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Рассматривается процесс адаптации численной модели нестационарной теплопроводности, реализованной при помощи методов конечных разностей. Для классического представления данных моделей в большинстве приложений и задач уже доказана алгоритмическая устойчивость, но в данном случае рассматривается задача, связанная с параметрической адаптацией уравнения нестационарной теплопроводности к нагреваемому веществу, выполненной при помощи решения смежной вариационной задачи. Основа данного подхода предполагает замену теплофизических параметров рассматриваемого уравнения на свободно настраиваемые параметры и их коррекцию («обучение модели») методом стохастического градиента. Чтобы избежать попадания в области неустойчивости при «обучении», необходимы ограничения на введенные настраиваемые параметры. В данной работе такие ограничения получены на основании доказанных условий устойчивости классической конечно-разностной модели нестационарной теплопроводности. В результате численного эксперимента было установлено, что предлагаемые ограничения позволяют в среднем увеличить количество устойчивых начальных условий на 14%, увеличить количество попаданий в устойчивые траектории на 61%. Также было проведено аналитическое сравнение порядков роста алгоритмической сложности классической и модифицированной модели. В результате расчетов было установлено, что обе модели имеют порядок роста О(n4), что было подтверждено численным экспериментом., This paper deals with the process of adaptation of a numerical model of nonstationary thermal conductivity implemented with the help of finite difference methods. The algorithmic stability has already been proved for the classical representation of these models in most applications and problems, but in this case we consider a problem related to the parametric adaptation of the equation of nonstationary heat conduction to the heated substance implemented by solving of the related variational problem. The basis of this approach implies replacement of thermophysical parameters of the equation in question by freely adjustable parameters and their adaptation ("model training") by a stochastic gradient method. Optimization of algorithmic equations that do not have an analytical form is associated with unstable initial conditions and "training" trajectories. To avoid falling into these regions we need to impose restrictions on the adjustable parameters. In this paper, such constraints are derived on the basis of proven stability conditions for the classical finite-difference model of non-stationary thermal conductivity. As a result of the numerical experiments, it is shown that the proposed constraints allow one to increase, on average, the number of stable initial conditions by 13%, as well as the number of experiments when stable trajectories are achieved - by 61%. In addition to this result, an analytical comparison of the growth orders of algorithmic complexity of the classical model and the modified one is also made. As a result of the calculations, it is found that both models have a growth order of O(n4), which is confirmed by numerical experiments., Управление большими системами: сборник трудов, Выпуск 101 2023, Pages 39-63

Smoothed particle hydrodynamics (SPH) method is a powerful tool for modeling solid mechanics problems, especially for large deformation problems. However, it suffers from boundary deficiency and difficulty of boundary condition treatment. In this work, a normalized SPH method is proposed to overcome these problems. The method is based on a newly developed normalized particle approximation. To derive this particle approximation, a normalized kernel approximation which is accurate for derivatives of linear functions everywhere in a problem domain is constructed, and all integral terms of the normalized kernel approximation including boundary terms are discretized by particle summations. The normalized particle approximation is free of matrix inversion, consequently attractive in computational stability and simplicity compared with other corrective particle approximations. Its approximation accuracy is demonstrated by calculating derivatives of test functions. Based on this particle approximation, the formulation of the normalized SPH method for transient solid mechanics problems is derived. Moreover, a direct method of treating traction boundary conditions is presented by making use of the boundary term of the normalized particle approximation. The accuracy and capability of the normalized SPH method are validated by the calculation of elastic wave propagation in solids and compared with commonly used SPH method.

Modified 3D Moving Particle Semi-Implicit (MPS) method is used to complete the numerical simulation of the fluid sloshing in LNG tank under multidegree excitation motion, which is compared with the results of experiments and 2D calculations obtained by other scholars to verify the reliability. The cubic spline kernel functions used in Smoothed Particle Hydrodynamics (SPH) method are adopted to reduce the deviation caused by consecutive two times weighted average calculations; the boundary conditions and the determination of free surface particles are modified to improve the computational stability and accuracy of 3D calculation. The tank is under forced multidegree excitation motion to simulate the real conditions of LNG ships, the pressures and the free surfaces at different times are given to verify the accuracy of 3D simulation, and the free surface and the splashed particles can be simulated more exactly.

A multi-step single-stage method is considered, which allows one to integrate stiff differential equations and systems of equations with high accuracy and low computational costs. The examples show that the proposed method is in solving stiff problems as good as the best available methods. The calculation results allow us to determine the absolute stability domains for the multi-step single-stage method, where it is possible to vary integration step within a wide range while maintaining the computational stability of the method.

Conditionspectrum measures the computational stability of solving a linear system. In this paper, ten comparative results involving $$\varepsilon $$ -conditionspectrum are presented. All these theorems generalize a well known eigenvalue theorem and simultaneously compare with an appropriate pseudospectra result.

When computing trajectories from model output, gridded winds are often temporally interpolated to a time step shorter than model output intervals to satisfy computational stability constraints. This study investigates whether trajectory accuracy may be improved for tropical cyclone (TC) applications by interpolating the model winds using advection correction (AC) instead of the traditional linear interpolation in time (LI) method. Originally developed for Doppler radar processing, AC algorithms interpolate data in a reference frame that moves with the pattern translation, or advective flow velocity. A previously developed trajectory AC implementation is modified here by extending it to three-dimensional (3D) flows, and the advective flows are defined in cylindrical rather than Cartesian coordinates. This AC algorithm is tested on two model-simulated TC cases, Hurricanes Joaquin (2015) and Wilma (2005). Several variations of the AC algorithm are compared to LI on a sample of 10 201 backward trajectories computed from the modeled 5-min output data, using reference trajectories computed from 1-min output to quantify position errors. Results show that AC of 3D wind vectors using advective flows defined as local gridpoint averages improves the accuracy of most trajectories, with more substantial improvements being found in the inner eyewall where the horizontal flows are dominated by rotating cyclonic wind perturbations. Furthermore, AC eliminates oscillations in vertical velocity along LI backward trajectories run through deep convective updrafts, leading to a ~2.5-km correction in parcel height after 20 min of integration.

Япарова Наталья Михайловна, канд. физ.-мат. наук, зав. кафедрой вычислительной математики и высокопроизводительных вычислений, Южно-Уральский государственный университет, г. Челябинск; iaparovanm@susu.ru. N.M. Yaparova, iaparovanm@susu.ru South Ural State University, Chelyabinsk, Russian Federation

The problem of forming sample estimates of the correlation matrix of observations that satisfy the criterion "computational stability – consistency" is considered. The variants in which the direct and inverse asymptotic forms of the correlation matrix of observations are approximated by various types of estimates formed from a sample of a fixed volume are investigated. The consistency of computationally stable estimates of the correlation matrix for their static regularization was analyzed. The contradiction inherent in the problem of regularization of the estimates with a fixed parameter is revealed. The dynamic regularization method as an alternative approach is proposed, which is based on the uniqueness theorem for solving the inverse problem with perturbed initial data. An optimal mean-square approximation algorithm has been developed for dynamic regularization of sample estimates of the correlation matrix of observations, using the law of monotonic decrease in the regularizing parameter with increasing sample size. An optimal dynamic regularization function was obtained for sample estimates of the correlation matrix under conditions of a priori uncertainty with respect to their spectral composition. The preference of this approach to the regularization of sample estimates of the correlation matrix under conditions of a priori uncertainty is proved, which allows to exclude the domain of computational instability from solving the inverse problem and obtain its solution in real time without involving prediction data and additional computational cost for finding the optimal value of the regularization parameter. The application of the dynamic regularization method is shown for solving the problem of detecting a signal at the output of an adaptive antenna array in a nondeterministic clutter and jamming environment. The results of a computational experiment that confirm the main conclusions are presented.

This work addresses the use and compare of machine vision software for image recognition compared to commercially available systems in automation industry. The aim is to compare computational stability and time-cost efficiency of integration of available algorithms against commercially sold systems. Three types of environments are selected for comparison - code development environment, graphical development environment and industrial machine vision system solution. The evaluation is based on the use of all three systems for the same types of applications and a comparison of their results - processing time, stability, development time and total development cost. Comparison applications are presence/absence, counting, dimensional measurement, search and character recognition.

To improve the performance of the blind equalizer (BE) in impulsive noise environments, a robust maximum likelihood algorithm (RMLA) is proposed for the communication systems using quadrature amplitude modulation signals. A novel robust maximum likelihood cost function based on the constant modulus algorithm is constructed to effectively suppress the influence of impulsive noise and ensure the computational stability. Theoretical analysis is presented to illustrate the robustness and good computational stability of the proposed algorithm under the impulsive noise ambient. Moreover, it is proved that the weight vector of the proposed BE can converge stably by LaSalle invariance principle. Simulation results are provided to further confirm the robustness and stability of the proposed RMLA.

We perform the analysis of investigations in the theory of branched continued fractions carried out for the last 20 years in the directions developed under the general supervision of Prof. V. Ya. Skorobohat’ko (18.07.1927–04.07.1996) including, in particular, the interpolation of functions of several variables by branched continued fractions, the determination of efficient criteria of convergence and computational stability of these fractions, the correspondence between multiple power series and functional branched continued fractions, the investigation of various classes of functional fractions, and the application of branched continued fractions.

Moving least square regression is an important local learning algorithm. In this paper, we consider a regularized moving least square regression algorithm in reproducing kernel Hilbert space. The localized representer theorem is different from the classical representer theorems for regularized kernel machines. It shows that, regularization not only ensures the computational stability, it is also necessary for the algorithm to preserve localization property. We also studied the learning performance of the regularized moving least square algorithm and conducted a rigorous error analysis. Compared with the unregularized method, convergence analysis of regularized moving least square regression requires more natural and much simpler conditions and achieves fast rates.

This paper describes higher accuracy of an improved SPICE model for a wide range of drain currents for a 1.2-kV, 36-A discrete SiC-MOSFET. The improved SPICE model is developed on the basis of our former Simplorer model by introducing new SPICE codes. Its accuracy and computational stability are validated by comparing the simulated switching waveforms with measurement and that from the manufacture's model up to the drain current of 30 A.

Япарова Наталья Михайловна, канд. физ.-мат. наук, зав. кафедрой вычислительной математики и высокопроизводительных вычислений, Южно-Уральский государственный университет, г. Челябинск; iaparovanm@susu.ru. N.M. Yaparova, iaparovanm@susu.ru South Ural State University, Chelyabinsk, Russian Federation Предложен метод определения температуры при нелинейном теплопереносе внутри цилиндрического тела с неизвестной начальной температурой. Исходными данными являются результаты измерений температурных функций вблизи поверхности тела. Рассматриваемая задача возникает при термической обработке изделий в камерных печах. Математическая модель процесса включает в себя нелинейное уравнение теплопроводности, учитывающее зависимость теплофизических свойств материала от температуры, и граничные условия, характеризующие процесс теплообмена на поверхности тела. Для определения температуры внутри тела предложен метод дискретной регуляризации, позволяющий последовательно находить температуру в направлении от поверхности к оси цилиндра. Вычислительная схема метода предполагает использование стабилизирующих функционалов, а также конечно-разностных уравнений для определения температурных значений по пространственной переменной, что позволяет уменьшить влияние неизвестных начальных условий. Использование стабилизирующих функционалов позволяет обеспечить устойчивость вычислительной процедуры относительно погрешности исходных данных. В статье приведены результаты исследований устойчивости вычислительной схемы. С целью проверки принципиальной возможности определения температуры и получения экспериментальных оценок температурных погрешностей проведен вычислительный эксперимент. В эксперименте выполнен расчет температурного поля цилиндра с неизвестной начальной температурой, проведен сравнительный анализ найденных температурных функций с тестовыми значениями и получены экспериментальные оценки уклонений вычисленных температур от тестовых значений. Результаты вычислительного эксперимента представлены в работе, согласуются с теоретическими оценками и подтверждают эффективность и надежность предлагаемой вычислительной схемы. The paper is devoted to the problem of determining the temperature inside the cylinder based on the surface measurements. The main characteristics of the material such as specific heat, thermal conductivity and material density depend on the temperature at each point of the body. This problem appears in complex heat treatment of products. The mathematical model of heat transfer is presented as nonlinear heat conduction equation with unknown initial condition. In this problem, both the Dirichlet and Neumann boundary conditions are given. The computational scheme of the method is based on the using of finite-difference equations and regularization technique. This approach allows to obtain the temperature inside cylindrical body and then to determine the unknown temperature on the cylinder axis. We investigate the stability of computational method and obtain the dependence of stability on the discretization steps and level error of the initial data. The reliability and the efficiency of the method were confirmed by results of computational experiments. Работа выполнялась при финансовой поддержке Министерства образования и науки Российской Федерации прикладных научных исследований в рамках базовой части Государственного задания «Разработка, исследование и реализация алгоритмов обработки данных динамических измерений пространственно-распределенных объектов», техническое задание 8.9692.2017/8.9 от 17.02.2017. The work was conducted with the financial support of the Ministry of Education and Science of the Russian Federation of applied scientific research within the framework of the basic part of the State task “Development, research and implementation of data processing algorithms for dynamic measurements of spatially distributed objects”, Terms of Reference 8.9692.2017/8.9 from 17.02.2017.

Kylin-2 is an advanced neutronics lattice code, developed by Nuclear Power Institute of China. Chebyshev Rational Approximation Method (CRAM) is used in Kylin-2 to solve the burnup chains. This method has the characters of high calculation accuracy and good computational stability, which is the most widely used numerical solution nowadays. However, when solving the large-scale matrices, the computing time would sharply increase, so the burnup calculation efficiency of Kylin-2 is not very high. In this study, an improved CRAM based on order reduction with Krylov Subspace Method (KSM), Combined Algorithm, has been used to solve burnup equations. According to the verification and analysis, comparing to CRAM, Kylin-2 code with Combined Algorithm has higher efficiency in burnup calculation while maintaining the accuracy.

Point kinetics equations (PKEs) is an significant model used to describe dynamic behaviors of neutron in nuclear reactors. How to cope with its stiffness problems is very important to solve PKEs accurately and efficiently. In this paper, a new semi-analytical algorithm based on Euler-Maclaurin Approximation (SAEMA) is developed to solve PKEs. SAEMA algorithm is applied and tested in different reactors with three typical reactivity insertion cases, including step, ramp and sinusoidal insertions. Firstly, the investigations on the computational stability are performed under different time step sizes. Secondly, the performances on computational accuracy are evaluated by comparing the results by SAEMA algorithm with the ones by analytical method and excellent CATS algorithm. Finally, by comparing the CPU time consumed by SAEMA algorithm with the physical time, the studies on the computational efficiency are also carried out. Just as results shows, SAEMA algorithm is reasonably an attractive way to solve the PKEs.

This paper discusses the merits of Generalized Perturbation Theory (GPT) for search mechanisms of relevance to an industrial code’s daily application context. Unavoidably, these search mechanisms are embedded in the code’s 3D whole-core iterative non-linear multi-physics coupled solution processes. Due to this, it is not so trivial to enable acceptable associated computational stability and run time performance. Relevant examples are the search mechanisms aimed at automated determination of specific control insertion depths. The latter are to enable specified target values for the axial power offset or core reactivity. Compared to executing those kinds of searches according to more primitive algorithms (such as the secant method), a co-usage of GPT was confirmed to enable a far better convergence efficiency and thereby also significant CPU run time reductions.

This paper presents correct results for a paper by Liu and Xing (2011) [1] that had errors, greatly simplifies its formulations, overcomes its numerical stability problem, and provides some new results. In their early paper, Liu and Xing obtained exact closed-form solutions for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method. The equations presented therein are correct, but the numerical values of the frequencies listed have some errors because the shear rigidities of the three-ply laminates were not correctly computed. In addition, the exact characteristic equations of the early paper can also be greatly simplified and the compact equations are presented herein. The characteristic equations in the early paper may have computational stability problem when plates become very thin. This paper overcome such computational difficulty. Some new exact frequency parameters are also included in this work.

A simulation framework based on Smoothed Particle Hydrodynamics (SPH) is introduced to model problems involving the interaction between flowing water and soil deformation. Changes in soil porosity and associated permeability are automatically adjusted within this framework. The framework’s capabilities are presented and discussed for three geotechnical problems caused by flowing water. The comparison between simulation results and experiments shows that SPH with the proposed concept is capable of quantitatively simulating the hydro-mechanical processes beyond limit state with satisfactory agreement. To improve the computational stability, a correction procedure and a new algorithm for the selection of the optimal time step are introduced.

As one of the most complex and vulnerable structures of body, the human knee joint should maintain dynamic equilibrium and stability in occupational and recreational activities. The evaluation of its stability and factors affecting it is vital in performance evaluation/enhancement, injury prevention and treatment managements. Knee stability often manifests itself by pain, hypermobility and giving-way sensations and is usually assessed by the passive joint laxity tests. Mechanical stability of both the human knee joint and the lower extremity at early stance periods of gait (0% and 5%) were quantified here for the first time using a hybrid musculoskeletal model of the lower extremity. The roles of muscle coactivity, simulated by setting minimum muscle activation at 0-10% levels and ACL deficiency, simulated by reducing ACL resistance by up to 85%, on the stability margin as well as joint biomechanics (contact/muscle/ligament forces) were investigated. Dynamic stability was analyzed using both linear buckling and perturbation approaches at the final deformed configurations in gait. The knee joint was much more stable at 0% stance than at 5% due to smaller ground reaction and contact forces. Muscle coactivity, when at lower intensities (