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Non-linear elastic micro-dilatation theory: matrix exponential function paradigm

INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 2015年 67-68卷 1-26页

作者： Ramezani, Hamidreza Jeong, Jena Univ Orleans Ecole Polytech ICMN CNRS Interfaces Confinement Mat & NanostructUMR F-45072 Orleans France Univ Paris Est Inst Rech Constructibilite ESTP F-94234 Cachan France

In the present paper, the micro-dilatation theory or void elasticity is extended to both large displacement and large dilatation. Firstly, the deformation gradient tensor has been freshly defined by means of the matrix exponential function. The newly defined relation for the deformation gradient has painstakingly investigated for the uniqueness, decomposition issues as well as objectivity and isotropy considerations. The relation of the displacement gradient and deformation gradient tensor is brought via the matrix logarithm function. The micro-dilatation theory constitutive laws are derived using the thermodynamic principles under the zero-centrosymmetric, weakly-centrosymmetric and fully-centrosymmetric cases. These cases have been derived and scrutinized by the numerical experiments. To achieve this assignment, the basic loadings are taken into account, e.g. the hydrostatic loading, simple traction and shear. Some conclusions and outlook pertaining to the above-mentioned cases and variable bulk density have thereafter discussed. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Micro-dilatation theory Porous media matrix exponential function matrix logarithm function Centrosymmetry Large deformation Pore-scale 3D-FEM

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Stability analysis of quasilinear systems on time scale based on a new estimation of the upper bound of the time scale matrix exponential function

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JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS 2024年第18期361卷

作者： Babenko, Serhii Defoort, Michael Djemai, Mohamed Nicaise, Serge Hrytsenko, Valerii Bohdan Khmelnytsky Natl Univ Cherkasy Dept Software Automated Syst Cherkassy Ukraine Polytech Univ Hauts De France CNRS LAMIH UMR 8201INSA Valenciennes France Quartz ENSEA Cergy France Polytech Univ Hauts De France CERAMATHS DMATHS Valenciennes France Polytech Univ Hauts De France CNRS FR 2037 Valenciennes France Bohdan Khmelnytsky Natl Univ Cherkasy Dept Automat & Comp Integrated Technol Cherkassy Ukraine

In this paper, the stability of quasilinear systems on time scale is analyzed based on a new estimation of the upper bound of the time scale matrix exponential function. First, some new upper bounds for the norm of the matrix exponential function eA(t, A ( t, t0) 0 ) are derived, where A is a regressive square matrix, t, t0 0 is an element of T , T being a time scale. The matrix exponential function generalizes the usual matrix exponential as well as the integer power of a matrix. It is shown that the obtained bounds are more accurate compared to the existing bounds of the norm of the matrix exponential function. One of the upper bounds is then used for stability investigation of quasilinear systems evolving on arbitrary time domains.

关键词： matrix exponential function Time scale theory Stability analysis Quasilinear systems

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Validity and error analysis of calculating matrix exponential function and vector product

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NONLINEAR ENGINEERING - MODELING AND APPLICATION 2024年第1期13卷

作者： Tu, Lihui Hong, Huanjie NingboTech Univ Sch Comp & Data Engn Ningbo 315100 Peoples R China

In this study, an efficient calculation method of matrix exponential function and vector product is proposed to reduce the calculation error. This method is based on the flexible exponential integral scheme;using B-series theory and two-color tree theory, the numerical series expression is obtained, and the exponential integral of the matrix exponential function is solved. By constructing a specific numerical integration scheme and analyzing its convergence under the theoretical framework of analytic semigroups, the convergence analysis of numerical schemes is studied and perfected. The numerical results show that the proposed method of matrix exponential function and vector product has a small error, which shows obvious advantages in reducing the calculation error.

关键词： matrix exponential function vector product efficient calculation numerical algorithm

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Precise integration method without inverse matrix calculation for structural dynamic equations

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Earthquake Engineering and Engineering Vibration 2007年第1期6卷 57-64页

作者：汪梦甫 F.T.K.Au Department of Civil Engineering Hunan University Department of Civil Engineering The University of Hong Kong

The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.

关键词： structural dynamics numerical integration inverse matrix calculation matrix exponential function Crout decomposed method

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Eventual cone invariance revisited

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LINEAR ALGEBRA AND ITS APPLICATIONS 2023年第1期675卷 274-293页

作者： Glueck, Jochen Hoelz, Julian Berg Univ Wuppertal Fak Math & Nat Wissensch Gaussstr 20 D-42119 Wuppertal Germany

We consider finite-dimensional real vector spaces X ordered by a closed cone X+ with non-empty interior and study eventual nonnegativity of matrix semigroups (etA)t & GE;0 with respect to this cone. Our first contribution is the observation that, for general cones, one needs to distinguish between different notions of eventual nonnegativity: (i) uniform eventual nonnegativity means that etA maps X+ into X+ for all sufficiently large each x E X+ the vector etAx is in X+ for all t larger than an x-dependent time t0;and (iii) weak eventual nonnegativity means that for each x E X+ and each functional x' in the dual cone X'+ the value (x', etAx) is in [0, oo) for all t larger than an x- and x'-dependent time t0. Until now, only the first of these notions has been studied in the literature. We demonstrate by examples that, somewhat surprisingly for finite-dimensional spaces, all three notions are different. Our second contribution is to show that typical Perron- Frobenius like properties remain valid under the weakest of the above ***, we study a strengthened form of the above mentioned concepts, namely eventual positivity. We prove that the uniform, individual and weak versions of this property are in contrast to the nonnegative case - equivalent, and that they can be characterized by spectral properties.& COPY;2023 Elsevier Inc. All rights reserved.

关键词： Eventual nonnegativity Eventual positivity Cones Ordered vector space matrix exponential function matrix semigroup Perron-Frobenius theorem Krein-Rutman theorem

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Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL)

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ENGINEERING WITH COMPUTERS 2019年第1期35卷 229-241页

作者： Kazem, Saeed Dehghan, Mehdi Amirkabir Univ Technol Dept Appl Math Fac Math & Comp Sci 424 Hafez Ave Tehran 15914 Iran

In this article, we apply the method of lines (MOL) for solving the time-fractional diffusion equations (TFDEs). The use of MOL yields a system of fractional differential equations with the initial value. The solution of this system could be obtained in the form of Mittag-Leffler matrix function. A direct method which computes the Mittag-Leffler matrix by applying its eigenvalues and eigenvectors analytically has been discussed. The direct approach has been applied on one-, two-, and three-dimensional TFDEs with Dirichlet, Neumann, and periodic boundary conditions as well.

关键词： Method of lines Time-fractional diffusion equations Mittag-Leffler function matrix exponential function Tridiagonal matrix Dirichlet Neumann and periodic boundary conditions

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Numerical procedures for a boundary value problem with a non-linear boundary condition

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APPLIED MATHEMATICS AND COMPUTATION 2004年第1期147卷 291-306页

作者： Dehghan, M Amirkabir Univ Technol Fac Math & Comp Sci Dept Math Appl Tehran 15914 Iran Inst Studies Theoret Phys & Math Tehran Iran

Numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial-boundary value problem for the two-dimensional parabolic partial differential equation (PDE) with a non-linear boundary condition. This class of parabolic PDEs plays a very important role in many branches of science and engineering. The non-linear condition is in the form of a double integral giving the specification of mass in the solution domain. Not only the problem has both Neumann and Dirichlet boundary conditions but the Dirichlet boundary condition is in a non-standard form. While sharing some common features with the one-dimensional models, the solution of two-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one-dimensional case. Due to the structure of the boundary conditions a I reduced one-dimensional parabolic equation for the new unknown upsilon(y, t) = integral(0)(1) u(x, y, t) dx can be formulated. The resulting problem has a non-local boundary condition. The new two-dimensional parabolic PDE with Neumann's boundary conditions will be solved numerically by using the method of lines semi-discretization approach. The space derivatives in the PDE are approximated by finite difference replacements. The solution of the resulting system of first-order linear ordinary differential. equations satisfies a recurrence relation which involves a matrix exponential function. The accuracy in time is controlled by choosing several subdiagonal Pade approximants to replace this matrix exponential term. Numerical techniques are developed to compute the required solution using a splitting method, leading to algorithms for sequential and parallel implementation. The algorithms are tested on a model problem from the literature. The article concludes with the results of some numerical experiments. (C) 2002 Elsevier Inc. All rights reserved.

关键词： Neumann's boundary conditions non-classic boundary value problems sequential and parallel algorithms stability parabolic partial differential equations rational approximant specification of mass matrix exponential function

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The precise integration method for semi-discretized equation in the dual reciprocity method to solve three-dimensional transient heat conduction problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2018年 95卷 160-166页

作者： Zhou, Fenglin You, Yulong Li, Guang Xie, Guizhong Li, Guang Hunan Univ Technol Coll Mech Engn Zhuzhou 412007 Peoples R China Zhengzhou Univ Light Ind Mech & Elect Engn Inst Zhengzhou 450002 Henan Peoples R China

A dual reciprocity precise integration method (DRPIM) is presented to solve 3-dimensional transient heat conduction problems. In the presented method, the transient heat conduction problem, which is an initial boundary value problem, is transformed into an initial value problem (IVP) through a dual reciprocity scheme. Then an analytical solution, which is described by terms of the matrix exponential function (MEF), to this IVP is applied. A precise integration method (PIM) is finally applied to compute the MEF accurately. The accuracy and stability of the presented method are demonstrated by three numerical examples.

关键词： Dual reciprocity method Precise integration method Transient heat conduction Boundary integral equation method matrix exponential function

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On the precise integration methods based on Pade approximations

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COMPUTERS & STRUCTURES 2009年第5-6期87卷 380-390页

作者： Wang, M. F. Au, F. T. K. Univ Hong Kong Dept Civil Engn Hong Kong Hong Kong Peoples R China Hunan Univ Yuelushan Dept Civil Engn Changsha 410082 Hunan Peoples R China

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can give precise numerical results approaching exact solutions at the integration points, but it is conditionally stable. By combining Pade approximation and the generalized Pade approximation of the matrix exponential function in precise integration, and by using three different types of quadrature formulae, a new generalized family of precise time step integration methods is developed to achieve unconditional stability and arbitrary order of accuracy. Numerical studies indicate that they are unconditionally stable algorithms with controllable numerical dissipation. They also demonstrate the validity and efficiency of these algorithms. (C) 2008 Elsevier Ltd. All rights reserved.

关键词： Generalized Pade approximation matrix exponential function Numerical integration Numerical stability Structural dynamics Unconditional stability

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A simple time domain calculation method for transient heat transfer models

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ENERGY AND BUILDINGS 2008年第9期40卷 1682-1690页

作者： Xu, Xinhua Wang, Shengwei Hong Kong Polytech Univ Dept Bldg Serv Engn Kowloon Hong Kong Peoples R China

This paper presents a simple time domain calculation method to derive thermal response factors and conduction transfer conduction (CTF) coefficients of finite differential models for estimating transient heat transfer through building structures. It is developed on the basis of converting the matrix exponential function, which is a part of the solution of the state equation established from the finite differential equations of building finite differential models, to matrix polynomial. The thermal response factors and CTF coefficients can be easily derived from the matrix polynomial with simple arithmetic and integral in time domain. This method avoids the time-consuming root-finding process of conventional methods and the computation of all the internal temperature of the finite differential model, while utilizing the advantage of the thermal response factors/CTF coefficients which relate the desired outputs at a moment to the previous inputs through a set of coefficients. Various case studies were conducted to validate the performance of this time domain calculation method in calculating the thermal response factors and CTF coefficients of various order finite differential models. (C) 2008 Elsevier B.V. All rights reserved.

关键词： transient heat transfer model finite differential model state space equation matrix exponential function time domain

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