1. GRID Parallel Simulations of a Meshless Method for Electromagnetic Transients
Guido Ala, Elisa Francomano, Antonino Spagnuolo
Università degli Studi di Palermo
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2. History SMOOTHED PARTICLE HYDRODYNAMICS (SPH) by Monaghan, Lucy, 1977
Used to solve Astrophysics and Fluid Dynamics problems
Application to Electromagnetics (SPEM)
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3. Basic Ideas Domain discretization Arbitrary particle distribution
Problem domain
No connectivity law is needed (Meshfree)
Particle representation
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4. Particle approximation of a function
Basic Ideas
Particle approximation of a function
The value of the function in a particle is expressed by using the values of near particles, weighted by a suitable function W (Smoothing Kernel)
Problem Domain
Support domain
Support Domains with the Smoothing Kernel Function
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5. + Formulation Integral representation of kernel approximation
particle approximation
Approximation of
Number of particles in the support of the particle in x
Smoothing length
Problem Domain
Smoothing kernel
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6. Dirac’s Function properties
Smoothing Kernel
Main properties
Normalization
Dirac’s Function properties
Compactness
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7. Smoothing Kernel Gaussian Kernel B-Spline Kernel
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8. Smoothing Kernel By using the curl’s theorem
If the support domain is fully into the problem domain
The differential operator is directly applied to the kernel function
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9. Consistency
If particles are irregularly distributed or there are domain zones with different distributions, the consistency condition for the kernel function may not be satisfied
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10. Variable distribution density of particles along the domain
Consistency
1-D wave equations
0 < x < 1
t > 0
Variable distribution density of particles along the domain
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11. With consistency restoration
No consistency
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12. Consistency
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13. Consistency
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14. Maxwell’s curl Equations
All EM phenomena can be described by mean of Maxwell’s curl equations and the constitutive relations of the media
1-D case
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15. Applications 2-D TM mode propagation in free space Simulation domain
80 x 80 cm2; 6400 particles
PML Layers: 7
Source: Electric field Ez (gaussian pulse at timestep 1)
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16. Applications
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17. Field values calculation
Applications
CPU time (ms)
Scalar
4 CPU parallel
dW evaluation
19217
5730
Field values calculation 30
180
Data transferring - 40
Total time
19247
5950
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18. To do Optimal domain partitioning strategy 3-D model (in progress)
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19. Bibliography
G.R. Liu, M.B. Liu “Smoothed Particle Hydrodynamics” World Scientific (2003)
G.R. Liu, “Meshfree Methods”, CRC Press (2003)
M. Lastiwka, N.Quinlan, M. Basa “Adaptive particle distribution for Smoothed Particle Hydrodynamics”
Belytschko T., Krongauz Y., Dolbow J. and Gerlach C., “On the completeness of meshfree methods”, International Journal for Numerical Methods in Engineering, 43, pp (1998)
Bonet J. and Lok T.-S. L., “Variational and momentum preservation aspects of Smooth Particle (1999) Hydrodynamics formulations”, Computer methods in applied mechanics and engineering, 180, pp
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20. Any Questions ? Thank you very much for your kind attention!
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