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Application of fast automatic differentiation for solving the inverse coefficient problem for the heat equation

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2016年第10期56卷 1743-1757页

作者： Zubov, V. I. Russian Acad Sci Dorodnicyn Comp Ctr Fed Res Ctr Comp Sci & Control Moscow 119333 Russia Moscow Inst Phys & Technol Dolgoprudnyi 141700 Moscow Oblast Russia

The problem of determining the thermal conductivity coefficient that depends on temperature is studied. The consideration is based on the initial-boundary value problem for the one-dimensional unsteady heat equation. The mean-root-square deviation of the temperature distribution field and the heat flux from the experimental data on the left boundary of the domain is used as the objective functional. An analytical expression for the gradient of the objective functional is obtained. An algorithm for the numerical solution of the problem based on the modern fast automatic differentiation technique is proposed. Examples of solving the problem are discussed.

关键词： heat conduction inverse coefficient problems gradient heat equation adjoint equations numerical algorithm

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EFFICIENT numerical METHOD FOR SOLUTION OF BOUNDARY VALUE PROBLEMS WITH ADDITIONAL CONDITIONS

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BRAZILIAN JOURNAL OF CHEMICAL ENGINEERING 2017年第3期34卷 873-883页

作者： Szukiewicz, Miroslaw K. Rzeszow Univ Technol Dept Chem & Proc Engn Al Powstancow Warszawy 6 PL-35959 Rzeszow Poland

An efficient numerical method for solution of boundary value problems with additional condition is presented. The approach is based on the shooting method but the procedure of seeking "the proper shot" allows one to satisfy "additional" boundary conditions. General considerations are illustrated by a real example. The computational example concerns the "dead zone" regime for the non-linear diffusion-reaction equation in heterogeneous catalysis. Accuracy and efficiency of the presented method confirm results obtained for a wide range of changes of process parameters, including the vicinity of a critical point. Calculations were performed with the use of the Maple (R) program.

关键词： numerical algorithm BVP with additional conditions dead zone CAS-type programs

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A finite volume method for numerical simulations of adiabatic shear bands formation

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COMMUNICATIONS IN NONLINEAR SCIENCE AND numerical SIMULATION 2021年 101卷 105858-105858页

作者： Muratov, R. V. Kudryashov, N. A. Ryabov, P. N. Natl Res Nucl Univ MEPhI Kashirskoe Shosse 31 Moscow Russia

The aim of this paper is to develop an effective finite volume method for numerical simulation of the adiabatic shear bands (ASB) formation processes. A formation of ASB happens at high-speed shear strains of ductile materials. A numerical simulation of such problems using Lagrangian approach is associated with some problems, the main one of which is a mesh distortion at large deformations. We use Eulerian approach to describe a motion of the non-linear elasto-plastic material. More specifically, we consider a modification of a well-known hypoelastic Wilkins model. In this paper we suggest a numerical method for modeling of high-speed shear deformations on two-dimensional meshes. The method is verified on the three test problems suggested by other authors. (c) 2021 Elsevier B.V. All rights reserved.

关键词： Mathematical model Adiabatic shear bands Strain localization Plasticity numerical algorithm Finite volume method

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A General Three-Dimensional numerical Technique for Determining the Contact Area of an Arbitrary Punch on an Elastic Half-Space

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INTERNATIONAL JOURNAL OF APPLIED MECHANICS 2016年第1期8卷

作者： Shen, Jing Jin Xu, Feng Yu Jiang, Guo Ping Nanjing Univ Posts & Telecommun Sch Automat Nanjing 210023 Jiangsu Peoples R China

The paper presents a numerical method for determining the contact area in three-dimensional elastostatic normal contact without friction. The method makes use of the theorem developed by Barber, the contact area is that over which the total indentation force achieves its maximum value. By approximating the punch by linear interpolation, the analytical expression for the indentation force is derived by virtue of the reciprocal theorem. The physical meaning of the parameter which determines the contact boundary is discussed, and its feasible range corresponding to the contact area is found. Then, the numerical algorithm for determining the parameter is developed and applied to solve several normal contact problems. The results show that the proposed numerical method possesses a good property on accuracy and convergency.

关键词： Contact problem variational condition reciprocal theorem numerical algorithm

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A robust numerical treatment of solid-phase diffusion in pseudo two-dimensional lithium-ion battery models

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JOURNAL OF POWER SOURCES 2023年 556卷

作者： Kim, Jinyong Mallarapu, Anudeep Santhanagopalan, Shriram Newman, John Natl Renewable Energy Lab Ctr Energy Convers & Storage Syst Golden CO 80401 USA

Solid-phase diffusion in active materials of lithium-ion batteries significantly affects charging and safety-related behavior of lithium-ion batteries. Therefore, it is essential to develop an efficient and robust numerical algorithm for solving solid-phase diffusion equations in physics-based battery models. In this work, we discuss the origins of numerical instabilities that can occur when solving the solid-phase diffusion equations using iterative methods. Then, in order to resolve such issues, we propose a simple numerical treatment to the surface flux term of dis-cretized solid-phase diffusion equations. To demonstrate its numerical robustness, the proposed method is implemented into a pseudo two-dimensional (P2D) physics-based battery model and simulations are conducted at wide ranges of operating conditions. Even with extremely poor initial guesses for the Li+ concentrations of the active materials, computations using the proposed method do not diverge and the their computational speeds are comparable to those with conventional initial guesses. Comprehensive tests of the proposed method are also performed with a dynamic current profile based on US06 driving profile and a multi-stage charging profile with very high initial C-rate (12C).

关键词： numerical algorithm Battery modeling Fast charging Battery solver Solid-state diffusion Pseudo-two-dimensional model

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Walk On Spheres algorithm for Helmholtz and Yukawa Equations via Duffin Correspondence

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METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 2017年第2期19卷 589-602页

作者： Yang, Xuxin Rasila, Antti Sottinen, Tommi Hunan First Normal Univ Dept Math Changsha 410205 Hunan Peoples R China Aalto Univ Dept Math & Syst Anal POB 11100 Espoo 00076 Finland Univ Vaasa Dept Math & Stat Fac Technol POB 700 Vaasa 65101 Finland

We show that a constant-potential time-independent Schrodinger equation with Dirichlet boundary data can be reformulated as a Laplace equation with Dirichlet boundary data. With this reformulation, which we call the Duffin correspondence, we provide a classical Walk On Spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the boundary value problem. We compare the obtained Duffin WOS algorithm with existing modified WOS algorithms.

关键词： Brownian motion Helmholtz equation Linearized Poisson-Boltzmann equation Monte Carlo simulation numerical algorithm Walk On Spheres algorithm Yukawa equation

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Application of the Fast Automatic Differentiation Technique for Solving Inverse Coefficient Problems

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2020年第1期60卷 15-25页

作者： Albu, A. F. Evtushenko, Yu G. Zubov, V., I Russian Acad Sci Fed Res Ctr Comp Sci & Control Dorodnicyn Comp Ctr Moscow 119333 Russia Moscow Inst Phys & Technol Dolgoprudnyi 141701 Moscow Oblast Russia

Results obtained by the authors in solving inverse coefficient problems are overviewed. The inverse problem under consideration is to determine a temperature-dependent thermal conductivity coefficient from experimental observations of the temperature field in the studied substance and (or) the heat flux on the surface of the object. The study is based on the Dirichlet boundary value problem for the nonstationary heat equation stated in the general -dimensional formulation. For this general case, an analytical expression for the cost functional gradient is obtained. The features of solving the inverse problem and the difficulties encountered in the solution process are discussed.

关键词： heat conduction inverse coefficient problems gradient heat equation numerical algorithm

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HITTING TIME FOR BESSEL PROCESSES-WALK ON MOVING SPHERES algorithm (WOMS)

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ANNALS OF APPLIED PROBABILITY 2013年第6期23卷 2259-2289页

作者： Deaconu, Madalina Herrmann, Samuel Univ Lorraine INRIA Nancy Grand Est IECN UMR 7502 F-54506 Vandoeuvre Les Nancy France Univ Lorraine INRIA Nancy Grand Est Project Team TOSCA F-54506 Vandoeuvre Les Nancy France Univ Bourgogne IMB UMR 5584 F-21078 Dijon France

In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation comes from mathematical finance when dealing with volatility models, but the results can also be used in optimal control problems. The aim here is to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox-Ingersoll-Ross process. The main tools we use are on one side, an adaptation of the method of images to this particular situation and on the other side, the connection that exists between Cox-Ingersoll-Ross processes and Bessel processes.

关键词： Bessel processes Cox-Ingersoll-Ross processes hitting time method of images numerical algorithm

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An improved technique to determine the controlling unstable equilibrium point in a power system

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS 1996年第4期43卷 313-323页

作者： Treinen, RT Vittal, V Kliemann, W IOWA STATE UNIV SCI & TECHNOL DEPT MATHAMESIA 50011

The accuracy of stability assessment provided by the transient energy function (TEF) method depends on the determination of the controlling unstable equilibrium point (UEP), The technique that currently determines the controlling UEP in the TEF method is based on the so-called exit point method and has also been recently labeled the BCU method, The exit point method consists of two basic steps, First, the exit point is approximated by the point theta(egsa), where the first maximum of the potential energy along the fault-on trajectory is encountered, Second, the minimum gradient point theta(mgp) along the trajectory from theta(egsa) is computed, The controlling UEP is then obtained by solving a system of nonlinear algebraic equations with theta(mgp) as an initial guess, It has been observed that this method lacks robustness in the sense that the following two problems may occur, 1) There may be no detection of the minimum gradient point theta(mgp) and hence, no determination of the controlling UEP, 2) if theta(mgp) is found, then based on the definition of the controlling UEP, it may not be in the domain of convergence of the controlling UEP for the particular solving algorithm used, Hence, another equilibrium point, possibly a stable equilibrium paint, not the controlling UEP will be located, This results in a flawed transient stability assessment, The result of this research has been the development of a new numerical technique for determining the controlling UEP, With an initial starting point that is close to the exit point this technique efficiently produces a sequence of points, An analytical foundation for this method is given which shows that under certain assumptions this sequence will converge to the controlling UEP, Hence this new technique exhibits a substantial improvement over the exit point method because of the following reasons: (1) the technique does not attempt to detect the point theta(mgp), (2) the technique can produce a point that is close to t

关键词： power system analysis computing power system stability power system transients BCU method IEEE 50-generator controlling unstable equilibrium point convergence exit point minimum gradient point nonlinear algebraic equations numerical algorithm power syste differential algebraic equations Power system analysis computing Power system stability Riccati equations exit point power systems power system transients Converge

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Systems of quasilinear conservation laws and algorithmization of variational principles

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2015年第9期55卷 1554-1566页

作者： Rykov, Yu. G. Feodoritova, O. B. Russian Acad Sci Keldysh Inst Appl Math Moscow 125047 Russia

A previously formulated new approach to the consideration of systems of quasilinear hyperbolic equations on the basis of variational principles is described in more detail in the case of special systems of three equations. It is shown that each field of characteristics can be represented as a solution of a variational problem. Moreover, the Rankine-Hugoniot relations at the corner points of the characteristics or at the intersections of the characteristics of a single family hold automatically. In the simplest case of the Hopf equation, a numerical algorithm is constructed on the basis of a variational principle.

关键词： hyperbolic system gas dynamics equations characteristics variational problem discontinuous solutions Rankine-Hugoniot relations numerical algorithm

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