Search results

The purpose of this paper is to develop and apply a high-order numerical method based on finite volume approximation for quadratic integrate-and-fire (QIF) neuron model with the help of population density approach.

The authors apply the population density approach for the QIF neuron model to derive the governing equation. The resulting mathematical model cannot be solved with existing analytical or numerical techniques owing to the presence of delay and advance. The numerical scheme is based along the lines of approximation: spatial discretization is performed by weighted essentially non-oscillatory (WENO) finite volume approximation (FVM) and temporal discretization are performed by strong stability-preserving explicit Runge–Kutta (SSPERK) method. Compared with existing schemes of orders 2 and 3 from the literature, the proposed scheme is found to be more efficient and it produces accurate solutions with few grid cells. In addition to this, discontinuity is added in the application of the model equation to illustrate the high performance of the proposed scheme.

The developed scheme works nicely for the simulation of the resulting model equation. The authors discussed the role of inhibitory and excitatory parts in variation of neuronal firing. The validation of the designed scheme is measured by its comparison with existing schemes in the literature. The efficiency of the designed scheme is demonstrated via numerical simulations.

It is expected that the present study will be a useful tool to tackle the complex neuron model and related studies.

The novel aspect of this paper is the application of the numerical methods to study the modified version of leaky integrate-and-fire neuron based on a QIF neuron. The model of the current study is inspired from the base model given in Stein (1965) and modified version in Kadalbajoo and Sharma (2005) and Wang and Zhang (2014). The applicability was confirmed by taking some numerical examples.

The purpose of this paper is to present two efficient immersed boundary methods (IBM) for simulation of thermal flow problems. One method is for given temperature condition (Dirichlet type), while the other is for given heat flux condition (Neumann type). The methods are applied to simulate natural and mixed convection problems to check their performance. The comparison of present results with available data in the literature shows that the present methods can obtain accurate numerical results efficiently.

The paper presents two efficient IBM solvers, in which the effect of thermal boundary to its surrounding fluid is considered through the introduction of a heat source/sink term into the energy equation. One is the temperature correction‐based IBM developed for problems with given temperature on the wall. The other is heat flux correction‐based IBM for problems with given heat flux on the wall. Note that in this solver, the offset of derivative condition is directly used to correct the temperature field.

As compared with existing solvers, the temperature correction‐based IBM determines the heat source/sink implicitly instead of pre‐calculated explicitly, so that the boundary condition for temperature is accurately satisfied. To the best of the authors' knowledge, the work of heat flux correction‐based IBM is the first endeavour for application of IBM to solve thermal flow problems with Neumann (heat flux) boundary condition. It was found that both methods presented in this work can efficiently obtain accurate numerical results for thermal flow problems.

The two methods presented in this paper are novel. They can effectively solve thermal flow problems with Dirichlet and Neumann boundary conditions.

The multi‐domain generalized differential quadrature method is applied in this paper to simulate the flows in Czochralski crystal growth. The effect of interface treatment on the numerical solution is studied through four types of interface approximations. The performance of those four interface approximations is validated by a benchmark problem suggested by Wheeler. It is demonstrated in this study that the multi‐domain GDQ approach is an efficient method which can obtain accurate numerical solutions by using very few grid points, and the overlapped interface approximation provides the most accurate numerical results.

A numerical model is presented in this paper to better describe the cavitated fluid flow phenomena in liquid‐lubricated Asymmetrical Herringbone Grooved Journal Bearings (HGJBs). An effective “follow the groove” grid transformation method is used in the present study to capture all the groove boundaries. A singularity at the groove edges is avoided with this approach. Symmetrical groove patterns as well as asymmetrical groove patterns can be accurately computed with the proposed method. The difficult problem of abrupt changes of oil film thickness in the liquid‐lubricated HGJB physical domain is modeled here through a series expansion approach. Results are comparable with available experimental and known numerical data from other investigators. Cavitation footprints, pressure distributions and their corresponding load characteristics are presented in this study. Effects of the critical transitional flow phenomena on the performance of the asymmetrical HGJBs are also determined through the present study.

The purpose of the current paper is to develop a numerical methodology, based on the immersed boundary-lattice Boltzmann computational framework, for the Neumann and Dirichlet boundary conditions in problems involving natural and forced convection heat transfer.

The direct forcing immersed boundary method is extended to study the heat transfer by incompressible flow within the thermal lattice Boltzmann method (LBM) computational framework. The direct forcing and heating immersed boundary-LBM introduces a heat source term to the thermal LBM to account for the heat transfer occurring at the immersed boundary. New numerical treatments for the Neumann type of boundary condition and for the calculation of the local Nusselt number are developed. The developed methodologies have been applied to flows around immersed bodies with natural and forced convection, including steady as well as unsteady flows.

Numerical experiments involving immersed bodies in natural and forced convection have been performed in order to assess the validity of the direct heating IB-LBM. The flow cases studied also include steady and transient flow phenomena. Flow velocity field and isotherms have been used for qualitative comparisons with existing, published results. The surface averaged Nusselt number, Strouhal number, and lift coefficient (for the unsteady flow cases) have been used for quantitative comparison with published results. The results show that there are satisfactory agreements, qualitatively and quantitatively, between the results obtained by using the present method and those previously published.

Limited application of immersed boundary to thermal flows within the LBM has been studied by researchers; the few past studies were limited to Dirichlet boundary conditions and/or using of feedback forcing and heating approaches. In the current paper, the direct forcing and heating approach was used which helps to eliminate the arbitrary constants used in the feedback approaches. The developed new numerical treatments for the Neumann type of boundary condition and for the calculation of the local Nusselt number eliminate the need to determine surface normal and temperature gradient in the normal direction for heat transfer calculation, which is particularly beneficial in cases with deforming or changing boundaries.

The current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it.

At first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed.

The results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach.

The local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.

This paper aims to examine how using lattice Boltzmann method (LBM) aids the study of the isothermal‐gas flow with slight rarefaction in long microtubes.

A revised axisymmetric lattice Boltzmann model is proposed to simulate the flow in microtubes. The wall boundary condition combining the bounce‐back and specular‐reflection schemes is used to capture the slip velocity on the wall. Appropriate relation between the Knudsen number and relax‐time constant is defined.

The computed‐slip velocity, average velocity and non‐linear pressure distribution along the microtube are in excellent agreement with analytical solution of the weakly compressible Navier‐Stokes equations. The calculated‐friction factors are also consistent with available experimental data. For simulations of slip flow in microtube, LBM is more accurate and efficient than DSMC method.

The laminar flow in circular microtube is assumed to be axisymmetric. The present LBM is only applied to the simulation of slip flows (0.01 < Kn₀<0.1) in microtube.

Lattice‐BGK method is a very useful tool to investigate the micro slip flows.

A revised axisymmetric D2Q9 lattice Boltzmann model is proposed to simulate the slip flow in axisymmetric microtubes.

The aim of the present study is to evaluate the accuracy and efficiency of Lattice‐BGK (LBGK) method application in simulation of the 3D flow through complex geometry. On the other hand, the steady flows through vascular tube with Reynolds number 10‐150 and different constriction spacing ratios are simulated.

The numerical method is based on the LGBK method with an incompressible D3Q19 model. To treat the curved boundary, the “bounce back” scheme combined with spatial interpolation of second order is applied.

The highly axisymmetric property in the direct 3D tube flow simulation is observed. Solutions obtained from LBGK method are quite consistent with that of finite volume method (FVM). The overall order of accuracy of these LBGK solutions is about 1.89. The LBGK incompressible D3Q19 model with the curved boundary treatment can handle the problems of 3D steady flow through complex geometry.

Investigating the flow in constricted vascular tubes with different stenose shape and higher Reynolds number is left for future work.

Lattice BGK method is the very useful tool to investigate the steady vascular flow.

Applying LBGK method with incompressible D3Q19 model to simulate the steady flow through complex geometry. The accuracy and efficiency of the present LBGK solver are examined.

The current work aims to present a parallel code using the open multi-processing (OpenMP) programming model for an adaptive multi-resolution high-order finite difference scheme for solving 2D conservation laws, comparing efficiencies obtained with a previous message passing interface formulation for the same serial scheme and considering the same type of 2D formulations laws.

The serial version of the code is naturally suitable for parallelization because the spatial operator formulation is based on a splitting scheme per direction for which the flux components are numerically computed by a Lax–Friedrichs factorization independently for each row or column. High-order approximations for numerical fluxes are computed by the third-order essentially non-oscillatory (ENO) and fifth-order weighted essentially non-oscillatory (WENO) interpolation schemes, assuming sparse grids in each direction. The grid adaptivity is obtained by a cubic interpolating wavelet transform applied in each space dimension, associated to a threshold operator. Time is evolved by a third order TVD Runge–Kutta method.

The parallel formulation is implemented automatically at compiling time by the OpenMP library routines, being virtually transparent to the programmer. This over simplifies any concerns about managing and/or updating the adaptive grid when compared to what is necessary to be done when other parallel approaches are considered. Numerical simulations results and the large speedups obtained for the Euler equations in gas dynamics highlight the efficiency of the OpenMP approach.

The resulting speedups reflect the effectiveness of the OpenMP approach but are, to a large extension, limited by the hardware used (2 E5-2620 Intel Xeon processors, 6 cores, 2 threads/core, hyper-threading enabled). As the demand for OpenMP threads increases, the code starts to make explicit use of the second logical thread available in each E5-2620 processor core and efficiency drops. The speedup peak is reached near the possible maximum (24) at about 22, 23 threads. This peak reflects the hardware configuration and the true software limit should be located way beyond this value.

So far no attempts have been made to parallelize other possible code segments (for instance, the ENO|-WENO-TVD code lines that process the different data components which could potentially push the speed up limit to higher values even further. The fact that the speedup peak is located close to the present hardware limit reflects the scalability properties of the OpenMP programming and of the splitting scheme as well. Consequently, it is likely that the speedup peak with the OpenMP approach for this kind of problem formulation will be close to the physical (and/or logical) limit of the hardware used.

This work is the result of a successful collaboration among researchers from two different institutions, one internationally well-known and with a long-term experience in applied mathematics for industrial applications and the other in a starting process of international academic insertion. In this way, this scientific partnership has the potential of promoting further knowledge exchange, involving students and other collaborators.

The proposed methodology (use of OpenMP programming model for the wavelet adaptive splitting scheme) is original and contributes to a very active research area in the past years, namely, adaptive methods for conservation laws and their parallel formulations, which is of great interest for the entire scientific community.

The purpose of this paper is to propose a new strong-form numerical method, called the free element method, for solving general boundary value problems governed by partial differential equations. The main idea of the method is to use a locally formed element for each point to set up the system of equations. The proposed method is used to solve the fluid mechanics problems.

The proposed free element method adopts the isoparametric elements as used in the finite element method (FEM) to represent the variation of coordinates and physical variables and collocates equations node-by-node based on the newly derived element differential formulations by the authors. The distinct feature of the method is that only one independently formed individual element is used at each point. The final system of equations is directly formed by collocating the governing equations at internal points and the boundary conditions at boundary points. The method can effectively capture phenomena of sharply jumped variables and discontinuities (e.g. the shock waves).

a) A new numerical method called the FEM is proposed; b) the proposed method is used to solve the compressible fluid mechanics problems for the first time, in which the shock wave can be naturally captured; and c) the method can directly set up the system of equations from the governing equations.

This paper presents a completely new numerical method for solving compressible fluid mechanics problems, which has not been submitted anywhere else for publication.