathrm{and}\mathcal{J}\left(u\right)={\int }_{\Omega }\alpha \sigma \left(\mathit{u}\right)·{\nabla }^s\mathit{u}.$$

(4)

To simplify the expression of the topological derivatives, we introduce the adjoint problems for displacements q and p as follows:

$$q\in \mathcal{V}(\Omega ):{\int }_{\Omega }\alpha \sigma \left(\mathit{q}\right)·{\nabla }^s\mathit{v}+{\int }_{\Omega }\rho k\mathit{q}·\mathit{v}=-2{\int }_{\Omega }\rho k\mathit{u}·\mathit{v},\forall \mathit{v}\in \mathcal{V}(\Omega ),$$

(5)

$$p\in \mathcal{V}(\Omega ):{\int }_{\Omega }\alpha \sigma \left(\mathit{p}\right)·{\nabla }^s\mathit{v}+{\int }_{\Omega }\rho k\mathit{p}·\mathit{v}=-2{\int }_{\Omega }\alpha \sigma \left(\mathit{u}\right)·{\nabla }^s\mathit{v},\forall \mathit{v}\in \mathcal{V}(\Omega ).$$

(6)

The topologically perturbed counterpart of problem (2) is expressed as follows: Find ${\mathit{u}}_{\epsilon }\in \mathcal{V}(\Omega )$, such that

$${\int }_{\Omega }{\alpha }_{\epsilon }\sigma \left({\mathit{u}}_{\epsilon }\right)·{\nabla }^s\mathit{v}+{\int }_{\Omega }{\rho }_{\epsilon }k{\mathit{u}}_{\epsilon }·\mathit{v}={\int }_{\Omega }{\beta }_{\epsilon }\mathit{f}·\mathit{v}\forall \mathit{v}\in \mathcal{V}(\Omega ),$$

(7)

where the coefficients ${\alpha }_{\epsilon }$, ${\rho }_{\epsilon }$, and ${\beta }_{\epsilon }$ are defined through Table 2 and Table 3. The associated shape functionals are then defined as

$${\mathcal{G}}_{\epsilon }\left(u_{\epsilon }\right)={\int }_{\Omega }{\rho }_{\epsilon }k{\parallel {\mathit{u}}_{\epsilon }\parallel }^2\mathrm{and}{\mathcal{J}}_{\epsilon }\left(u_{\epsilon }\right)={\int }_{\Omega }{\alpha }_{\epsilon }\sigma \left({\mathit{u}}_{\epsilon }\right)·{\nabla }^s{\mathit{u}}_{\epsilon }.$$

(8)

2.1.1. Existence of the Topological Derivative

Equations (2) and (7) introduce the shape functionals for the original and perturbed domains. Based on these, the existence of the associated topological derivative can be stated as follows:

Lemma 1.

Let u and $u_{\epsilon }$ be solutions to the original (2) and perturbed (7) problems, respectively. Then, the estimate $\parallel {\mathit{u}}_{\epsilon }{-\mathit{u}\parallel }_{H^1(\Omega ;R^2)}=O\left(\epsilon \right)$ holds true.

Proof. 

Let us subtract (2) from (7). By setting $\mathit{v}={\mathit{u}}_{\epsilon }-\mathit{u}$ as the test function, after some simple analytical work, there is

$$\begin{array}{c}{\int }_{\Omega }{\alpha }_{\epsilon }\sigma ({\mathit{u}}_{\epsilon }-\mathit{u})·{\nabla }^s({\mathit{u}}_{\epsilon }-\mathit{u})+{\int }_{\Omega }{\rho }_{\epsilon }k{\parallel {\mathit{u}}_{\epsilon }-\mathit{u}\parallel }^2=\hfill \\ {\int }_{B_{\epsilon }}(1-{\gamma }_{\alpha })\alpha \sigma \left(\mathit{u}\right)·{\nabla }^s({\mathit{u}}_{\epsilon }-\mathit{u})+{\int }_{B_{\epsilon }}(1-{\gamma }_{\rho })\rho k\mathit{u}·({\mathit{u}}_{\epsilon }-\mathit{u})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\int }_{B_{\epsilon }}(1-{\gamma }_{\beta })\beta \mathit{f}·({\mathit{u}}_{\epsilon }-\mathit{u}),\hfill \end{array}$$

(9)

where the contrasts introduced in Table 2 and Table 3 are considered. The Cauchy–Schwarz inequality yields

$${\int }_{\Omega }{\alpha }_{\epsilon }\sigma ({\mathit{u}}_{\epsilon }-\mathit{u})·{\nabla }^s({\mathit{u}}_{\epsilon }-\mathit{u})+{\int }_{\Omega }{\rho }_{\epsilon }k\parallel {\mathit{u}}_{\epsilon }{-\mathit{u}\parallel }^2⩽C_1\epsilon {\parallel {\mathit{u}}_{\epsilon }-\mathit{u}\parallel }_{H^1(\Omega ;R^2)},$$

(10)

where the elliptic regularity of function u is used. The coercivity of the bilinear form on the left-hand side of Equation (10) results

$$c\parallel {\mathit{u}}_{\epsilon }{-\mathit{u}\parallel }_{H^1(\Omega ;R^2)}^2⩽{\int }_{\Omega }{\alpha }_{\epsilon }\sigma ({\mathit{u}}_{\epsilon }-\mathit{u})·{\nabla }^s({\mathit{u}}_{\epsilon }-\mathit{u})+{\int }_{\Omega }{\rho }_{\epsilon }k{\parallel {\mathit{u}}_{\epsilon }-\mathit{u}\parallel }^2$$

(11)

which leads to the result

$$\parallel {\mathit{u}}_{\epsilon }{-\mathit{u}\parallel }_{H^1(\Omega ;R^2)}⩽C\epsilon ,$$

(12)

with the constant $C=C_1/c$ independent of the small parameter $\epsilon$. □

2.1.2. Topological Sensitivities

The following fourth-order polarization tensor associated with the elasticity model is introduced as

$$\mathbb{P}={\displaystyle \frac{1}{2}}{\displaystyle \frac{1-{\gamma }_{\alpha }}{1+{\gamma }_{\alpha }{\delta }_2}}\left((1-{\delta }_2)\mathbb{I}+{\displaystyle \frac{1}{2}}({\delta }_1-{\delta }_2){\displaystyle \frac{1-{\gamma }_{\alpha }}{1+{\gamma }_{\alpha }{\delta }_1}}\mathbf{I}\otimes \mathbf{I}\right).$$

(13)

The constants ${\delta }_1$ and ${\delta }_2$ are given by

$${\delta }_1={\displaystyle \frac{{\mu }_l+{\lambda }_l}{{\mu }_l}},{\delta }_2={\displaystyle \frac{3{\mu }_l+{\lambda }_l}{{\mu }_l+{\lambda }_l}},$$

(14)

where ${\mu }_l$ and ${\lambda }_l$ are Lame’s coefficients, both considered constants everywhere. In this case, the plane stress assumption is applied as

$${\mu }_l={\displaystyle \frac{E}{2(1+\nu )}}{\lambda }_l={\displaystyle \frac{\nu E}{1-{\nu }^2}}.$$

(15)

The two main results of this section are stated here and are analogous to the work of [11]. The proofs of Theorems 1 and 2 are in the Appendix A.

Theorem 1.

Let $\mathcal{G}\left(u\right)$ be the shape functional defined by (4)-left; then, its associated topological derivative is given by

$$D_T\mathcal{G}=\alpha \mathbb{P}\sigma \left(\mathit{u}\right)·{\nabla }^s\mathit{q}-(1-{\gamma }_{\rho })\rho k\mathit{u}·(\mathit{u}+\mathit{q})+(1-{\gamma }_{\beta })\beta \mathit{f}·\mathit{q}a.e.\mathit{in}\Omega$$

(16)

where $\mathit{q}$ is the adjoint state solution of (5).

Theorem 2.

Let $\mathcal{J}\left(u\right)$ be the shape functional presented in (4)-right; then, its topological derivative is given by

$$D_T\mathcal{J}=\alpha \mathbb{P}\sigma \left(\mathit{u}\right)·{\nabla }^s(\mathit{u}+\mathit{p})-(1-{\gamma }_{\rho })\rho k\mathit{u}·\mathit{p}+(1-{\gamma }_{\beta })\beta \mathit{f}·\mathit{p}a.e.\mathit{in}\Omega$$

(17)

where $\mathit{p}$ is the adjoint solution of problem (6).

2.2. Eigenvalue Problem

The eigenvalue problem for the plane structures model of clamped free vibration can be stated as follows: Find u and $\lambda$, such that

$$\left\{\begin{array}{ccccc}\hfill -\mathrm{div}\left(\alpha \sigma \right(\mathit{u}\left)\right)& \hfill =& \lambda \rho \mathit{u}\hfill & \mathrm{in}& \Omega ,\\ \hfill \mathit{u}& \hfill =& 0\hfill & \mathrm{on}& \partial \Omega .\end{array}\right.$$

(18)

One can define the associated first eigenvalue as

$${\lambda }_1=\underset{\mathit{u}\in H_0^1(\Omega ;{\mathbb{R}}^2),\mathit{u}\ne 0}{inf}{\displaystyle \frac{{\int }_{\Omega }\alpha \sigma \left(\mathit{u}\right)·{\nabla }^s\mathit{u}}{{\int }_{\Omega }\rho {\parallel \mathit{u}\parallel }^2}},$$

(19)

with u being the solution of (18).

Note that $D_T$ obeys the quotient rule for differentiation and uses the functions described in Theorems 1 and 2. In particular, since the topological derivative obeys the basic rules of the differential calculus (including the quotient rule for differentiation), the rigorous justification for these kind of results can be found in the book by [15] (Ch 9). The topological derivative of

$$J\left(\mathcal{D}\right):={\lambda }_1,$$

(20)

is given by

$$D_{T}J={\displaystyle \frac{-\alpha \mathbb{P}\sigma \left(\mathit{u}\right)·{\nabla }^s\mathit{u}+(1-{\gamma }_{\rho })\rho {\lambda }_1{\parallel \mathit{u}\parallel }^2}{{\int }_{\Omega }\rho {\parallel \mathit{u}\parallel }^2}}.$$

(21)

3. Numerical Results

In this section, several numerical examples are presented to demonstrate the capability of the topological derivative concept in delivering optimal topologies for the problems addressed here. Based on the results from the numerical examples presented, the effectiveness of the proposed methodology for optimizing the first eigenvalue under various boundary conditions and loading scenarios will be evaluated.

In all examples, the topological optimization problem focuses on maximizing the first eigenvalue while adhering to a volume constraint. The convergence to a local minimum is demonstrated through various numerical examples. We apply a topology optimization algorithm that employs an evolution equation for the level-set function, based on $D_T$ (for further details, see [14]).

First Eigenvalue Maximization

Three cases are presented: Cases A, B, and C. We introduce a hold-all domain $\mathcal{D}\subset R^2$ such that $\Omega \subseteq \mathcal{D}$. The hold-all domain is discretized using linear triangular finite elements, resulting in an initial grid with uniform mesh spacing.

In Case A, the hold-all domain $\mathcal{D}$ is a rectangle $(0,3)\times (0,1),{\mathrm{m}}^2$, clamped on the left side along the contours ${\mathrm{\Gamma }}_{D_1}=0\times (0,0.1)$ and ${\mathrm{\Gamma }}_{D_2}=0\times (0.9,1.0)$, with a concentrated mass $m=0.4$ located at the point $(3,0.5)$, as depicted in Figure 1a. Figure 1b shows the displacement in the initial domain.

In Case B, the hold-all domain $\mathcal{D}$ is a rectangle $(0,6)\times (0,1),{\mathrm{m}}^2$, clamped at the point $(0,0)$ and simply supported at the point $(6,0)$. The concentrated masses $m=0.7$ are located at the points $(1,1)$ and $(5,1)$, and an additional concentrated mass $m_1=1.4$ is located at the coordinate $(3,1)$, as depicted in Figure 2a. The displacement is shown in Figure 2b.

In Case C, the hold-all domain $\mathcal{D}$ is a rectangle $(0,4)\times (0,1),{\mathrm{m}}^2$. Concentrated masses $m=m_i$, where $i=1,2,\dots ,11$ with $\sum m_i=70$, are uniformly distributed along the upper side of the rectangle (see Figure 3a). Figure 3b shows the displacement in the initial domain.

Linear triangular finite elements were employed for discretization in all the cases presented.

Young’s modulus is set as $E=210$ GPa and contrast parameters as ${\gamma }_{\alpha }={\gamma }_{\rho }={10}^{-3}$. Information on the initial grid, number of elements and number of nodes in the initial mesh, Poisson’s ratio $\nu$, and penalty parameter $\mu$ are expressed in

Table 4. Considered cases.

Cases	Grid	Elements	Nodes	$\mathit{\nu }$	$\mathit{\mu }$
A	$30\times 90$	10,800	5521	0.25	1.4
B	$30\times 180$	21,600	11,011	0.3	0.7
C	$10\times 40$	1600	851	0.3	1.5

, for each case.

It should be noted that mesh refinement procedures were employed as part of the optimization scheme to improve the boundary resolution of the final topology. Table 4 provides details on the final mesh resulting from the refinement process.

To enhance accuracy and achieve smoother topologies, four levels of uniform mesh refinement were implemented during the iterative process for Cases A and B, and five levels for Case C.

The mesh refinement levels resulted in 2,764,600 elements and 1,384,321 nodes for Case A, 5,529,600 elements and 2,768,161 nodes for Case B, and 409,600 elements and 205,601 nodes for Case C.

The optimal topologies for Cases A, B, and C are illustrated in Figure 4, Figure 5, Figure 6 and Figure 7, respectively. The effect of mesh refinement on the optimized topology for Case A is evident (see Figure 5). The final topology for Case B (Figure 6) exhibited asymmetry due to the boundary conditions, which included clamping at point (0,0) and simple support at point (6,0). In this example, the initial mesh comprises 19,200 elements (40 × 120 grid) and 9761 nodes, with all other parameters and boundary conditions consistent with those described for Case A.

Figure 8 introduces the normalized first eigenvalue history ${\lambda }_1/{\lambda }_1^0$ as the iterative process has evolved while Figure 9 depicts the normalized first eigenvalues’ history ${\lambda }_1/{\lambda }_2$.

It can be observed that Case C achieved a higher ratio of ${\lambda }_1/{\lambda }_2$ compared to Cases A and B.

However, during the optimization process, no coincident eigenvalues were observed. Figure 10 and Figure 11 illustrate the evolution of the shape functional and volume, respectively.

To evaluate the influence of the parameter $\mu$ on the final design, a set of four cases is considered. Variations in the parameter $\mu$ are detailed in

Table 5. Penalty parameter.

	Case D1	Case D2	Case D3	Case D4
$\mu$	$0.5$	$1.0$	$1.5$	$2.0$

.

For Cases D1, D2, D3, and D4, the hold-all domain $\mathcal{D}$ is a unit square $(0,1)\times (0,1),{\mathrm{m}}^2$, clamped at the points $(0,0)$ and $(0,1)$. A concentrated mass $m=0.1$ is applied at the point $(1,1)$, as shown in Figure 12a, with the initial displacement depicted in Figure 12b.

The final topologies for each case are presented in Figure 13a–d. Figure 14 and Figure 15 show the history of the shape functional and volume fraction, respectively.

The history of the normalized first eigenvalue, ${\lambda }_1/{\lambda }_1^0$ (where ${\lambda }_1^0$ is its initial value), is illustrated in Figure 16, showing the evolution until the process is halted.

Figure 17 depicts the history of the normalized first eigenvalue ratio, ${\lambda }_1/{\lambda }_2$. As before, no presence of multiple eigenvalues was observed.

4. Conclusions

In this work, the associated topological derivative of the first eigenvalue for plane structure problems was used for topology optimization. The convergence to a local minimum was achieved by combining the level set method with the analytical formula for the topological derivative ($D_T$). While coincident eigenvalues are often encountered in this type of problem, they were not observed in the present work. The algorithm converged successfully for all numerical examples, and no special methods were needed to handle multiple eigenvalues. The results highlight the effectiveness of the $D_T$ concept in deriving optimal topologies for eigenvalue problems in plane structures.