Reaction–diffusion systems for the microscopic cellular model of the cardiac electric field

The paper deals with a mathematical model for the electric activity of the heart at microscopic level. The membrane model used to describe the ionic currents is a generalization of the phase‐I Luo–Rudy, a model widely used in 2‐D and 3‐D simulations of the action potential propagation. From the mathematical viewpoint the model is made up of a parabolic reaction diffusion system coupled with an ODE system. We derive existence and some regularity results. Copyright © 2006 John Wiley & Sons, Ltd.     

Modeling the capacity of a novel flow-energy harvester

The performance of a new type of flow energy harvester based on oscillating foils is investigated through numerical modeling by using two methods, a 2D thin-plate model and a 3D nonlinear boundary-element model. The fluid–structure interaction problem involved in the dynamics of a heaving/pitching foil coupled with an actuation/energy harvesting system in this device is examined. The 2D analysis allows us to simulate dynamics of the flapping-foil system over a large range of parameters and to identify areas of special interests (e.g., high energy output or high efficiency). In the vicinity of these areas the 3D model can accurately predict the performance of the system. By examining the power extraction capacity and efficiency of the system at various geometric, mechanical, and kinematic parameters, the optimal performance of the system is determined. In addition, the performance is found to be enhanced by the presence of a solid ground, as well as the thickness of the foil (at certain frequencies).     

Multilayered model in optics and quantum mechanics

Three types of dispersion equations are analyzed that describe the eigenvalues of the effective refractive index of a multilayer plane optical waveguide and the energy eigenvalues of a quantum particle placed in a piecewise constant potential field. The first equation (D1) is derived by setting to zero the determinant of the system of linear equations produced by matching the solutions in the layers. The second equation (D2) is obtained using the well-known method of characteristic matrices. The third equation has been obtained in the general case by the author and is known as a multilayer equation. Simple relations between the three equations are established. It is shown that the set of roots of D2 exactly coincides with the set of eigenvalues of the multilayer problem, while the roots of D1 and the multilayer equation contain those equal to the refractive index in the optical case (or to the potential in the quantum case) in internal layers of the system, which may be superfluous. Examples are presented.     

线性不确定系统的鲁棒D型迭代学习控制

研究一类线性不确定系统的鲁棒D型迭代学习控制问题.首先针对一类线性标称控制对象,建立其迭代学习控制的二维模型;然后基于获得的二维模型,利用二维系统稳定性理论,获得系统在迭代初态与期望初态一致和不一致两种情形下的D型迭代学习控制律的存在条件和设计方法;进一步,将所得结果推广至控制对象包含不确定性的情形.所得结果以线性矩阵不等式(LMI)的形式给出,可以方便地利用Matlab中的LMI工具箱求解.最后,数值仿真实例验证了本文所提方法的有效性.     

Analysis of a Quantum Subband Model for the Transport of Partially Confined Charged Particles

This paper is devoted to the analysis of a quantum subband model, which is presented as an alternative to the standard 3D Schr?dinger-Poisson system for modeling the transport of electrons strongly confined along one direction. This subband model is composed of quasistatic 1D Schr?dinger equations in the direction of the confinement, coupled to 2D time-dependent Schr?dinger equations describing the transport in the non-confined directions. Selfconsistent electrostatic interactions are also taken into account via the Poisson equation. This system is studied in the framework of the strong partial confinement asymptotics introduced in the article “Adiabatic approximation of the Schr?dinger-Poisson system with a partial confinement”, by Ben Abdallah, Méhats and Pinaud (SIAM J. Math. Anal. 36 (2005), 986–1013).     

On the nonlinear stability and the existence of selective decay states of 3D quasi-geostrophic potential vorticity equation

In this article, we study the dynamics of large-scale motion in atmosphere and ocean governed by the 3D quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible-irreversible coupling.     

Analysis of a Quantum Subband Model for the Transport of Partially Confined Charged Particles

This paper is devoted to the analysis of a quantum subband model, which is presented as an alternative to the standard 3D Schr?dinger-Poisson system for modeling the transport of electrons strongly confined along one direction. This subband model is composed of quasistatic 1D Schr?dinger equations in the direction of the confinement, coupled to 2D time-dependent Schr?dinger equations describing the transport in the non-confined directions. Selfconsistent electrostatic interactions are also taken into account via the Poisson equation. This system is studied in the framework of the strong partial confinement asymptotics introduced in the article “Adiabatic approximation of the Schr?dinger-Poisson system with a partial confinement”, by Ben Abdallah, Méhats and Pinaud (SIAM J. Math. Anal. 36 (2005), 986–1013).     

Computation of dynamical phase transitions in solids

This paper deals with the dynamics of phase boundaries in a nonlinear elastic two-phase material. We consider the elasticity system in 1D and the equations of anti-plane shear motion in 2D, where effects of viscosity and capillarity are neglected. These first-order conservation laws allow to represent phase boundaries as shock-like sharp interfaces. However, in contrast to what is known for homogeneous materials, the entropy inequality does not select a unique solution, and an additional criterion, the so-called kinetic relation, is required.Based on a scheme introduced by Hou, Rosakis and LeFloch [T. Hou, Ph. Rosakis, P.G. LeFloch, A level-set approach to the computation of twinning and phase-transition dynamics, J. Comput. Phys. 150 (1999) 302–331] we focus on the numerical solution of a specific model system. Using a level-set technique to enforce the kinetic relation on the discrete level leads to a reformulation of the original system in the form of a system of conservation laws coupled to a Hamilton–Jacobi equation for each phase boundary. The numerical method for the reformulated system is constructed for unstructured meshes (in 2D), and a self-adaptive algorithm is introduced.In the 1D-case we show that the reformulated system has a solution that corresponds to exact dynamical phase boundaries of the elasticity system which obey the kinetic relation. To validate the method in 2D, we present computations on the interaction of a plane wave with a phase boundary. The efficiency of the adaptation mechanism is demonstrated by an example showing the development of microstructures by twinning.     

一类四元数矩阵三次方程的可解性(英文)

确立了某类分块矩阵[M（₁1） M₁₂ XM₂₁ Y M₂₃Z M₃₂ M₃₃]的最大秩公式,其中,X,Y和Z是三个受限于四元数线性矩阵方程A₁X=C₁,XB₁=C₂,A₂Y=D₁,YB₂=D₂,A₃Z=E₁,ZB₃=E₂的变量矩阵.作为该公式的一项应用,我们推导出上述矩阵方程解集等同于某类四元数三次矩阵方程组A₁X=C₁,XB₁=C₂,A₂Y=D₁,YB₂=D₂,A₃Z=E₁,ZB₃=E₂,XYZ=J解集的条件.     

Analysis of nonlinear systems arise in thermoelasticity using fractional natural decomposition scheme

The pivotal aim of the present study is to employ fractional natural decomposition method (FNDM) to find the solution for a nonlinear system arising in thermoelasticity. The considered coupled system is generalised many physical phenomena associated with the material with elastic characters and its temperature and also which is called a Cauchy problem. We consider the coupled system by incorporating the Caputo fractional operator and investigate three distinct cases for different initial values to illustrate the applicability and efficiency of the FNDM. With respect to fractional order, we capture the behaviour of the achieved solution cited in three different cases and exemplified with the aid of 2D and 3D plots for the particular value of the parameters in the model. Moreover, some interesting behaviours of the projected model are confirms the prominence of the employed fractional operator while analysing the nonlinear coupled equations exemplifying real-world problems and also shows the capability of the considered algorithm.     

A novel 2D partial unwinding adaptive Fourier decomposition method with application to frequency domain system identification

This paper proposes a two‐dimensional (2D) partial unwinding adaptive Fourier decomposition method to identify 2D system functions. Starting from Coifman in 2000, one‐dimensional (1D) unwinding adaptive Fourier decomposition and later a type called unwinding AFD have been being studied. They are based on the Nevanlinna factorization and a maximal selection. This method provides fast‐converging rational approximations to 1D system functions. However, in the 2D case, there is no genuine unwinding decomposition. This paper proposes a 2D partial unwinding adaptive Fourier decomposition algorithm that is based on algebraic transforms reducing a 2D case to the 1D case. The proposed algorithm enables rational approximations of real coefficients to 2D system functions of real coefficients. Its fast convergence offers efficient system identification. Numerical experiments are provided, and the advantages of the proposed method are demonstrated.     

Comparison of hyperelastic 2‐D and 3‐D model foams at large deformations

The influence of the modeling dimension on the determination of effective properties for hyperelastic foams is investigated by means of regular 2‐D and 3‐D model foams. For calculating the effective stress‐strain relationships of both microstructures, a strain energy based homogenization procedure is employed. The results from numerical analyses show that with a 2‐D model foam the basic deformation mechanisms of the 3‐D model can be captured. Nevertheless, due to the distinct quantitative deviations found from the homogenization analyses, 3‐D modeling approaches should be used if quantitative predictions for the effective material properties are required. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

狭窄流域内气泡破裂现象数值模拟

基于边界积分法建立狭窄流域内气泡破裂数值模型,开发相应的计算程序,分别模拟对称破裂与非对称破裂两类典型工况并与相应实验结果进行对比,计算值与实验值吻合很好,表明三维数值模型的有效性．从狭窄流域内气泡运动的基本现象入手,基于开发的程序系统地研究气泡的对称破裂与非对称破裂,在已有数值研究成果和实验数据基础上,提出气泡破裂的可行性准则,研究分裂后子气泡的动力学特性,并分析距离参数对气泡破裂特性及子气泡动力学行为的影响,总结相应规律,旨在为相关气泡破裂特性研究提供参考．     

The ODEs forms of linearly elastic shell model and its numerical computing

In this paper, we successfully use 1D model to approximate the 3D problems. Firstly the PDEs (Partial Differential Equations) forms of Koiter’s Model for 2D linear elastic shell is proposed on special curvature coordinate system, i.e., spherical-coordinate system. Then the ODEs (Ordinary Differential Equations) forms of Koiter’s Model for 2D linear elastic shell is proposed under the assumption that the shell is axis-symmetric. Finally, we do numerical experiments to verify validity and accuracy of 1D models.     

Analysis of finite element approximations of a phase field model for two-phase fluids

This paper studies a phase field model for the mixture of two immiscible and incompressible fluids. The model is described by a nonlinear parabolic system consisting of the nonstationary Stokes equations coupled with the Allen-Cahn equation through an extra phase induced stress term in the Stokes equations and a fluid induced transport term in the Allen-Cahn equation. Both semi-discrete and fully discrete finite element methods are developed for approximating the parabolic system. It is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model. Error estimates are derived for the semi-discrete method, and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law. Numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach.

     

Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

We consider numerical approximations for a modified phase field crystal model with a strong nonlinear vacancy potential. Based on the invariant energy quadratization approach and stabilized strategies, we develop linear, unconditionally energy stable numerical schemes using the first-order Euler method, the second-order backward differentiation formulas and the second-order Crank–Nicolson method, respectively. We rigorously prove the unconditional energy stability, the mass conservation of these three numerical schemes and carry out error estimates in time for the first-order numerical scheme. Various numerical experiments in 2D and 3D are carried out to validate the accuracy, energy stability, mass conservation, and efficiency of the proposed schemes.     

Integrated hydrodynamic model for simulation of river-lake-sluice interactions

In this paper, an integrated model for simulating the hydrodynamic process of river-lake-sluice (RLS) systems is presented. It includes a novel one-dimensional (1D) and two-dimensional (2D) coupling method called the coupling-zone iteration-correction (CZIC) method, and an improved numerical algorithm for the sluice problem. The 1D river-network model and the 2D lake model are coupled by establishing a coupling region, and iterative correction is carried out to ensure the accurate transfer of hydraulic parameters. The convergence conditions of the CZIC method are discussed theoretically, and the proper spatial step of the coupling zone is adopted according to different inflow conditions to ensure stable computation. In order to deal with the transition of flow regimes during the gate operation, a method for calculating the discharge capacity is presented. In addition, a general difference coefficient of the river reach is deduced for hydrodynamic calculation with sluices included. Simulations on open channels demonstrate that (1) simulated values of the CZIC method are consistent with the results of the full 2D model; (2) the sluice solving algorithm can stably handle the flow transition between the orifice flow and weir flow. Furthermore, the developed integrated model is applied to the middle and lower reaches of the Huaihe River, including the Hongze Lake and fifteen sluices. Numerical simulation results reproduced the hydrodynamic process during the flood season of 2007 accurately and efficiently. The errors of the present model are also compared with that of the MIKE model, and the results show that the proposed methods perform better than MIKE, especially in rising and flood periods. Therefore, it seems likely that the developed integrated model will work well in hydrodynamic modelling of large-scale complex RLS systems.     

A compact local one‐dimensional scheme for solving a 3D N‐carrier system with Neumann boundary conditions

We consider a mathematical model for thermal analysis in a 3D N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On the pattern of space division by territories

Summary In this paper the tesselation of territories is discussed. When a mass of animals with teritoriality is introduced at random simultaneously into a finite two-dimensional (2D) region, territories are gradually formed and finally settle to a steady state. A model calculation for this process is carried out, where a Voronoi polygon is assumed as a territory. Comparisons are made between the model calculation and Barlow's observation on an artificial population of mouthbreeder fish. Differences between 1D and 2D case are also discussed. The Institute of Statistical Mathematics