Transient analysis in industrial chip design leads to very large systems
of differential-algebraic equations (DAEs). The numerical solution of these
DAEs strongly depends on the so called index of the DAE. In general,
the higher the index of the DAE is, the more sensitive the numerical
solution will be to errors in the computation. So, it is advisable to use
mathematical models with small index or to reduce the index.
This paper presents an index reduction method that uses information
based on the topology of the circuit. In addition, we show that the
presented method retains structural properties of the DAE.
The numerical simulation of very large scale integrated
circuits is an important tool in the development of new
industrial circuits. This topic has received increasing attention
within the last years. The main problem in circuit simulation is
that the model equations lead to differential algebraic equations
(DAEs). One known property of circuit DAEs is that they
may have an index larger than one, i.e., they may contain socalled
hidden constraints. The increased index has numerous
disadvantages on the numerical treatment of circuit DAEs.
The determination of these hidden constraints can be done
investigating the circuit topology. Until now, this information
has only been used for the consistent initialization of the circuit
equations. A recent approach has been to reduce the index of the
circuit DAE in order to improve their numerical behaviour. This
paper will give graph theoretical methods that lead to constraints
in a favorable formulation. Furthermore, the index reduction via
minimal extension will be performed for circuit DAEs, using these
constraints.
Element-based Topological Index Reduction for Differential-Algebraic Equations in Circuit Simulation
(2005)
The numerical simulation of very large scale integrated
circuit is an important tool in the development of new
industrial circuits. In the course of the last years, this topic has
received increasing attention. Common modeling approaches for
circuits lead to differential-algebraic systems (DAEs). In circuit
simulation, these DAEs are known to have index 2, given some
topological properties of the network. This higher index leads
to several undesirable effects in the numerical solution of the
DAEs. Recent approaches try to lower the index to improve
the numerical behaviour. These methods usually involve costly
algebraic transformations of the differential-agebraic equations.
Especially, for large scale circuit equations, these transformations
become too costly to be efficient.
We will present methods that change the topology of the network
itself, while replacing certain elements in oder to obtain a
network that leads to a DAE of index 1. This procedure can
be performed prior to the actual numerical simulation. The
decreasing of the index usually leads to significantly improved
numerical behaviour.