Computational Science Research in Support of Petascale Electromagnetic Modeling Page: 4 of 5
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With finite-element spacial discretization , ft E(x, T) dT = Z e(t) - Ni(x), and the
implicit Newmark-beta scheme  for temporal discretization, the following linear system is
solved for each time step:
(M + 3(At)2K) en+1
(2M - (1 - 2/3)(At)2K) en - (M + 3(At)2K) en-1
- (At)2 (,3fn+l + (1 - 2#3)fn + Qfn-1)
where matrix K and M are the same as in Eq(4) and vector f =f Ni - J dQ is the discretized
representation of the current density. Note that the four discretized vectors en, en-1 fn,
and fn-1, need to be transferred from the current window configuration to the next window
configuration when the particle beam moves out of the window.
A window is defined with its front and back boundaries
perpendicular to the velocity of the particle beam. Inside
the window, the finite-element basis function order p of
tetrahedral elements is set to be nonzero value while outside
of the window p is zero. This effectively makes the number of
degrees of freedom (DOF) to be zero outside of the window,
therefore reducing the computational efforts. A padding
zone is put between the front of the beam and the front
boundary of the window so that the particle beam will
stay in the window for a while. By changing the size of
the padding zone, the window size is adjusted. When the
particle beam moves out the padding zone, the window will
move forward with a given distance. The vectors en, en-1
fn, and fn-1 as shown in Eq (8) are transferred element-wise
according to the changes of the finite-element order p's. If p
of a tetrahedral element increases, zeroes are filled for those
additional coefficients. On the other hand, if p decreases,
the corresponding coefficients are dropped.
The short-range wakefield of a coupler shown in Figure 2
is calculated with the moving window through p-refinement
on a curvilinear tetrahedral mesh with 13 million elements.
During the simulation, the window moved 5 times with each
window having 2.37 million, 1.08 million, 1.02 million, 1.02
million, 1.50 million, and 1.78 million elements, respectively.
The run-time of the simulation using the moving window
technique is only one-tenth of that using all 13 million
Figure 3. A picture of a mesh
for the coupler region.
it I., ~ - -
Figure 4. Wakefield comparison
with moving h-refined meshes and
with a uniform mesh.
A series of windows with padding zones are defined in the vicinity of the moving particle beam.
The corresponding series of meshes are generated by scientists from SciDAC ITAPS Center with
a dense mesh region inside each window and a coarse mesh region outside the window. Figure 3
shows one such mesh for the coupler region shown in Figure 2. When the particle beam moves
out of one window and into the next window, the vectors en and en-1 are transferred onto the
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Lee, L.-Q.; Akcelik, V; Ge, L; Chen, S; Schussman, G; Candel, A et al. Computational Science Research in Support of Petascale Electromagnetic Modeling, article, June 20, 2008; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc900946/m1/4/: accessed February 23, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.