Diana Estévez Schwarz

Finding Beneficial DAE Structures in Circuit Simulation

Circuit simulation is a standard task for the computer-aided design of electronic circuits. The transient analysis is well understood and realized in powerful simulation packages for conventional circuits. But further developments in the production engineering lead to new classes of circuits which may cause difficulties for the numerical integration. The dimension of circuit models can be quite large (105 equations). The complexity of the models demands a higher abstraction level. In this paper, we analyze electric circuits with respect to their structural properties. We discuss the relevant subspaces of the resulting differential algebraic equations (DAEs) and present algorithms for calculating the index as well as consistent initial values.

The computation of consistent initial values for nonlinear index-2 differential–algebraic equations

The computation of consistent initial values for differential–algebraic equations (DAEs) is essential for starting a numerical integration. Based on the tractability index concept a method is proposed to filter those equations of a system of index-2 DAEs, whose differentiation leads to an index reduction. The considered equation class covers Hessenberg-systems and the equations arising from the simulation of electrical networks by means of Modified Nodal Analysis (MNA). The index reduction provides a method for the computation of the consistent initial values. The realized algorithm is described and illustrated by examples.

Diagnosis of singular points of properly stated DAEs using automatic differentiation

The reliability of numerical results provided by conventional numerical integration methods for ODEs or DAEs depends on the properties of the problem. In particular, since the solution may not be unique at singular points, arbitrary solutions may be obtained. For DAEs, such singularities may occur, if the structure or the dimension of the spaces related to the DAE change. Moreover, even though a numerical singularity is not given in a strict mathematical sense, the numerical behavior may be analogous for sensitive problems. In this contribution, we aim at a characterization of this sensitivity, considering the condition number of a suitable matrix related to the DAE that is constructed using automatic differentiation. We show how this approach, which builds on the projector based analysis, can be applied to properly stated DAEs of index up to 2.

A new projector based decoupling of linear DAEs for monitoring singularities

For higher index differential-algebraic equations (DAEs) some components of the solution depend on derivatives of the right-hand side. In this context, two main results are pointed out here. On the one hand, a description of the different types of undifferentiated components involved in the DAE is obtained by a projector-based decoupling. To this end, we define a new decoupling based on the number of inherent differentiations of the right-hand side that are required to determine each component. On the other hand, we introduce characteristic values that characterize the robustness of our numerically determined index-classification and decoupling as well as a meaningful indicator that permit the diagnosis of singular points.

Diagnosis of singular points of structured DAEs using automatic differentiation

At a singular point of a DAE, the IVP fails to have a unique solution. Hence, numerical integration methods cannot provide reasonable results. Unfortunately, common error control strategies do not always detect these circumstances and arbitrary solutions may be given to the user without warnings of any kind.Automatic (or Algorithmic) Differentiation (AD) opens new possibilities to realize an analysis of DAEs and to monitor assumptions required for the existence and uniqueness of IVPs. We show how the diagnosis of singular points can be performed for structured quasi-linear DAEs up to index 2. Our approach uses the projector based analysis for DAEs employing AD. The resulting method is illustrated by several examples, with particular emphasis on simple electrical circuits containing controlled sources.

Projector based integration of DAEs with the Taylor series method using automatic differentiation

Automatic (or Algorithmic) Differentiation (AD) opens new possibilities to analyze and solve DAEs by projector based methods. In this paper, we present a new approach to compute consistent initial values and integrate DAEs up to index two, considering the nonlinear DAE in each time-step as a nonlinear system of equations for Taylor expansions. These systems will be solved by the Newton-Kantorowitsch method, whereas the resulting linear systems are decoupled using the splitting techniques related to the tractability index concept. This approach provides a description of the inherent ODE that allows an application of the classical Taylor series method to the integration of initial value problems. Linear and nonlinear DAEs with index up to two are examined and solved numerically.

Consistent initialization for DAEs in Hessenberg form

For nonlinear DAEs, we can hardly make a reasonable statement unless structural assumptions are given. Many results are restricted to explicit DAEs, often in Hessenberg form of order up to three. For the DAEs resulting from circuit simulation, different beneficial structures have been found and exploited for the computation of consistent initial values. In this paper, a class of DAEs in nonlinear Hessenberg form of arbitrary high order is defined and analyzed with regard to consistent initialization. For this class of DAEs, the hidden constraints can be systematically described and the consistent initialization can be determined step-by-step solving linear subproblems, an approach hitherto used for the DAEs resulting from circuit simulation. Finally, it is shown that the DAEs resulting from mechanical systems fulfill the defined structural assumptions. The algorithm is illustrated by several examples.For nonlinear DAEs, we can hardly make a reasonable statement unless structural assumptions are given. Many results are restricted to explicit DAEs, often in Hessenberg form of order up to three. For the DAEs resulting from circuit simulation, different beneficial structures have been found and exploited for the computation of consistent initial values. In this paper, a class of DAEs in nonlinear Hessenberg form of arbitrary high order is defined and analyzed with regard to consistent initialization. For this class of DAEs, the hidden constraints can be systematically described and the consistent initialization can be determined step-by-step solving linear subproblems, an approach hitherto used for the DAEs resulting from circuit simulation. Finally, it is shown that the DAEs resulting from mechanical systems fulfill the defined structural assumptions. The algorithm is illustrated by several examples.

A new approach for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization

This article describes a new algorithm for the computation of consistent initial values for differential-algebraic equations (DAEs). The main idea is to formulate the task as a constrained optimization problem in which, for the differentiated components, the computed consistent values are as close as possible to user-given guesses. The generalization to compute Taylor coefficients results immediately, whereas the amount of consistent coefficients will depend on the size of the derivative array and the index of the DAE. The algorithm can be realized using automatic differentiation (AD) and sequential quadratic programming (SQP). The implementation in Python using AlgoPy and SLSQP has been tested successfully for several higher index problems.

Monitoring Singularities While Integrating DAEs

Modern simulation tools for ODEs/DAEs allow a direct input of equations that are solved at the push of a button. However, if the mathematical assumptions that guarantee the correctness of the solution are not given, then no reliable results can be expected. Automatic (or algorithmic) differentiation (AD) opens new possibilities to analyze and solve ODEs/DAEs. In this paper, we outline how the index determination, the computation of consistent initial values, the integration and the diagnosis of singular points can be reliably carried out for DAEs up to index 3. The approach uses the projector based analysis for DAEs employing AD.

Discovering Singular Points in DAE Models

The quality of numerical simulations of DAE models depends on basic assumptions that exclude singularities. In practice, most algorithms used in simulation tools do not check the assumed properties of the DAE. Consequently, since the solution may not be unique at singular points, arbitrary solutions may be obtained. For DAEs, such singularities may occur if the structure or the dimension of the spaces related to the DAE change. Moreover, even though a numerical singularity is not given in a strict mathematical sense, the numerical behavior may be analogous for sensitive problems. We aim at a characterization of this sensitivity, considering the condition number of a suitable matrix related to the DAE. Illustrative examples are given.