Topology optimization of multiple-barrier synchronous reluctance motors with initial random hollow circles

Abstract

Multiple air barriers and iron webs of the synchronous reluctance motor (SynRM) are vital to achieving higher average torque and a lower ripple torque. However, it is challenging to determine the optimum multiple barriers because the optimization results are significantly affected by the initial flux path and moreover are sensitive to a penalization scheme in the topology optimization. Interestingly, in nature, the hollow circular patterns of the trabecular bone at infancy gradually adapt to the optimal (also very complicated) trabecular architecture, which provides a characteristic lightweight design. Inspired by the above observation, this study proposed a novel topology optimization for multiple-barrier SynRMs, which initially started from random hollow circles. To distinguish between iron and air, the dq-axis-dependent penalization scheme is also proposed instead of a conventional spatially constant penalization scheme. Using the adjoint variable method, the analytical sensitivity was derived to consider a nonlinear B-H curve in the topology optimization. By optimizing various SynRMs with different filtering radii, this study investigated the relationship between multiple barriers and motor performance.

Appendices

Appendix 1. Finite element analysis for electromagnetic problems

To solve electromagnetic problems, (3) is discretized and then converted to (22). In (22), the subscripts i and j refer to the ith and jth nodes, respectively. Under constant Q, (22) can be solved to determine a nodal vector potential (A_z,_j) at the jth node after the error vector (${\mathtt{\varepsilon}}_i^{\left(\varpi \right)}$) converges within tolerance, as follows:

$$ \sum \limits_{j=1}^n\left({K}_{ij}^{\left(\varpi \right)}{\mathbf{A}}_{\boldsymbol{z},\boldsymbol{j}}^{\left(\varpi \right)}\right)-{\mathbf{Q}}_i={\boldsymbol{\upvarepsilon}}_i^{\left(\varpi \right)}, for\kern0.15em i=1,2,\dots, n $$

(22)

where ${\mathbf{K}}^{\left(\varpi \right)}=\left[\begin{array}{cccc}{K}_{11}^{\left(\varpi \right)}& {K}_{12}^{\left(\varpi \right)}& \cdots & {K}_{1n}^{\left(\varpi \right)}\\ {}{K}_{21}^{\left(\varpi \right)}& {K}_{22}^{\left(\varpi \right)}& \cdots & {K}_{2n}^{\left(\varpi \right)}\\ {}\cdots & \cdots & \ddots & \cdots \\ {}{K}_{n1}^{\left(\varpi \right)}& {K}_{n2}^{\left(\varpi \right)}& \cdots & {K}_{nn}^{\left(\varpi \right)}\end{array}\right]$;${\mathbf{A}}_z^{\left(\varpi \right)}={\left[{\mathbf{A}}_{\mathrm{z},1}^{\left(\varpi \right)},{\mathbf{A}}_{\mathrm{z},2}^{\left(\varpi \right)},\cdots, {\mathbf{A}}_{\mathrm{z},n}^{\left(\varpi \right)}\right]}^{\mathrm{T}}$; Q = [Q₁, Q₂, ⋯, Q_n]^T; ${\boldsymbol{\upvarepsilon}}^{\left(\varpi \right)}={\left[{\boldsymbol{\upvarepsilon}}_1^{\left(\varpi \right)},{\boldsymbol{\upvarepsilon}}_2^{\left(\varpi \right)},\cdots, {\boldsymbol{\upvarepsilon}}_n^{\left(\varpi \right)}\right]}^{\mathrm{T}}$; ϖ is the iteration number of the Newton-Raphson method; n and m_max are the total numbers of nodes and elements, respectively, used in the FE analysis; K^(ϖ) is a n × n system matrix; N_{m, i} and N_{m, j} are the shape functions at the ith and jth node, respectively, in the mth element; |G_m| is the Jacobian of the mth element, and u_m and v_m are the local coordinates of the mth element. The system matrix (K^(ϖ)) includes information on the nonlinear reluctivity and mesh connectivity, as follows:

$$ {K}_{ij}^{\left(\varpi \right)}=\sum \limits_{m=1}^{m_{\mathrm{max}}}{\int}_{\Omega_m}{v}_m{\overline{N}}_{m,i,j}\left|{G}_m\right|{du}_m{dv}_m $$

(23)

where ${\overline{N}}_{m,i,j}=\left(\frac{\partial {N}_{m,i}}{\partial x}\frac{\partial {N}_{m,j}}{\partial x}+\frac{\partial {N}_{m,i}}{\partial y}\frac{\partial {N}_{m,j}}{\partial y}\right)$ and ν_m = ν_m(‖B_m‖). The electric forcing function (Q_i) at the ith node is expressed as

$$ {\mathbf{Q}}_i=\sum \limits_{m=1}^{m_{\mathrm{max}}}\underset{\Omega_m}{\int }{\mathbf{J}}_{z,i}{N}_{m,i}\left|{G}_m\right|{du}_m{dv}_m $$

(24)

Then, the error vector (${\boldsymbol{\upvarepsilon}}_i^{\left(\varpi \right)}$) at the ith node can be represented, as follows:

$$ {\boldsymbol{\upvarepsilon}}_i^{\left(\varpi \right)}=\sum \limits_{l=1}^n\left(\left({K}_{il}^{\left(\varpi \right)}+{\tilde{K}}_{il}^{\left(\varpi \right)}\right)\left({\mathbf{A}}_{\mathrm{z},\mathtt{l}}^{\left(\varpi +1\right)}-{\mathbf{A}}_{\mathrm{z},\mathtt{l}}^{\left(\varpi \right)}\right)\right) $$

(25)

where ${\tilde{K}}_{il}^{\left(\varpi \right)}=\sum \limits_{j=1}^n\frac{\partial {K}_{ij}^{\left(\varpi \right)}}{\partial {\mathbf{A}}_{z,l}^{\left(\varpi \right)}}{\mathbf{A}}_{\mathrm{z},j}^{\left(\varpi \right)}$. Note that the n × n matrix (${\tilde{\mathbf{K}}}^{\left(\varpi \right)}$) represents the response of the system matrix (K^(ϖ)) with respect to the vector potential (${\mathbf{A}}_{z,j}^{\left(\varpi \right)}$).

Appendix 2. Adjoint variable for Maxwell’s stress tensor-based electromagnetic torque

The adjoint variable method has been widely used to solve structural problems (Jasbir and Edward 1979) and electromagnetic problems (Gitosusastro et al. 1989; Park et al. 1992). If (22) converges within a tolerance, the total derivative of (22) with respect to the design variable (ρ_e) can be expressed, as follows:

$$ {\left.\frac{\partial }{\partial {\rho}_e}\left(\sum \limits_{j=1}^n{K}_{ij}{\mathbf{A}}_{z,j}\right)\right|}_{{\mathbf{A}}_z=\mathrm{const}}+\sum \limits_{l=1}^n\left({\left.\frac{\partial }{\partial {\mathbf{A}}_{z,l}^{\mathrm{T}}}\left(\sum \limits_{j=1}^n{K}_{ij}{\mathbf{A}}_{z,j}\right)\right|}_{\boldsymbol{\uprho} =\mathrm{const}}\frac{\partial {\mathbf{A}}_{z,l}}{\partial {\rho}_e}\right)=0\ \mathrm{for}\ i=1,2,\dots, n. $$

(26)

Note that, in this study, the coil area is not included as the design domain. Therefore, $d{\mathbf{Q}}_i/d{\rho}_e$ becomes zero. Then, the first term in (26) can be re-expressed using the matrix (D = [D₁, D₂, ⋯, D_n]^T), as

$$ {\displaystyle \begin{array}{l}{\left.\frac{\partial }{\partial {\rho}_e}\left(\sum \limits_{j=1}^n{K}_{ij}{\mathbf{A}}_{z,j}\right)\right|}_{{\mathbf{A}}_z=\mathrm{const}}\\ {}={\left.\frac{\partial }{\partial {\nu}_e^{\left(\mathrm{SIMP}\right)}}\left(\sum \limits_{j=1}^n{K}_{ij}{\mathbf{A}}_{z,j}\right)\right|}_{{\mathbf{A}}_z,{\rho}_e=\mathrm{const}}\frac{\partial {\nu}_e^{\left(\mathrm{SIMP}\right)}}{\partial {\rho}_e}={\mathbf{D}}_i\end{array}} $$

(27)

where $\frac{\partial {\left(\sum \limits_{j=1}^n{\mathbf{K}}_{ij}{A}_{z,j}\right)}_{A_z,{\rho}_e=\mathrm{const}}}{\partial {\nu}_e^{\left(\mathrm{SIMP}\right)}}={\left.\sum \limits_{j=1}^n\sum \limits_{e=1}^{m_{\mathrm{max}}}{\int}_{\Omega_e}{\overline{N}}_{e,i,j}{\boldsymbol{A}}_{z,j}\left|{G}_e\right|{du}_e{dv}_e\right|}_{{\boldsymbol{A}}_z,{\rho}_e=\mathrm{const}}$ and $\frac{\partial {v}_e^{\left(\mathrm{SIMP}\right)}}{\partial {\rho}_e}=p{\rho}_e^{p-1}\left({v}^{\left(\mathrm{iron}\right)}\left(\left\Vert {\mathbf{B}}_e\right\Vert \right)-{v}^{\left(\mathrm{air}\right)}\right)+{\rho}_e^p\frac{\partial {v}^{\left(\mathrm{iron}\right)}\left(\left\Vert {\mathbf{B}}_e\right\Vert \right)}{\partial {\rho}_e}$. The second term in (26) can also be re-expressed using ${\tilde{K}}_{il}^{\left(\varpi \right)}$ which is obtained from the FE analysis, as

$$ \sum \limits_{l=1}^n\left({\left.\frac{\partial }{\partial {\boldsymbol{A}}_{z,l}^{\mathrm{T}}}\left(\sum \limits_{j=1}^n{K}_{ij}{\mathbf{A}}_{z,j}\right)\right|}_{\boldsymbol{\rho} =\mathrm{const}}\frac{\partial {\mathbf{A}}_{z,l}}{\partial {\rho}_e}\right)=\sum \limits_{l=1}^n\left({\left.\left({K}_{il}+{\tilde{K}}_{il}\right)\right|}_{\boldsymbol{\uprho} =\mathrm{const}}\frac{\partial {\mathbf{A}}_{z,l}}{\partial {\rho}_e}\right) $$

(28)

Through the above procedure, (26) can be simplified as follows:

$$ \sum \limits_{l=1}^n\left({\left.\left({K}_{il}+{\tilde{K}}_{il}\right)\right|}_{\boldsymbol{\uprho} =\mathrm{constant}}\frac{\partial {\mathbf{A}}_{z,l}}{\partial {\rho}_e}\right)=-{\mathbf{D}}_i\ \mathrm{for}\ i=1,2,\dots, n. $$

(29)

In general, the computation of ∂A_{z, l}/∂ρ_e in (29) requires high computing cost. To reduce such computing cost, an adjoint variable (λ_i) is multiplied to both sides of (29) as

$$ {\lambda}_i\sum \limits_{l=1}^n\left({\left.\left({K}_{il}+{\tilde{K}}_{il}\right)\right|}_{\boldsymbol{\uprho} =\mathrm{constant}}\frac{\partial {\mathbf{A}}_{z,l}}{\partial {\rho}_e}\right)=-{\lambda}_i\ {\mathbf{D}}_i $$

(30)

In this study, the adjoint variable is defined as follows:

$$ \sum \limits_{i=1}^n{\left.\left({K}_{il}+{\tilde{K}}_{il}\right)\right|}_{\boldsymbol{\uprho} =\mathrm{constant}}{\lambda}_i={\left.\frac{\partial T\left({\theta}_{k,}\;\boldsymbol{\uprho} \right)}{\partial {\mathbf{A}}_{z,l}}\right|}_{\boldsymbol{\uprho} =\mathrm{constant}} $$

(31)

where T(θ_k, ρ) is Maxwell’s stress tensor (MST)–based torque. Using the adjoint variable method, the torque response with respect to the design variable can be easily calculated without the heavy computation of ∂A_{z, l}/∂ρ_e, as

$$ \sum \limits_{l=1}^n{\left.\frac{\partial T\left({\theta}_{k,}\;\boldsymbol{\uprho} \right)}{\partial {\mathbf{A}}_{z,l}}\right|}_{\boldsymbol{\uprho} =\mathrm{constant}}\frac{\partial {\mathbf{A}}_{z,l}}{\partial {\rho}_e}=-{\boldsymbol{\lambda}}^{\mathrm{T}}\mathbf{D} $$

(32)

where λ = [λ₁, λ₂, ⋯, λ_n]^T is an n × 1 adjoint variable vector.

Note that the integration boundary for the MST is not included as the design domain. Therefore, ∂T(θ_k, ρ)/∂ρ_e becomes zero. Finally, the derivative of the torque with respect to the design variable is expressed, as follows:

$$ \frac{dT\left({\theta}_{k,}\;\boldsymbol{\uprho} \right)}{d{\rho}_e}=\sum \limits_{l=1}^n{\left.\frac{\partial T\left({\theta}_{k,}\;\boldsymbol{\uprho} \right)}{\partial {\mathbf{A}}_{z,l}}\right|}_{\boldsymbol{\uprho} =\mathrm{constant}}\frac{\partial {\mathbf{A}}_{z,l}}{\partial {\rho}_e}=-{\boldsymbol{\lambda}}^{\mathbf{T}}\mathbf{D} $$

(33)