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SECTION I

INTRODUCTION

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 Formula$A$ 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

MAGNETOSTATIC PROBLEM

Let Formula$\Omega$ be a polyhedral domain with the boundary Formula$\partial \Omega$, and let Formula$n$ be the unit normal vector to the boundary. Assume that the boundary Formula$\partial \Omega$ consists of two disjoint parts Formula$\Gamma_{E}$ and Formula$\Gamma_{N}$. Then we consider the following magnetostatic problem with the Coulomb gauge condition [5], [6]: Formula TeX Source $$\eqalignno{& {\rm rot}\,(\nu \,{\rm rot}A)=J \quad {\rm in}\ \Omega, & \hbox{(1a)}\cr & {\rm div}\,A=0 \quad {\rm in}\ \Omega, & \hbox{(1b)}\cr & A\times n=0 \quad {\rm on}\ \Gamma_{E}, & \hbox{(1c)}\cr & ({\nu \,{\rm rot}\,A})\times n=0\quad {\rm on}\ \Gamma_{N}, & \hbox{(1d)}\cr & A\cdot n=0\quad {\rm on}\,\Gamma_{N}, & \hbox{(1e)} }$$ where Formula$A$ [Wb/m] is the magnetic vector potential, Formula$J$ [A/m2] is the electric current density, and Formula$\nu$ [m/H] is the magnetic reluctivity. Finally, we assume that Formula TeX Source $${\rm div}\,J=0\quad {\rm in}\ \Omega, \quad J\cdot n=0 \quad {\rm on}\ \Gamma_{N}. \eqno{\hbox{(2)}}$$

Let Formula$L^{2}(\Omega)$ be a space of functions defined in Formula$\Omega$ and square summable in Formula$\Omega$ with its inner product Formula$(\cdot, \cdot)$ and let Formula$H^{1}(\Omega)$ be a space of functions in Formula$L^{2}(\Omega)$ with derivatives up to the first order. We define Formula$V$ and Formula$Q$ by Formula 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 Formula$p$: Find Formula$(A, p)\in V\times Q$ such that, for any Formula$({A^{\ast}, p^{\ast}})\in V\times Q$ Formula 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 Formula$p=0$ in Formula$\Omega$, because the electric current density is divergence free.

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

By the elimination of the Lagrange multiplier Formula$p_{h}$ formally, (5) can be written as follows: Find Formula$A_{h} \in V_{h}$ such that, for any Formula$A_{h}^{\ast} \in V_{h}$ Formula 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 Formula$\nu$ is given, we can solve it by a method for solving singular systems [15].

SECTION III

DOMAIN DECOMPOSITION METHOD

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

The polyhedral domain Formula$\Omega$ is partitioned into non-overlapping subdomains: Formula TeX Source $$\Omega =\bigcup_{i=1}^{{\rm N}} {\Omega^{(i)}}, \eqno{\hbox{(8)}}$$ where the superscript Formula$(i)$ corresponds to the subdomain Formula$\Omega^{(i)}$. By reordering, the linear system (7) is rewritten as follows: Formula 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 Formula$I, B$ correspond to edges in the interior of subdomains and on the interface boundary. Formula$R_{B}^{(i)}$ 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: Formula 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 Formula$(K_{II}^{(i)})^{\dagger}$ is a generalized inverse of Formula$K_{II}^{(i)}$. 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.

Figure 1
Fig. 1. TEAM Workshop Problem 20 (I).
SECTION IV

NUMERICAL EXAMPLES

A. Models

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 Formula$1/(4 \pi\times 10^{{-7}})$ [m/H] in the region of air and coil, and 100 [m/H] in the region of center pole and yoke.

Figure 2
Fig. 2. TEAM Workshop Problem 20 (II).

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 Formula$(0 \leq x, 0 \leq y)$.

Table 1
TABLE I MESH INFORMATION OF TEAM WORKSHOP PROBLEM 20

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).

B. Accuracy Check

We consider the Formula$(\alpha-\beta)$ 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.

Figure 3
Fig. 3. Comparison plane.
Table 2
TABLE II Formula${\rm B}_{\rm z}$ ACCURACY CHECK

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.

Figure 4
Fig. 4. Accuracy checks for the TEAM 20 model (I).
Figure 5
Fig. 5. Accuracy checks for the TEAM 20 model (II).

C. Large-Scale Analysis

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 Formula$10^{-5}$. 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.

Figure 6
Fig. 6. Convergence histories of 100 million DOF of the TEAM 20 model.
SECTION V

CONCLUSION

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.

Footnotes

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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Hiroshi Kanayama

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Masao Ogino

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Shin-Ichiro Sugimoto

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Seigo Terada

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