The following example is mainly intended for a comparative study of performance and memory management of AMIGOS. Although such comparisons are not really expressive they give an idea about the performance of the system. As a comparative tool an existing highly optimized `hand coded' finite element solver (STAP - SMART THERMAL ANALYSIS PROGRAM [Bau94][Sta97][Sab98]) has been chosen since internal knowledge about STAP is available and performance differences can be estimated better.

STAP is a simulation tool which allows transient and static thermal analysis of interconnects under electrical stress. A preconditioned conjugate gradient method is implemented to solve the large linear system in order to obtain the distributions of potential and temperature. To achieve an efficient utilization of computer memory, a compressed matrix format for the sparsely occupied stiffness matrix is used.

A two layer interconnect structure has been chosen as benchmark example where temperature and potential distributions are calculated.

The model for the numerical calculation of joule self heating effects
is based on two partial differential equations. First,
the heat-conduction equation

c_{p}
- div^{ . }grad (T)
= p |
(4.28) |

has to be solved to calculate the temperature distribution

For stationary problems the heat-conduction equation can be simplified to

div^{ . }grad (T)
= - p |
(4.29) |

The power loss density

p = ^{ . }grad () |
(4.30) |

that is calculated by solving the Euler Equation

div^{ . }grad ()
= 0 |
(4.31) |

The

(T) = ^{ . } |
(4.32) |

In this model the dependence of
on the temperature *T* is
determined by the coefficient .
is the conductivity
at the reference temperature *T*_{0} of 300K.

The developed analytical model of AMI looks like:

MODEL PowerLoss_Temperature = [U,T]; # the solution vector { ####### The Three-Dimensional Finite Element Discretization ######### N(xsi,eta,zta) = [ 1 - xsi - eta - zta , xsi , eta , zta ]; dN = [[ -1 , 1 , 0 , 0 ] [ -1 , 0 , 1 , 0 ] [ -1 , 0 , 0 , 1 ]]; x = [[X,Y,Z]]; dxdxsi = dN * x; detJ = |dxdxsi|; dxsidx = Inv(dxdxsi); dNdX = dxsidx*dN; i = 1..4; ######################### The Parameters ############################ Param alpha = 0.0043; Param gam_el = 4.17; Param gam_300 = 222.0E-6; gam(T) = gam300/(1+alpha*(T-300)); gam_th = 1/4 * Sum{i}(gam(T[i])); ################### The Differential Equations ###################### K(gamma) = dNdX^T * gamma * dNdX; # laplace operator in # dependence of the value of # gamma p1[i] = 0; # define a zero-vector of # size 1x4 E = dNdX*U; # the electrical field p2 = gam_el*N(1/4,1/4,1/4)^T*(E^T*E); # the power loss density K = [[ K(gam_el) , O] # disctretized laplace [ O , K(gam_th)]]; # operators for both # quantities ############ The Residuum Vector and its Jacobian Matrix ############ residuum = detJ*(K*[[U][T]]-[[p1][p2]]); jacobian = D([[U][T]],residuum^T)^T; }

The comparisons (Fig. 4.26 and Fig. 4.27) were made calculating 14863 nodes building 73728 tetrahedrons (the calculated results are shown in Fig. 4.28, Fig. 4.29 and Fig. 4.30). The example was set up so that both systems needed three iterations to converge. Due to the different approaches to solve the differential equation systems within the two simulators it turned out that AMIGOS gains on performance with increasing non linearity. This is an effect due to the implicit dependence definition of the quantities using the analytical model interface. This approach has the disadvantage that the memory consumption of AMIGOS is much higher. Furthermore, STAP exploits the symmetry of the resulting sparse matrix for compression methods which is not used by AMIGOS for the sake of flexibility. Fig. 4.26 depicts the performance and memory comparisons related to the STAP solver.

As expected AMIGOS cannot compete with optimized algorithms specialized to definite problems, although the demonstrated comparisons are satisfying and a lot of effort has been invested for high performance. However, a factor that can not be beaten by any other `hand-coded' algorithm shows Fig. 4.27. The development time of a new model has been compared where both scientists (developer of AMIGOS and developer of STAP) are already highly familiar with their algorithms thus the break-in period is negligible. The goal was to implement a transient electric simulation which was in scope of feasibility for the STAP system otherwise the differences in the result would be much higher.

1998-12-11