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Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability

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Abstract

Piezoelectric actuators have received substantial attention among the industry and academia due to quick responses, such as high output force, high stiffness, high accuracy, and precision. However, the design of piezoelectric actuators always suffers from the emergence of several localized hinges with only one-node connection, which have difficulty satisfying manufacturing and machining requirements (from the over- or under-etching devices). The main purpose of the current paper is to propose a robust isogeometric topology optimization (RITO) method for the design of piezoelectric actuators, which can effectively remove the critical issue induced by one-node connected hinges and simultaneously maintain uniform manufacturability in the optimized topologies. In RITO, the isogeometric analysis replacing the conventional finite element method is applied to compute the unknown electro elastic fields in piezoelectric materials, which can improve numerical accuracy and then enhance iterative stability. The erode—dilate operator is introduced in topology representation to construct the eroded, intermediate, and dilated density distribution functions by non-uniform rational B-splines. Finally, the RITO formulation for the design of piezoelectric materials is developed, and several numerical examples are performed to test the effectiveness and efficiency of the proposed RITO method.

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Abbreviations

CAMD:

Continuous approximation of material distribution

DDF:

Density distribution function

FEM:

Finite element method

IGA:

Isogeometric analysis

ITO:

Isogeometric topology optimization

MEMS:

Micro-electro-mechanical system

MMA:

Method of moving asymptotes

NURBS:

Non-uniform rational B-splines

OC:

Optimality criteria

PEMAP:

Piezoelectric material with penalization

PEMAP-P:

Piezoelectric material with penalization and polarization

PZT:

Lead zirconate titanate

RITO:

Robust isogeometric topology optimization

SIMP:

Solid isotropic material penalization

B u :

Strain—displacement matrix

c E :

Stiffness tensor in constant electrical field

D :

Electrical displacement

e :

Number of the finite element

e :

Piezoelectric coefficient matrix

E :

Electrical field

E 0, e 0, ε 0 :

Stiffness, electromechanical coupling, and dielectric coefficients of piezoelectric solids, respectively

E mjn, e min, ε min :

Minimum values of stiffness, electromechanical coupling, and dielectric coefficients of the voids, respectively

f :

Global force imposed at the design domain

f b :

Body force

fd:

Dummy load

f e :

Force in the eth finite element

f s :

Surface traction

G 1 :

Volume constraint for the eroded, intermediate, and dilated topologies

G 2 :

Volume constraint in the intermediate design

G (Φ):

Volume constraint function

h :

Thickness of the piezoelectric plate

i :

Number of control point in the first parametric direction

j :

Number of control point in the second parametric direction

J :

Objective function

J 1 :

Jacobi matrix from the parametric space to physical space

J 2 :

Jacobi matrix from the bi-unit parent element space to parametric space

k uu :

Spring stiffness at the output location

k uu :

Mechanical stiffness matrix

k uφ :

Piezoelectric coupling matrix

k φφ :

Dielectric stiffness matrix

m, n :

Total number of control points in the parametric directions η and ξ, respectively

M j,q :

B-spline basis functions in the second parametric direction

N i,p :

B-spline basis functions in the first parametric direction

p :

Degree of NURBS basis functions in the first parametric direction

p uu, p uφ, p φφ, p po :

Penalization parameters for stiffness, piezoelectricity, dielectric, and polarization, respectively

q :

Degree of NURBS basis functions in the second parametric direction

q c :

Surface charge density accumulated on the electrodes

q e :

Charge density in the eth finite element

\(R_{i,j}^{p,q}\) :

NURBS basis functions in 2D

S :

Mechanical strain

T :

Mechanical stress

u :

Displacement field

u e :

Displacement field in the eth finite element

u i,j :

Displacement at the (i,j)th control point

u out :

Output displacement at the specified locations of the design domain

V max :

Maximum material consumption

ω i,j :

Positive weight at the (i,j)th control point

ξ, ζ :

First and second parametric directions, respectively

ε S :

Permittivity coefficient matrix

φ :

Electric potential

φ e :

Electric potential in the eth finite element

Ωe :

Physical design domain of the eth IGA element

\(\widetilde\Omega \) :

Bi-unit parent element

φ :

Control design variable

φ min :

Positive integer to avoid the occurrence of numerical singularity

\({{\hat \varphi }_{{\rm{eo}}}},{{\hat \varphi }_{{\rm{id}}}},{{\hat \varphi }_{{\rm{do}}}}\) :

Eroded, intermediate, and dilated control design variables, respectively

\({{\tilde \varphi }_{i,j}}\) :

(i, j)th smoothed control design variable

β, η :

First and second parameters in the threshold projection, respectively

η eo, η id, η do :

Different values of the parameter η to define the erode, intermediate and dilate operators in threshold projection, respectively

λ :

Adjoint vector of the dummy load

Φ:

Density distribution function

Φeo, Φid, Φdo :

Eroded, intermediate, and dilated DDFs, respectively Φiso Value of the iso-contour of the DDF

Φtop :

Structural topology

Φ eotop ,Φ idtop ,Φ dotop :

Eroded, intermediate, and dilated topologies, respectively

ψ :

Second type of design variables for the polarization

ψ:

Continuous function for the second type of design variable

Ψeo, Ψid, Ψdo :

Eroded, intermediate, and dilated continuous functions for the second type of design variable, respectively

Ψ eotop ,Ψ idtop ,Ψ dotop :

Optimized distributions of the polarization in three eroded, intermediate, and dilated designs, respectively

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 52105255), the National Key R&D Program of China (Grant No. 2020YFB1708300), the Tencent Foundation or XPLORER PRIZE, the Knowledge Innovation Program of Wuhan-Shuguang, and the Open Fund of Key Laboratory for Intelligent Nano Materials and Devices of the Ministry of Education NJ2020003 (Grant No. INMD-2021M02).

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Authors and Affiliations

  1. Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Jie Gao & Zhi Yan

  2. Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan, 430074, China

    Jie Gao & Zhi Yan

  3. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China

    Mi Xiao, Liang Gao & Hao Li

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  1. Jie Gao
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  2. Mi Xiao
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  3. Zhi Yan
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  4. Liang Gao
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Correspondence to Zhi Yan or Liang Gao.

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Gao, J., Xiao, M., Yan, Z. et al. Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability. Front. Mech. Eng. 17, 27 (2022). https://doi.org/10.1007/s11465-022-0683-5

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