The numerical method for the optimal supporting position and related optimal control for the catalytic reaction system

This paper considers the numerical approximation for the optimal supporting position and related optimal control of a catalytic reaction system with some control and state constraints, which is governed by a nonlinear partial differential equations with given initial and boundary conditions. By the Galerkin finite element method, the original problem is projected into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then the control parameterization method is applied to approximate the control and reduce the original system to an optimal parameter selection problem, in which both the position and related control are taken as decision variables to be optimized. This problem can be solved as a nonlinear optimization problem by a particle swarm optimization algorithm. The numerical simulations are given to illustrate the effectiveness of the proposed numerical approximation method.     

Double Diffusive Magnetohydrodynamic (MHD) Mixed Convective Slip Flow along a Radiating Moving Vertical Flat Plate with Convective Boundary Condition

In this study combined heat and mass transfer by mixed convective flow along a moving vertical flat plate with hydrodynamic slip and thermal convective boundary condition is investigated. Using similarity variables, the governing nonlinear partial differential equations are converted into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved using a semi-numerical/analytical method called the differential transform method and results are compared with numerical results. Close agreement is found between the present method and the numerical method. Effects of the controlling parameters, including convective heat transfer, magnetic field, buoyancy ratio, hydrodynamic slip, mixed convective, Prandtl number and Schmidt number are investigated on the dimensionless velocity, temperature and concentration profiles. In addition effects of different parameters on the skin friction factor, , local Nusselt number, , and local Sherwood number are shown and explained through tables.     

A numerical study of the Belousov-Zhabotinskii reaction using Galerkin finite element methods

The Belousov-Zhabotinskii reaction has been modelled by Field and Noyes [5] as a pair of nonlinear parabolic equations. Previous studies of these, both theoretical and numerical, have assumed wave solutions travelling with constant velocity leading to a simplification of the mathematical model in the form of a system of ordinary differential equations. In the present study a finite element Galerkin method is used directly on the original parabolic system for a range of parameter values.     

A multiscale approach for modelling wave propagation in an arterial segment

A mathematical model of blood flow through an arterial vessel is presented and the wave propagation in it is studied numerically. Based on the assumption of long wavelength and small amplitude of the pressure waves, a quasi-one-dimensional (1D) differential model is adopted. It describes the non-linear fluid-wall interaction and includes wall deformation in both radial and axial directions. The 1D model is coupled with a six compartment lumped parameter model, which accounts for the global circulatory features and provides boundary conditions. The differential equations are first linearized to investigate the nature of the propagation phenomena. The full non-linear equations are then approximated with a numerical finite difference method on a staggered grid. Some numerical simulations show the characteristics of the wave propagation. The dependence of the flow, of the wall deformation and of the wave velocity on the elasticity parameter has been highlighted. The importance of the axial deformation is evidenced by its variation in correspondence of the pressure peaks. The wave disturbances consequent to a local stiffening of the vessel and to a compliance jump due to prosthetic implantations are finally studied.     

A review on spatial aggregation methods involving several time scales

This article is a review of spatial aggregation of variables for time continuous models. Two cases are considered. The first case corresponds to a discrete space, i.e. a set of discrete patches connected by migrations, which are assumed to be fast with respect to local interactions. The mathematical model is a set of coupled ordinary differential equations (O.D.E.). The spatial aggregation allows one to derive a global model governing the time variation of the total numbers of individuals of all patches in the long term. The second case considers a continuous space and is a set of partial differential equations (P.D.E.). In that case, we also assume that diffusion is fast in comparison with local interactions. The spatial aggregation allows us again to obtain an O.D.E. governing the total population density, which is obtained by integration all over the spatial domain, at the slow time scale. These aggregations of variables are based on time scales separation methods which have been presented largely elsewhere and we recall the main results. We illustrate the methods by examples in population dynamics and prey–predator models.     

脉冲-扩散周期竞争系统解的渐近行为

研究了在周期变化环境中具有扩散及种群密度可能发生突变的两竞争种群动力系统的数学模型．模型由反应扩散方程组以及初边值及脉冲条件组成．文章建立了研究模型的上下解方法,获得了一些比较原理．利用脉冲常微分方程的比较定理以及利用相应的脉冲常微分方程的解控制和估计所讨论模型的解,研究了系统模型的解的渐近性质．     

A finite element beam-model for efficient simulation of large-scale porous structures

This paper presents a new method for the generation of a beam finite element (FE) model from a three-dimensional (3D) data set acquired by micro-computed tomography (micro-CT). This method differs from classical modeling of trabecular bone because it models a specific sample only and differs from conventional solid hexahedron element-based FE approaches in its computational efficiency. The stress-strain curve, characterizing global mechanical properties of a porous structure, could be well predicted (R(2)=0.92). Furthermore, validation of the method was achieved by comparing local displacements of element nodes with the displacements directly measured by time-lapsed imaging methods of failure, and these measures were in good agreement. The presented model is a first step in modeling specific samples for efficient strength analysis by FE modeling. We believe that with upcoming high-resolution in-vivo imaging methods, this approach could lead to a novel and accurate tool in the risk assessment for osteoporotic fractures.     

Methods for parameter identification in oscillatory networks and application to cortical and thalamic 600 Hz activity.

Directed information transfer in the human brain occurs presumably by oscillations. As of yet, most approaches for the analysis of these oscillations are based on time-frequency or coherence analysis. The present work concerns the modeling of cortical 600 Hz oscillations, localized within the Brodmann Areas 3b and 1 after stimulation of the nervus medianus, by means of coupled differential equations. This approach leads to the so-called parameter identification problem, where based on a given data set, a set of unknown parameters of a system of ordinary differential equations is determined by special optimization procedures. Some suitable algorithms for this task are presented in this paper. Finally an oscillatory network model is optimally fitted to the data taken from ten volunteers.     

Numerical simulation of blood flow in the left ventricle and aortic sinus using magnetic resonance imaging and computational fluid dynamics

Understanding cardiac blood flow patterns has many applications in analysing haemodynamics and for the clinical assessment of heart function. In this study, numerical simulations of blood flow in a patient-specific anatomical model of the left ventricle (LV) and the aortic sinus are presented. The realistic 3D geometry of both LV and aortic sinus is extracted from the processing of magnetic resonance imaging (MRI). Furthermore, motion of inner walls of LV and aortic sinus is obtained from cine-MR image analysis and is used as a constraint to a numerical computational fluid dynamics (CFD) model based on the moving boundary approach. Arbitrary Lagrangian–Eulerian finite element method formulation is used for the numerical solution of the transient dynamic equations of the fluid domain. Simulation results include detailed flow characteristics such as velocity, pressure and wall shear stress for the whole domain. The aortic outflow is compared with data obtained by phase-contrast MRI. Good agreement was found between simulation results and these measurements.     

Continuum simulations of acetylcholine diffusion with reaction-determined boundaries in neuromuscular junction models

The reaction-diffusion system of the neuromuscular junction has been modeled in 3D using the finite element package FEtk. The numerical solution of the dynamics of acetylcholine with the detailed reaction processes of acetylcholinesterases and nicotinic acetylcholine receptors has been discussed with the reaction-determined boundary conditions. The simulation results describe the detailed acetylcholine hydrolysis process, and reveal the time-dependent interconversion of the closed and open states of the acetylcholine receptors as well as the percentages of unliganded/monoliganded/diliganded states during the neuro-transmission. The finite element method has demonstrated its flexibility and robustness in modeling large biological systems.     

A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease

We have previously described a new approach to planning treatments for cardiovascular disease, Simulation-Based Medical Planning, whereby a physician utilizes computational tools to construct and evaluate a combined anatomic/physiologic model to predict the outcome of alternative treatment plans for an individual patient. Current systems for Simulation-Based Medical Planning utilize finite element methods to solve the time-dependent, three-dimensional equations governing blood flow and provide detailed data on blood flow distribution, pressure gradients and locations of flow recirculation, low wall shear stress and high particle residence. However, these methods are computationally expensive and often require hours of time on parallel computers. This level of computation is necessary for obtaining detailed information about blood flow, but likely is unnecessary for obtaining information about mean flow rates and pressure losses. We describe, herein, a space-time finite element method for solving the one-dimensional equations of blood flow. This method is applied to compute flow rate and pressure in a single segment model, a bifurcation, an idealized model of the abdominal aorta, in three alternate treatment plans for a case of aorto-iliac occlusive disease and in a vascular bypass graft. All of these solutions were obtained in less than 5 min of computation time on a personal computer.     

头部组织三维核磁共振电阻抗成像算法的仿真研究

文章给出了一种基于核磁共振技术的三维阻抗成像（电导率分布）重构算法,并将该方法应用于人体头部组织电导率分布重构上。该代数重构方法是利用高分辨率的核磁共振成像系统对成像物体进行三维构建和不同组织的边界区分,根据核磁共振系统中测量得到的磁感应强度Bx和By分量并结合有限元数值计算得到的电流密度分布J组成非线性矩阵,通过迭代求解此非线性矩阵,来解决三维电导率分布的重构问题。在三层球头模型（包括头皮、颅骨和大脑）上分别进行的仿真实验结果表明,该算法具有较强的抗噪声能力和较好的收敛性,重构的头部电导率分布图像具有较高的精确性。     

Modelling the effect of temperature on the range expansion of species by reaction-diffusion equations

The spatial dynamics of range expansion is studied in dependence of temperature. The main elements population dynamics, competition and dispersal are combined in a coherent approach based on a system of coupled partial differential equations of the reaction-diffusion type. The nonlinear reaction terms comprise population dynamic models with temperature dependent reproduction rates subject to an Allee effect and mutual competition. The effect of temperature on travelling wave solutions is investigated for a one dimensional model version. One main result is the importance of the Allee effect for the crossing of regions with unsuitable habitats. The nonlinearities of the interaction terms give rise to a richness of spatio-temporal dynamic patterns. In two dimensions, the resulting non-linear initial boundary value problems are solved over geometries of heterogeneous landscapes. Geo referenced model parameters such as mean temperature and elevation are imported into the finite element tool COMSOL Multiphysics from a geographical information system. The model is applied to the range expansion of species at the scale of middle Europe.     

Computational models to predict stenosis growth in carotid arteries: which is the role of boundary conditions?

This work addresses the problem of prescribing proper boundary conditions at the artificial boundaries that separate the vascular district from the remaining part of the circulatory system. A multiscale (MS) approach is used where the Navier-Stokes equations for the district of interest are coupled to a non-linear system of ordinary differential equations which describe the circulatory system. This technique is applied to three 3D models of a carotid bifurcation with increasing stenosis resembling three phases of a plaque growth. The results of the MS simulations are compared to those obtained by two stand-alone models. The MS shows a great flexibility in numerically predicting the haemodynamic changes due to the presence of a stenosis. Nonetheless, the results are not significantly different from a stand-alone approach where flows derived by the MS without stenosis are imposed. This is a consequence of the dominant role played by the outside districts with respect to the stenosis resistance.     

CFD simulation of flow through heart: a perspective review

This work addresses the problem of prescribing proper boundary conditions at the artificial boundaries that separate the vascular district from the remaining part of the circulatory system. A multiscale (MS) approach is used where the Navier–Stokes equations for the district of interest are coupled to a non-linear system of ordinary differential equations which describe the circulatory system. This technique is applied to three 3D models of a carotid bifurcation with increasing stenosis resembling three phases of a plaque growth. The results of the MS simulations are compared to those obtained by two stand-alone models. The MS shows a great flexibility in numerically predicting the haemodynamic changes due to the presence of a stenosis. Nonetheless, the results are not significantly different from a stand-alone approach where flows derived by the MS without stenosis are imposed. This is a consequence of the dominant role played by the outside districts with respect to the stenosis resistance.     

Numerical methods and parameter estimation of a structured population model with discrete events in the life history

We consider two numerical methods for the solution of a physiologically structured population (PSP) model with multiple life stages and discrete event reproduction. The model describes the dynamic behaviour of a predator-prey system consisting of rotifers predating on algae. The nitrate limited algal prey population is modelled unstructured and described by an ordinary differential equation (ODE). The formulation of the rotifer dynamics is based on a simple physiological model for their two life stages, the egg and the adult stage. An egg is produced when an energy buffer reaches a threshold value. The governing equations are coupled partial differential equations (PDE) with initial and boundary conditions. The population models together with the equation for the dynamics of the nutrient result in a chemostat model. Experimental data are used to estimate the model parameters. The results obtained with the explicit finite difference (FD) technique compare well with those of the Escalator Boxcar Train (EBT) method. This justifies the use of the fast FD method for the parameter estimation, a procedure which involves repeated solution of the model equations.     

A generalized finite difference method for modeling cardiac electrical activation on arbitrary, irregular computational meshes

A generalized finite difference (GFD) method is presented that can be used to solve the bi-domain equations modeling cardiac electrical activity. Classical finite difference methods have been applied by many researchers to the bi-domain equations. However, these methods suffer from the limitation of requiring computational meshes that are structured and orthogonal. Finite element or finite volume methods enable the bi-domain equations to be solved on unstructured meshes, although implementations of such methods do not always cater for meshes with varying element topology. The GFD method solves the bi-domain equations on arbitrary and irregular computational meshes without any need to specify element basis functions. The method is useful as it can be easily applied to activation problems using existing meshes that have originally been created for use by finite element or finite difference methods. In addition, the GFD method employs an innovative approach to enforcing nodal and non-nodal boundary conditions. The GFD method performs effectively for a range of two and three-dimensional test problems and when computing bi-domain electrical activation moving through a fully anisotropic three-dimensional model of canine ventricles.     

Systematization of a set of closure techniques

Approximations in population dynamics are gaining popularity since stochastic models in large populations are time consuming even on a computer. Stochastic modeling causes an infinite set of ordinary differential equations for the moments. Closure models are useful since they recast this infinite set into a finite set of ordinary differential equations. This paper systematizes a set of closure approximations. We develop a system, which we call a power p closure of n moments, where 0≤p≤n. [Keeling, 2000a] and [Keeling, 2000b] approximation with third order moments is shown to be an instantiation of this system which we call a power 3 closure of 3 moments. We present an epidemiological example and evaluate the system for third and fourth moments compared with Monte Carlo simulations.