2008 IEEE EMC Symposium


A mixed nodal-mesh formulation
 of the PEEC method based on
   efficient graph algorithms
   Giu...
Outline
Brief PEEC theory review

Numerical problems for low-frequency EFIE (PEEC, MoM)

Existing solution methods
 • EFIE...
EMC problems




   Conductors and dielectrics involved
   Large number of unknowns
PEEC fundamentals-Mixed-potential approach




             is the flight time from r’ to r
PEEC equivalent circuit - example




                                                 l2
                    l1




 KVL ...
Numerical problems for low-frequency EFIE

                                    is valid at all frequencies s

 Inside cond...
Mixed Potential Problems
                                                   ×A N p × Ne
                       1          ...
EFIE loop-star solution
                                                                Frequency vs Iteration




       ...
Nodal Analysis (NA) solution




  s and 1/s terms: problems at low frequency
Mesh Analysis (MA) solution




  s and 1/s terms: problems at low frequency
Modified Nodal Analysis
          (MNA) solution




              s terms




MNA keeps the basis functions for currents ...
Equivalent circuit and the corresponding graph
                                          Target: exploiting the
          ...
Independent cycles and loops
            Grid                            Spanning tree




                               ...
BFS visit of a graph
White: node not reached by the visit
Grey: node reached by the visit but not all its neighbors have b...
Compute_Loops
If the actual visited grey (becoming black) node is
adjacent to another grey node, then a loop has been
iden...
Compute_Internal_Loops - 1
• The internal independent loops must cover all the internal
  edges of the grid
• The internal...
Compute_Internal_Loops - 2
• Perform a BFS visit on the face
  nodes;
• For each edge of the trees so found
  we add a loo...
Computational complexity
• Both Compute_Loops and Compute_Internal_Loops
  have O( |V| ) worst case time, where V is the s...
Memory saving
Internal egdes                        Total egdes       m
Internal nodes
                                   ...
Example: single conductor with rectangular
   cross-section (skin effect problem)
                                 nz = 3
...
Numerical results-1




Frequency-range- 1 Hz-10 GHz
MNA assumed as reference
Electrical ports terminated on 50 Ω resistan...
Numerical results-1




          NA vs MNA          MA vs MNA

                 MNMA vs MNA
          MNMA behaves as MNA...
Numerical results-1




              Condition number

Again, MNMA behaves as MNA at low frequency
providing smaller cond...
Numerical results-2




Frequency-range- 1 Hz-5 GHz
MNA assumed as reference
Electrical ports terminated on 50 Ω resistanc...
Numerical results-2




          NA vs MNA          MA vs MNA

                 MNMA vs MNA
          MNMA behaves as MNA...
Numerical results-2




              Condition number

As before, MNMA behaves as MNA at low frequency
providing smaller ...
Numerical results-2




Using NA and MA accuracy is lost below 100 kHz
The proposed approach makes the PEEC method suitabl...
Conclusions
• This paper has presented a novel hybrid nodal-mesh
  formulation of the PEEC method.
• The solenoidal nature...
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Computing Loops

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In this paper a new mixed nodal-mesh formulation of the PEEC
method is proposed. Based on the hypothesis that charges reside
only on the surface of conductors and that current density is
solenoidal inside them, a novel scheme is developed fully
exploiting the physical properties of charges and currents. It
comes out that the presented approach allows to reduce the number
of unknowns while preserving the accuracy. An elegant and
efficient algorithm, based on graph theory, is proposed to
automatically search independent loops on three dimensional
rectangular grids such as those arising in volumetric PEEC
formulation. The method is validated through numerical results
that confirm the accuracy of the proposed formulation from
DC-to-daylight and its capability to provide memory saving.

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Computing Loops

  1. 1. 2008 IEEE EMC Symposium A mixed nodal-mesh formulation of the PEEC method based on efficient graph algorithms Giulio Antonini, Daniele Frigioni Giuseppe Miscione UAq EMC Laboratory University of L’Aquila, ITALY {antonini,frigioni}@ing.univaq.it
  2. 2. Outline Brief PEEC theory review Numerical problems for low-frequency EFIE (PEEC, MoM) Existing solution methods • EFIE Loop-star solution • PEEC MNA-based solution Proposed strategy • Graph theory • Loops finding • Computational complexity Numerical results Conclusions
  3. 3. EMC problems Conductors and dielectrics involved Large number of unknowns
  4. 4. PEEC fundamentals-Mixed-potential approach is the flight time from r’ to r
  5. 5. PEEC equivalent circuit - example l2 l1 KVL and KCL are enforced to each loop (l1,l2) and each node (1,2,3)
  6. 6. Numerical problems for low-frequency EFIE is valid at all frequencies s Inside conductors and dielectrics ρ(r,s)=0, the current density J(r,s) is solenoidal • Standard basis functions doesn’t exploit such property • Traditional MPIE technique, employing RWG basis functions breaks down due to the singularity of the matrix produced by the method of moments (MoM) projection solution method. • For a closed structure with Ne edges and Np patches, the RWG based MoM can be represented as: P M = jω L + jω where L and P are the vector and scalar potential components. As the frequency reduces, the scalar potential matrix of rank Np, dominates over the vector potential matrix of rank Ne and therefore the condition number of the matrix increases quadratically with decreasing frequencies.
  7. 7. Mixed Potential Problems ×A N p × Ne 1 T + A Ne × N p × P Z = jω jωL Ne Np Ne Rank=Ne Rank=Np Effects Beyond Machine Precision 1. Fast Solver Convergence Suffers 2. Direct Solver Result Suffers Courtesy of Gope, Ruehli, Jandhyala
  8. 8. EFIE loop-star solution Frequency vs Iteration Number of iteration 300 250 200 Loop-Star 150 Basis Rearrangement 100 50 Star basis 0 Loop basis 10 08 06 04 E+ E+ E+ E+ 00 00 00 00 9. 9. 9. 9. frequency (Hz) • Loop basis for solenoidal current (Magneto-static) • Star basis for curl-free current (Electrostatic) • Frequency scaling for improved spectral property • Number of iterations does not scale with frequency • Loop finding is not easy Courtesy: Slide by Swagato Chakraborty
  9. 9. Nodal Analysis (NA) solution s and 1/s terms: problems at low frequency
  10. 10. Mesh Analysis (MA) solution s and 1/s terms: problems at low frequency
  11. 11. Modified Nodal Analysis (MNA) solution s terms MNA keeps the basis functions for currents and charges separated, Accuracy at low frequency is preserved.
  12. 12. Equivalent circuit and the corresponding graph Target: exploiting the solenoidal nature of volume currents A 3-D hexahedral grid is generated 3-D equivalent circuit
  13. 13. Independent cycles and loops Grid Spanning tree Independent set of loops Independent set of cycles (cycles with length 4) Loops currents as unknonws for interior volumes
  14. 14. BFS visit of a graph White: node not reached by the visit Grey: node reached by the visit but not all its neighbors have been visited Black: node reached by the visit along with all its neighbors
  15. 15. Compute_Loops If the actual visited grey (becoming black) node is adjacent to another grey node, then a loop has been identified
  16. 16. Compute_Internal_Loops - 1 • The internal independent loops must cover all the internal edges of the grid • The internal edges can be covered by running Compute_Loops on the subgraph induced by the internal nodes • The remaining edges must be covered by choosing loops such that the subgraph induced by the external edges belonging to a loop is acyclic
  17. 17. Compute_Internal_Loops - 2 • Perform a BFS visit on the face nodes; • For each edge of the trees so found we add a loop; • The last internal loops are found by properly choosing some of the edge nodes and building the internal loops associated to them. • The unknowns are: Internal loops currents External edges currents External nodes
  18. 18. Computational complexity • Both Compute_Loops and Compute_Internal_Loops have O( |V| ) worst case time, where V is the set of nodes of the considered graph; • The unique algorithm known in the literature for the same problem is and requires O(|V|² log(|V|)) worst case time, being based on the Dijkstra algorithm.
  19. 19. Memory saving Internal egdes Total egdes m Internal nodes Total nodes n External edges External nodes Independent internal loops MNMA number of unknowns MNA number of unknowns Unknowns saving Significant for thick objects and electrically large problems Power systems (transformers, electrical machines) Interconnects, skin-effect modeling
  20. 20. Example: single conductor with rectangular cross-section (skin effect problem) nz = 3 nx = 5 nz = 15 z ny = 10 nx=150 y n = nx x ny x nz = 6750 ni = 1924 Saved unknowns = 2 ni = 3848 x Bus with 16 conductors Saved unknowns = 2 ni = 3848 x 16 = 61568
  21. 21. Numerical results-1 Frequency-range- 1 Hz-10 GHz MNA assumed as reference Electrical ports terminated on 50 Ω resistances
  22. 22. Numerical results-1 NA vs MNA MA vs MNA MNMA vs MNA MNMA behaves as MNA at low frequency but at a reduced cost in terms of unknowns (8.33 %)
  23. 23. Numerical results-1 Condition number Again, MNMA behaves as MNA at low frequency providing smaller condition number than NA and MA.
  24. 24. Numerical results-2 Frequency-range- 1 Hz-5 GHz MNA assumed as reference Electrical ports terminated on 50 Ω resistances • mixed nodal mesh analysis (MNMA): 3212 unknowns
  25. 25. Numerical results-2 NA vs MNA MA vs MNA MNMA vs MNA MNMA behaves as MNA at low frequency but at a reduced cost in terms of unknowns (7.59 %)
  26. 26. Numerical results-2 Condition number As before, MNMA behaves as MNA at low frequency providing smaller condition number than NA and MA.
  27. 27. Numerical results-2 Using NA and MA accuracy is lost below 100 kHz The proposed approach makes the PEEC method suitable to be used for: • power electronics modeling • electrical machines (transformers, motors) modeling • broadband modeling in conjunction with macromodeling techniques (AFS)
  28. 28. Conclusions • This paper has presented a novel hybrid nodal-mesh formulation of the PEEC method. • The solenoidal nature of currents inside conductors is fully exploited by adopting as unknowns potentials to infinity of surface nodes, mesh currents for volumes and external edges. • The identification of interior loops is achieved through an efficient graph algorithm whose complexity is linear with the number of nodes. • The numerical results have proved that the proposed scheme always requires less unknowns than MNA while preserving the accuracy at low frequencies. • Thus, the proposed approach is appealing for low-frequency applications and for broadband modeling as well.

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