Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability

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.

Key words： piezoelectric actuator; isogeometric topology optimization; uniform manufacturability; robust formulation; density distribution function

Nomenclature

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
Variables
{\mathit{B}}_u	Strain‒displacement matrix
{\mathit{c}}^{\mathrm{E}}	Stiffness tensor in constant electrical field
\mathit{D}	Electrical displacement
e	Number of the finite element
\mathit{e}	Piezoelectric coefficient matrix
\mathit{E}	Electrical field
E_0,e_0,{\epsilon }_0	Stiffness, electromechanical coupling, and dielectric coefficients of piezoelectric solids, respectively
E_{min}, e_{min}, {\epsilon }_{min}	Minimum values of stiffness, electromechanical coup-ling, and dielectric coefficients of the voids, respectively
f	Global force imposed at the design domain
{\mathit{f}}_{\mathrm{b}}	Body force
{\mathit{f}}_{\mathrm{d}}	Dummy load
{\mathit{f}}_e	Force in the eth finite element
{\mathit{f}}_{\mathrm{s}}	Surface traction
G_1	Volume constraint for the eroded, intermediate, and dilated topologies
G_2	Volume constraint in the intermediate design
G\left(\mathrm{\Phi }\right)	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
{\mathit{J}}_1	Jacobi matrix from the parametric space to physical space
{\mathit{J}}_2	Jacobi matrix from the bi-unit parent element space to parametric space
k_{uu}	Spring stiffness at the output location
{\mathit{k}}_{uu}	Mechanical stiffness matrix
{\mathit{k}}_{u\varphi }	Piezoelectric coupling matrix
{\mathit{k}}_{\varphi \varphi }	Dielectric stiffness matrix
m, n	Total number of control points in the parametric directions \eta and \xi, 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_{u\varphi }, p_{\varphi \varphi }, p_{\mathrm{p}\mathrm{o}}	Penalization parameters for stiffness, piezoelectricity, dielectric, and polarization, respectively
q	Degree of NURBS basis functions in the second parametric direction
{\mathit{q}}_{\mathrm{c}}	Surface charge density accumulated on the electrodes
{\mathit{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
ue	Displacement field in the eth finite element
ui,j	Displacement at the (i,j)th control point
uout	Output displacement at the specified locations of the design domain
Vmax	Maximum material consumption
{\omega }_{i,j}	Positive weight at the (i,j)th control point
\xi ,\zeta	First and second parametric directions, respectively
{\mathit{\epsilon }}^{\mathrm{S}}	Permittivity coefficient matrix
ϕ	Electric potential
ϕe	Electric potential in the eth finite element
{\mathrm{\Omega }}_e	Physical design domain of the eth IGA element
\tilde{\mathrm{\Omega }}	Bi-unit parent element
\phi	Control design variable
{\phi }_{min}	Positive integer to avoid the occurrence of numerical singularity
{\hat{\phi }}_{\mathrm{e}\mathrm{o}},{\hat{\phi }}_{\mathrm{i}\mathrm{d}}, {\hat{\phi }}_{\mathrm{d}\mathrm{o}}	Eroded, intermediate, and dilated control design variables, respectively
{\tilde{\phi }}_{i,j}	\left(i,j\right)th smoothed control design variable
\beta, \eta	First and second parameters in the threshold projection, respectively
{\eta }_{\mathrm{e}\mathrm{o}},{\eta }_{\mathrm{i}\mathrm{d}},{\eta }_{\mathrm{d}\mathrm{o}}	Different values of the parameter \eta to define the erode, intermediate and dilate operators in threshold projection, respectively
\lambda	Adjoint vector of the dummy load
\mathrm{\Phi }	Density distribution function
{\mathrm{\Phi }}_{\mathrm{e}\mathrm{o}},{\mathrm{\Phi }}_{\mathrm{i}\mathrm{d}}, {\mathrm{\Phi }}_{\mathrm{d}\mathrm{o}}	Eroded, intermediate, and dilated DDFs, respectively
{\mathrm{\Phi }}_{\mathrm{i}\mathrm{s}\mathrm{o}}	Value of the iso-contour of the DDF
{\mathrm{\Phi }}_{\mathrm{t}\mathrm{o}\mathrm{p}}	Structural topology
{\mathrm{\Phi }}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{e}\mathrm{o}},{\mathrm{\Phi }}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{i}\mathrm{d}}, {\mathrm{\Phi }}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{d}\mathrm{o}}	Eroded, intermediate, and dilated topologies, respectively
\psi	Second type of design variables for the polarization
\mathrm{\Psi }	Continuous function for the second type of design variable
{\mathrm{\Psi }}_{\mathrm{e}\mathrm{o}},{\mathrm{\Psi }}_{\mathrm{i}\mathrm{d}},{\mathrm{\Psi }}_{\mathrm{d}\mathrm{o}}	Eroded, intermediate, and dilated continuous functions for the second type of design variable, respectively
{\mathrm{\Psi }}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{e}\mathrm{o}},{\mathrm{\Psi }}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{i}\mathrm{d}}, {\mathrm{\Psi }}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{d}\mathrm{o}}	Optimized distributions of the polarization in three eroded, intermediate, and dilated designs, respectively

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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