SANDIA REPORT
SAND2015-0339
Unlimited Release
Printed March 2014
Aleph code electrostatic solver
verification
Matthew T. Bettencourt
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Albuquerque, New Mexico 87185 and Livermore, California 94550
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2
SAND2015-0339
Unlimited Release
Printed March 2014
Aleph code electrostatic solver verification
Matthew T. Bettencourt
Electromagnetic Theory
Sandia National Laboratories
P.O. Box 5800, MS 1168
Albuquerque, NM 87185-1168
mbetten@sandia.gov
Abstract
Aleph is an electrostatic particle-in-cell code which uses the finite element method to solve for
the electric potential and field based on external potentials and discrete charged particles. The
field solver in Aleph was verified for two problems and matched the analytic theory for finite
elements. The first problem showed the mesh-refinement convergence for a nonlinear field with
no particles within the domain. This matched the theoretical convergence rates of second order
for the potential field and first order for the electric field. Then the solution for a single particle
in an infinite domain was compared to the analytic solution. This also matched the theory of first
order convergence in both the potential and electric fields for both problems over a refinement
factor of 16. These solutions give confidence that the field solver and charge weighting schemes
are implemented correctly.
3
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Contents
Summary
8
1 Introduction
11
2 Results
15
Solution to Laplace's equation: the charge-free solution
15
Solution to Poisson's equation: solutions with charge
19
3 Conclusion
25
References
26
Appendix
A Input deck
27
5
List of Figures
1
Contour plot and line-out of potential and electric fields. As can be seen the bulk
of the differences are at the boundaries for the electric fields and the potential field
is nearly exact.
8
The effect of regualization on the electric field magnitude is shown with a regualization factor (5x = 0.01 in dimensionless units.
12
Two and three dimensional mesh used in the convergence study. The resolution
shown represents the second mesh in the refinement study.
16
2.2 Top: 3D Potential field (left) and corresponding 2D electric fields (right). Bottom:
Line out of 2D field between points (-0.1, -1.0) to (0.1, 1.0). The exact and fine
solutions overlap.
17
1.1
2.1
2.3
Convergence plots of the two-dimensional potential (left) and the electric (right)
fields in three norms for the two-dimensional grid shown in Fig. 2.1
18
Convergence plots of the three-dimensional potential (left) and the electric (right)
fields in three norms for the three-dimension grid shown in Fig. 2.1
18
Top': 3D (left) and 2D (right) potential fields for a single particle with infinite
domain boundary conditions. Bottom: Line out of 2D field between points (0.0,
-1.0) to (0.0, 1.0). The exact and fine solutions overlap.
19
Relative error in the potential field normalized by the local value of the potential
field
21
2.7
Relative error in the electric field normalized by the local value of the electric field
22
2.8
Convergence plots of the two-dimensional potential (left) and the electric field
(right) in three norms for a single particle in the two-dimension grid shown in Fig.
2.1
23
Convergence plots of the three-dimensional potential (left) and the electric field
(right) in three norms for a single particle in the three-dimension grid shown in
Fig. 2.1.
23
2.4
2.5
2.6
2.9
6
List of Tables
1
Summary of mesh-refinement convergence results for homogeneous and in-homogeneous
problems in both 2D and 3D. Theory states that the homogeneous problem should
converge as h2, where h is mesh size, for the potential field and as h for the electric
field, while both fields should be first order for the in-homogeneous problem. .... 9
2.1
Grid resolution requested to Cubit and calculated from the mesh file for both two
and three dimensions
15
2.2 Two-dimensional convergence rates for the potential and electric field in the L1,L2
and L., norms
16
2.3
2.4
2.5
Three-dimensional convergence rates for the potential and electric field in the L1,
L2 and L. norms
17
Two-dimensional convergence rates for the potential and electric field for a single
particle in the L., L1, and L2 norms. The first point was neglected in the convergence rate because it was not in the asymptotic regime.
20
Three-dimensional convergence rates for the potential and electric field for a single
particle in the L1,L2 and L.norms.
20
7
Summary
The electrostatic finite-element solver within the Aleph code was tested against two different
verification problems with analytic solution. The first was the solution to a homogeneous problem forced only by boundary conditions, Fig 1, and the second was an in-homogeneous problem
consisting of a single point charge. Both of these problems were tested in both two and three dimensions on a non-trivial mesh. These problems were tested against the theoretical convergence
order and were found to match theory to a sufficient level of accuracy. Due to the delta function
nature of the in-homogeneous problem convergence was only first order in the L1 norm as predicted. The convergence results are summarized in Table 1. These tests give great confidence in
several parts of the Aleph code; the core field solver, the application of boundary conditions, the
gradient operator which computes the electric field, and the charge weighting scheme. These tests
were conducted with Aleph SVN version 4403 compiled with gcc-4.4.7 and Cubit version 14.1.
These examples are in the Aleph repository at alephlexamples/verification/field_solve.
14001330120011001000900800700-Simlatbn E
-51,7xilatbn V
-ExactE
-ExactV
— Drererce E
-Drererce
600500900300200/
100-
o
-100-
-300
Figure 1. Contour plot and line-out of potential and electric
fields. As can be seen the bulk of the differences are at the boundaries for the electric fields and the potential field is nearly exact.
8
Table 1. Summary of mesh-refinement convergence results for
homogeneous and in-homogeneous problems in both 2D and 3D.
Theory states that the homogeneous problem should converge as
h2, where h is mesh size, for the potential field and as h for the
electric field, while both fields should be first order for the inhomogeneous problem.
Norm
L1
L2
L.
Homogeneous Problem
2D
3D
Potential Electric Potential Electric
Field
Field
Field
Field
1.91
2.04
1.79
1.67
2.02
1.64
1.98
1.59
1.85
1.21
1.74
1.08
9
In-Homogeneous Problem
2D
3D
Potential Electric Potential Electric
Field
Field
Field
Field
1.28
1.23
1.07
1.03
1.17
0.20
0.67
0.31
0.17
0.30
0.00
0.46
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Chapter 1
Introduction
Aleph is an electrostatic particle-in-cell(PIC)used in simulating low density plasma processes
in complex geometries. It uses an unstructured finite element method (FEM)[1] to solve the
electrostatic field and a collection of Lagrangian particles to represent the electrons, ions, and
uncharged particles in the simulation.
The electric field is defined as the negative gradient of the electric potential 0,
Ë = —vo.
(1.1)
Furthermore, the divergence of the electric field is proportional to the charge density,
V •E = 12 = —V20.
(1.2)
E0
Thus,the electric potential is defined as a solution to Poisson's equation with the source term being
proportional to the charge density. For Dirichlet boundary conditions the values on the boundary
represent the voltage at that location. For Neumann boundary conditions, the value represents
_a•ff.
Solving the system without charges reverts to simply solving the Laplace equation with a prescribed boundary condition. One can perform a separation of variables technique to determine
the form of the solution. In one dimension only linear solutions are valid solutions to Laplace's
equation. In two-dimensions a more complicated solution is allowed in the form of
02D(x,y)= clx+c2y+[c3 sin(Xx)+c4cos(Xx)][c5 sinh(Xx)+c6 cosh(Xx)],
(1.3)
where A, can be chosen to satisfy the boundary conditions. In three dimensions one can see that
03D(x,y,z) = clx+ c2y+c3z+ [c4 sin(Xx)+c5 cos(A,x)][c6 sinh(Xx)+c7 cosh(Xx)],
(1.4)
is a valid solution.
Once a charged particle is introduced the equation is no longer homogeneous and we are now
solving Poisson's equation and one requires a different analytic solution. For this one uses the
Green's function to compute the solution. The Green's function in two and three dimensions are
G(Y)=
r l'71(1 Y 1),
11
1
G(Y)= 47E 1 y r
(1.5)
Therefore, the analytic potential for point charges qi located at xi is the solution of
1 N
V2 = —Eqiö(x— Yi),
E0 i=1
(1.6)
1 N
0(X)= —
Eo i=t
(1.7)
E
— A).
The solution technique for Aleph, and all PIC codes, is to use a charge which is not a delta
function, but has a finite size and shape [2][3]. Aleph defaults to a charge with a constant density
and the shape ofthe current element. This weights equal charge density to the nodes for the element
containing the particle of
where Vj is the volume of elementjwhich contains particle i. Because
of this particle smoothing it can be shown that the charge density approaches the delta function
linearly as AxD, where D is the dimensionality of the problem with a solution proportional to xpl .
Thus, the solution is expected to converge as Ax1. Furthermore, in two and three dimensions the
potential becomes singular at the particle, which could cause an extremely large error if the particle
is near a nodal location. Because of this we will be utilizing a regularized potential [4] which is of
the form
(1.8)
GR(Y)= G(.X'+ 3x).
The effect of this is shown for the electric field magnitude in Fig. 1.1. This work will use a
regularization factor of half a typical length of an element edge.
100
• Unregularized
• Regualized
50
o
-50
1000.2
-0.1
0.1
02
x
Figure 1.1. The effect of regualization on the electric field magnitude is shown with a regualization factor Sx = 0.01 in dimensionless units.
For this work we implemented a special infinite boundary condition. When point charges are
present this boundary condition mimics infinite free space by setting a Dirichlet condition of the
12
regularized Green's function (1.8). Note that the Green's function and regularization factor are
only used in computing this specialized boundary condition and the analytic solution to compare
against the output of Aleph.
13
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Chapter 2
Results
The solution for the electrostatic field can be broken into two parts. The first is the solution
to the homogeneous part of (1.2)(Laplace's equation) forced only by the boundary values. This
tests the basic FEM solution technique. Then one can solve the in-homogeneous problems with
non-trivial charge density. Both of these solutions have analytic solutions to compare against and
are addressed below.
The meshes for the simulations provided below were created using Cubit version 14.1. The
results below are based on the average edge length of a tetrahedron for three-dimensions and
triangle for two-dimensions. This is slightly different than the input spacing requested of Cubit.
Table 2.1 shows the input and output mesh spacings.
Table 2.1. Grid resolution requested to Cubit and calculated from
the mesh file for both two and three dimensions
Cubit Input 2D Edge Length 3D Edge Length
0.1000
0.0794
0.0907
0.0500
0.0437
0.0467
0.0229
0.0250
0.0237
0.0125
0.0119
0.0125
0.0061
0.0063
N/A
Solution to Laplace's equation: the charge-free solution
To perform the analysis a non-trivial geometry and meshes were created, Fig. 2.1, then a specific Dirichlet boundary condition (1.3),(1.4) was applied to all the nodes on the external surfaces.
Then the potential field was solved for the internal nodes and the electric field was computed from
the potential field and projected to the nodes through a solid angle averaging. The results for this
solve are shown in Fig. 2.2 for the potential and electric fields. Both the potential and electric
fields were compared to the analytic result and an order of convergence was calculated through
successive mesh refinements.
15
Figure 2.1. Two and three dimensional mesh used in the convergence study. The resolution shown represents the second mesh in
the refinement study.
For two dimensions one can see in Fig. 2.3 that the order for the potential is approximately
second order in all the standard norms, whereas the error for the electric field is between first and
second order, consistent with finite element theory for linear basis functions used in the code. The
exact convergence rates are given in Table 2.2. A convergence rate p means that en « AxP, where
en is the error in the n norm defined as
)
n
=i1VPIn
(Elisi
(2.1)
en =
MCLOPD
•
Table 2.2. Two-dimensional convergence rates for the potential
and electric field in the L1,L2 and Lo, norms.
Norm Potential Field
1.91
L1
2.02
L2
L.
1.85
Electric Field
2.04
1.64
1.21
Similar tests were done in three dimensions with similar results as shown in Fig 2.4 and Table
2.3.
As can be seen the L2 error for the Laplace equation is nearly identical to 2.0 for the potential
and between 1 and 2 for the electric field. This is as expected and confirms the numerical accuracy
of the solver.
16
1500
— E exact
— V exact
—E coarse
—Vcoarse
— E One
— V rine
14001300120011001000900500700000-
400300200100-
-100200
0.1
a2
0,3
0.4
0,5
0.0
0.7
22
2,9
12
1,3
12
Figure 2.2. Top: 3D Potential field (left) and corresponding 2D
electric fields (right). Bottom: Line out of 2D field between points
(-0.1, -1.0) to (0.1, 1.0). The exact and fine solutions overlap.
Table 2.3. Three-dimensional convergence rates for the potential
and electric field in the L1, L2 and L. norms.
Norm
L1
L2
L.
Potential Field
1.79
1.98
1.74
17
Electric Field
1.67
1.59
1.08
1 .B
1 .9
0
10
-2
10
3
10
i.
0
L.17
—
—
—
—
—
—
—
101,—
—
L Infinity nonn
LI Norm
L2 Norm
Second Order Ref
6
1 2
_4
10
10
10
L Infinity Norm
L1 Nonn
L2 Nomi
First Order Ref
Second Order Ref
w
5
10-
3
io
6
1
10
le
2
10
10
1
Grid Spacing
Grid Spacing
Figure 2.3. Convergence plots of the two-dimensional potential (left) and the electric (right) fields in three norms for the twodimensional grid shown in Fig. 2.1.
0
10
:—
—
—
—
1. —
10
101
—
—
—
102 —
6
1
14
L Infinity Norm
L1 Nrom
L2 Nonn
Second Order Ref
L Infinity Norm
L1 Norm
L2 Norm
First Order Ref
Second Order Ref
3
102
10
4
10
10
1
10 -2
10
10
Grid Spacing
Grid Spacing
Figure 2.4. Convergence plots of the three-dimensional potential
(left) and the electric (right) fields in three norms for the threedimension grid shown in Fig. 2.1.
18
1
Solution to Poisson's equation: solutions with charge
To compute the convergence of the fields created by a single particle in a domain the same
geometry was used as in the previous section. A single particle was placed in the domain to
induce a field. Now the boundary conditions were set to the analytic potential value for the particle
introduced to the geometry, then the field was solved. The solution is shown in Fig. 2.5.
le+12
— Eflne
— V flre
— Eeeact
— V exact
— Ecoarse
— V coarse
90+11
80+11
7e+11
60+11
50+11
4e+11
3e+11
2e+11
1 e+11
1 e+11
0.1
0.2
0.3
0.4
0,5
0.6
0.7
0.8
0.9
1.1
1.2
1.3
1.4
1.5
1.6
1,8
1.9
Figure 2.5. Top': 3D (left) and 2D (right) potential fields for a
single particle with infinite domain boundary conditions. Bottom:
Line out of 2D field between points (0.0, -1.0) to (0.0, 1.0). The
exact and fine solutions overlap.
An identical convergence analysis as shown previously was performed. These results are shown
in Figures 2.8, 2.9, and Tables 2.4, 2.5. As one can see, the convergence in the L1 and L2 norms are
as theory predicts, first order; however, the results are much more noisy. Second the convergence
19
in the Lc° norm is nearly zero. This is as expected because of the nature of the exact solution. If
one inserts a charge whose size is proportional to an element size, as constant charge weighting
dictates, one gets a solution which has a singularity which grows as th. This produces a solution
which has a Fourier transform with constant magnitude for all wave numbers representable on a
mesh. Thus, as one refines the mesh there is always locally a contribution to the error which is
constant. This results in a zeroth order convergence in the Loo norm. However, this effect is purely
local and thus allows lower order norms to converge as expected, at an order higher than zeroth
order convergence. This argument is analogous to one made for boundary conditions in [5].
The high level of noise in the convergence plots is because the particle placement within an
element has a great deal to do with the error in the run and no effort to center the particle in the
element was made. This is extremely evident in the Le. norms which locally can be negative. As
can be seen in Figs. 2.6, 2.7 the error is very localized. This is caused by the inability of the FEM
to resolve a function of the form r+3 as r —> 0. This error causes the large error in the higher order
norms.
Table 2.4. Two-dimensional convergence rates for the potential
and electric field for a single particle in the L., Li, and L2 norms.
The first point was neglected in the convergence rate because it
was not in the asymptotic regime.
Norm Potential Field
1.28
L1
1.17
L2
L.
0.30
Electric Field
1.07
0.20
0.00
Table 2.5. Three-dimensional convergence rates for the potential
and electric field for a single particle in the Li,L2 and Los norms.
Norm
L1
L2
Lc°
Potential Field
1.23
0.67
0.17
20
Electric Field
1.03
0.31
0.46
Pseudocolor
Var: V
EL0.1400
— 0.1050
I
0.07002
0.03501
2.570e-07
Max: 0.1400
Min: 2.570e-07
Figure 2.6. Relative error in the potential field normalized by the
local value of the potential field.
21
Pseudocolor
Var: E_nd angle_smooth_magnitude
-0.579
E
—0.4184
1
0.2789
0.1395
7.069e-07
Max: 0.5579
Min: 7.069e-07
Figure 2.7. Relative error in the electric field normalized by the
local value of the electric field.
22
0
10
o
10
101
-2
t. 10
o
w
6
1 2
w
—
—
—
—
—
10-4
L Infinity Norm
L1 Norm
L2 Norm
First Order Ref
Second Order Ref
-2
10
10-
10
le
10 1
—
—
—
—
—
3
L Infinity Norm
L1 Norm
L2 Norm
First Order Ref
Second Order Ref
-2
10
10
1
Grid Spacing
Grid Spacing
Figure 2.8. Convergence plots of the two-dimensional potential
(left) and the electric field (right)in three norms for a single particle in the two-dimension grid shown in Fig. 2.1.
0
10
o
10
1
10-
..
wE
,1
w
-2
10
—
—
—
—
—
—
—
—
—
—
L Infinity Norm
L1 Norm
L2 Norm
First Order Ref
Second Order Ref
101
10 -2
10
10
Grid Spacing
Grid Spacing
Figure 2.9. Convergence plots of the three-dimensional potential (left) and the electric field (right) in three norms for a single
particle in the three-dimension grid shown in Fig. 2.1.
23
L Infinity Norm
L1 Norm
L2 Norm
First Order Ref
Second Order Ref
1
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Chapter 3
Conclusion
The field solver in Aleph was verified for two problems and matched the analytic theory for
finite elements. The first problem showed the mesh-refinement convergence for a nonlinear field
with no particles within the domain. This matched the theoretical convergence rates of second
order for the potential field and first order for the electric field. Then the solution for a single
particle in an infinite domain was compared to the analytic solution. This also matched the theory
of first order convergence in both the potential and electric fields. These solutions give confidence
that the field solver and charge weighting schemes are implemented correctly.
25
References
[1] Gilbert Strang and George J. Fix. An analysis of the finite element method, volume 212.
Prentice-Hall Englewood Cliffs, 1973.
[2] Charles K. Birdsall and A. Bruce Langdon. Plasma physics via computer simulation. CRC
Press, 2005.
[3] Roger W. Hockney and James W. Eastwood. Computer simulation using particles. Taylor &
Francis Group, 1988.
[4] A. J. Christlieb, R. Krasny, and J. P. Verboncoeur. A treecode algorithm for simulating electron
dynamics in a Penning—Malmberg trap. Computer physics communications, 164(1):306-310,
2004.
[5] P. Colella. Volume of fluid methods for partial differential equations. In Proceedings of the
International Conference on Godunov Methods, Oct 18-22 1999. Oxford, UK.
26
Appendix A
Input deck
These tests have been committed to the Aleph repository and added to the regression tests run
prior to check-in. Below is the input deck used for testing the code. More information such as the
error calculation routines can be found in the repository.
# INPUT DECK FOR FIELD VERIFICATION TESTS
units = SI
Input mesh file name = happy.g
timestep size = 4e-50
total number of timesteps = 1
apply electrostatic model to all, interpolation=constant, window = discrete, size = 1,
stride = 1
Exodus output file name = output.exoll
exodus output stride = 1
output electric field with smoothing = none
output electric field with smoothing = solid_angle_average
# Comment next line for for particle tests BC for voltage on nodelist_l is verification_dirichlet_bc
# Uncomment these next four lines for particle tests #BC for voltage on nodelist_1 is
infinite boundary_condition
#regularization factor = 0.0125
#particle weighting = 1
#initial H+ weight = 6.241509341896704e+18, T=000.0, vx = 0, vy = 0.00, vz = 0, inside =
point, x=-.0, y=.4, z=0.38
27
v1.40
28
Sandia National laboratories
29