Mathematical Analysis and Simulation of Field Models in Accelerator Circuits

This article proposes modifications of the Parareal algorithm for its application to higher index differential algebraic equations (DAEs). It is based on the idea of applying the algorithm to only the differential components of the equation and the computation of corresponding consistent initial conditions later on. For differential algebraic equations with a special structure as, e.g. given in flux-charge modified nodal analysis, it is shown that the usage of the implicit Euler method as a time integrator suffices for the Parareal algorithm to converge. Both versions of the Parareal method are applied to numerical examples of nonlinear index 2 differential algebraic equations.

Transient simulation becomes a bottleneck for modern IC designs due to large numbers of transistors, interconnects and tight design margins. For modified nodal analysis (MNA) formulation, we could have differential algebraic equations (DAEs) which consist ordinary differential equations (ODEs) and algebraic equations. Study of solving DAEs with conventional multi-step integration methods has been a research topic in the last few decades. We adopt matrix exponential based integration method for circuit transient analysis, its stability and accuracy with DAEs remain an open problem. We identify that potential stability issues in the calculation of matrix exponential and vector product (MEVP) with rational Krylov method are originated from the singular system matrix in DAEs. We then devise a robust algorithm to implicitly regularize the system matrix while maintaining its sparsity. With the new approach, $\varphi$ functions are applied for MEVP to improve the accuracy of results. Moreover our framework no longer suffers from the limitation on step sizes thus a large leap step is adopted to skip many simulation steps in between. Features of the algorithm are validated on large-scale power delivery networks which achieve high efficiency and accuracy.

Reachability analysis is a fundamental problem for safety verification and falsification of Cyber-Physical Systems (CPS) whose dynamics follow physical laws usually represented as differential equations. In the last two decades, numerous reachability analysis methods and tools have been proposed for a common class of dynamics in CPS known as ordinary differential equations (ODE). However, there is lack of methods dealing with differential algebraic equations (DAE), which is a more general class of dynamics that is widely used to describe a variety of problems from engineering and science, such as multibody mechanics, electrical circuit design, incompressible fluids, molecular dynamics, and chemical process control. Reachability analysis for DAE systems is more complex than ODE systems, especially for high-index DAEs because they contain both a differential part (i.e., ODE) and algebraic constraints (AC). In this paper, we propose a scalable reachability analysis for a class of high-index large linear DAEs. In our approach, a high-index linear DAE is first decoupled into one ODE and one or several AC subsystems based on the well-known Marz decoupling method utilizing admissible projectors. Then, the discrete reachable set of the DAE, represented as a list of star-sets, is computed using simulation. Unlike ODE reachability analysis where the initial condition is freely defined by a user, in DAE cases, the consistency of the initial condition is an essential requirement to guarantee a feasible solution. Therefore, a thorough check for the consistency is invoked before computing the discrete reachable set. Our approach successfully verifies (or falsifies) a wide range of practical, high-index linear DAE systems in which the number of state variables varies from several to thousands.

This paper proposes the implementation of dynamic power systems as a system of differential algebraic equations (DAEs) and to solve it by means of DAE solvers, because it is easier and more natural to write the equations as a system of DAEs than to write them as a system of ordinary differential equations (ODEs). Problem-solving environments have recently included routines in their packages to solve DAEs. One of these routines has been used to simulate a power system composed of a nonlinear, three-phase, three-legged transformer, a load and all their possible connections. The magnetic transformer relations and the connections give some of the algebraic equations of the DAE system.

Index-analysis for a method of lines discretising multirate partial differential algebraic equations

The modeling of a system composed by a gas phase and organic aerosol particles, and its numerical resolution are studied. The gas-aerosol system is modeled by ordinary differential equations coupled with a mixed-constrained optimization problem. This coupling induces discontinuities when inequality constraints are activated or deactivated. Two approaches for the solution of the optimization-constrained differential equations are presented. The first approach is a time splitting scheme together with a fixed-point method that alternates between the differential and optimization parts. The ordinary differential equations are approximated by the Crank-Nicolson scheme and a primal-dual interior-point method combined with a warm-start strategy is used to solve the minimization problem. The second approach considers the set of equations as a system of differential algebraic equations after replacing the minimization problem by its first order optimality conditions. An implicit 5th-order Runge-Kutta method (RADAU5) is then used. Both approaches are completed by numerical techniques for the detection and computation of the events (activation and deactivation of inequality constraints) when the system evolves in time. The computation of the events is based on continuation techniques and geometric arguments. Moreover the first approach completes the computation with extrapolation polynomials and sensitivity analysis, whereas the second approach uses dense output formulas. Numerical results for gas-aerosol system made of several chemical species are proposed for both approaches. These examples show the efficiency and accuracy of each method. They also indicate that the second approach is more efficient than the first one. Furthermore theoretical examples show that the method for the computation of the activation is of second order for the first approach and exact for the second one.

The mathematical foundations of physical systems modeling languages

Rational general solutions of systems of first-order algebraic partial differential equations

A method is proposed to transform a system of differential algebraic equations (D.A.E.) to a system of ordinary differential equations (O.D.E.) which can be solved relatively easily by standard numerical techniques.Two examples, including a model of an absorption tower, are given to illustrate the utility of the method. The example problems reveal that this easily implemented technique offers significant savings in CPU time compared to the numerical solution of the untransformed D.A.E.s particularly when the algebraic equations are nonlinear.Furthermore, it appears to be faster and/or more reliable than other numerical schemes which have been recently developed for equations of this type.

EASY-FIT is an interactive software system to identify parameters in explicit model functions, steady-state systems, Laplace transformations, systems of ordinary differential equations, differential algebraic equations, or systems of one-dimensional time-dependent partial differential equations with or without algebraic equations. Proceeding from given experimental data, i.e. observation times and measurements, the minimum least squares distance of measured data from a fitting criterion is computed, that depends on the solution of the dynamical system. The software system is implemented in form of a Microsoft Access database running under MS-Windows 95/98/NT/2000. The underlying numerical algorithms are coded in Fortran and are executable independently from the interface. Model functions are either interpreted and evaluated symbolically by a Fortran-similar modeling language, that allows in addition automatic differentiation of nonlinear functions, or by user-provided Fortran subroutines.

Multibody dynamics comprises methodologies for the design and analysis of mechanical systems, with Kalman filters being the principal state estimation methods. The Kalman filters are generally formulated for unconstrained systems, the dynamics of which are described by ordinary differential equations in state-space models. However, multibody systems are constrained systems, and the commonly used method to describe their dynamics involves using differential algebraic equations (DAEs) comprising differential and algebraic equations. The differential equations also include the Lagrange multipliers. Hence, incorporating multibody systems, described by DAEs into the scheme of Kalman filters, cannot be achieved straightforwardly, which facilitates different strategies being addressed. This study develops a novel method for converting DAEs into a state-space model. A transition model of the time derivatives of Lagrange multipliers and a Lagrange multiplier constraint vector are devised and used in the state and output equations, respectively. The continuous- and discrete-time extended Kalman filters (CEKF and DEKF) are constructed using the proposed state-space model, and state estimations are simulated on the benchmark planar four- and five-bar linkages. Further, a demonstration of system observability is conducted, and sensitivity to the initial state estimates is studied. These tests demonstrate that the proposed state-space model achieves observable systems and that both the CEKF and DEKF, constructed using the proposed state-space model, can estimate the states with a wide range of initial conditions.

In his algebraic theory of differential equations, J. F. Rittf has developed a decomposition theory for systems of algebraic differential equations by introducing the idea of irreducible systems and proving that every system is equivalent to one and essentially only one finite set of irreducible systems.The analogous theorem for algebraic difference equations was given by Ritt and Doob.fThe purpose of the present paper is to derive a decomposition theorem for algebraic mixed difference equations; i.e., equations which contain algebraically one or more unknown functions y(x), their "transforms" y(x), y(x+l), y(x+2), ---, and the derivatives of those transforms.

System of fractional differential algebraic equations with applications

Differential elimination with Dixon resultants

Sliding motion of discontinuous dynamical systems described by semi-implicit index one differential algebraic equations