© S. Zhang et al., Published by EDP Sciences, 2025

1 Introduction

Optimization in structural engineering is a methodology grounded in numerical algorithms that integrates principles from engineering mechanics and computational mathematics. It enables systematic adjustment and optimization of structural parameters, boundary shapes, and even material distributions, all under various physical constraints and manufacturing conditions, to achieve optimal performance objectives [1]. Among its various forms, topology optimization represents an advanced paradigm that goes beyond mere boundary refinement or dimensional tuning. It operates directly on the material layout and connectivity within the entire design domain. This approach offers exceptional design flexibility and innovation potential, allowing for the generation of novel structural configurations that transcend the limitations of designer experience, thereby significantly enhancing mechanical performance, reliability, and weight efficiency [2].

Driven by the growing demand in high-end manufacturing, topological optimization techniques such as density-based methods [3], level set methods [4], and evolutionary methods [5] have been continuously developed. Research has progressively evolved from traditional stiffness-based problems to encompass strength, stability, dynamic response, microstructural studies based on energy homogenization, and multi-physics problems [6].

In engineering applications with strict dynamic performance requirements, such as offshore platforms, aircraft and satellite payload compartments, the position of the center of mass plays a decisive role in determining stability and dynamic performance [7–8]. In practical engineering, overall center of mass constraints are incorporated into both component layout and wing frame topology optimization to ensure that the Boeing 757 wing and its subsystems meet the balance and stability requirements of the aircraft; this consideration is particularly critical, as Bakhtiarinejad et al. have reported that without effective control of the center of mass, systems may suffer from attitude deviations, abnormal vibrations, or control instability during service, thereby severely compromising functional reliability and operational safety [8]. Thus far, only a small body of research in topology optimization has considered the regulation of a structure's overall center of mass. Cao et al. proposed a unidirectional scheme for center of mass control utilizing the level-set methodology, which was subsequently implemented for the optimization of thin-walled components, including the regulation of the moment of inertia in connecting rods and the suppression of flutter in trapezoidal rudder wings [9]. Liang et al. incorporated center of mass control within a finite volume method framework [10]. Nevertheless, existing studies on center-of-mass control in structural topology optimization remain inadequate: on the one hand, the absence of precise strategies for complex structures limits the ability to freely regulate the mass-center position; on the other hand, the lack of dynamic adjustment mechanisms during the optimization process prevents an effective balance between performance objectives and mass-center constraints, thereby inevitably leading to performance degradation.

This study aims to establish a level-set-based topology optimization framework with explicit center-of-mass constraints, enabling precise control of mass-center position and adaptive regulation to balance structural performance with spatial mass distribution. The primary innovations of this study are summarized below:

– A topology optimization scheme employing the level-set technique is constructed, which incorporates both volume and center of mass constraints to achieve simultaneous enhancement of structural efficiency and spatial configuration.

– A relaxation strategy for the center of mass constraint is introduced, which enhances the algorithm's search flexibility by allowing tolerance-based deviation and enables a balanced optimization between structural performance and spatial positioning while maintaining the directional evolution of the center of mass.

– The efficacy of the proposed method is verified using two- and three-dimensional computational experiments, which reveal the controllability and robustness of the center of mass across different coordinate directions. Moreover, the results confirm its capacity to drive the center of mass toward predefined target regions with reliable convergence.

– The developed approach provides a flexible and effective tool for spatial structural designs that require specific center of mass distributions, showing strong potential for engineering applications and broader adoption.

This paper is organized in the subsequent sections. Section 2 outlines the mathematical formulation of a topology optimization framework with center of mass regulation, which encompasses both sensitivity-driven strict and relaxed strategies, together with the procedure of optimization. Section 3 presents validation through a series of two- and three-dimensional numerical experiments, which provide an in-depth analysis of the method's influence on the evolution of structural configurations. The role of the relaxation mechanism in tuning both structural layout and compliance performance is also explored using a three-dimensional cantilever beam example. Finally, Section 4 concludes the study, discussing its engineering applicability and outlining future research directions.

2 Level set-based topology optimization method with center of mass control

2.1 Optimization mathematical model

For topology optimization employing the level-set approach, the geometry of the structure is implicitly captured using a level-set function, which evolves according to the progression of the zero-level interface. Originally proposed by Osher and Sethian in 1988 [11] to tackle interface evolution and front-capturing problems, the level-set approach was later generalized to structural optimization by Sethian and Wiegmann [12] as well as by Osher and Santosa [13], thus forming a unified framework that integrates geometric representation with optimization procedures. Beginning in 2003, Wang et al. [14] further combined the level set method with shape derivative theory, resulting in a more comprehensive framework for topology optimization. Unlike density-based methods such as SIMP [3] and ESO/BESO [15], the level-set approach characterizes structural boundaries with smoothness and continuity, which renders it particularly effective for applications demanding geometric precision, boundary smoothness, and adaptability to topological variations.

Under the level-set framework, variations in the structural boundary are implicitly characterized through the zero contour of an associated higher-dimensional function ϕ (x, t) [16]. For the two-dimensional scenario, it can be formulated as follows:

$$\{\begin{array}{c}\hfill \varphi (x,t)>0\forall x\in \mathrm{\Omega }\partial \mathrm{\Omega }\hfill \\ \hfill \varphi (x,t)=0\forall x\in \partial \mathrm{\Omega }\hfill \\ \hfill \varphi (x,t)<0\forall x\in \text{D}/\mathrm{\Omega }\hfill \end{array}.$$(1)

Figure 1 illustrates that D represents the overall design space containing an arbitrary interior point, whereas ∂Ω designates the solid boundary

The objective of the optimization is to minimize structural compliance. Following earlier studies [17–19], a center-of-mass constraint is incorporated into the optimization process, which allows for accurate positioning control. Both volume and center of mass are imposed jointly as constraints. The corresponding optimization formulation is expressed as given below:

$\begin{array}{c}\text{Minimize}J\left(u,\varphi \right)={\displaystyle \underset{D}{\int }}\left(ϵ\left(u\right):C:ϵ\left(u\right)\right)H\left(\varphi \right)d\mathrm{\Omega }\hfill \\ s.t.\{\begin{array}{c}\begin{array}{cc}a\left(u,v,\varphi \right)=l\left(v,\varphi \right)\hfill & \forall v\in U\hfill \end{array}\hfill \\ \begin{array}{cc}u=u_0\hfill & in{\mathrm{\Gamma }}_u\hfill \end{array}\hfill \\ \begin{array}{c}\begin{array}{c}\begin{array}{cc}C:ϵ\left(u\right)\cdot n=\tau \hfill & in{\mathrm{\Gamma }}_{\tau }\hfill \end{array}\hfill \\ G\left(\varphi \right)={\displaystyle \underset{D}{{\displaystyle \int }}}H\left(\varphi \right)d\mathrm{\Omega }-V_{max}\le 0\hfill \end{array}\hfill \\ \begin{array}{c}{g}_x\left(\varphi \right)=R_x\left(\varphi \right)-R_{x,t}=0\hfill \\ g_y\left(\varphi \right)=R_y\left(\varphi \right)-R_{y,t}=0\hfill \\ g_z\left(\varphi \right)=R_z\left(\varphi \right)-R_{z,t}=0\hfill \end{array}\hfill \end{array}\hfill \end{array}\hfill \end{array}$(2)

where J (u, ϕ) is the objective function, u the displacement field, ϵ the train tensor, Cthe constitutive elasticity tensor, H (ϕ) the Heaviside function. In weak variational form, a (⋅) = l (⋅) is the linear elastic equilibrium equation. v denotes the virtual displacement field within the spaceUof admissible kinematic displacements.u₀ and τ are the given displacement and traction on the Dirichlet and the Newman boundary conditions in Γ_u and Γ_τ respectively. n the normal direction along the boundary. G denotes the volume constraint. V_max is the prescribed volume limit within the design region D. g_x, g_y, and g_z are the constraints of the center of mass in x, y, and z direction respectively, whereR_x (ϕ) , R_y (ϕ) , andR_z (ϕ) represent the current coordinates of the positions of the center of mass, and R_x,t, R_y,t, andR_z,tdenote the target coordinates.

The Heaviside function H (ϕ)is introduced to improve the accuracy and robustness of derivative calculations in subsequent sensitivity analyses [20,21]. It is defined as the following form:

$$H_{ϵ}\left(\varphi \right)=\frac{1}{2}\left(1+\frac{\varphi }{\sqrt{{\varphi }^2+{ϵ}^2}}\right)$$(3)

where ϵ is the numerical smoothing coefficient to adjust the intensity of the variation of the function H_ϵ (ϕ).

Fig. 1 Illustration of the 2D level set function.

2.2 Exact control method for the position of the center of mass

To enforce the mass-center constraint during the evolution of the level-set framework, it is necessary to explicitly derive the gradients of the mass-center coordinates relative to the level-set description. Here, the derivation in the X-direction is presented as an illustrative example, and the Y- and Z-components follow analogously. By applying the Heaviside function, the continuous expression for the mass-center position is approximated as:

$${\text{R}}_{\text{x}}(\varphi )=\frac{{{\displaystyle \int }}_{\mathrm{\Omega }}{\text{x\hspace{0.17em}H}}_{ϵ}(\varphi )\hspace{0.17em}\text{d}\mathrm{\Omega }}{{{\displaystyle \int }}_{\mathrm{\Omega }}{\text{H}}_{ϵ}(\varphi )\hspace{0.17em}\text{d}\mathrm{\Omega }}=\frac{{\text{N}}_{\text{x}}(\varphi )}{\text{M}(\varphi )}$$(4)

where x denotes the coordinate of the point along the X-axis. N_x (ϕ) andM (ϕ) are defined as follows:

$$N_x\left(\varphi \right)={\displaystyle \underset{\mathrm{\Omega }}{{\displaystyle \int }}}x\hspace{0.17em}{H}_{ϵ}\left(\varphi \right)\hspace{0.17em}d\mathrm{\Omega }$$(5)

$$M\left(\varphi \right)={\displaystyle \underset{\mathrm{\Omega }}{{\displaystyle \int }}}{H}_{ϵ}\left(\varphi \right)\hspace{0.17em}d\mathrm{\Omega }$$(6)

By employing the quotient rule on the continuous formulation for the center of mass position given in equation (4). When taken with respect to ϕ, its partial derivative is expressed as:

$$\frac{\partial R_x}{\partial {\varphi }_i}=\frac{x_i-R_x}{M\left(\varphi \right)}\cdot {\delta }_{ϵ}\left({\varphi }_{\text{i}}\right)$$(7)

where the subscript i inx_iand ϕ_idenotes the i th node, and δ_ϵ (ϕ_i) represents the Delta function—the derivative of the Heaviside function. Consequently, the constraint sensitivity is formulated as:

$$\frac{g_{\text{x}}(\varphi )}{\partial {\varphi }_{\text{i}}}=\frac{{\text{x}}_{\text{i}}{\text{-R}}_{\text{x}}}{\text{M}(\varphi )}·{\delta }_{ϵ}({\varphi }_{\text{i}}).$$(8)

Finally, the sensitivity of constraint g_x (ϕ) with respect to design variable ϕ_i is presented in equation (8). In the subsequent optimization iterations, this derivative is utilized as guidance to drive the structural boundary evolution, ensuring that the center of mass progressively converges toward the specified target position.

2.3 Relaxation-based optimization method driven by center of mass sensitivity

In practical optimization processes involving center of mass control, strict or exact enforcement of the target position can prematurely constrain the search direction, thereby limiting the design space and potentially leading to suboptimal results. For design scenarios where exact center of mass positioning is not required, this section introduces a component-wise tolerance relaxation mechanism. When the bounds on the center of mass position are relaxed, greater search flexibility is introduced into the algorithm, enabling a balanced compromise between center of mass regulation and structural performance.

In this method, a component-wise tolerance threshold τ_j = [τ_x, τ_y, τ_z] is assigned for each direction (j ∈ {x, y, z}). When the absolute deviation of the position of the center of mass |R_j (ϕ) − R_j,t| exceeds its corresponding tolerance τ_j, the sensitivity of the centre of mass is considered during the optimization. Conversely, if the deviation |R_j (ϕ) − R_j,t| is less than or equal to τ_j, the sensitivity is excluded from the evolution to allow greater flexibility. To implement this adaptive mechanism, a sensitivity selection function e_j for centre of mass control is introduced, and its mathematical expression is given as follows:

$$\frac{g_j\left(\varphi \right)}{\partial {\varphi }_i}={\text{e}}_j\cdot \frac{x_i-R_j}{\text{M}\left(\varphi \right)}\cdot {\delta }_{ϵ}\left({\varphi }_i\right).$$(9)

Finally, within the relaxation scheme, the constraint sensitivity relative to the design parameter is expressed as:

$${\text{e}}_j=\{\begin{array}{cc}\hfill 1,\hfill & \hfill \text{if}\left|{\text{R}}_j-{\text{R}}_{j,t}\right|>{\tau }_j\hfill \\ \hfill 0,\hfill & \hfill \text{otherwise}\hfill \end{array}.$$(10)

2.4 Level set function update considering center of mass sensitivity

Within the shape derivative framework [22], the assessment of boundary propagation velocity in level-set-based topology optimization relies on the distribution of strain energy density together with the Lagrange multiplier λ. The obtained velocity fieldV_n is then applied in optimization procedures driven by gradients, for instance, the steepest descent approach. To concurrently enforce both volume and mass-centre constraints, this study employs the augmented Lagrangian technique [23,24], whereby an augmented Lagrangian function incorporating these constraints is constructed; the multipliers act as penalty terms that dynamically adjust constraint enforcement, enabling the optimization process to minimize compliance while simultaneously driving material distribution and the center of mass toward their prescribed targets, thus achieving a unified balance between structural performance and spatial control. The velocity fieldV_n is constructed and subsequently adopted to update the level-set representation:

$$V_n=ϵ\left(u\right):C:ϵ\left(u\right)-{\lambda }_V-{\lambda }_C\cdot {\displaystyle {\displaystyle \sum }_{j\in \left\{x,y,z\right\}}}\frac{g_j\left(\varphi \right)}{\partial {\varphi }_i}$$(11)

where λ_V refers to the Lagrange multiplier corresponding to the volume fraction restriction [16], while λ_C designates the multiplier associated with the centre of mass constraint. Their values are updated through the augmented Lagrangian scheme given below:

$${\lambda }_{\text{c}}^{(\text{k+1})}={\lambda }_{\text{c}}^{(\text{k})}\text{+}{\gamma }_{\text{c}}\cdot {(\text{|g}}_{\text{x}}({\varphi }^{(\text{k})}{)\text{|+|g}}_{\text{y}}({\varphi }^{(\text{k})}{)\text{|+|g}}_{\text{z}}({\varphi }^{(\text{k})})\text{|})$$(12)

where γ_c is the penalty parameter.

Consequently, the level set function ϕ can be evolved according to the expression given below:

$${\varphi }^{k+1}={\varphi }^k+\mathrm{\Delta }t\cdot V_n$$(13)

where Δt is the time step size.

As indicated by the above formulation, the method utilizes the velocity field to integrate compliance, center of mass, and volume information, thereby enabling comprehensive regulation of structural performance, mass distribution, and material utilization. During the level set function update process, the strain energy density reflects the structure's response to external loads and serves as the primary driving factor for compliance minimization. The Lagrange multipliers act as penalty terms that dynamically adjust the volume constraint, ensuring that the material usage remains within the prescribed limit. Meanwhile, the sensitivity of the center of mass introduces spatial information into the optimization, enabling the method to not only enhance structural performance but also actively regulate the global spatial distribution, thereby exerting control over the structural center of mass.

3 Numerical implementation

3.1 Precise control of center of mass in 2D structures

In this study, a classical cantilever beam example is employed to illustrate the effectiveness of the proposed center-of-mass control approach in two-dimensional topology optimization. Figure 2 depicts the computational domain, defined as a 60 × 30 rectangle, which is discretized into an identical grid of finite elements. The material properties are set with a Young's modulus of 1 and a Poisson's ratio of 0.3. A concentrated load of F = −100 is applied at the midpoint of the right edge, whereas the left edge is clamped. The volume fraction is restricted to 0.5. The design domain spans from coordinate (0, 0) at the lower-left corner to (60, 30) at the upper-right corner. The prescribed center of mass is specified at (30, 12). Figure 3 shows the resulting topology, where the blue rectangle designates the target location of the center of mass and the red dot indicates the actual outcome.

Fig 2 2D cantilever beam model for center of mass control optimization.

Fig. 3 Optimization process of center of mass control for the 2D cantilever beam.

3.2 Precise control of center of mass in 3D structures

Example 1 — Cantilever Beam

In this example, a three-dimensional cantilever beam is analyzed. The design region, defined as a cuboid of dimensions 50 × 20 × 10, is partitioned into an identical grid consisting of 50 × 20 × 10 finite elements. The constitutive material is assumed to possess a Young's modulus of 1 and a Poisson's ratio of 0.3. A volume fraction limit of 0.2 is prescribed. A concentrated vertical force of magnitude F = −1 is applied at the midpoint of the right-hand surface, whereas the opposite surface on the left is constrained as fixed. Figure 4 presents this case study: subfigure (a) illustrates the setup of the optimization model, and subfigure (b) shows the compliance minimization result without enforcing center of mass control.

Figure 5 presents the outcomes of topology optimization applied to a cantilever beam under varying constraints on the center of mass. The six subfigures are organized in sequence (from left to right and top to bottom) to depict different prescribed center of mass positions, providing insight into their influence on the structural evolution. Distinct geometric adaptations can be recognized across the optimized layouts corresponding to different targets.

A comparison along the horizontal direction between Figures 5a–5f shows that, as the Y-coordinate of the center of mass target rises from 8 to 14, the overall centroid shifts upward. As a result, the lower region of the structure becomes thinner, while the upper region thickens to accommodate this vertical displacement.

Likewise, a vertical comparison across Figures 5a–5f demonstrates that, when the X-coordinate of the target increases from 15 to 20, the material distribution gradually migrates from the fixed support toward the free end. This adjustment effectively moves the overall centroid closer to the prescribed location. It can also be observed that the control of the center of mass is predominantly achieved through redistribution of material along the longitudinal axis, rather than pronounced changes in the cross-sectional shape.

Figure 6 presents the optimization outcome of the cantilever beam when the center of mass is simultaneously restricted along the X, Y, and Z axes, with the target location defined as (15, 10, 8). The associated plot depicts the variation of the distance between the actual and designated center of mass during the optimization procedure. A continuous decrease in this distance can be observed, indicating that the structure's center of mass gradually approaches the specified target point.

Figure 7 presents the evolution of the minimum compliance objective, showing its convergence behavior together with the changes in structural configuration. Table 1 provides the specified target coordinates of the center of mass and the associated minimum compliance results for each optimization scenario.

Fig. 4 Optimization model and result of the cantilever beam.

Fig. 5 Optimization results of cantilever beam under different target position of center of mass.

Fig. 6 Optimization result of cantilever beam with simultaneous control of position of center of mass coordinate (x, y, z) and distance evolution curve to target position.

Fig. 7 Convergence curve of compliance and structural configuration evolution of the cantilever beam.

Example 2 — Simply Supported Beam

The design domain, measuring 50 × 20 × 10, is divided into 50 × 20 × 10 finite elements for analysis. The material is characterized by a Young's modulus of 1 and a Poisson's ratio of 0.3, while the maximum allowable volume fraction is constrained to 0.2. A point load of F= −1 is applied in the vertical downward direction at the midpoint of the upper surface, whereas the two ends of the lower boundary are fixed supports. Figure 8 depicts the configuration of this case study: subfigure (a) shows the optimization setup, and subfigure (b) demonstrates the result for the standard compliance minimization problem without incorporating center of mass constraints.

Figure 9 shows the topology optimization outcomes for the simply supported beam under various prescribed center of mass positions. In Figure 10, the center of mass is constrained simultaneously in the X, Y, and Z directions, with target coordinates set as (26, 10, 8). The figure also provides the curve describing the distance variation between the current and target center of mass during the optimization process. Figure 11 presents the convergence trend of the compliance function alongside the evolution of the structural form. Finally, Table 2 reports the specified target coordinates of the center of mass and the corresponding minimum compliance values obtained for this case.

Analysis of the above optimization results reveals that introducing the position of center of mass control leads to an increase in the compliance objective value compared to cases without such constraints. This is because, to satisfy the specified center of mass requirements, the optimization algorithm actively adjusts the internal material distribution, which can affect the structural stiffness to some extent. Moreover, by specifying different target center of mass positions, the optimization process is capable of generating a diverse set of structural configurations, thereby offering designers greater flexibility and a broader design space for balancing structural performance with spatial layout requirements.

Fig. 8 Optimization model and result of the simply supported beam.

Fig. 9 Optimization results of the simply supported beam under different target positions of the center of mass.

Fig. 10 Optimization result of the simply supported beam with simultaneous control of the position of center of mass coordinates (x, y, z) and distance evolution curve to target position.

Fig. 11 Optimization results of center of mass control for the cantilever beam under different modes.

3.3 Relaxation-based optimization of center of mass control

The computational domain, with dimensions of 50 × 20 × 10, is discretized into an identical grid consisting of 50 × 20 × 10 finite elements. The material is modeled with a normalized elastic modulus of 1 and a Poisson ratio of 0.3, while the allowable material volume is limited to 40%. A concentrated vertical load of F= −1 is imposed at the midpoint of the right boundary, and the left edge of the domain is fully constrained. In this case, the prescribed center of mass is set to (25,12,6), subject to a component-wise tolerance condition. The optimization results are reported in Figure 11: subfigure (a) illustrates the outcome under strict center of mass control, whereas subfigure (b) shows the configuration obtained using the tolerance relaxation strategy. Table 3 provides a summary of the results under these different control schemes.

The optimization results presented above indicate that both relaxed optimization modes for center of mass control effectively expand the search space of the algorithm. As shown in Table 3, compared to the exact control strategy, the relaxed mode yields structures with reduced compliance values. Fundamentally, the component-wise tolerance relaxation strategy provides the algorithm with a feasible design domain shaped as a unit cube, allowing greater positional deviation of the center of mass. This enlarged search space enables more diverse structural configurations. Figure 11 highlights that the structure generated under relaxed control (Fig. 11b) demonstrates higher symmetry, in contrast to the one obtained with strict control (Fig. 11a). Overall, adopting a relaxed center of mass control approach enhances design flexibility, allowing the control range to be tuned for specific engineering applications and enabling a balanced compromise between performance and spatial arrangement.

4 Conclusion and outlook

In order to overcome the insufficient treatment of center of mass control in topology optimization, this study introduces a level-set-based framework that integrates a dedicated control mechanism. A sensitivity-driven constraint function is adopted to formulate a dual-constraint model simultaneously addressing both volume and center of mass requirements. To achieve coordinated optimization under these conditions, the augmented Lagrangian technique is employed. Furthermore, a component-wise tolerance relaxation scheme is incorporated to broaden the search domain and increase the diversity of attainable solutions. Benchmark tests in two- and three-dimensional cases validate the robustness of the method, highlighting its controllability, stability, and adaptability in regulating the center of mass. This framework therefore offers a scalable tool for the design of engineering structures in which mass sensitivity and dynamic behavior are critical, underscoring its potential for practical deployment and further investigations.

In aerospace, the center of mass control is a certificated requirement shaping longitudinal stability and controllability. In offshore engineering, overall platform stability depends on the center of mass relative to buoyancy/métacentric criteria; co-optimizing topside equipment placement and supporting structure with explicit center of mass targets can reduce motions under environmental loading. In lightweight vehicle design—especially EVs with concentrated battery mass—the center of mass height and distribution strongly affect rollover propensity and handling, so incorporating the center of mass objectives into topology/layout optimization can lower SSF-related risk while preserving structural performance.

In this context, possible extensions of this research can be considered in two aspects. First, accounting for self-weight effects in the control process would enhance applicability to realistic engineering situations. Second, the framework could be further tested in complex design problems. For example, in offshore floating platform engineering, maintaining center of mass stability is essential to guarantee structural reliability under environmental loading. The proposed strategy may be employed for floating foundation cases to regulate mass distribution under complex conditions, thereby facilitating safer design and improved performance of floating support systems.

Acknowledgments

The authors wish to express their sincere gratitude to the State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment for providing research facilities and technical support. We also deeply appreciate the financial support from the National Key Research and Development Program, the National Natural Science Foundation of China, and the Guangdong Basic and Applied Basic Research Foundation, which made this work possible.

Funding

This research was supported by the National Key Research and Development Plan (2024YFB3814704), the National Natural Science Foundation of China (12572136), the Guangdong Basic and Applied Basic Research Foundation (2023A1515012830), and the State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment (GZ24110).

Conflicts of interest

The author hereby declares that there is no conflict of interest in the content or conclusions presented in this work.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Author contribution statement

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

References

All Tables

All Figures

	Fig. 1 Illustration of the 2D level set function.
In the text

	Fig 2 2D cantilever beam model for center of mass control optimization.
In the text

	Fig. 3 Optimization process of center of mass control for the 2D cantilever beam.
In the text

	Fig. 4 Optimization model and result of the cantilever beam.
In the text

	Fig. 5 Optimization results of cantilever beam under different target position of center of mass.
In the text

	Fig. 6 Optimization result of cantilever beam with simultaneous control of position of center of mass coordinate (x, y, z) and distance evolution curve to target position.
In the text

	Fig. 7 Convergence curve of compliance and structural configuration evolution of the cantilever beam.
In the text

	Fig. 8 Optimization model and result of the simply supported beam.
In the text

	Fig. 9 Optimization results of the simply supported beam under different target positions of the center of mass.
In the text

	Fig. 10 Optimization result of the simply supported beam with simultaneous control of the position of center of mass coordinates (x, y, z) and distance evolution curve to target position.
In the text

	Fig. 11 Optimization results of center of mass control for the cantilever beam under different modes.
In the text