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Constructing the field of values of decomposable and general matrices using the ZNN based path following method

numerical LINEAR ALGEBRA WITH APPLICATIONS 2023年第6期30卷 e2513-e2513页

作者： Uhlig, Frank Auburn Univ Dept Math & Stat Auburn AL USA Auburn Univ Dept Math & Stat Auburn AL 36849 USA

This paper describes and develops a fast and accurate path following algorithm that computes the field of values boundary curve partial differential delta F(A) for every conceivable complex or real square matrix A. It relies on the matrix flow decomposition algorithm that finds a proper block-diagonal flow representation for the associated hermitean matrix flow F-A(t)=cos(t)H+sin(t)K under unitary similarity if that is possible. Here F-A(t) is the 1-parameter-varying linear combination of the real and skew part matrices H=(A+A*)/2 and K=(A-A*)/(2i) of A. For indecomposable matrix flows, F-A(t) has just one block and the ZNN based field of values algorithm works with F-A(t) directly. For decomposing flows F-A(t), the algorithm decomposes the given matrix A unitarily into block-diagonal form U*AU=diag(Aj) with j>1 diagonal blocks A(j) whose individual sizes add up to the size of A. It then computes the field of values boundaries separately for each diagonal block A(j) using the path following ZNN eigenvalue method. The convex hull of all sub-fields of values boundary points partial differential F(A(j)) finally determines the field of values boundary curve correctly for decomposing matrices A. The algorithm removes standard restrictions for path following FoV methods that generally cannot deal with decomposing matrices A due to possible eigencurve crossings of FA(t). Tests and numerical comparisons are included. Our ZNN based method is coded for sequential and parallel computations and both versions run very accurately and fast when compared with Johnson's Francis QR eigenvalue and Bendixson rectangle based method and compute global eigenanalyses of F-A(tk) for large discrete sets of angles t(k) is an element of[0,2 pi] more slowly.

关键词： block-diagonal matrix decomposable matrix field of values matrix flow numerical algorithm single parameter varying matrices time-varying matrix flow unitary similarity

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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 continuous fracture front tracking algorithm with multi layer tip elements (MuLTipEl) for a plane strain hydraulic fracture

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JOURNAL OF PETROLEUM SCIENCE AND ENGINEERING 2022年第0期217卷

作者： Dontsov, E. V. ResFrac Corp 555 Bryant St 185 Palo Alto CA 94301 USA

The problem of a plane strain hydraulic fracture propagating in a layered formation is considered. Fracture toughness, in-situ stress, and leak-off coefficient are assumed to vary by layer, while the elastic properties are kept constant throughout the domain for simplicity. The purpose of this study is to develop a numerical algorithm based on a fixed mesh approach, which is able to solve the above problem accurately using elements which can even be larger than the layer size. In order to do this, the concept of fictitious tip stress is first introduced for determining the fracture front location. In this technique, an additional stress is applied to the tip element to suppress the opening and to mimic the width corresponding to the actual fracture front location. A theoretical basis for this concept has been established and it is further calibrated for piece-wise constant elements. Once the ability to track the crack front location is developed, the effect of layers is included by varying properties as a function of front location. Several numerical examples benchmarking the numerical solution, as well as highlighting capabilities of the algorithm to tackle multiple thin layers accurately are presented.

关键词： Hydraulic fracturing numerical algorithm Thin layers Plane strain

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CaTSM: A Pseudo-Spectral Thermodynamically Consistent Model of Compressible Flows

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JOURNAL OF ADVANCES IN MODELING EARTH SYSTEMS 2022年第11期14卷 e2022MS003112-e2022MS003112页

作者： Brown, Justin M. Dewar, William Radko, Timour Naval Postgrad Sch Dept Oceanog Monterey CA 93943 USA Florida State Univ Dept Earth Ocean & Atmospher Sci Tallahassee FL USA

We introduce a pseudo-spectral algorithm that includes full compressible dynamics with the intent of simulating near-incompressible fluids, CaTSM (Compressible and Thermodynamically consistent Spectral Model). A semi-implicit scheme is used to model acoustic waves in order to evolve the system efficiently for such fluids. We demonstrate the convergence properties of this numerical code for the case of a shock tube and for Rayleigh-Taylor instability. A linear equation of state is also presented, which relates the specific volume of the fluid linearly to the potential temperature, salinity, and pressure. This permits the results to be easily compared to a Boussinesq framework in order to assess whether the Boussinesq approximation adequately represents the relevant exchange of energy to the problem of interest. One such application is included, that of the development of a single salt finger, and it is shown that the energetic behavior of the system is comparable to the typical canonical development of the problem for oceanographic parameters. However, for more compressible systems, the results change substantially even for low-Mach number flows.

关键词： compressible fluids incompressible fluids numerical algorithm microscale

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Computing Contraction Metrics: Comparison of Different Implementations 24

Computing Contraction Metrics: Comparison of Different Imple...

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24th International Symposium on Mathematical Theory of Networks and Systems (MTNS)

作者： Giesl, P. Hafstein, S. Mehrabinezhad, I Univ Sussex Dept Math Falmer BN1 9QH England Univ Iceland Fac Phys Sci Dunhagi 5 IS-107 Reykjavik Iceland

We discuss the implementation of a novel demanding algorithm to compute contraction metrics for nonlinear systems. We compare MATLAB- and C++-implementations and study the effect of parallelizing the code. Further, we explore the use of different low-level math-libraries for the C++ -implementation and the use of an undocumented code-path that considerably speeds up MATLAB code on AMD's Ryzen processors. Copyright (C) 2021 The Authors.

关键词： Contraction metric Riemannian metric Lyapunov function numerical algorithm Mesh-free collocation

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Angular Values of Nonautonomous and Random Linear Dynamical Systems: Part I-Fundamentals

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SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS 2022年第2期21卷 1245-1286页

作者： Beyn, Wolf-Juergen Froyland, Gary Huels, Thorsten Bielefeld Univ Dept Math D-33501 Bielefeld Germany Univ New South Wales Sch Math & Stat Sydney NSW 2052 Australia

We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than pi/2. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.

关键词： nonautonomous dynamical systems random dynamical systems angular value ergodic average principal angles of subspaces numerical algorithm

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On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations

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COMPUTATIONAL & APPLIED MATHEMATICS 2018年第4期37卷 4966-4976页

作者： Assanova, A. T. Kadirbayeva, Zh. M. Inst Math & Math Modeling Dept Differential Equat 125 Pushkin Str Alma Ata 050010 Kazakhstan

A linear two-point boundary value problem for a system of loaded differential equations with impulse effect is investigated. Values in the previous impulse points are taken into consideration in the conditions of impulse effect. The considered problem is reduced to an equivalent multi-point boundary value problem for the system of ordinary differential equations with parameters. A numerical implementation of parametrization method is offered using the Runge-Kutta method of 4th-order accuracy for solving the Cauchy problems for ordinary differential equations. The constructed numerical algorithms are illustrated by examples.

关键词： Loaded differential equation Impulse effect Parametrization method numerical algorithm

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Double Scalar Variables Plastic-Damage Model for Concrete

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JOURNAL OF ENGINEERING MECHANICS 2022年第2期148卷 04021143-04021143页

作者： Lu, Dechun Meng, Fanping Zhou, Xin Wang, Guosheng Du, Xiuli Beijing Univ Technol Key Lab Urban Secur & Disaster Engn Minist Educ Beijing 100124 Peoples R China

A framework of the plastic-damage model with double scalar variables is established in nominal stress space under the small deformation assumption. In the damaged part, a damage tensor composed of double scalar variables is presented to comprehensively characterize the isotropic damage behavior in three-dimensional (3D) conditions. The damage laws of Young's modulus and shear modulus are proposed to capture their different damage characteristic observed in the test. For one-dimensional (1D) and 3D conditions, the applicability of single scalar and double scalar damage variables is discussed. The macroscopic damage difference between these two damage variables when describing damage under 3D conditions is analyzed. In the plastic part, the plastic strain increment is determined by two parts of magnitude and direction. The magnitude is obtained by the consistency condition, and the flow direction is defined by the nonorthogonal flow rule that can satisfactorily reproduce the dilatancy behavior of concrete. The proposed model is implemented by the explicit Runge-Kutta (RK) method with the fifth-order accuracy and the Pegasus method. The performance of the model is assessed by the comparison results between the model and the cyclic loading and unloading test data under different stress paths.

关键词： Concrete Plastic-damage model Damage variables Flow rule numerical algorithm

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numerical investigation of the wave force on a partially immersed rectangular structure: Long waves over a flat bottom

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OCEAN ENGINEERING 2021年 221卷 108540-108540页

作者： Gusev, Oleg, I Khakimzyanov, Gayaz S. Chubarov, Leonid B. Fed Res Ctr Informat & Computat Technol Academician MA Lavrentiev Ave 6 Novosibirsk 630090 Russia

This paper describes the results from a numerical estimation of the force exerted by long surface waves on a fixed and partially immersed rectangular structure. The topic is connected with the need of making decisions on the design, placement and operation of hydrotechnical structures of this kind. The wave-structure interaction is simulated in the framework of a nonlinear potential flow model. The results obtained allow to determine the dependence of the hydrodynamic force on the length and amplitude of a single wave, the length and submergence of the structure. We compare the results obtained for different initial wave shapes, and consider the differences between the forces acting on a vertical wall and a partially immersed structure. In particular, it is shown that at relatively small submergence of the structure, longer waves affect it less, while at large submergence and in the case of wave-wall interaction, the opposite behaviour is observed.

关键词： Long surface wave Partially immersed structure Wave force Potential flow model numerical algorithm Calculation results

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Calibration of a multi-mobile coil magnetic manipulation system utilizing a control-oriented magnetic model

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MECHATRONICS 2022年 84卷 102774-102774页

作者： Osinde, Nahashon O. Etievant, Maxime Byiringiro, Jean Bosco Andreff, Nicolas Univ Bourgogne Franche Comte FEMTO ST Inst 15B Ave Montboucons F-25000 Besancon France Dedan Kimathi Univ Technol Mechatron Engn Nyeri Kenya Dedan Kimathi Univ Technol Siemens Mechatron Certificat Ctr Nyeri Kenya

In this paper, we address the calibration of a family of magnetic manipulation systems composed of several coils that are moved around by serial robot manipulators. We show in this paper that the calibration of the whole system ultimately results in calibrating the manipulator and coil separately up to an unknown rigid transformation. For calibration of the coil, we propose to use a model that has not been used so far in the literature;a control-oriented model which is sufficiently accurate and computes the magnetic field in real time. A protocol for calibrating the magnetic manipulation system using the Nelder-Mead algorithm to estimate the model parameters is presented. Calibration was performed through simulations and validated experimentally on a physical system. It was observed that the root mean square error was reduced by 37% after calibration of the physical system, indicating an improvement in accurately estimating the magnetic model.

关键词： Calibration Multi-mobile coil Magnetic manipulation Microrobotics numerical algorithm

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