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

Abstract

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.

Graphical abstract

Keywords

nodal line / topology optimization / structural dynamics design / material-field series-expansion

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Nomenclature

||·||2	2-norm
C	Correlation matrix
Cd(r)	Vector {\left\{C\left(\mathit{r},{\mathit{r}}_1\right),C\left(\mathit{r},{\mathit{r}}_2\right),...,C\left(\mathit{r},{\mathit{r}}_{N_{\text{MFP}}}\right)\right\}}^{\text{T}}
E	Young’s modulus
E0	Young’s modulus of the considered solid material
Emin	Small values of Young’s modulus for avoiding single-element matrices
f_{m_q}	Nodal line measurement function
g_{m_q}	Constraint function
K^{\prime }	Number of current nodal lines
K	Structural stiffness matrix
lc	Given correlation length
L_1^{\ast },L_2^{\ast },...,L_K^{\ast }	Desired nodal lines
L_{k,m_q}^{}	Current nodal line
Lm	Corresponding nodal lines
M	Mass matrix
NMFP	Number of material points
p	Penalty parameter
r	Coordinate vector
ri	Material points
{\mathit{r}}_{k,j}^{\ast }	Desired node
{\mathit{r}}_{k,j,m_q}^{}	Current node
\mathit{u}, \ddot{\mathit{u}}	Vectors of the degrees of freedom and accelerations, respectively
α	Weighting factor
β	Parameter that controls the smoothness of the mapping
ρ	Mass density
ρ0	Mass density of the considered solid material
ρmin	Small values of mass density for avoiding single-element matrices
η	Vector of the undetermined coefficients or design variables
μ1, μ2	Poisson’s ratios
λm	mth eigenvalue
ωm	mth eigenfrequency
\phi \left(\mathit{r}\right)	Continuous material-field function
Φ	Corresponding eigenvectors
Φm	Corresponding eigenmode
χ(r)	Structural topology
Ωdes	Design domain
ζ(ωi)	User-specified function for some specific order eigenfrequencies
Δj	Allowable deviation of the current nodal lines from the desired nodal lines
\mathbf{\Lambda }	Diagonal matrix composed of the first N largest eigenvalues

Acknowledgements

This work was supported financially by the Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2022A1515240059), the National Natural Science Foundation of China (Grant No. 52275237), and the Shenzhen Stability Support Key Program in Colleges and Universities of China (Grant No. GXWD20220817133329001).

Conflict of Interest

The authors declare that they have no conflict of interest.

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