The marching squares algorithm [36] extracts iso-density contours by examining the values within each element and locating contour segments according to a specified threshold τ. Fig. 3a illustrates an example density field, which serves as the input for contours extraction. When the scalar values at the two endpoints of a line segment, denoted by v1 and v2, satisfy the condition v1<τ<v2, the intersection position s along the segment is computed via linear interpolation(4)s=x1+τ−v1v2−v1(x2−x1),where x1 and x2 denote the coordinates of the two vertices of the segment.

However, directly applying the marching squares algorithm may result in incomplete extraction of the outer boundary, as illustrated in Fig. 3b. This issue arises from incomplete sign changes across the threshold in scalar field adjacent to the bounding box. To accurately preserve the overall dimensional characteristics of the structure while automatically extracting the complete outer boundary, a virtual element padding strategy is introduced. Specifically, the material distribution near the design domain boundary is typically either fully solid (ρ≈1) or void (ρ≈0). By extending the optimized density field with an additional layer of virtual elements assigned the value 2τ−1, the scalar values at the two sides of a boundary become v2=1 and v1=2τ−1. Substituting v2 and v1 into Eq. (4) yields (5)s=x1+τ−(2τ−1)1−(2τ−1)(x2−x1)=x1+1−τ2(1−τ)(x2−x1)=12(x1+x2).

In this expression, x1 and x2 denote the locations of the virtual element and the boundary element in the optimized density field, respectively. The term 12(x1+x2) represents the midpoint between the two elements, ensuring that the intersection point is precisely aligned with the boundary of the design domain. As a result, the complete outer boundary can be correctly extracted, as shown in Fig. 3c.

In summary, the proposed strategy enables contours extraction at arbitrary iso-density levels within inner hole regions, while guaranteeing accurate alignment with the design domain and continuity of the outer boundary contours.

Specifically, consider a two-dimensional sequence of points {Pi}i=1N associated with a given contour, where Pi=(xi,yi). To smooth the boundary point sequence, a sliding window of size Nw is introduced. The two endpoints are excluded from the smoothing process and retain their original positions, i.e., P^i=Pi for i=1 and i=N, where P^i denotes the smoothed point. All remaining boundary points are smoothed using a local mean filter strategy to suppress geometric noise (6)P^i=1|Wi|∑j∈WiPj,Wi=[max{0,i−⌊Nw2⌋}, min{N,i+⌊Nw2⌋+1}],where Wi denotes the index range of the sliding window and |Wi| is its cardinality. After the points are smoothed, a filtering strategy is further applied to reduce the number of points. Starting from the first point, subsequent points are retained only if their distance from the previously retained one is not smaller than the filter radius rf. In contrast, the outer contour undergoes smoothing only, without applying the distance-based filtering, which results in a relatively dense point distribution to ensure higher geometric fidelity for interpolation.

In this study, a NURBS interpolation method based on global geometric constraints [37] is employed to reconstruct the preprocessed contour points into smooth, explicitly parameterized boundary representations that are well suited for shape optimization and CAD-based design operations. Given a set of points Pi∈R2 (i=0,1,…,N), a p-degree NURBS curve is constructed through three standard steps. The resulting NURBS curve is expressed as (7)C(u)=∑j=0NNj,p(u)Qj,where Qj denote the control point to be determined, and Nj,p(u) are the p-degree NURBS basis functions defined over the knot vector [u0,u1,…,um]. This formulation exactly interpolates the point sequence by solving a system of linear equations.

In the first step, a parameter value ui is assigned to each data point Pi, thereby establishing a correspondence between the point sequence and the parameter domain. The parameter values are computed using a chord-length-based scheme, (8)ui={0,i=0,ui−1+|Pi−Pi−1|∑j=1N|Pj−Pj−1|,i=1,…,N−1,1,i=N,and are subsequently normalized to the interval [0,1].

In the second step, the knot vector is constructed using the averaging method. With end knots of multiplicity p+1, the total number of knots is given by m=N+p+2, and the internal knots are defined as (9)uk={0,k=0,…,p,1p∑j=k+1−pkuj,k=p+1,…,m−p−1,1,k=m−p,…,m.

In the final step, the control points are determined by enforcing the interpolation conditions C(ui)=Pi, which leads to the following linear system (10)AQ=P,where Ai+1,j+1=Nj,p(ui) is the coefficient matrix, Q=[Q0,…,QN]T is the control point vector and P are the points for interpolation. Solving this system yields a uniquely determined set of control points, resulting in a smooth curve with Cp−1 continuity.

As shown in Fig. 4, the design domain in the shape optimization stage is defined as the area enclosed by the outer boundary, while the solid domain is obtained through a Boolean operation between the design domain and the void domain. The shape optimization problem is formulated as(11)Find:r=(r1,…,rm)∈Rm,min:C=FTU,s.t.{K(r)U=F,V(r)V0≤V¯.δ=0.

Compared with the topology optimization formulation in Eq. (1), both problems share the same objective of minimizing structural compliance. However, the design variables are changed from the element density ρ to the polar radius vector r of the inner boundary control points. In addition to the volume fraction constraint, a topological homeomorphism constraint defined by δ=0 is imposed to preserve topological equivalence, which will be discussed in Section 4.3.

Taking the contour in Fig. 5a as an example, boundary self-intersection may arise during shape optimization if the control points are directly manipulated in Cartesian coordinates, as shown in Fig. 5b. To avoid this issue, the Cartesian coordinates are transformed into polar coordinates according to(12)ri=(xi−xc)2+(yi−yc)2,θi=atan2⁡(yi−yc,xi−xc),as illustrated in Fig. 5c. Here, (xi,yi) denote the Cartesian coordinates of the control points, while (13)xc=1n∑i=1nxi,yc=1n∑i=1nyi,denote the geometric center of the control points for a single contour. The function atan2⁡(⋅,⋅) represents the four-quadrant inverse tangent. To ensure consistency in the angular representation, the angle θi is normalized to the interval [0,2π) as (14)θi←(θi+2π)mod(2π).

Thus, the control points can be represented by the triplet (15)(xc, yc, {(θi, ri)}i=1n).

The polar radii of all holes are then assembled to form the design variable vector r=(r1,…,rm)∈Rm, thereby restricting the movement of each control point to its radial direction. Fig. 5d demonstrates the effectiveness of this approach in enabling shape adjustments while preventing boundary self-intersection.

The optimization objective C is solved using finite element method, frequent remeshing of the grids following each geometric update would significantly reduce computational efficiency and pose considerable implementation challenges. Therefore, FG-FEM is adopted for structural response analysis. In this approach, the structure is represented using a level set function (LSF) Φ, which links the geometric description with the analysis model. A LSF is introduced over the design domain Ω⊂Rd (d=2), defined as (16){ΦΩ(x)=0,∀x∈∂Ω,ΦΩ(x)>0,∀x∈Ω∖∂Ω,ΦΩ(x)<0,∀x∈D∖Ω.

Here, x denotes an arbitrary point in space, and the structural boundary ∂Ω is defined as the zero level set of ΦΩ. A positive value of ΦΩ(x) indicates that the point lies inside the solid domain, whereas a negative value corresponds to the void domain. The corresponding composite LSF of the solid domain is defined as (17)ΦΩ(x)=min{{φk}k=1Nv,ΦD},where Nv denotes the number of inner holes, with φk and ΦD representing the LSFs of the void domain and the design domain, respectively. The construction of the corresponding LSFs is described sequentially.

In the optimization model, the minimum distance required to prevent interference between adjacent holes cannot be prescribed a priori. Consequently, directly imposing simple bound constraints [rmin,rmax] on design variables would restrict the feasible design space and result in overly conservative design limits. To address this issue, an interference detection and elimination strategy is proposed, in which potential boundary interference during geometric evolution is quantitatively identified and resolved, thereby enforcing the topological homeomorphism constraint while allowing full exploration of high-performance structural shapes.

Building on the FG-FEM, consider a domain discretized into Ny×Nx finite elements. Let φk denote the LSF associated with the k-th hole, constructed from the boundary curve Ck and evaluated at an arbitrary grid node (X,Y). According to Eq. (20), φk(X,Y)<0 if and only if the node lies inside Ck. The values of the LSFs for Nv holes evaluated on the discretized grid nodes are stored in a three-dimensional array φ∈R(Ny+1)×(Nx+1)×Nv, such that (24)φi,j,k=φk(Xj,Yi),i=1,…,Ny+1, j=1,…,Nx+1, k=1,…,Nv.

The coverage multiplicity χ on the grid node (Xj,Yi) is then defined as (25)χi,j=∑k=1NvI(φi,j,k<0),where I(⋅) denotes the indicator function. A value χi,j≥2 means that the node is simultaneously covered by at least two holes, signifying the occurrence of geometric interference. The total interference measure, defined in Eq. (11) as the sum of coverage multiplicities over all nodes, is given by (26)δ=∑i=1Ny+1∑j=1Nx+1χi,jI(χi,j≥2).

The quantity δ serves as a measure of geometric interference and is employed to guide the interference elimination procedure. Specifically, δ>0 indicates the occurrence of boundary interference, whereas δ=0 indicates that no boundary interference occurs, thereby enforcing the topological homeomorphism constraint throughout the structural evolution process.

Fig. 9 illustrates a representative example in which interference between inner holes violates the topological homeomorphism constraint, together with the corresponding interference elimination process. At the initial phase, geometric interference is detected, with 14 nodes concurrently covered by two holes. During the interference elimination process, the polar radii gradually contract, resulting in a progressive decrease in the interference values. Eventually, the interference is completely eliminated, the polar radii stabilize, and the topological homeomorphism constraint is satisfied (δ=0).

The sensitivity of the compliance with respect to the design variable ri is derived as (27)∂C∂ri=∂FT∂riU+FT∂U∂ri.

Since the external load F is independent of the design variables, and recalling the equilibrium equation KU=F, differentiation with respect to ri yields (28)∂K∂riU+K∂U∂ri=0,from which the displacement sensitivity is obtained as (29)∂U∂ri=−K−1∂K∂riU,substituting Eqs. (29) into (27), the compliance sensitivity with respect to ri is obtained as (30)∂C∂ri=FT∂U∂ri=−UT∂K∂riU=−βE0∑e=1Ne(ρeβ−1∂ρe∂riueTk0ue).

The sensitivity of the volume with respect to ri is expressed as (31)∂V∂ri=∑e=1Ne∂ρe∂ri.

Based on Eq. (22), the sensitivity of the element density ρe with respect to ri can be written as(32)∂ρe∂ri=14∑n=14∂H(ΦΩ(xne))∂ri,which is difficult to evaluate analytically due to the implicit dependence of the LSF on the design variables.

The sensitivity of the interference measure δ with respect to the design variable ri is not analytically differentiable. Thus the sensitivities of both the interference measure and the element density are computed numerically using the central finite difference scheme, which is widely adopted in sensitivity analysis for its simplicity and numerical stability [41]. The finite difference approximation is applied as follows:(33)∂δ∂ri≈ΔδΔri=δ(ri+Δr)−δ(ri−Δr)2Δr,∂ρe∂ri≈ΔρeΔri=ρe(ri+Δr)−ρe(ri−Δr)2Δr,where Δr denotes the finite difference step size used in the sensitivity evaluation.

Fig. 10 illustrates the detailed procedure of the shape optimization framework. The optimization process starts with the initialization of the design variables and algorithmic parameters. Since the reconstructed geometry is obtained from topology optimization results and the boundary is initialized in the intermediate density region, the subsequent optimization process may suffer from premature convergence. To alleviate this issue, small random perturbations are introduced during the initialization stage. Specifically, for each hole, a set of random noise values with the same dimension as the polar radii variables is generated from a uniform distribution within the interval [−2Δr,2Δr], where Δr is the step size used in the central finite difference scheme (Eq. (33)). The perturbations are added to the polar radii variables to slightly modify the initial design configuration.

After initialization, the density field of the design domain is evaluated based on the current boundary representation. The interference measure δ is then calculated to determine whether boundary interference occurs. If δ≠0, an interference elimination procedure is activated. In this stage, the sensitivity of δ with respect to the design variables is computed, and the design variables are updated to reduce the interference until δ=0. This process guarantees that the reconstructed geometry preserves the topological homeomorphism of the topology optimization result.

Once the interference has been removed (δ=0), the algorithm proceeds to the shape optimization stage. The sensitivities of the objective function (compliance C) and the volume constraint V are computed, and the design variables are updated using the Method of Moving Asymptotes (MMA) [42]. After each update, the density field of the solid domain and the interference measure δ are recalculated to verify whether new boundary interference has been introduced during the optimization process.

The entire procedure is embedded within an outer iterative loop. The optimization proceeds until convergence is achieved or the maximum number of iterations is reached. Convergence is evaluated using two criteria: the difference between the current constraint value and its prescribed value, and the maximum change in the design variables between two consecutive iterations. The optimization terminates when both quantities fall below their prescribed thresholds, namely 10−4 for the constraint deviation and 10−2 for the design variable changes.

The contour points shown in Fig. 11 are extracted from the topology optimization results using a threshold value of τ=0.6. With the proposed virtual element padding strategy, a closed and continuous outer contour is obtained. Building upon this capability, the strategy enables the extraction of contours at arbitrary threshold levels, while ensuring the outer contour remains properly aligned with the design domain.

The contour points obtained after smoothing and filtering are shown in Fig. 12, where the smoothing window size and filtering radius are set to Nw=3 and rf=3, respectively. Through the sparse sampling procedure, the total number of contour points is reduced from 445 to 99. In addition, the outer contour points are locally densified near the corner regions of the design domain to prevent excessive curvature distortion when high-order NURBS curves undergo sharp turning. As a result, the contour points exhibit enhanced smoothness and geometric consistency, providing a stable and well-conditioned input for subsequent interpolation. For comprehensive investigations into the influence of the reconstruction parameters (τ, Nw, and rf) on mechanical performance, the reader is referred to Appendix A.

The interpolated NURBS boundaries and corresponding control points are shown in Fig. 13, where a third-order (p=3) closed interpolation is achieved by appending the midpoint of the endpoints to both ends of the filtered point sequence, resulting in a total of 105 design variables. Compared with the topology optimization results, the reconstructed model provides a clear and parametric description. The main geometric features of both the outer and inner contours are accurately captured, yielding smooth and continuous boundaries that help reduce stress concentrations. Moreover, the sparse and well organized distribution of control points leads to a reduced-dimensional design variable space, which significantly reduces the computational cost of shape optimization.

Through the shape optimization step, the inner holes are systematically adjusted. As shown in Fig. 14, the optimized geometry exhibits only moderate changes relative to the initial reconstruction, indicating that the shape optimization primarily serves to recover mechanical optimality and constraint consistency rather than to alter the overall structural layout.

The shape optimization history and quantitative results shown in Fig. 15 provide clear insight into the convergence behavior and effectiveness of the proposed optimization strategy. As shown in Fig. 15a, nonzero interference values are detected during the early iterations, which trigger the interference elimination procedure. During this phase, the optimization temporarily prioritizes the reduction of interference rather than the minimization of structural compliance. Once the interference is fully eliminated (δ=0), the algorithm automatically switches to the shape optimization procedure, during which the volume fraction rapidly converges to the target value and remains tightly controlled. Meanwhile, the compliance decreases smoothly, indicating stable convergence without reintroducing geometric interference, with the process terminating after 134 iterations. This behavior demonstrates that the volume constraint is effectively enforced in the early stage of the optimization, after which the algorithm focuses on further reducing structural compliance. Fig. 15a shows that the reconstructed geometry prior to shape optimization exhibits deviations of 3.8% in compliance and 3.2% in volume fraction relative to the topology optimization results, indicating that the geometry reconstruction process introduces noticeable performance discrepancies. After the shape optimization stage, these deviations are reduced to 0.3% and 0%, respectively, demonstrating the effectiveness of the proposed framework in preserving mechanical performance. Fig. 15b visualizes the interference reduction process, showing that the initially intersecting boundaries are gradually separated to satisfy the topological homeomorphism constraints.

Fig. 16 shows the cantilever beam, Michell beam, and half-MBB beam, each with a distinct geometric scale. The corresponding results of topology optimization, geometry reconstruction, and shape optimization are presented in Fig. 17. It should be noted that, in the Michell beam case, the final optimized structure exhibits a certain degree of asymmetry. This behavior arises from the asymmetric distribution of control points generated during the geometry reconstruction stage. Since the polar radii are adopted as design variables, each control point is constrained to move only along its radial direction, and symmetry in the final optimized structure is therefore not always guaranteed.

The detailed mechanical performance values are summarized in Table 2, and the corresponding relative deviations with respect to the topology optimization results are illustrated in Fig. 18. The cantilever beam exhibits the largest discrepancies, with a compliance increase of up to 25.6% and a volume fraction reduction of 16.0% prior to shape optimization. After the shape optimization stage, the compliance deviations are substantially reduced for all cases, remaining within a narrow range of approximately 0.5%–2.6%, while the volume fraction deviations are fully eliminated in every example. The proposed framework effectively mitigates compliance deviation and consistently restores the target volume fraction, demonstrating its robustness and effectiveness across different structural configurations and geometric scales.

A quadcopter frame is considered as a representative engineering example. Topology optimization is first performed to provide the basis for geometry reconstruction using both heuristic, visually guided methods and the proposed framework. The reconstructed designs are then comparatively evaluated through computer-aided engineering (CAE) simulations, and the superior design is further fabricated by 3D printing.

The initial structure of the frame is shown in Fig. 19a and is subsequently partitioned according to its functional requirements, as illustrated in Fig. 19b, where the rib region is defined as the design domain. The topology optimization results are presented in Fig. 19c, which serves as the basis for geometry reconstruction. It can be observed that the boundaries of the frame are rough, making the results difficult to be directly applied in practical engineering scenarios. Fig. 19d presents the reconstructed geometry obtained through heuristic methods.

Using the reconstructed geometry based on heuristic methods as a reference, contour points are uniformly sampled along the straight segments of the contours and used as inputs for the proposed two-stage framework, as illustrated in Fig. 20a. Fig. 20b shows the optimized boundaries obtained under compliance minimization without increasing the structural volume. Fig. 20c illustrates the boundary control points that are imported into CAD software and used to sketch the optimized boundaries onto the initial structure. Fig. 20d then presents the reconstructed geometry generated by subsequent hole-cutting operations.

As shown in Fig. 21a, under linear elastic assumptions and identical loading conditions, the model reconstructed using the proposed framework exhibits reduced deformation compared with the heuristic reconstruction, with the maximum displacement decreasing from 10.448 to 9.342 mm. This reduction can be attributed to the fact that the shape optimization procedure targets compliance minimization, which redistributes material toward regions of high strain energy and better defines the primary load paths, thereby enhancing structural stiffness. In addition, the stress distribution is improved after the optimization. As shown in Fig. 21b, the optimized structure exhibits more regions with lower stress levels. The maximum stress decreases from 6488.2 to 4573.1 MPa, indicating that the proposed framework also leads to a noticeable improvement in stress performance by generating smoother structural boundaries.

Subsequently, the optimized frame is fabricated using 3D printing, and the resulting physical prototype is shown in Fig. 22.

Overall, the numerical examples and the engineering case demonstrate that the proposed framework can effectively reconstruct topology optimization results while preserving the structural performance. Moreover, the reconstructed geometry is fully compatible with standard CAD, CAE, and CAM workflows, thereby facilitating its integration into practical engineering design and industrial manufacturing processes.