SECTION I

TO TREAT models with complex structures, Finite Element Analysis (FEA) with large-scale meshes is one of the most effective methods. In addition, the Domain Decomposition Method (DDM) is one of the efficient parallel algorithms for FEA in the distributed memory parallel computers [1], [2], [3]. We have introduced the DDM algorithm for 3-dimensional nonlinear magnetostatic problems with the magnetic vector potential $A$ as an unknown function [4], [5], [6], and successfully solved a 100 million degrees of freedom problem [7].

In a non-overlapping DDM, the whole computational domain is decomposed into subdomains, and then the problem to be solved is also decomposed into subdomain-interior (subdomain) problems and a subdomain-interface (interface) problem. Generally DDM solves the interface problem using iterative methods [8], [9], [10] with solving subdomain problems, which means to perform FEA in each subdomain. Since the magnetostatic problem with our formulation produces real symmetric linear equations, we have employed the Conjugate Gradient (CG) method [11] for the interface problem. However, since the coefficient matrix of large-scale analysis becomes ill-conditioned, it suffers from low convergence rate. To solve this issue, we propose DDM algorithms based on the Conjugate Residual (CR) method [12] or the MINimal RESidual (MINRES) method [13]. In this paper, we are mainly concerned with MINRES because it shows monotonical decreasing of residual norms [14]. Though the original finite element equation that we employ is singular, we use the direct method [9] to solve subdomain problems, introducing the Lagrange multiplier [7].

The formulation of magnetostatic problems is described in Section 2. The DDM algorithm is discussed in Section 3. Section 4 shows some numerical examples.

SECTION II

Let $\Omega$ be a polyhedral domain with the boundary $\partial \Omega$, and let $n$ be the unit normal vector to the boundary. Assume that the boundary $\partial \Omega$ consists of two disjoint parts $\Gamma_{E}$ and $\Gamma_{N}$. Then we consider the following magnetostatic problem with the Coulomb gauge condition [5], [6]:
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$$\eqalignno{& {\rm rot}\,(\nu \,{\rm rot}A)=J \quad {\rm in}\ \Omega, & \hbox{(1a)}\cr & {\rm div}\,A=0 \quad {\rm in}\ \Omega, & \hbox{(1b)}\cr & A\times n=0 \quad {\rm on}\ \Gamma_{E}, & \hbox{(1c)}\cr & ({\nu \,{\rm rot}\,A})\times n=0\quad {\rm on}\ \Gamma_{N}, & \hbox{(1d)}\cr & A\cdot n=0\quad {\rm on}\,\Gamma_{N}, & \hbox{(1e)} }$$ where $A$ [Wb/m] is the magnetic vector potential, $J$ [A/m^{2}] is the electric current density, and $\nu$ [m/H] is the magnetic reluctivity. Finally, we assume that
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$${\rm div}\,J=0\quad {\rm in}\ \Omega, \quad J\cdot n=0 \quad {\rm on}\ \Gamma_{N}. \eqno{\hbox{(2)}}$$

Let $L^{2}(\Omega)$ be a space of functions defined in $\Omega$ and square summable in $\Omega$ with its inner product $(\cdot, \cdot)$ and let $H^{1}(\Omega)$ be a space of functions in $L^{2}(\Omega)$ with derivatives up to the first order. We define $V$ and $Q$ by TeX Source $$\eqalignno{V& \equiv \{v\in ({L^{2}(\Omega)})^{3};\cr & \quad {\rm rot}\ v\in ({L^{2}(\Omega)})^{3},\ v\times n=0\ {\rm on}\ \Gamma_{E}\}, & \hbox{(3a)}\cr Q& \equiv \{{q\in H^{1}(\Omega); q=0\ {\rm on}\ \Gamma_{E}}\}. & \hbox{(3b)} }$$

Now, a weak formulation of (1) is constructed by the introduction of the Lagrange multiplier $p$: Find $(A, p)\in V\times Q$ such that, for any $({A^{\ast}, p^{\ast}})\in V\times Q$ TeX Source $$\eqalignno{& ({\nu\ {\rm rot}\ A, {\rm rot}\ A^{\ast}}) +({{\rm grad} p, A^{\ast}})= ({J, A^{\ast}}), & \hbox{(4a)}\cr & ({A, {\rm grad}\ p^{\ast}})=0. & \hbox{(4b)} }$$ It is easy to see $p=0$ in $\Omega$, because the electric current density is divergence free.

Let us decompose $\Omega$ into a union of tetrahedra. We approximate $A$ by the simplest Nedelec elements of simplex type, and $p$ by the conventional piecewise linear tetrahedral elements. Denoting finite element spaces corresponding to $V$ and $Q$ by $V_{h}$ and $Q_{h}$, respectively, we have the following finite element approximation: Find $({A_{h}, p_{h}})\in V_{h} \times Q_{h}$ such that, for any $({A_{h}^{\ast}, p_{h}^{\ast}})\in V_{h} \times Q_{h}$ TeX Source $$\eqalignno{& ({\nu\ {\rm rot}\ A_{h}, {\rm rot}\ A_{h}^{\ast}}) +({{\rm grad} p_{h}, A_{h}^{\ast}}) =({\mathtilde{{J}}_{h}, A_{h}^{\ast}}), & \hbox{(5a)}\cr & ({A_{h}, {\rm grad}\ p_{h}^{\ast}})=0, & \hbox{(5b)} }$$ where $\mathtilde{{J}}_{h}$ is a corrected electric current density with consideration of the continuity [6]. Therefore, it is easy to see $p_{h} =0$ in $\Omega$.

By the elimination of the Lagrange multiplier $p_{h}$ formally, (5) can be written as follows: Find $A_{h} \in V_{h}$ such that, for any $A_{h}^{\ast} \in V_{h}$ TeX Source $$({\nu\ {\rm rot}\ A_{h}, {\rm rot}\ A_{h}^{\ast}}) =({\mathtilde{{J}}_{h}, A_{h}^{\ast}}). \eqno{\hbox{(6)}}$$ Although (6) is singular even if $\nu$ is given, we can solve it by a method for solving singular systems [15].

SECTION III

First, we describe DDM algorithm [4], [5]. Let us denote the finite element equation (6) by the matrix form as follows: TeX Source $$Ku=f, \eqno{\hbox{(7)}}$$ where $K$ denotes the coefficient matrix, $u$ the unknown vector, and $f$ the known vector. In this section, for simplicity, we assume that $\nu$ is given.

The polyhedral domain $\Omega$ is partitioned into non-overlapping subdomains: TeX Source $$\Omega =\bigcup_{i=1}^{{\rm N}} {\Omega^{(i)}}, \eqno{\hbox{(8)}}$$ where the superscript $(i)$ corresponds to the subdomain $\Omega^{(i)}$. By reordering, the linear system (7) is rewritten as follows: TeX Source $$\left[\matrix{{K_{II}^{(1)}} & 0 & \cdots & 0 & {K_{IB}^{(1)} R_{B}^{(1)}} \cr 0 & \ddots & & \vdots & \vdots \cr \vdots & & \ddots & 0 & \vdots \cr 0 & \cdots & 0 & {K_{II}^{({{\rm N}})}} & {K_{IB}^{({{\rm N}})} R_{B}^{({{\rm N}})}} \cr {R_{B}^{(1)T} K_{IB}^{(1)T}} & \cdots & \cdots & {R_{B}^{({{\rm N}})T} K_{IB}^{({{\rm N}})T}} & \displaystyle\sum_{i=1}^{{\rm N}} {R_{B}^{(i)T} K_{BB}^{(i)} R_{B}^{(i)}}} \right] \left[\matrix{{u_{I}^{(1)}} \cr \vdots \cr \vdots \cr {u_{I}^{({{\rm N}})}} \cr {u_{B}}} \right] =\left[\matrix{{f_{I}^{(1)}} \cr \vdots \cr \vdots \cr {f_{I}^{({{\rm N}})}} \cr \displaystyle\sum_{i=1}^{{\rm N}} {R_{B}^{(i)T} f_{B}^{(i)}}} \right], \eqno{\hbox{(9)}}$$where the subscripts $I, B$ correspond to edges in the interior of subdomains and on the interface boundary. $R_{B}^{(i)}$ maps the global degrees of freedom (DOF) of the interface to the local DOF of the subdomain interface. Equation (9) leads to linear systems as follows: TeX Source $$\eqalignno{& K_{II}^{(i)} u_{I}^{(i)} =f_{I}^{(i)} -K_{IB}^{(i)} R_{B}^{(i)} u_{B},\quad i=1,\ldots,{\rm N}, & \hbox{(10)}\cr & \left\{\sum_{i=1}^{{\rm N}} {R_{B}^{(i)T} \left\{{K_{BB}^{(i)} -K_{IB}^{(i)T} \left({K_{II}^{(i)}} \right)^{\dagger}K_{IB}^{(i)}} \right\} R_{B}^{(i)}} \right\} u_{B} \cr & \quad =\sum_{i=1}^{{\rm N}} {R_{B}^{(i)T} \left\{{f_{B}^{(i)} -K_{IB}^{(i)T} \left({K_{II}^{(i)}} \right)^{\dagger}f_{I}^{(i)}}\right\}}, & \hbox{(11)} }$$ where $(K_{II}^{(i)})^{\dagger}$ is a generalized inverse of $K_{II}^{(i)}$. We call (10) subdomain problems, and (11) the interface problem. In this paper, (11) are solved by iterative methods and (10) is solved by the direct method after the introduction of the Lagrange multiplier [7]. Here, the Krylov subspace methods require the coefficient matrix-vector multiplication in each iterative procedure, however, such calculation for (11) can be replaced by solving subdomain problems. Therefore in our DDM algorithm, subdomain problems are also solved by the direct method at each iteration for solving the interface problem.

SECTION IV

TEAM Workshop Problem 20 [16] is considered. This model consists of a center pole, a yoke and a coil, see Figs. 1 and 2. The length unit in both figures is [mm]. The center pole and the yoke are made of SS400 and the coil is made of polyimide electric wire. The electric current in the coil is 1 000 [A]. The magnetic reluctivity is, for simplicity, a positive constant in each element, the values are $1/(4 \pi\times 10^{{-7}})$ [m/H] in the region of air and coil, and 100 [m/H] in the region of center pole and yoke.

Table I shows numbers of elements and DOF without considering the Lagrange multiplier in subdomains. These are modeled as one fourth models of Fig. 1 $(0 \leq x, 0 \leq y)$.

In this section, we compare convergence behavior of the interface problem between different iterative methods for solving the interface problem. A simplified diagonal scaling preconditioner [4] is used for the interface problem. All computations are performed by a PC cluster consisting of Intel Core i7 920 (2.66 GHz/Quad Core).

We consider the $(\alpha-\beta)$ plane shown in Fig. 3 as the comparison plane of the magnetic flux density z- component. An accuracy check is carried out by comparison of computational solutions and measured results. Measured values I and II are gotten from Okayama University and Doshisha University, respectively. Comparable results have been gotten. It is done in Table II. We can see that more accurate results are obtained as the mesh size is decreased.

The plots in Fig. 4 represent the relative error between our computation and observation of Okayama University. The horizontal axis is the average length of edge. Also, the vertical axis is the relative error. Through these results, we can see that larger the mesh size, bigger the relative error. Fig. 5 is for observation of Doshisha University. The blue line (with +) represents the relative error by our linear computation using MINRES. The red line (with ⋅) is for our previous nonlinear computation with weak convergence criterion of CG, considering B-H curves [7]. It is noted that slopes of two direct lines in both logarithmic scale figures are 1.

Next, large-scale analysis with 100 million DOF is performed. CG or MINRES is used for the interface problem, and the direct method is for subdomain problems. Convergence criterion of the interface problem is $10^{-5}$. Although Fig. 6 shows convergence histories with relative preconditioned residual norms, CG does not converge. On the other hand, we have successfully solved the large mesh case by MINRES. In these cases, computation is stopped when the maximum iteration count (4 000) is reached. It took about 50 hours in the MINRES case.

SECTION V

Very often, CG has generally been used to solve the magnetostatic problem. However, it suffers from low convergence rate, or no convergence. In this paper, especially for large-scale analysis, DDM algorithms with CR or MINRES are proposed. These methods show stable convergence compared with CG and using MINRES, we have successfully solved a large-scale problem with 100 million DOF.

Corresponding author: H. Kanayama (e-mail: kanayama@mech.kyushu-u.ac.jp).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

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