Numerical Examples
4.3 Case 3: A magnetostic problem of a magnetic sphere
4.3.1 Problem statement
For the magnetostatic problem, the theoretical framework is presented in Fig. 4.28.
A(x) is the magnetic vector potential (unit: Vsm−1), B(x) is the magnetic flux density (unit: Wb), H(x) is the magnetic field intensity (unit: Am−1), the J (x) is the free cur-rent density (unit: Am−2), µ is the magnetic permeability (unit: NA−2). The governing equations of B(x) and H(x) are
∇ × H(x) = J(x), (4.47)
and
∇ · B(x) = 0. (4.48)
For the linear isotropic medium, the constitutive equation is
B(x) = µH(x). (4.49)
Introducing the vector potential, we have
B(x) =∇ × A(x), (4.50)
where A(x) must satisfy
∇ · A(x) = 0. (4.51)
Substituting Eqs. (4.49) and (4.51) into Eq. (4.47), we obtain
∇ × (∇ × A(x)) = µJ(x). (4.52)
Combining Eq. (4.51) with Eq. (4.52), we obtain a nonhomogeneous Laplace equation
− ∆A(x) = µJ(x). (4.53)
In this case, we consider no free current density in our numerical examples.
4.3.2 Numerical results and discussions
In order to investigate the validity of the quaternion BEM, we consider a unit sphere that is a uniformly magnetized sphere in an external uniform field B0(x). The problem is sketched in Fig. 4.29. The radius of sphere is R. The relative permeability (µr) is the ratio of the permeability of a magnetized medium µ2to the permeability of free space µ1
µr = µ2
µ1. (4.54)
In terms of quaternion algebra, the external uniform field B0(x) is expressed as
B0(x) = Bci3. (4.55)
To satisfy Eqs. (4.50) and (4.51), there are four cases which can generate the same external uniform field B0(x):
The cases A, B and C have been considered by Tsuboi and Tanaka [36]. The interface conditions are
where ∂Ω stands for the interface.
Exact solutions of the B1 and B2field for the four cases are
B1(x) = B11(x)i1+ B21(x)i2+ B31(x)i3,
Exact solutions of the A1 and A2 field for the four cases are
We used 162 nodal points with 320 elements as shown in Fig. 4.30 to solve the magnetostatic problems. On using quaternion algebra, the relation between Aj(x) and
Bj(x) is
Bj(x) =−D(x)Aj(x), j = 0, 1, 2. (4.68)
where the subscript j indicates the jth field. Discretizing the quaternion valued BIEs, we
have [
Notices that the scalar parts in the four quaternion valued functions, A1(x), A2(x), B1(x) and B2(x) are all equal to zero. Deleting them, Eqs. (4.69) and (4.70) can be simplified
as [
For the interface conditions in Eq. (4.57) to Eq. (4.59), we have [
[
We combine Eqs. (4.73), (4.74), (4.78), and (4.79) to obtain the linear algebraic equation
for the case A, whereas Fig. 4.32 shows (A1, A2) with nearly singularity alleviated. For comparison the exact solution is shown in Fig. 4.33. We also plot the distribution of A2 on the x1-axis for the case A and good agreement is seen when compared with the exact solution in Fig. 4.34. The solution 1 was solved by the original quaternion valued BIE and solution 2 by alleviating the singular integral quaternion valued BIE. The results for
the cases B, C and D are shown in Figs. 4.35 to 4.46. Figures 4.47 shows the magnetic flux density field (B1, B3) on the plane x2 = 0 for all cases, whereas Fig. 4.48 with nearly singularity alleviated. The exact solution of the magnetic flux density for all cases is shown in Fig. 4.49. In Fig. 4.50, we also plot the distributions of the B3 component on the x1-axis. Good agreements are observed when compared with the exact solution.
Figure 4.51 summarizes our linear element results that more nodal points and elements we use, better results we have.
Figure 4.1: Sketch of eccentric spheres
Figure 4.2: The mesh distribution of eccentric spheres
(a) Constant element (b) Linear element
Figure 4.3: Potential contour on the plane x3 = 0 for a concentrated source at point (2.5,0,0) using 324 nodal points and 640 triangular elements
(a) Constant element (b) Linear element
Figure 4.4: Potential contour on the plane x3 = 0 for a concentrated source at point (2.5,0,0) with nearly singularity alleviated using 324 nodal points and 640 triangular ele-ments
(a) Constant element (b) Linear element
Figure 4.5: Potential contour on the plane x3 = 0 for a concentrated source at point (2.5,0,0) using 804 nodal points and 1600 triangular elements
(a) Constant element (b) Linear element
Figure 4.6: Potential contour on the plane x3 = 0 for a concentrated source at point (2.5,0,0) with nearly singularity alleviated using 804 nodal points and 1600 triangular elements
Figure 4.7: Analytical solution on the plane x3 = 0 for a concentrated source at point (2.5,0,0) using bispherical coordinates
(a) Constant element (b) Linear element
Figure 4.8: Potential contour on the plane x3 = 0 for a concentrated source at point (0,0,2.5) using 324 nodal points and 640 triangular elements
(a) Constant element (b) Linear element
Figure 4.9: Potential contour on the plane x3 = 0 for a concentrated source at point (0,0,2.5) with nearly singularity alleviated using 324 nodal points and 640 triangular ele-ments
(a) Constant element (b) Linear element
Figure 4.10: Potential contour on the plane x3 = 0 for a concentrated source at point (0,0,2.5) using 804 nodal points and 1600 triangular elements
(a) Constant element (b) Linear element
Figure 4.11: Potential contour on the plane x3 = 0 for a concentrated source at point (0,0,2.5) with nearly singularity alleviated using 804 nodal points and 1600 triangular elements
Figure 4.12: Analytical solution on the plane x3 = 0 for a concentrated source at point (0,0,2.5) using bispherical coordinates
Figure 4.13: Geometry and boundary conditions of a cube
T
∇T k∇T
F
∇
k
∇·
Figure 4.14: The framework of heat conduction
Figure 4.15: Geometry of heat conduction in functionally graded material
(a) Constant element (b) Linear element Figure 4.16: Temperature distribution on the plane x2 = 0 for β = 0
(a) Constant element (b) Linear element
Figure 4.17: Temperature distribution on the plane x2 = 0 with nearly singularity allevi-ated for β = 0
Figure 4.18: Exact solution of temperature distribution on the plane x2 = 0 for β = 0
(a) Constant element
(b) Linear element
Figure 4.19: Temperature distribution on the line (x , 0.5, 0.5) for β = 0
(a) Constant element (b) Linear element Figure 4.20: Temperature distribution on the plane x2 = 0 for β = 1.5
(a) Constant element (b) Linear element
Figure 4.21: Temperature distribution on the plane x2 = 0 plane with nearly singularity alleviated for β = 1.5
Figure 4.22: Exact solution of temperature distribution on the plane x2 = 0 for β = 1.5
(a) Constant element
(b) Linear element
Figure 4.23: Temperature distribution on the line (x1, 0.5, 0.5) for β = 1.5
(a) Constant element (b) Linear element Figure 4.24: Temperature distribution on the plane x2 = 0 for β = 10
(a) Constant element (b) Linear element
Figure 4.25: Temperature distribution on the plane x2 = 0 plane with nearly singularity alleviated for β = 10
Figure 4.26: Exact solution of temperature distribution on the plane x2 = 0 for β = 10
(a) Constant element
(b) Linear element
Figure 4.27: Temperature distribution on the line (x1, 0.5, 0.5) for β = 10
A
∇ · A = 0
B
∇ · B = 0
J
∇ · J = 0
H
∇ × B=µJ
∇ × A = B
B = µH
∇ × H = J
−∆A = µJ
Figure 4.28: The framework of magnetostatics
Figure 4.29: A magnetized sphere in an external uniform magnetic field
Figure 4.30: Mesh distribution of triangular elements for the sphere
(a) Constant element (b) Linear element Figure 4.31: Vector field (A1, A2) for case A on the plane x3 = 0
(a) Constant element (b) Linear element
Figure 4.32: Vector field (A1, A2) for case A on the plane x3 = 0 with nearly singularity alleviated
Figure 4.33: Exact solution of (A1, A2) for case A on the plane x3 = 0
(a) Constant element
(b) Linear element
Figure 4.34: Distribution of A2on the x1-axis for case A
(a) Constant element (b) Linear element Figure 4.35: Vector field (A1, A2) for case B on the plane x3 = 0 plane
(a) Constant element (b) Linear element
Figure 4.36: Vector field (A1, A2) for case B on the plane x3 = 0 plane with nearly singularity alleviated
Figure 4.37: Exact solution of (A1, A2) for case B on the plane x3 = 0
(a) Constant element
(b) Linear element
Figure 4.38: Distribution of A2 on the x1-axis for case B
(a) Constant element (b) Linear element Figure 4.39: Vector field (A1, A2) for case C on the plane x3 = 0
(a) Constant element (b) Linear element
Figure 4.40: Vector field (A1, A2) for case C on the plane x3 = 0 plane with nearly singularity alleviated
Figure 4.41: Exact solution of (A1, A2) for case C on the plane x3 = 0
(a) Constant element
(b) Linear element
Figure 4.42: Distribution of A2 on the x1-axis for case C
(a) Constant element (b) Linear element Figure 4.43: Vector field (A1, A2) for case D on the plane x3 = 0
(a) Constant element (b) Linear element
Figure 4.44: Vector field (A1, A2) for case D on the plane x3 = 0 with nearly singularity alleviated
Figure 4.45: Exact solution of (A1, A2) for case D on the plane x3 = 0
(a) Constant element
(b) Linear element
Figure 4.46: Distribution of A2on the x1-axis for case D
(a) Constant element (b) Linear element Figure 4.47: Vector field (B1, B3) for all cases on the plane x2 = 0
(a) Constant element (b) Linear element
Figure 4.48: Vector field (B1, B3) for all cases on the plane x2 = 0 with nearly singularity alleviated
Figure 4.49: Exact solution of (B1, B3) for all cases on the plane x2 = 0
(a) Constant element
(b) Linear element
Figure 4.50: Distribution of B3 on the x1-axis for all cases
Figure 4.51: Distribution of B3 on the x1-axis for all cases at different nodal points and linear elements