Topology optimization using material-field series expansion and Kriging-based algorithm

Topology optimization is a mathematical method that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximizing the performance of the system. Recently, this technique has experienced a surge in interest credit to its vital input as a tool for novel design in structural and multidisciplinary engineering problems. To this end, several topology optimization techniques have been developed. These techniques are mainly gradient based and suffer from one drawback; i.e. they require information about the sensitivity of objective or constraint functions with respect to a large number of design variables. Consequently, the challenge of developing a non-gradient topology optimization (NGTO) method has attracted the interest of many researchers. NGTO-based approaches have become highly sought after not only because they help circumvent some of the drawbacks encountered in gradient based methods, but also since they offer better global-searching ability and run efficiently by calculating multiple samples simultaneously on parallel computers. Noteworthy reports have established that the difficulty with NGTO lies mainly in the huge dimensionality of the design space.

Generally, the existing NGTO approaches reported in the literature, where the reported number of finite element evaluations typically exceed 20,000 even for very coarsely discretized topology optimization problems, are regarded as being hopelessly inefficient for topology optimization problems. Therefore, to address this issue, researchers from the Dalian University of Technology in China: Dr. Yangjun Luo, Dr. Jian Xing and Professor Zhan Kang developed a new NGTO method based on a reduced-dimensionality model established recently using a material-field series expansion (MFSE). Their goal in the study was to present an efficient non-gradient approach to the topology optimization of structures when no information is available about design sensitivity. Their work is currently published in the research journal, Computer Methods in Applied Mechanics and Engineering.

In their work, the problem of topology optimization was constructed as a constrained minimization model with the series expansion coefficients as the design variables, based on the material-field series expansion. The researchers further used the Kriging-based optimization algorithm incorporating two infill criteria to solve the optimization problem. In addition, the researchers proposed a special strategy using a self-adjusting design domain and remodeling the surrogate function to improve the searching efficiency of the Kriging-based algorithm.

The authors reported that the reduced series expansion of the material field reduced the number of design variables greatly, and the optimized solution could be obtained using the modified surrogate-based algorithm with a self-adaptive strategy for the design domain. Better still, the study reported that the KG-MFSE method was applied successfully to compliance-minimization problems, a fluid problem, and a nonlinear soft-robot structure.

In summary, the study by Dalian University of Technology scientists presented an effective NGTO topology optimization method based on the material-field series expansion and the Kriging surrogate model. Remarkably, the results with clear topology were reported to be free of numerical difficulties such as checkerboard patterns and mesh dependency. In a statement to Advances in Engineering, Dr. Yangjun Luo further commented that their new technique for topology optimization problems is straightforward and effective because it provides convenient access to finite element software and parallel computers.