Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability
Jie GAO, Mi XIAO, Zhi YAN, Liang GAO, Hao LI
Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability
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 onenode connection, which have difficulty satisfying manufacturing and machining requirements (from the over or underetching 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 onenode 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 nonuniform rational Bsplines. 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.
piezoelectric actuator / isogeometric topology optimization / uniform manufacturability / robust formulation / density distribution function
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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  Microelectromechanical system 
MMA  Method of moving asymptotes 
NURBS  Nonuniform rational Bsplines 
OC  Optimality criteria 
PEMAP  Piezoelectric material with penalization 
PEMAPP  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 
${E}_{0},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}{e}_{0},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}{\epsilon}_{0}$  Stiffness, electromechanical coupling, and dielectric coefficients of piezoelectric solids, respectively 
${E}_{min}$, ${e}_{min}$, ${\epsilon}_{min}$  Minimum values of stiffness, electromechanical coupling, 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 $e$th 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 biunit 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 
${M}_{j,q}$  Bspline basis functions in the second parametric direction 
${N}_{i,p}$  Bspline 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 $e$th 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 
${\omega}_{i,j}$  Positive weight at the (i,j)th control point 
$\xi ,\phantom{\rule{thickmathspace}{0ex}}\zeta $  First and second parametric directions, respectively 
${\mathit{\epsilon}}^{\mathrm{S}}$  Permittivity coefficient matrix 
ϕ  Electric potential 
ϕ_{e}  Electric potential in the eth finite element 
${\mathrm{\Omega}}_{e}$  Physical design domain of the eth IGA element 
$\stackrel{~}{\mathrm{\Omega}}$  Biunit 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 
${\stackrel{~}{\phi}}_{i,j}$  $\left(i,j\right)$th smoothed control design variable 
$\phantom{\rule{thickmathspace}{0ex}}\beta $, $\eta $  First and second parameters in the threshold projection, respectively 
${\eta}_{\mathrm{e}\mathrm{o}},\phantom{\rule{thickmathspace}{0ex}}{\eta}_{\mathrm{i}\mathrm{d}},\phantom{\rule{thickmathspace}{0ex}}{\eta}_{\mathrm{d}\mathrm{o}}$  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 
${\mathrm{\Phi}}_{\mathrm{e}\mathrm{o}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}{\mathrm{\Phi}}_{\mathrm{i}\mathrm{d}}$, ${\mathrm{\Phi}}_{\mathrm{d}\mathrm{o}}$  Eroded, intermediate, and dilated DDFs, respectively 
${\mathrm{\Phi}}_{\mathrm{i}\mathrm{s}\mathrm{o}}$  Value of the isocontour of the DDF 
${\mathrm{\Phi}}_{\mathrm{t}\mathrm{o}\mathrm{p}}$  Structural topology 
${\mathrm{\Phi}}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{e}\mathrm{o}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}{\mathrm{\Phi}}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{i}\mathrm{d}}$, ${\mathrm{\Phi}}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{d}\mathrm{o}}$  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 
${\mathrm{\Psi}}_{\mathrm{e}\mathrm{o}},\phantom{\rule{thickmathspace}{0ex}}{\mathrm{\Psi}}_{\mathrm{i}\mathrm{d}},\phantom{\rule{thickmathspace}{0ex}}{\mathrm{\Psi}}_{\mathrm{d}\mathrm{o}}$  Eroded, intermediate, and dilated continuous functions for the second type of design variable, respectively 
${\mathrm{\Psi}}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{e}\mathrm{o}},\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{1pt}{0ex}}{\mathrm{\Psi}}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{i}\mathrm{d}}$, ${\mathrm{\Psi}}_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{d}\mathrm{o}}$  Optimized distributions of the polarization in three eroded, intermediate, and dilated designs, respectively 
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