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Problems of developing a mathematical core for modeling the dynamics of technical systems
Authors
	Zhuk D.M.
	Kozhevnikov D.Yu.
	Manichev V.B.
Date of publication
	2020
DOI
	10.31114/2078-7707-2020-4-31-38
Abstract
	The article discusses the main statements and scientific results of many years of work of the authors in the field of dynamic systems modeling, which are the basis for the development of the library of mathematical programs MZK-a library in C for solving algebraic and differential equations with the maximum possible computer accuracy. It is planned to use the MZK library as a mathematical core for the software package for mathematical modeling of heterogeneous dynamic systems PA10 (Program for Analysis version 10). It is shown that to solve systems of very high dimensions, it is necessary to take into account the errors of rounding numbers when performing elementary arithmetic operations. The most important element of computer-aided design (CAD) systems for technical objects, including microelectronics products, is mathematical modeling of objects, processes and phenomena in the time domain. The reliability and accuracy of mathematical modeling of dynamic systems depends on the mathematical core, which is a block for solving systems of differential-algebraic equations (DAU), a special case of which are systems of ordinary differential equations (odes) in normal Cauchy form, resolved with respect to derivatives. To solve these systems, implicit integration methods are used, which are reduced to solving systems of nonlinear algebraic equations (NAU) and ultimately linear algebraic equations (LAU). After writing the math core, the developer must test it to identify errors and determine whether the math core meets the requirements. Testing of programs for analyzing the dynamics of technical systems is carried out in two directions: testing on specific known practical problems (electrical, mechanical, hydraulic circuits, etc.) and testing of the mathematical core (systems of differential-algebraic equations (DAU). Since the main characteristics (accuracy and speed of the solution) depend on the mathematical core, its testing should confirm the reliability and efficiency of the CAD developed. Currently, the first direction is developed quite widely, especially for microelectronics CAD (ECAD/EDA systems) [2], and the second is rarely used, despite the fact that it is necessary for the further development of existing CAD systems and the development of new ones, so the authors have performed a number of works on the second direction [3]-[5]. The article deals with the main statements and scientific results of the authors ' long-term work in the field of dynamic systems modeling, which are the basis for the development of the library of mathematical programs MZK-a library in C for solving algebraic and differential equations with the maximum possible computer accuracy. It is planned to use the MZK library as a mathematical core for the software package for mathematical modeling of heterogeneous dynamic systems PA10 (PA10) [1]. The ODE system solvers in PA10 will exceed the accuracy and reliability of the ODE system solvers in General-purpose electronic circuit simulation programs of the SPICE type (for example, the Gear and Trapezoid programs in the Multisim software package). It is shown that to solve systems of very high dimensions, it is necessary to take into account the errors of rounding numbers when performing elementary arithmetic operations, which are not taken into account in the above-mentioned General-purpose electronic circuit modeling programs. When solving DAE systems of high and ultra-high dimensions, implicit integration methods should be used, applying at all stages of the calculation algorithms with increased bit depth, in order to exclude the influence of rounding errors on the final result of calculations. When solving NAE systems, you should use the Newton method with the maximum possible analytical calculation of the Jacobian elements, and for numerical calculation, use even-order accuracy methods, calculating the increment using the formula 2√eps*abs(x_i), where eps is the machine accuracy of calculating the addition operation, x_i is the current value of the corresponding variable. When solving LAE systems, it is necessary to take into account the sparsity of the corresponding matrices according to the principle – store only non-zero elements and perform calculations only with non-zero elements, setting an appropriate minimum barrier for them, below which all numbers are considered zero. It is necessary to implement the implicit Euler method to obtain an always known accurate and reliable solution of the corresponding DAE systems with aperiodic solutions and the implicit trapezoidal method to obtain an always known accurate and reliable solution of the corresponding DAE systems with fast oscillating solutions.
Keywords
	computer-aided design of electronic circuits, mathematical modeling, dynamical systems, ordinary differential equations, differential-nonlinear algebraic equations, linear algebraic equations
Library reference
	Zhuk D.M., Kozhevnikov D.Yu., Manichev V.B. Problems of developing a mathematical core for modeling the dynamics of technical systems // Problems of Perspective Micro- and Nanoelectronic Systems Development - 2020. Issue 4. P. 31-38. doi:10.31114/2078-7707-2020-4-31-38
URL of paper
	http://www.mes-conference.ru/data/year2020/pdf/D027.pdf