Achieving desired nodal lines in freely vibrating structures via material-field series-expansion topology optimization

Accurately controlling the nodal lines of vibrating structures with topology optimization is a highly challenging task.The major difficulties in this type of problem include a large number of design variables,the highly nonlinear and multi-peak characteristics of iteration,and the changeable orders of eigenmodes.In this study,an effective material-field series-expansion(MFSE)-based topology optimization design strategy for precisely controlling nodal lines is proposed.Here,two typical optimization targets are established:(1)minimizing the difference between structural nodal lines and their desired positions,and(2)keeping the position of nodal lines within the specified range while optimizing certain dynamic performance.To solve this complex optimization problem,the structural topology of structures is first represented by a few design variables on the basis of the MFSE model.Then,the problems are effectively solved using a sequence Kriging-based optimization algorithm without requiring design sensitivity analysis.The proposed design strategy inherently circumvents various numerical difficulties and can effectively obtain the desired vibration modes and nodal lines.Numerical examples are provided to validate the proposed topology optimization models and the corresponding solution strategy.

A MATLAB code for the material-field series-expansion topology optimization method

This paper presents a MATLAB implementation of the material-field series-expansion(MFSE)topology optimization method.The MFSE method uses a bounded material field with specified spatial correlation to represent the structural topology.With the series-expansion method for bounded fields,this material field is described with the characteristic base functions and the corresponding coefficients.Compared with the conventional density-based method,the MFSE method decouples the topological description and the finite element discretization,and greatly reduces the number of design variables after dimensionality reduction.Other features of this method include inherent control on structural topological complexity,crisp structural boundary description,mesh independence,and being free from the checkerboard pattern.With the focus on the implementation of the MFSE method,the present MATLAB code uses the maximum stiffness optimization problems solved with a gradientbased optimizer as examples.The MATLAB code consists of three parts,namely,the main program and two subroutines(one for aggregating the optimization constraints and the other about the method of moving asymptotes optimizer).The implementation of the code and its extensions to topology optimization problems with multiple load cases and passive elements are discussed in detail.The code is intended for researchers who are interested in this method and want to get started with it quickly.It can also be used as a basis for handling complex engineering optimization problems by combining the MFSE topology optimization method with non-gradient optimization algorithms without sensitivity information because only a few design variables are required to describe relatively complex structural topology and smooth structural boundaries using the MFSE method.