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A Survey of the L1 Scheme in the Discretisation of Time-Fractional Problems

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Numerical mathematics(Theory,Methods and Applications) 2022年第4期15卷 1173-1192页

作者： Martin Stynes Applied and computational mathematics division Beijing Computational Science Research CenterBeijing 100193China

A survey is given of convergence results that have been proved when the L1 scheme is used to approximate the Caputo time derivative D_(t)^(α)(where 0<α<1)in initial-boundary value problems governed by D_(t)^(α)u−△u=f and similar equations,while taking into account the weak singularity that is present in typical solutions of such *** aspects of these analyses are outlined,such as global and local convergence bounds and the techniques used to derive them,fast implementation of the L1 scheme,semilinear problems,multi-term time derivatives,α-robustness,a posteriori error analysis,and two modified L1 schemes that achieve better *** fifty references are provided in the bibliography,more than half of which are from the period 2019-2022.

关键词： L1 scheme fractional derivative survey

A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR SECOND-ORDER ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

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Journal of computational mathematics 2016年第5期34卷 532-548页

作者： Qian Zhang Ran Zhang Department of mathematics Jilin University Changchun China Applied and computational mathematics division Beijing Computational Science Research Center Beijing China

In this paper, we present a weak Galerkin （WG） mixed finite element method for solving the second-order elliptic equations with Robin boundary conditions. Stability and a priori error estimates for the coupled method are derived. We present the optimal order error estimate for the WG-MFEM approximations in a norm that is related to the L^2 for the flux and H1 for the scalar function. Also an optimal order error estimate in L^2 is derived for the scalar approximation by using a duality argument. A series of numerical experiments is presented that verify our theoretical results.

关键词： Second-order elliptic equations Robin boundary conditions Weak Galerkin,Weak divergence.

A combination of damage locating vector method （DLV） and differential evolution algorithm （DE） for structural damage assessment

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Frontiers of Structural and Civil engineering 2018年第1期12卷 92-108页

作者： T. NGUYEN-THOI A. TRAN-VIET N. NGUYEN-MINH T. VO-DUY V. HO-HUU division of computational mathematics and engineering Institute for Computational Science Ton Duc Thang University Hochiminh City Vietnam division of Construction Computation Institute for Computational Science Ton Duc Thang University Hochiminh City Vietnam Faculty of Civil engineering Ton Duc Thang University Hochiminh City Vietnam

In this study, a two-stage method is presented for identifying multiple damage scenarios. In the first stage, the damage locating vector （DLV） method using normalized cumulative energy （nce） is employed for damage localization in structures. In the second stage, the differential evolution algorithm （DE） is used for damage severity of the structures. In addition, in the second stage, a modification of an available objective function is made for handing the issue of symmetric structures. To verify the effectiveness of the present technique, numerical examples of a 72-bar space truss and a one-span steel portal frame are considered. In addition, the effect of noise on the performance of the identification results is also investigated. The numerical results show that the proposed combination gives good assessment of damage location and extent for multiple structural damage cases.

关键词： damage assessment damage locating vector method (DLV) differential evolution (DE) multiple damagelocation assurance criterion (MDLAC) mode shape error function

An efficient block variant of robust structured multifrontal factorization method

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Chinese Physics B 2013年第8期22卷 225-232页

作者：左宪禹莫则尧谷同祥 School of Computer and Information engineering Henan University Laboratory of computational Physics Institute of Applied Physics and Computational Mathematics

Based on the two-dimensional three-temperature （2D3T） radiation diffusion equations and its discrete system, using the block diagonal structure of the three-temperature matrix, the reordering and symbolic decomposition parts of the RSMF method are replaced with corresponding block operation in order to improve the solution efficiency. We call this block form method block RSMF （in brief, BRSMF） method. The new BRSMF method not only makes the reordering and symbolic decomposition become more effective, but also keeps the cost of numerical factorization from increasing and ensures the precision of solution very well. The theoretical analysis of the computation complexity about the new BRSMF method shows that the solution efficiency about the BRSMF method is higher than the original RSMF method. The numerical experiments also show that the new BRSMF method is more effective than the original RSMF method.

关键词： HSS structure low-rank property multifrontal method two-dimensional three-temperature radia-tive diffusion equations

Thermo-field Dynamics of the Casimir Effect and Its Quantum Deformation

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Chinese Physics Letters 2001年第6期18卷 740-742页

作者： JING Hui XIE Bing-Hao CHEN Jing-Ling Theoretical Physics division Nankai Institute of MathematicsNankai UniversityTianjin 300071 Laboratory of computational Physics Institute of Applied Physics and Computational MathematicsBeijing 100088

The Casimir effect of the deformed cavity field at finite temperature is investigated by generalizing the thermo field dynamics formalism into q-deformed *** has been shown that the impact of q-deformation on the Casimir force only manifests in the finite temperature case and the expression for the ideal pure vacuum remains unchanged,which just coincides with the suggestions of Man’ko et al.[***.A 176(1993)173]about the nature of q-oscillators as the nonlinear vibrations of electromagnetic field.

关键词： field deformed Casimir

Damage assessment of laminated composite beam structures using damage locating vector （DLV） method

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Frontiers of Structural and Civil engineering 2015年第4期9卷 457-465页

作者： T. VO-DUY division of computational mathematics and engineering (CME) Institute for Computational Science (INCOS) Ton Duc ThangUniversity Hochiminh City Vietnam division of Construction Computation (DCC) Institute for Computational Science (INCOS) Ton Duc Thang University HochiminhCity vietnam Faculty of Civil engineering Ton Duc Thang University Hochiminh City Vietnam

In this paper, the damage locating vector (DLV) method using normalized cumulative energy (nce) is employed to locate multiple damage sites in laminated composite beam structures. Numerical simulations of two laminated composite beams are employed to investigate several damage scenarios in which the degradation of elements is modeled by the reduction in the longitudinal Young’s modulus and transverse Young’s modulus of beam layers. The results show that the DLV method gives good performance for this kind of structure.

关键词： damage locating vector method (DLV) laminated composite beam structure normalized cumulative energy structural health monitoring (SHM)

Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method

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Frontiers of Structural and Civil engineering 2019年第2期13卷 324-336页

作者： T.VO-DUY V.HO-HUU T.NGUYEN-THOI division of computational mathematics and engineering Institute for Computational ScienceTon Due Thang UniversityHo Chi Minh CityVietnam Faculty of Civil engineering Ton Due Thang UniversityHo Chi Minh CityVietnam

In the present study, the free vibration of laminated functionally graded carbon nanotube reinforced composite beams is analyzed. The laminated beam is made of perfectly bonded carbon nanotubes reinforced composite (CNTRC) layers. In each layer, single-walled carbon nanotubes are assumed to be unifonnly distributed (UD) or functionally graded (FG) distributed along the thickness direction. Effective material properties of the two-phase composites, a mixture of carbon nanotubes (CNTs) and an isotropic polymer, are calculated using the extended nile of mixture. The first-order shear deformation theory is used to formulate a governing equation for predicting free vibration of laminated functionally graded carbon nanotubes reinforced composite (FG?CNTRC) beams. The governing equation is solved by the finite element method with various boundary conditions. Several numerical tests are perfbnned to investigate the influence of the CNTs volume fractions, CNTs distributions, CNTs orientation angles, boundary conditions, length-to-thickness ratios and the numbers of layers on the frequencies of the laminated FG-CNTRC beams. Moreover, a laminated composite beam combined by various distribution types of CNTs is also studied.

关键词： free vibration analysis laminated FG-CNTRC beam finite element method first-order shear deformation theory composite material

LIOUVILLE RESULTS FOR STABLE SOLUTIONS OF QUASILINEAR EQUATIONS WITH WEIGHTS

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Acta Mathematica Scientia 2019年第2期39卷 357-368页

作者： Phuong LE Vu HO Department of Mathematical Economics Banking University of Ho Chi Minh City division of computational mathematics and engineering Institute for Computational ScienceTon Duc Thang University Faculty of mathematics and Statistics Ton Duc Thang University

This paper is devoted to the quasilinear equation ■where p > 2,Ω is a(bounded or unbounded) domain of R^N,w_1,w_2 are nonnegative continuous functions and f is an increasing function. We establish a Liouville type theorem for nontrivial stable solutions of the equation under some mild assumptions on Ω,w_1, w_2 and f, which extends and unifies several results on this topic.

关键词： quasilinear equations stable solutions nonexistence Liouville theorems

An extended cell-based smoothed discrete shear gap method （XCS-FEM-DSG3） for free vibration analysis of cracked Reissner-Mindlin shells

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Frontiers of Structural and Civil engineering 2015年第4期9卷 341-358页

作者： M. H. NGUYEN-THOI division of computational mathematics and engineering (CME) Institute for Computational Science (INCOS) Ton Duc ThangUniversity Hochiminh city Vietnam Faculty of Civil engineering Ton Duc Thang University Hochiminh city Vietnam

A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) was recently proposed and proven to be robust for free vibration analyses of Reissner-Mindlin shell. The method improves significantly the accuracy of the solution due to softening effect of the cell-based strain smoothing technique. In addition, due to using only three-node triangular elements generated automatically, the CS-FEM-DSG3 can be applied flexibly for arbitrary complicated geometric domains. However so far, the CS-FEM-DSG3 has been only developed for analyzing intact structures without possessing internal cracks. The paper hence tries to extend the CS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells by integrating the original CS-FEM-DSG3 with discontinuous and crack–tip singular enrichment functions of the extended finite element method (XFEM) to give a so-called extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3). The accuracy and reliability of the novel XCS-FEM-DSG3 for free vibration analysis of cracked Reissner-Mindlin shells are investigated through solving three numerical examples and comparing with commercial software ANSYS.

关键词： cracked Reissner-Mindlin shell free vibration analysis cell-based smoothed discrete shear gap method (CS-FEM-DSG3) extended cell-based smoothed discrete shear gap method (XCS-FEM-DSG3) smoothed finite element methods (SFEM)