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.

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. 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. EMC problems
Conductors and dielectrics involved
Large number of unknowns

4. PEEC fundamentals-Mixed-potential approach
is the flight time from r’ to r

5. PEEC equivalent circuit - example
l2
l1
KVL and KCL are enforced to each loop (l1,l2) and each node (1,2,3)

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. 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. 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. Nodal Analysis (NA) solution
s and 1/s terms: problems at low frequency

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

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. 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. 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. 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. Compute_Loops
If the actual visited grey (becoming black) node is
adjacent to another grey node, then a loop has been
identified

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

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. Numerical results-1
Condition number
Again, MNMA behaves as MNA at low frequency
providing smaller condition number than NA and MA.

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. 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. Numerical results-2
Condition number
As before, MNMA behaves as MNA at low frequency
providing smaller condition number than NA and MA.

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. 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.