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Finite Element Method

Finite Element Method. To be added later. Inductance. Given a set of k conductors, compute the k  k impedance matrix Z(  ). V1. V2. I1. I2. Partial Inductance. For any two pieces of interconnect, the partial inductance. k. l. Application. Partial inductance assumes Unit current

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Finite Element Method

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  1. Finite Element Method • To be added later ELEN 689

  2. Inductance • Given a set of k conductors, compute the kk impedance matrix Z() V1 V2 I1 I2 ELEN 689

  3. Partial Inductance • For any two pieces of interconnect, the partial inductance k l ELEN 689

  4. Application • Partial inductance assumes • Unit current • Current return at infinity • It works OK for thin conductors and known current distribution • It does not work for large plate or if current distribution is unknown ELEN 689

  5. Compute Inductance • Send 1A current in one conductor and 0A current through other conductors, then potential drop gives impedance V1 V2 1 0 ELEN 689

  6. Boundary Element Method • Laplace integral equation where J(r) is current density,  is conductivity, and (r) is potential drop across volume r ELEN 689

  7. Discretization • Partition conductors into n filaments I5 I1 I6 I2 I7 I3 I1 I6 I4 ELEN 689

  8. Incident Matrix B n2 f5 f1 f6 f2 n1 n3 f7 f3 f8 f4 n filaments m nodes ELEN 689

  9. Linear Systems • Linear system for current and potential • I is filament current vector •  is filament potential drop vector • R is a diagonal matrix of filament DC resistance: ELEN 689

  10. Linear System (cont’d) • L is the partial inductance matrix • In addition, Kirchoff’s Law must be satisfied where Id is the external current ELEN 689

  11. n2 I5 I1 I6 I2 n1 n3 I7 I3 I8 I4 Example ELEN 689

  12. Rewrite Linear System • Note that =BV, where V is the node potential • Large system; R, B: sparse; L: dense • Solution methodology • Iterative methods • Pre-conditioners are critical ELEN 689

  13. Problem • The original system is hard to solve: • Some algorithms (FastHenry) solved it anyway • We need a better formulation ELEN 689

  14. Solenoidal Basis Method • Linear system • Solenoidal basis • Basis for current that satisfies Kirchoff’s law: • Reduced system ELEN 689

  15. Intuition • Any current vector I satisfying Kirchoff’s law and boundary condition can be written as the sum of two parts: • A unit current from external node to external node • A linear combination of loop currents ELEN 689

  16. Example ELEN 689

  17. Mesh Currents • Filament current vector I can be written as the sum of a particular current Ip and a linear combination of mesh currents 1A 1A + = Ip 1A 1A ELEN 689

  18. New Formulation • After some manipulation, the problem is changed to the following: • Solve Im from ZmIm=Vm, where • Zm is mesh-to-mesh impedance matrix • Im is mesh current vector, and • Vm is a vector of voltage drop on the Ip path, due to unit current at each mesh • Solution of Im gives potential drop between external nodes, which is one row of Z() ELEN 689

  19. What is Pre-conditioning? • When matrix A is in “bad” shape, i.e., A has a large condition number, then iterate methods to solve Ax=b take a long time to converge • If we can find a matrix M, called the pre-conditioner, such that (MA) is in “good” shape, then solving (MA)x=Mb can be very fast • Ideally, if M=A-1 then we are done ELEN 689

  20. Preconditioning • Reduced system • Pre-conditioners ELEN 689

  21. Hierarchical Approximations • Both L and M are dense and large • Hierarchical method used to compute matrix-vector products with both L and • Used for fast decaying Greens functions, such as 1/r (r : distance from origin) • Reduced accuracy at lower cost ELEN 689

  22. Avoiding Complex Numbers • Reduced system • Separate real and complex components ofthe system • Solve this system by iterative method ELEN 689

  23. Extract R, C and L together • Existence of C affects the accuracy of above method • Most accurate approach is to extract R, C and L all in one equation • Introduce current variables normal to the conductor surface and relate it to charge • Expensive. Necessary in the future? ELEN 689

  24. Assignment #2 (Due 3/6) • 1. Use FEM to solve the capacitance problem. • 2. For the hierarchical algorithm discussed on 1/28, assume the two panels (A and H) are of size 2x4, and the distance between them is 1. Assuming the partition is A=C+E+F+G and H=M+N+L+J, give the block entry matrix. ELEN 689

  25. Assignment #3 (Due 3/13) • 1. Use the solenoidal algorithm to perform inductance extraction for a pair of conductors: x2+y21, 0z10 and (x-10)2+y21, 0z10. • 2. Download and compile FastHenry, and compare with the above results http://rleweb.mit.edu/vlsi/codes.htm . Hand in printout of input file and output ELEN 689

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