NASA/TP—2000–210194
Accuracy Improvement in Magnetic Field Modeling for an Axisymmetric Electromagnet Andrew V. Ilin and Franklin R. Chang Díaz Advanced Space Propulsion Laboratory Lyndon B. Johnson Space Center Houston, Texas Yana L. Gurieva and Valery P. Il’in Institute of Computational Mathematics and Mathematical Geophysics SD RAS Novosibirsk, Russia
December 2000
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NASA/TP—2000–210194
Accuracy Improvement in Magnetic Field Modeling for an Axisymmetric Electromagnet Andrew V. Ilin and Franklin R. Chang Díaz Advanced Space Propulsion Laboratory Lyndon B. Johnson Space Center Houston, Texas Yana L. Gurieva and Valery P. Il’in Institute of Computational Mathematics and Mathematical Geophysics SD RAS Novosibirsk, Russia
National Aeronautics and Space Administration Lyndon B. Johnson Space Center Houston, Texas 77058
December 2000
Acknowledgments This research was partially sponsored by NASA’s Lyndon B. Johnson Space Center.
Available from: NASA Center for AeroSpace Information 7121 Standard Drive Hanover, MD 21076-1320 301-621-0390
National Technical Information Service 5285 Port Royal Road Springfield, VA 22161 703-605-6000
This report is also available in electronic form at http://techreports.larc.nasa.gov/cgi-bin/NTRS
Contents Figures.................................................................................................................................................iv Tables ...................................................................................................................................................v Acronyms ..........................................................................................................................................vii Nomenclature ....................................................................................................................................vii Abstract................................................................................................................................................1 1.
Introduction.................................................................................................................................1
2.
VASIMR system.........................................................................................................................2
3.
The fundamental magnetostatic problem ..................................................................................4
4.
Solution strategy .........................................................................................................................5 4.1 Domain discretization ..................................................................................................................5 4.2 Equation discretization.................................................................................................................6 4.3 Solving the algebraic system ...................................................................................................... 10
5.
Numerical experiments for a single coil ..................................................................................10 5.1 1D tests...................................................................................................................................... 10 5.2 Coil complexity.......................................................................................................................... 11 5.3 Deriving an exact solution and calculating the numerical error................................................... 11 5.4 Finite difference grid ................................................................................................................. 13 5.5 Domain size and boundary conditions ..................................................................................... 14
6.
Numerical experiments for a multi-coil system ......................................................................16
7.
Computational complexity.......................................................................................................20
8.
Conclusion ................................................................................................................................22
References .........................................................................................................................................24 Appendix 1: Compact finite difference scheme of the fourth order ..............................................25 Appendix 2: Finite volume approximations ....................................................................................26 Appendix 3: Computation of boundary conditions.......................................................................29 Appendix 4: 1D test ..........................................................................................................................32 Appendix 5: Calculation of the magnetic field on the symmetry line............................................37 Appendix 6: Convergence in the boundary condition calculation.................................................37
iii
Figures Figure 1. The VASIMR system............................................................................................................2 Figure 2. Projected design of the first spaceflight low-power VASIMR thruster. ...........................3 Figure 3. Trimetric view with an actual photo of the present VASIMR configuration at the ASPL........................................................................................................................................4 Figure 4. Definition of the magnetic flux u(r, z).................................................................................5 Figure 5. Defining the computational domain Ω for the axisymmetric single coil with rectangular cross section c and generating a rectangular grid for the problem....................6 Figure 6. 9-point and 5-point stencils with local numeration of the nodal points. ............................7 Figure 7. Difficult vs. easy coils from the mesh-generation point of view. ....................................11 Figure 8. A magnetic field near an “easy” coil. . ...............................................................................11 Figure 9. Magnetic field Bz (red line) and relative errors δB of the magnetic field B for the single coil “c1”, calculated on the symmetry line r = 0 by 5FD2N (blue lines) and 9FD4U (green lines) methods using three different mesh sizes: 64 × 64, 128 × 128 and 256 × 256. .......................................................................................................................12 Figure 10. An example for the uniform (a) and nonuniform (b) finite difference meshes for “c1” coil. ................................................................................................................................13 Figure 11. Maximal relative error δ of the 5-point scheme 5FD2N, as a function of the domain size R3. .....................................................................................................................15 Figure 12. Illustration of the principle of superposition. .................................................................16 Figure 13. Magnetic field for the 14-coil magnetoplasma rocket configuration. .............................17 Figure 14. a) Sample uniform meshes used for the 9FD4U method (top half of the picture); area around coils is not covered by the 9FD4U method; b) sample nonuniform mesh used for the 5FD2N method (bottom half of the picture)...................................................19 Figure 15. Relative error of the magnetic field for the set of 14 coils calculated on the symmetry line by 5FD2N and 9FD4U methods. ................................................................20 Figure 16. Numerical error δ vs. number of arithmetic operation Q needed for magnetic field calculation in c1-domain.......................................................................................................21 Figure 17. Magnetic field in the present VASIMR configuration calculated by the numerical code together with superimposed photographs of plasma source and plasma exhaust. ................................................................................................................................................23 Figure 18. Local numbering used in the compact 9-point finite difference scheme (17) and two boxes around mesh node (i , j), used in finite volume discretization (21)..................25 Figure 19. One-dimensional magnetic field for the infinite solenoid with internal radius R1 = 0.5 and external radius R2 = 1.0 . .........................................................................................33 Figure 20. Relative error of the 1D calculation using 5-point scheme (6) on nonuniform mesh. .....................................................................................................................................33 Figure 21. Minimization of the numerical error with respect to the relative mesh step increment. ..............................................................................................................................35 Figure 22. One-dimensional magnetic field for the infinite wire with radius Z2 = 0.5 ....................36
iv
Tables Table 1. Comparison of the 9-point finite difference fourth-order on uniform grid approximation with 5-point finite difference second-order on nonuniform mesh approximation. Symbols L and J are used to show pros and cons for each scheme. .....9 Table 2. Description of the multi-coil magnetic configuration considered for the RTD VASIMR thruster shown in Figure 13. Here J is the total current in each coil. The current density function ρ in each coil is computed as ρ = J / (R2 – R1)(Z2 – Z1).............18 Table 3. Errors of numerical integration for coil “c1” ......................................................................38 Table 4. Errors of computing the elliptic integrals for coil “c1” ......................................................38
v
vi
Acronyms ASPL FDM FEM FVM ICRF ICRH JSC MB MFlops RAM RTD VASIMR 1D, 2D 5FD2N 9FD4U
- Advanced Space Propulsion Laboratory - finite difference method - finite element method - finite volume method - ion cyclotron resonance frequency - ion cyclotron resonance heating - Johnson Space Center - megabytes - million float point operations - random access memory - radiation and technology demonstrator - Variable Specific Impulse Magnetoplasma Rocket - one- and two-dimensional - 5-point finite difference second-order on nonuniform mesh technique - 9-point finite difference fourth-order on uniform mesh technique
Nomenclature A = (0, Aθ, 0) Ah A(l) B = (Br, 0, Bz) Bza, Bzh c, c1, c2 cp e f = µrρ H hri, hzj hr, hz Iij i, j J K, E k Lr, Lz, Lrh, Lzh M(.) m Nr, Nz n n(ε) P, Q p0, …p8, f0 p00, ...p88 0 < R1 < R2
- vector magnetic potential - a matrix in algebraic system for the discretized problem - a local balance matrix in finite volume discretization - vector magnetic induction - analytical and numerical values of z-component of magnetic induction - test coils - coefficients in Tailor expansion - residual function - right hand side in the magnetostatic equation - distance from the coil to the domain boundary - variable grid step sizes for nonuniform mesh - constant grid step sizes for uniform mesh - finite volume integral - mesh indices - total current in the coil - full elliptic integrals of the first and second orders - order of accuracy - 1D differential operators and their grid approximations - elliptic integral function - number of quadrature points in numerical integration - numbers of mesh steps in r- and z-directions - length of the finite Tailor expansion - number of iterations needed to solve the algebraic system - the volumes of required computer memory and arithmetic operations - coefficients of finite difference scheme - entries of the local balance matrix in finite volume approximation - internal and external radii of the coil vii
R3, Z0, Z3 (r, θ, z) r i , zj Sr = {(r’, θ, z): r’ 0 is a relative increment of the grid-step. This means that in each subdomain k , the length L( k ) can be expressed through a number of mesh steps Nkr as following:
L( k ) = hmin + hmin (1 + h) + hmin (1 + h)2 + :::hmin (1 + h)N ,1 = hmin (1 + h)N , 1 =h: r k
r k
The 1D adaptive mesh generation has the following three steps. 1) For given Nr number of points and for given relative mesh increment h, create nonuniform mesh in the domain . This requires us to nd 6 unknowns: the number of grid steps in subdomains N1r; N2r; N3r ; N4r; N5r and the minimal mesh size hmin . The unknowns must satisfy the following 6 equations: 5 X hmin ((1 + h)N , 1)=h = L( k); k = 1; :::5; Nkr = Nr ; (52) r k
k=1
The following iteration process works eciently to resolve the above system: a) Start with initial guess for N5r ; b) Calculate hmin = L( 5)h=((1 + h)N5 , 1); c) Calculate P Nkr = ln(1 + L( k )h=hmin)= ln(1 + h) for k = 1; :::4; P d) Check: if 5k=1 Nkr < Nr , then increase N5r by one and return to b). If 5k=1 Nkr > Nr , then reduce N5r by one and return to b) until convergence. 2) Calculate error (51) of the numerical solution for the generated mesh, using analytical solution u(r) (49). 3) Optimize mesh by minimizing the error with respect to the relative mesh increment h. It was found that the function (h) is a convex smooth function with one minimum (see Figure 21), which can be found using the following iteration process: a) Start with minimal h1 , maximal h4 and intermediate h3 : h1 < h3 < h4 and calculate corresponding errors 1 ; 3 and 4 , using step p 2); b) Calculate another intermediate h2 = h1 h3 with corresponding error 2 ; c) If 2 > 3 , then assign h1 = h4 ; h4 = h2 and return to step b), else assign h4 = h1 ; h1 = h3 ; h3 = h2 and return to step b). The described nonuniform mesh in radial direction can be eciently used for solving 2D problems, because the 2D solution at the vertical symmetry plane z = 0 behaves similarly to the 1D solution (48). Now let us describe the method of generating an adaptive nonuniform mesh in the axial domain. To do this we use a similar approach as used for the radial grid. Consider behavior of the magnetic eld of a single coil along the middle radius line r = (R1 + R2)=2. For values of z inside the coil: z < Z2, radial component Br goes up from zero to its maximal value at the coil edge. Outside the coil, Br goes down, approaching zero at large z . It is hard to nd a 1D magnetostatic problem in r
34
150
error of 1−D solution
100
50
δ1 0
δ2
h
0
δ3
h
0.1
0.2
0.3
δ4
h
0.4
0.5 0.6 mesh increment
h
0.7
0.8
0.9
1
Figure 21: Minimization of the numerical error with respect to the relative mesh step increment. Cartesian coordinates with a solution having that behavior, but a solution of the 1D problem for an in nite cylindrical wire with uniform current behaves in a very similar way:
8 > < B (z ) = > :
z; 0 z Z ; 2 22 Z2 ; Z z Z ; 2 3 2z
(53)
That solution is displayed in Figure 22. Corresponding magnetic potential A(z ) has the following analytical formula
8 Z 2 , z2 Z 2 Z > < 2 4 + 22 ln Z3 ; 0 z Z2; 2 A(x) = > Z 2 Z3 : 2 ln Z2 z Z3 ; 2 z ;
which can be found as a solution to the following 1D Poisson problem , 1 @ z @A = ; @A (0) = 0; A(Z ) = 0
(54)
(55) 3 @z with a piece-wise uniform charge density function = 1 for 0 z Z2 and = 0 otherwise. In formulas (53, 54, 55) we used variable z for the radial coordinate because the formulas are used to generate an adaptive z -mesh as described below. The computational domain = [0; Z3] was divided into two subdomains: 1 = [0; Z2]; 2 = [Z2; Z3]. The grid was created to be ne next to the only singular point: z = Z2 . In 1 the grid step size satis es to the following relation: hi+1 = hi =(1 + h), in 2 : hi+1 = hi (1 + h), where h > 0 is
z @z
@z
35
0.4
0.2
B
0
−0.2
−0.4
−0.6
−0.8
−1 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Figure 22: One-dimensional magnetic eld for the in nite wire with radius Z2 = 0:5. a relative increment of the grid-step. It means that in each subdomain k , the length L( k ) can be expressed through a number of mesh steps Nkr as following:
L( k ) = hmin + hmin (1 + h) + hmin (1 + h)2 + :::hmin (1 + h)N ,1 = hmin (1 + h)N , 1 =h: r k
r k
The 1D adaptive mesh generation has the following three steps. 1) For given Nr number of points and for given relative mesh increment h, create a nonuniform mesh in the domain . This requires us to nd 3 unknowns: the number of grid steps in subdomains N1r; N2r and the minimal mesh size hmin . The unknowns must satisfy the following 3 equations:
hmin ((1 + h)N , 1)=h = L( k); k = 1; 2; N1r + N2r = Nr ; r k
(56)
The following iteration process works eciently to resolve the above system: a) Start with initial guess for N2r ; b) Calculate hmin = L( 2)h=((1 + h)N2 , 1); c) Calculate N1r = ln(1 + L( 1)h=hmin )= ln(1 + h); d) Check: if N1r + N2r < Nr , then increase N2r by one and return to b). If N1r + N2r > Nr , then reduce N2r by one and return to b) until convergence. 2) Calculate error (51) of the numerical solution for the generated mesh, using the analytical solution u(r) (54). 3) Optimize the mesh by minimizing the error with respect to the relative mesh increment h. The minimization is done using the same method as for generating the adaptive r-mesh, described above. r
36
Appendix 5: Calculation of the magnetic eld on the symmetry line Accurate calculation of the magnetic eld Bz on the symmetry line requires use of both the ne grid in the radial direction and the high-order nite di erence approximation. Taking into consideration the zero conditions for magnetic ux: u(0; z ) = 0 and the magnetic potential: A(0; z ) = 0, the following analytical formula can be used for computing the axial magnetic eld: 1 @u @ 2u a B (0; z ) = lim = (0; z ): z
r!0
r @r
@r2
The formula can be approximated be the following second-order nite di erence expression: (57) Bzh (0; zj ) = (2hur1);j2 + O(h2); 1 or the fourth-order nite di erence approximation: " r r 2 hr 2 # 2 h + h Bzh (0; zj ) = hr (2hr + hr ) 1 hr 2 u1;j , (hr +1 hr ) u2;j + O(h4): (58) 2 1 2 1 1 2 These approximations are derived from the Taylor expansion of u(r; z ) near the axis under the condition u1;j = 0, as well as the smoothness and symmetry property of the solution. Let us note that for a test function u = r2 + ar4, the exact value is Bza = 2 and its approximations Bzh of the second and fourth orders equal 2(1 + ah2r ) and 2 respectively.
Appendix 6: Convergence in the boundary condition calculation The numerical error of the magnetic eld depends on several factors: the distance H = minfZ3 , Z2 ; R3,R2 g of the domain boundary from the coil, the number m of quadrature points in the numerical integration (43), the number n of the roundo in the series (42) and the characteristic mesh-step (R3 , R2)=m. We consider the in uence of these parameters on the example of the rectangular coil c1. We use three variants of the domain boundary at a di erent distance from the coil. These boundaries are de ned by the following computational domains ( = f0 r R3; 0 z Z3g):
1 : R3 = Z3 = 2; 2 : R3 = Z3 = 4; 3 : R3 = Z3 = 8: Here we take into account that, due to the symmetry property of the eld for the single coil, the Neumann boundary condition @u=@z = 0 is used at z = 0 and the computational domain includes the half of the coil for z > 0. We need to distinguish these domains because the value of s2 decreases when the distance H increases. So in the series (42) we can take a smaller number of terms n and decrease the computational costs. Moreover, the underintegral function in (43) is smoother in this case. So it is possible to use a smaller number m in the quadrature formulas. In Table 3 we present the relative errors ( a (m)(R3; zj ) ) u ( R ; z ) , u 3 j ^ = max ^j = j ua (R3; zj ) of the boundary values for di erent integers m, distances R3 , and suciently big n = n. Errors of numerical integration are denoted by r and s for the rectangular quadrature formula (43) and the Simpson's quadrature formula (44) respectively. Actually, we de ne ua(R3; zj ) = u(m)(R3; zj ); 37
where m is the minimal value of m which satis es the condition u(m)(R3; zj ) , u(m=2)(R3; zj ) max = 10,13: j u(m)(R3; zj ) It means that, in some sense, u(m) (R3; zj ) is an \almost exact" numerical solution at the boundary. Here m = 200, n = 32 were taken. m 1 2 4 8 16 , 4 , 6 , 7 , 8
1 : s 1.110 8.210 5.710 3.710 2.310,9
1 : r 0.045 0.011 0.0028 0.00069 0.00017 , 5 , 6
2 : s 2.710 1.710 1.110,7 6.710,9 4.210,10
2 : r 0.038 0.0095 0.0024 0.00060 0.00015
3 : s 5.610,6 3.510,7 2.210,8 1.410,9 8.610,11
3 : r 0.036 0.0091 0.0023 0.00057 0.00014 Table 3. Errors of numerical integration for coil \c1" The table shows that the error is of order O(m,2 ) for the central rectangle and O(m,4) for the Simpson's quadratures and the last ones give an essentially smaller error. For a big distance H the quadratures are more precise. Let us note that the calculations using Gauss quadratures provide considerably higher eciency. Its error for m = 7 satis es the inequality g < 10,14 for all three types of the computational domain. Table 4 includes the errors ) ( a (n)(R3; zj ) u ( R ; z ) , u 3 j = max j = j ua (R3; zj ) of the boundary values for a di erent number of series terms in (42). The quadrature used is Simpson's one. Here ua (R3; zj ) is the \-asymptotically" exact numerical solution with m = 200; n = 64. n 2 4 8 16 32
1 0.44 0.22 0.072 0.011 5.910,4
2 0.19 0.045 0.0034 3.510,5 8.410,9
3 0.067 0.0056 5.810,5 1.210,8 2.010,15 Table 4. Errors of computing the elliptic integrals for coil \c1" Note that n = 32 is not enough for 1 if high accuracy is required.
38
