Finite Element Method

Finite Element Method • To be added later ELEN 689

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Example ELEN 689

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

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

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

Preconditioning • Reduced system • Pre-conditioners ELEN 689

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

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

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

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

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

Finite Element Method