The local level set method (LLSM) is higher than the LSMs with global models in computational efficiency, because of the use of narrow-band model. The computational efficiency of the LLSM can be further increased by avoiding the reinitialization procedure by introducing a distance regularized equation (DRE). The numerical stability of the DRE can be ensured by a proposed conditionally stable difference scheme under reverse diffusion constraints. Nevertheless, the proposed method possesses no mechanism to nucleate new holes in the material domain for two-dimensional structures, so that a bidirectional evolutionary algorithm based on discrete level set functions is combined with the LLSM to replace the numerical process of hole nucleation. Numerical examples are given to show high computational efficiency and numerical stability of this algorithm for topology optimization.

Topology optimization is a numerical iterative procedure for making an optimal layout of a structure or the best distribution of material in the conceptual design stage [

With an implicit local level set model, the computational efficiency of the local level set method (LLSM) [

In the conventional LSM [

The aim of this work is to solve the aforementioned numerical issues that still exist in the LLSM for topology optimization of two-dimensional structures. A bidirectional evolutionary algorithm based on the discrete level set functions (DLSFs) is proposed to find a stable topological solution first and then combined with the LLSM to further evolve the local details of the topology and shape of the structure. Transforming the DLSFs into the local level set function of the LLSM is achieved by iteratively solving the DRE. After that, the DRE is incorporated into the LLSM to avoid the reinitialization procedure. A difference scheme under reverse diffusion constraints is formulated for the DRE to improve its numerical stability. Typical examples are given to show the effectiveness of the proposed algorithm in terms of convergence, computational efficiency, and numerical stability.

In the local level set method (LLSM) [

Narrow-band model and local level set function in LLSM.

It can be seen from Figure

A two-dimensional structural model is built in the work region

The values of elemental densities can be derived from the DLSFs. If the

The structural stiffness design has been widely investigated in numerous literatures for topological sensitivity analysis. The standard notion [

Similar to the bubble-method [

Based on an interpolation function proposed by Shepard [

Over the last two decades, many topology description models have been developed for topology optimization of structures, which can roughly be classified into two categories, the material distribution model and the boundary description model [

It is assumed that the volume in the

A parameter

It is assumed that the DLSF

Finally, a stable topological solution is obtained when the following convergence criterion is satisfied:

The majority of logical steps of the bidirectional evolutionary algorithm are presented in Figure

Flow chart depicting logical steps of the bidirectional evolutionary algorithm.

In the distance regularized level set evolution (DRLSE) [

In the original DRLSE, the energy density

It is easy to verify the boundedness of the diffusion rate

If

The diffusion effect can be divided into two parts: the forward diffusion for

With the diffusion rate

It is noted that the common difference schemes for the DRE with parts of the negative diffusion rates are incapable of remaining stable during an iterative process, according to the stability definition of the difference equation. In our numerical experiments,

It has been verified by our numerical experiments that the evolution of level set can remain bounded stability even after a large number of iterations for solving (

First, (

Then the four flow functions are defined as

It can be seen that the four diffusion rates in (

If

It can be seen that the absolute values of

The procedure for the LLSM with the DRE consists of two main parts, transforming the models of discrete level set functions into the local level set function and solving the difference schemes of the LLSE associated with the DRE. The final DLSFs corresponding to the stable topological solution can be transformed into the LLSF within the initial narrow-band

Flow chart depicting logical steps of the LLSM.

With the classical level set model [

For a number of level set-based approaches [

The reader is referred to the article [

The normal velocity field can be naturally extended to the whole domain using the so-called “ersatz material” approach, which fills the void areas with a weak material and then the material density is assumed to be piecewise constant in each element and is adequately interpolated in those elements cut by the zero level set (the shape boundary) [

In this section, two widely researched examples, the cantilever beam and the arch bridge, are presented in the context of structural minimum compliance design to demonstrate the characteristics of the proposed method. Some of the system parameters using the same values are defined as follows.

Young’s elasticity modulus for the solid material is

Shown in Figure

Design domain of the first cantilever beam and its boundary conditions.

In the design domain as shown in Figure

Topologies of zero level set at different steps: (a) Step

The corresponding level set surface at different steps: (a) Step

Convergent histories of the objective function and the constraint.

This study has also investigated the influence of different initial models of structure on the final design. Figure

Design domain of the second cantilever beam and its boundary conditions.

Topologies of zero level set in the two cases of the second cantilever beam.

The case starts from the full-material initial configurations

The case starts from the least-material initial configurations

Figure

Mesh-independent solutions of the second cantilever beam: (a)

The design domain of an arch-bridge model with a size

Design domain of the arch bridge.

This example focuses on the new characteristic of the proposed algorithm for improving the convergence of the bidirectional evolutionary algorithm using the LLSM. The structural topologies corresponding to the zero level set are depicted in Figure

Topologies of zero level set at different steps starting from the case of

Convergent histories of the objective function and the constraint.

Starting from different initial models, the final models obtained by the bidirectional evolutionary algorithm and the LLSM, respectively, are shown in Figure

Topologies of zero level set for the initial model, and the final models obtained by the bidirectional evolutionary algorithm and the LLSM, respectively: (a) case 1,

It can be seen from Figures

The LLSM is intended to remarkably increase the computational efficiency of the conventional LSMs using global models. To overcome the issue of hole nucleation of the LLSM, a bidirectional evolutionary algorithm is combined with the LLSM. This proposed algorithm has been used successfully in topology optimization of two-dimensional (2-D) structures, and it is easy to be extended to 3-D structures. The main features of this algorithm unknown to the conventional LSMs and the LLSM can be summarized as follows:

The discrete level set functions can be efficiently transformed into the local level set function by iteratively solving the distance regularized equation (DRE).

The DRE can be used instead of the reinitialization equation to further increase the computational efficiency of the LLSM.

A conditionally stable difference scheme under reverse diffusion constraints is formulated to ensure the numerical stability of the DRE.

If the stable topological solutions of the bidirectional evolutionary algorithm are inconsistent, the LLSM can achieve at least the consistent local optimal solution for the different cases of initial models.

High computational efficiency and numerical stability of the proposed algorithm have been verified by three typical numerical examples.

The authors declare that there is no conflict of interests regarding the publication of the paper.

The financial support from National Natural Science Foundation of China (no. 51278218) is gratefully acknowledged.