In the proposed method, the FEA of solid regions is prioritized. Weak elements are temporarily removed in this step. When FEA of the solid regions is completed, FEA of the weak regions is performed with the displacement solution of solid regions as the Dirichlet boundary conditions. The FEA procedure is shown in Fig. 2.

Since the element densities can only be one or ρmin, the elements and corresponding node degrees of freedom (DOFs) in the solid regions and weak regions can be obtained according to the ordering rules of DOFs. Then, we can establish FE formulae of solid regions and weak regions. When considering the static small deformation problem, the FE formula of solid regions is as follows: (2)Ksus=fswhere Ks is the global stiffness matrix of the solid regions and fs is the load vector of the solid regions. The boundary conditions between the solid regions and the weak regions are zero Newman boundaries. Compared with the original FE model, this model removes weak elements.

In regard to the finite deformation problem, the nonlinear FE formula must be considered: (3)r(us)=ps−fs(us)=0where r is the residual force; ps is the external load vector of the solid regions; and fs is the load vector of the internal nodes in the solid regions at equilibrium, and its expression is as follows: (4)fs=∑MsCeTfewhere Ms is the set of solid elements; Ce is a matrix that transforms the modal force vector of the element to the globally nodal force vector; and fe is the nodal force vector of the element.

The most frequently applied scheme for solving nonlinear systems is the Newton–Raphson algorithm. It is based on a Taylor series development of Eq. (3) at a known state uks: (5)r(uks+Δuk,λ)=r(uks,λ)+Dr(uks,λ)Δuk+δwhere the parameter λ denotes the load level for which the solution has to be determined; DrΔuk characterizes the directional derivative of r at uks; and the vector δ is the residuum of the Tayler series. More details about the Newton–Raphson scheme can be found in the literature [42].

After the displacement vector us is obtained by solving Eqs. (2) or (5), we need to solve the displacement vector uv. To ensure the stability of the analysis, it can be solved simply by the linear FE formula.

In the FE model of weak regions, Dirichlet boundary conditions are imposed on the interface to maintain the continuity of the displacement field. The implementation process is as follows: first, the components of the displacement vector us need to be rearranged. The DOFs of noninterface nodes are placed in the front as uss, and the DOFs of interface nodes are placed in the back as uc. The rearranged displacement vector u~s is as follows: (6)u~s=[uss,uc]T 

The global stiffness matrix, displacement vector and load vector of the weak regions are also divided into two parts according to the noninterface nodes and the interface nodes. The updated FE formula is: (7)[KvvKvcKcvKcc][uvuc]=[fvfc]where Kvv is the stiffness matrix assembled by DOFs of noninterface nodes in weak regions and Kcc is the stiffness matrix assembled by DOFs of interface nodes. Kvc and Kcv are the coupling stiffness matrices, and they are symmetric.

Since uc has been obtained by Eq. (6), it can be moved to the right, and Eq. (7) can be rearranged: (8)Kvvuv=fv−Kvcuc

When the load-independent design problem is considered, the weak regions usually have no external load so that fv is a zero vector. Eq. (8) can be simplified as follows: (9)Kvvuv=−Kvcuc

The displacement vector uv is obtained by solving Eq. (9).

In the case of small deformation problems, the sensitivity number αe of an element that determines the element to be added or removed has been derived by Alonso et al. [43]: (10)αe=−ueΔKeuewhere ue is the element displacement vector and ΔKe is the variation in the elemental stiffness matrix. Since the densities of elements can only be one or ρmin, the sensitivity values have the following two forms: (11){αiR=ueiKeiuei,iftheith element is removedαiA=−ueiKeiuei,iftheithelement is added

When addressing small deformation problems, according to Huang et al. [21], the structural compliance can be replaced by an incremental form. The sensitivity number of the ith element is defined as (12){αiR=Eni,iftheith element is removedαiA=−Eni,iftheith element is addedwhere Eni is the strain energy of the ith element in its equilibrium state.

It should be noted that the sensitivity numbers in Eqs. (11) and (12) are both expressed by the element strain energy (or its negative value). However, this expression has different meanings for linear and nonlinear structures. For an elastoplastic structure, it denotes the total elastic and plastic strain energies, while for a linear structure, it is only for the elastic strain energy.

Two sensitivity thresholds αthR and αthV are chosen to determine material variation: solid elements with sensitivities less than αthV are converted to weak elements; weak elements with sensitivities greater than αthV are converted to solid elements (Fig. 3). Assuming that the volume to be removed is ΔVR and the volume to be added is ΔVA, the amount of material changed in an iteration is (13)ΔV=ΔVA−ΔVR

ΔV can be limited to a definite value by adjusting the two thresholds. One can refer to reference [43] for more details about material iteration.

The above strategies can effectively avoid the adverse effects caused by weak elements and maintain the integrity of sensitivities. However, in the execution process, structural instability may occur due to some special structural features. To ensure the robustness of the algorithm, these structural features must be solved.

Hinged elements often appear in solid regions when material is almost removed (Fig. 4). These structures are usually unstable. In computational geometry, such one-point connected structures belong to nonmanifold structures, and the point is named the nonmanifold point. A more common form of nonmanifold point in topology optimization is the checkerboard pattern. The checkerboard pattern is not expected due to its overestimated stiffness and difficulty in manufacturing [44]. Poulsen [45] proposed a simple and implicit scheme based on a structural grid to prevent checkerboard patterns and one-node hinges. However, we need direct control to ensure that these situations are avoided in each iteration.

In SERA, the initial structure is usually made entirely of solid elements. In the early optimization process, solid elements are gradually transformed into weak elements. This may lead to a situation in which some solid elements are completely separated from the main structure, forming an island (Fig. 10). In the original SERA method, an island is surrounded by weak elements that have low stiffness to support it so that the island can be constrained to a certain extent and the condition number is not too large to affect the stability of the numerical calculation. However, in the proposed method, the solid regions are analyzed without weak materials. The island is free of any constraints, which adversely affects the numerical stability.

To accommodate islands, we need to identify them first. The recognition of the nonmanifold point mentioned above is the recognition of elements, while an island is a region composed of an uncertain number of elements. Although the domain is divided in SERA, the numbers of solid regions and weak regions are still unknown. We introduce the fire-burning method [40] to group solid material by regions, and then the boundary conditions are used to identify whether a certain solid region is an island. Since an island has no effect on the structure stiffness, we group it into weak regions for FEA.

The condition number of a matrix is a well-known measure of ill conditioning. For an n×n matrix K, its condition number is: (15)κ(K)=||K−1||||K||where ||⋅|| is any matrix norm. If the matrix K is singular, we usually regard the condition number as infinite [47].

When considering the static small deformation problem, the design domain is discretized by the FE mesh, and the governing equation can be expressed by a linear equation: (16)Ku=fwhere u and f are the displacement vector and the load vector, respectively, and K is the global stiffness matrix that is assembled by element stiffness matrices.

In SERA, the element stiffness matrix is expressed as follows: (17)Ke=ρeK0where ρe=1 or ρmin means that the element is a solid element or a weak element and K0 is the stiffness matrix of the solid element. It is easy to evaluate from Eqs. (15)–(17) that the condition number of K is related to the ratio of the maximum and minimum elemental density in the design domain. By convention, the density of the solid element is usually set to one. If the density of the weak element decreases, the condition number of the corresponding stiffness matrix will increase.

A simple example can illustrate this. Fig. 13 shows an optimal structural topology of the short beam example. The density of the solid element is one, and the density of the weak element varies from 10⁻⁹ to 10⁻³⁰. The condition number of the global stiffness matrix changes as Fig. 14 shows. According to the curve, the logarithm of the weak elemental density is proportional to the logarithm of the condition number of K.

Since the condition number of the global stiffness matrix is a direct parameter to measure numerical stability, it can be seen from the above that two cases can lead to ill-conditioned systems: (1)

An extremely small ratio of the minimum and maximum densities.

The method proposed in Section 2 can completely solve the first case. The second case can be solved by the strategies introduced in Section 3.

The load and support conditions of the short beam example are shown in Fig. 14. The design domain is discretized by 100 × 50 rectangular elements. The upper limit of the volume fraction is 0.3, and the filter radius is 0.04.

In SERA, the optimal structure can usually be obtained faster by deleting materials than by adding materials. Therefore, the initial structures in the subsequent examples are all composed of solid elements. In the first half of the optimization process, the solid elements are gradually rejected until the volume constraint is met, and then the same amounts of solid elements and weak elements are increased or decreased until convergence.

In this example, four values, 10⁻⁹, 10⁻¹⁸, 10⁻²⁷, and 10⁻³⁶, are selected as the design values ρmin of the weak elements, and the original SERA method is also conducted for comparison. The iteration curves in Figs. 16 and 17 show that the order of the condition numbers almost increases inversely with the decrease in ρmin, which confirms the conclusion in Section 4.1. The iteration path also changes as ρmin changes. Occasionally, the condition number or objective function will increase dramatically in a step (shown in Figs. 16c, 16d and 17b–17d), which may greatly affect the iterative process and the optimal topology. Fig. 17 shows the structural topology and displacement field in the abnormal step. There are nonmanifold points and islands in the position framed by the dotted lines in Figs. 18a and 18c and abnormally large displacement values in the corresponding positions in Figs. 18b and 18d. According to Figs. 16e–16h and 17e–17h, after solving the nonmanifold points and islands, the condition number can be controlled from 10⁴ to 10⁷ by the proposed method and is not affected by ρmin.

Figs. 19–22 show the optimal topologies and corresponding objective values. In summary, the optimal results obtained by the original SERA method are affected by ρmin. Once ρmin drops to 10⁻³⁶, the optimal topology changes prominently. Figs. 17 and 19 show that the optimal topologies are almost unchanged as ρmin varies. Theoretically, the proposed method allows ρmin to be arbitrarily decreased with no effect on the optimization.

To verify the computational efficiency of the proposed method, the computational time is counted. The original SERA consumed 9.8 and 18.4 s for computing the two examples in Fig. 15; the proposed method consumed 18.0 and 23.2 s for computing the two examples. Therefore, the proposed method is less efficient than the original SERA. From Fig. 23, the time consumed by the proposed method in each iteration has a large oscillation, while in the original method, the time consumed in each iteration is relatively stable. It is well known that the most time-consuming process in optimization iteration is FEA. Since the number of elements in the two methods is the same, the time consumed by FEA is basically fixed. However, there is another time-consuming process in the proposed method: nonmanifold point processing. Since nonmanifold points do not appear at every step, nonmanifold point processing is also not performed at every step, which is the reason for the oscillation of the time-consumption curve.

In this example, we also verify the stability of the proposed method under different mesh densities. Three mesh densities are considered: 1) coarse, consisting of 100 × 50 rectangular elements; 2) medium, consisting of 200 × 100 rectangular elements; 3) fine, consisting of 300 × 150 rectangular elements. The loading position is shown in Fig. 15a. The design value of the weak material is 10⁻³⁶. The results for the three cases are shown in Fig. 24.

The optimal structural topologies are almost the same for the three mesh densities. Only the boundary smoothness and the size of local holes are slightly different. Therefore, the robustness of the proposed method is not affected by the mesh density. However, there is another factor to consider as the mesh density increases: in the proposed method, the nonmanifold point processing is time-consuming. Once the elements increase, the nonmanifold points may increase, which will reduce computational efficiency. To adapt to large-scale computation, we consider how to improve the efficiency of nonmanifold point processing in future work.

To verify the effectiveness of the proposed method for solving excessive mesh deformation under finite deformation, we considered an example proposed by Yoon et al. [46] shown in Fig. 25. The design domain is discretized by 60 × 60 rectangular elements, and the concentrated forces f1=0.02 and f2=0.03 are loaded. The elastic modulus E of solid is one, and Poisson’s ratio ν is 0.3. The design value of weak material ρmin is 10−9.

For better comparison, we analyzed the structure for the following three options: (1) geometric linearity or geometric nonlinearity; (2) inclusion or exclusion of weak regions; and (3) with or without a region partition strategy. The results are shown in Fig. 26. The corresponding settings and convergence for Figs. 26a–26e are shown in Table 1.

When no weak region is included (Figs. 26a and 26b), convergence can be achieved by a geometric linear or geometric nonlinear model. However, the maximum deformation of the former is larger than that of the latter. In addition, the rods on the upper side of the structure in Fig. 26a have become significantly thicker, which is obviously unreasonable. When considering weak regions, the result of the solid regions remains the same under the geometric linear model (Fig. 26c). However, in the geometric nonlinear model, the analysis process does not converge due to excessive mesh deformation (Fig. 26d). After adopting region recognition, nonlinear theory and linear theory are applied to the solid regions and weak regions, respectively. As shown in Figs. 26b and 26e, the results are exactly the same in the solid regions, which illustrates that the proposed method guarantees that excessive mesh deformation in the weak regions has no influence on nonlinear analysis so that convergence can be achieved.

Now, we apply the topology optimization method with a region partitioning strategy to this problem. The design domain and boundary conditions are still consistent with the model in Fig. 25. The filter radius r = 0.25. The volume fraction constraint is 0.28. Two initial structures (Figs. 27a and 27c) are tested for optimization: in one, the design area is completely covered by solid elements with a volume fraction of one; the other is the same as the model in Fig. 25, with a volume fraction of 0.28.

As shown in Figs. 27b and 27d, there are some differences in the optimal structures. The first optimal structure is identical to that obtained by Lahuerta et al. [48]. The second one, which is close to the first one, is significantly worse. The reason is that the second initial structure is much different from the optimal structure and falls into a local solution. However, under the first initial structure, there is no excessive deformation in the optimization process, and nonconvergence caused by mesh distortion did not occur. To verify the proposed algorithm, the second initial structure is optimized, and the mesh deformation is captured during the optimization process, as shown in Fig. 28. Although excessive mesh deformation exists in the first three steps, the optimization iteration is not adversely affected.

The design domain and boundary condition of the cantilever beam are shown in Fig. 29. The design domain is discretized by 120 × 30 rectangular elements. The magnitudes of the concentrated load are 0.0002, 0.0005, 0.0007 and 0.001. The initial volume fraction is 1.0, and the volume fraction constraint is 0.5. The design value ρmin of the weak material is 10−9, and the filter radius r is 0.014.

In this example, we verify the difference in optimization results under geometric linear and geometric nonlinear assumptions based on a region partitioning strategy. The effects of different loads on the topology optimization results are examined simultaneously. The results are shown in Fig. 30. On the first two lines in Fig. 30, we compare the optimal structures and objective values for different loads in linear and nonlinear solutions and obtained the following conclusions: (1) As expected, under the assumption of geometric linearity, the value of concentrated force does not affect the optimal topology; (2) Under the assumption of geometric nonlinearity, the optimal topology varies substantially with the load value. When the load value is very large, its result is notably different from the geometric linear solution. (3) The objective function values in geometric linear solutions and geometric nonlinear solutions are different. When the load value f is 0.001, the objective value of the geometric linear solution is only 17% of that of the geometric nonlinear solution.

The third line in Fig. 30 shows the final deformation mesh of the geometric nonlinear solution. When the load is 0.001, the weak elements around the loading point are clearly excessively deformed (Fig. 31). Due to the introduction of the region partitioning strategy, the weak regions are solved by a linear algorithm so that nonconvergence does not occur and the optimal structure can be obtained.