a highly accurate high-order validated method to solve 3d...
TRANSCRIPT
A highly accurate high-order validated
method to solve 3D Laplace equation
M.L.Shashikant, Martin Berz and Kyoko MakinoMichigan State University, E Lansing, MI ,USA
December 19, 2004
Approach
We present an approach based on high-order quadrature and a high-orderfinite element method to find a validated solution of the Laplace equationwhen derivatives of the solution are specified on the boundary
∆ψ¡−→r ¢ = 0 in volume Ω
³R3´
∇ψ ¡−→r ¢ = −→f ¡−→r ¢ on surface ∂Ω ³R3´
Where do we want to use this approach?
In accelerator/spectrometer magnets where the magnet manufacturer pro-vides only discrete field data in the volume of interest
MAGNEX: A large acceptance MAGNetic spectrometer for EXcyt beams, at the Labora-
tori Nazionali del Sud - Catania (Italy).
(Fringe fields, high aspect ratio, discrete data)
ObjectiveIntroduction
TheoryExamples
MAGNEX Dipole Magnet
Shashikant Manikonda, Martin Berz, Kyoko Makino SCAN 2004
What do we expect from this method/tool ?
• Provide validated local expansion of the field ( ψ ¡−→r ¢ and ∂nxiψ ¡−→r ¢)• Highly accurate (work for case with high aspect ratio)
• Computationally inexpensive
• Provide information about the field quality and if possible reduce noisein experimentally obtained field data
Note about Laplace Equation
• Existence and uniqueness of the solution for 3D case can be shown
using Green’s formula
• Integral kernels that provides interior fields in terms of the boundaryfields or source are smoothing
Interior fields will be analytic even if the field/source on the surface
data fails to be differentiable
• Analytic closed form solution can be found for few problems with
certain regular geometries where separation of variables method can
be applied
Numerical methods to solve Laplace equation
• Finite Difference, Method of weighted residuals and Finite elementmethods
— Numerical solution as data set in the region of interest
— Relatively low approximation order
— Prohibitively large number of mesh points and careful meshing re-
quired
• Boundary integral methods or Source based field models
— Field inside of a source free volume due to a real sources outsideof it can be exactly replicated by a distribution of fictitious sources
on its surface. Error due to discretization of the source falls off
rapidly as the field point moves away from the source.
∗ Image charge method· Choose planes/grids to place point charges (or Gaussian dist)
· Solve a large least square fit problem to find the charges
· Lot of guess work and computation time involved in gettingthe solution
∗ Methods using Helmholtz’ theorem· Helmholtz’ theorem is used to find electric or magnetic field
directly from the surface field data
· In our approach we make use of the Taylor model frame workto implement this
Helmholtz’ theorem
Any vector field−→B that vanishes at infinity can be written as the sum of
two terms, one of which is irrotational and the other, solenoidal
−→B (x) = ∇×At (x) +∇φn (x)
φn (x) =1
4π
Z∂Ω
n (xs) ·−→B (xs)
|x− xs| ds− 1
4π
ZΩ
∇ ·−→B (xv)
|x− xv| dV
At (x) = − 1
4π
Z∂Ω
n (xs)×−→B (xs)
|x− xs| ds+1
4π
ZΩ
∇×−→B (xv)
|x− xv| dV
For a source free volume we have, ∇×−→B (xv) = 0 and ∇ ·−→B (xv) = 0
Volume integral terms vanish, φn (x) and At (x) are completely determined
from the normal and the tangential magnetic field data on surface ∂Ω
φn (x) =14π
R∂Ω
n(xs)·−→B (xs)|x−xs| ds At (x) = − 1
4π
R∂Ω
n(xs)×−→B (xs)|x−xs| ds
B is Electric or Magnetic field∂Ω is a surface which bounds volume Ωxs and xv denote points on ∂Ω and within Ω
∇ denote the gradient with respect to xvn is a unit normal vector pointing away from ∂Ω
Implementation using Taylor Models
• Split domain of integration ∂Ω in to smaller regions Γi
• Expand them to higher orders in surface variables rs and the volume
variables r
— Expanded in rs about the center of each surface element
— Expanded in r about the center of each volume element
— Field is chosen to be constant over each surface element
Z xNx
x0
Z yNy
y0g (x, y) dxdy =
ix=Nx−1,iy=Ny−1,kx=∞,ky=∞Xix=0,iy=0,kx=0,ky=0
h2kx+1x
(2kx + 1)! · 22kxh2ky+1y
(2ky + 1)! · 22ky
g2kx,2kyÃxix+1 + xix
2,yiy+1 + yiy
2
!
We can obtain:
Scalar potential φn (r) if we choose g(x, y) = ns · f(rs)|r−rs|Vector potential At (r) if we choose g(x, y) = ns × f(rs)
|r−rs|
• — Benefits
∗ The dependence on the surface variables are integrated over sur-face sub-cells Γi, which results in a highly accurate integration
formula
∗ The dependence on the volume variables are retained, whichleads to a high order finite element method
∗ By using sufficiently high order, high accuracy can be achievedwith a small number of surface elements
• Depending on the accuracy of the computation needed, we choosestep sizes, order of expansion in r (x, y, z) and order of expansion in
rs (xs, ys, zs)
Validated Integration in COSY
xiuZxil
f (x) dxi ∈³Pn,∂−1f
³x|xi=xiu−xi0
´− Pn,∂−1f
³x|xi=xil−xi0
´, In,∂−1f
´
This method has following advantages:
* No need to derive quadrature formulas with weights, support points xi,and an explicit error formula* High order can be employed directly by just increasing the order of theTaylor model limited only by the computational resources* Rather large dimensions are amenable by just increasing the dimensionalityof the Taylor models, limited only by computational resources
An Analytical Example: the Bar Magnet
x1 ≤ x ≤ x2, |y| ≥ y0, z1 ≤ z ≤ z2
As a reference problem we consider the magnetic field of rectangular iron
bars with inner surfaces (y = ±y0) parallel to the mid-plane (y = 0)
From this bar magnet one can obtain analytic solution for the magnetic
field B (x, y, z) of the form
By (x, y, z) =B04π
Xi,j
(−1)i+j⎡⎣arctan
⎛⎝ Xi · ZjY+ ·R+ij
⎞⎠+ arctan⎛⎝ Xi · ZjY− ·R−ij
⎞⎠⎤⎦Bx (x, y, z) =
B04π
Xi,j
(−1)i+j⎡⎣ln
⎛⎝Zj +R−ijZj +R+ij
⎞⎠⎤⎦Bz (x, y, z) =
B04π
Xi,j
(−1)i+j⎡⎣ln
⎛⎝Xj +R−ijXj +R+ij
⎞⎠⎤⎦
where i, j = 1, 2 ,
Xi = x− xi, Y± = y0 ± y, Zi = z − zi
and R± =³X2i + Y 2j + Z2±
´12
We note that only even order terms exist in the Taylor expansion of this
field about the origin.
-0.003-0.002
-0.001 0
0.001 0.002
0.003
X
-0.004-0.002
0 0.002
0.004
Y
0
2
4
6
8
10
12
BY
By component of the field on the mid-plane.
Results
1. To study the dependency of the Interval part of the potentials and Bfield on the surface element length
• All of the volume is considered as just one volume element
• Examine contributions of each surface element towards the totalintegral
— Expansion is done at r = (.1, .1, .1) and
— Plot of interval width VS surface element length for scalar po-tential
— Plot of interval width VS surface element length for vector po-tential (x component)
• Plot of interval width VS Order for different surface element lengthfor x component of Magnetic field
-22
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3
LOG
10(In
terv
al W
idth
)
LOG2(Surface Element Length)
Interval Width VS Surface Element Length for Scalar Potential
Order 8Order 7Order 6Order 5Order 4Order 3Order 2
Figure 1: Integration over single surface element (for φ)
-22
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3
LOG
10(In
terv
al W
idth
)
LOG2(Surface Element Length)
Interval Width VS Surface Element Length for X Vector Potential
Order 8Order 7Order 6Order 5Order 4Order 3Order 2
Figure 2: Integration over single surface element (for Ax)
-11
-10
-9
-8
-7
-6
-5
-4
4 4.5 5 5.5 6 6.5 7 7.5 8
LOG
(Inte
rval
Wid
th)
ORDER
Interval Width VS Order for different stepsize
Step size = 0.0166Step size = 0.0192Step size = 0.0147
Figure 3: Interval width VS Order (for different step size)
2 Study the dependency of the Polynomial part and Interval part of the
B field on the volume element length
• The surface element length is locked at 1/128
• Plot of the error calculated for the polynomial part VS the volumeelement length
• Plot of interval width VS volume element length for y componentof Magnetic field
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
-7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3
LOG
(Inte
rval
Wid
th)
LOG2(Length of Volume Element)
Interval Width VS Length of Volume Element for Scalar Potential
Order 8Order 7Order 6
Figure 4: Interval width VS Volume element length (for φ)
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
-7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3
LOG
(Inte
rval
Wid
th)
LOG2(Length of Volume Element)
Interval Width VS Length of Volume Element for X Component of Vector Potential
Order 8Order 7Order 6
Figure 5: Interval width VS Volume element length (for Ax)
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
LOG
(Ave
rage
err
or)
Volume element length
Error VS Volume element length
Order 3Order 5Order 7Order 9
Figure 6: Error VS Volume element length for polynomial part (for By)
-10
-9.5
-9
-8.5
-8
-7.5
-7
-6.5
-6
-5.5
-5
-1.95 -1.9 -1.85 -1.8 -1.75 -1.7 -1.65 -1.6
LOG
(Inte
rval
Wid
th)
LOG(Surface Element Length)
Interval Width VS Surface Element Length for Y Component of Magnetic Field
Order 8Order 7Order 6
Figure 7: Interval width VS Volume element length for By
Summary
• Helmholtz’ theorem implemented using the Taylor Model tools providea promising approach to find local expansion of the field in the volume
of interest
• Accuracy achieved is very high compared to conventional numericalfield solvers
• Provides a good way to check the field quality
• This method can be extended for PDE’s