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

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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.

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Abbreviations

‖·‖2 :

2-norm

C :

Correlation matrix

C d(r):

Vector \({\{C({\boldsymbol{r}},{{\boldsymbol{r}}_1}),\,\,C({\boldsymbol{r}},{{\boldsymbol{r}}_2}), \ldots,C({\boldsymbol{r}},{{\boldsymbol{r}}_{{N_{{\rm{MFP}}}}}})\} ^{\rm{T}}}\)

E :

Young’s modulus

E 0 :

Young’s modulus of the considered solid material

E min :

Small values of Young’s modulus for avoiding single-element matrices

\({f_{{m_q}}}\) :

Nodal line measurement function

\({g_{{m_q}}}\) :

Constraint function

K′ :

Number of current nodal lines

K :

Structural stiffness matrix

l c :

Given correlation length

\(L_1^ {\ast},L_2^{\ast}, \ldots,L_K^ {\ast}\) :

Desired nodal lines

\({L_{k,{m_q}}}\) :

Current nodal line

L m :

Corresponding nodal lines

M :

Mass matrix

N MFP :

Number of material points

p :

Penalty parameter

r :

Coordinate vector

r i :

Material points

\({\boldsymbol{r}}_{k,j}^ \ast \) :

Desired node

\({{\boldsymbol{r}}_{k,j,{m_q}}}\) :

Current node

\({\boldsymbol{u}},{\boldsymbol{\ddot u}}\) :

Vectors of the degrees of freedom and accelerations, respectively

α :

Weighting factor

β :

Parameter that controls the smoothness of the mapping

ρ :

Mass density

ρ 0 :

Mass density of the considered solid material

ρ min :

Small values of mass density for avoiding single-element matrices

η :

Vector of the undetermined coefficients or design variables

μ 1, μ 2 :

Poisson’s ratios

λ m :

mth eigenvalue

ω m :

mth eigenfrequency

φ (r):

Continuous material-field function

Φ :

Corresponding eigenvectors

Φ m :

Corresponding eigenmode

χ(r):

Structural topology

Ωdes :

Design domain

ζ(ω i):

User-specified function for some specific order eigenfrequencies

Δj :

Allowable deviation of the current nodal lines from the desired nodal lines

Λ :

Diagonal matrix composed of the first N largest eigenvalues

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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. GXWD202208171333 29001).

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Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Yi Yan, Xiaopeng Zhang & Jiaqi He

  2. School of Mechanical Engineering, Dalian University of Technology, Dalian, 116024, China

    Dazhi Wang

  3. School of Science, Harbin Institute of Technology, Shenzhen, 518055, China

    Yangjun Luo

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  1. Yi Yan
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  2. Xiaopeng Zhang
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Correspondence to Yangjun Luo.

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Yan, Y., Zhang, X., He, J. et al. Achieving desired nodal lines in freely vibrating structures via material-field series-expansion topology optimization. Front. Mech. Eng. 18, 42 (2023). https://doi.org/10.1007/s11465-023-0758-y

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