This paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size H and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size h. The error estimates derived show that if we choose h = O (|logh|1/2H3), then the two-level method we provide has the same H1 and L2 convergence orders of the velocity and the pressure as the one-level stabilized method. However, the L2 convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.

In this paper, we consider a fully discrete local discontinuous Galerkin (LDG) finite element method for a time-fractional Korteweg-de Vries (KdV) equation. The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space. We show that our scheme is unconditionally stable and convergent through analysis. Numerical examples are shown to illustrate the efficiency and accuracy of our scheme.

The equation of state (EOS) plays a crucial role in hyperbolic conservation laws for the compressible fluid. Whereas, the solid constitutive model with elastic-plastic phase transition makes the analysis of the solid Riemann problem more difficult. In this paper, one-dimensional elastic-perfectly plastic solid Riemann problem is investigated and its exact Riemann solver is proposed. Different from previous works treating the elastic and plastic phases integrally, we resolve the elastic wave and plastic wave separately to understand the complicate nonlinear waves within the solid and then assemble them together to construct the exact Riemann solver for the elastic-perfectly plastic solid. After that, the exact solid Riemann solver is associated with the fluid Riemann solver to decouple the fluid-solid multi-material interaction. Numerical tests, including gas-solid, water-solid high-speed impact problems are simulated by utilizing the modified ghost fluid method (MGFM).

The lattice Boltzmann equation (LBE) is considered as a promising approach for simulating flows of liquid and gas. Most of LBE studies have been devoted to regular square LBE and few works have focused on the rectangular LBE in the simulation of gaseous microscale flows. In fact, the rectangular LBE, as an alternative and efficient method, has some advantages over the square LBE in simulating flows with certain computational domains of large aspect ratio (e.g., long micro channels). Therefore, in this paper we expand the application scopes of the rectangular LBE to gaseous microscale flow. The kinetic boundary conditions for the rectangular LBE with a multiple-relaxation-time (MRT) collision operator, i.e., the combined bounce-back/specular-reflection (CBBSR) boundary condition and the discrete Maxwell's diffuse-reflection (DMDR) boundary condition, are studied in detail. We observe some discrete effects in both the CBBSR and DMDR boundary conditions for the rectangular LBE and present a reasonable approach to overcome these discrete effects in the two boundary conditions. It is found that the DMDR boundary condition for the square MRT-LBE can not realize the real fully diffusive boundary condition, while the DMDR boundary condition for the rectangular MRT-LBE with the grid aspect ratio a≠1 can do it well. Some numerical tests are implemented to validate the presented theoretical analysis. In addition, the computational efficiency and relative difference between the rectangular LBE and the square LBE are analyzed in detail. The rectangular LBE is found to be an efficient method for simulating the gaseous microscale flows in domains with large aspect ratios.

The accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in one space dimension. The governing equations consist of two components, namely a hydraulic part described by the shallow water equations and a sediment part described by the Exner equation. Based on different formulations of the morphodynamic equations, we propose a family of three finite volume methods. The numerical fluxes are reconstructed using a modified Roe’s scheme that incorporates, in its reconstruction, the sign of the Jacobian matrix in the morphodynamic system. A well-balanced discretization is used for the treatment of the source terms. The method is well-balanced, non-oscillatory and suitable for both slow and rapid interactions between hydraulic flow and sediment transport. The obtained results for several morphodynamic problems are considered to be representative, and might be helpful for a fair rating of finite volume solution schemes, particularly in long time computations.

In this paper, we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional. We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.

One of the useful methods for offshore oil exploration in the deep regions is the use of tension leg platforms (TLP). The effective mass fluctuating of the structure which due to its vibration can be noted as one of the important issues about these platforms. With this description, dynamic analysis of these structures will play a significant role in their design. Differential equations of motion of such systems are nonlinear and providing a useful method for its analysis is very important. Also, the amount of added mass coefficient has a direct effect on the level of nonlinearity of partial differential equation of these systems. In this study, Homotopy analysis method has been used for closed form solution of the governing differential equation. Linear springs have been used for modeling the stiffness of this system and the effects of torsion, bending and damping of water have been ignored. In the study of obtained results, the effect of added mass coefficient has been investigated. The results show that increasing of this coefficient decreases the bottom amplitude of fluctuations and the system frequency. The obtained results from this method are in good agreement with the published results on the valid articles.

An implicit discontinuous Galerkin method is introduced to solve the time-domain Maxwell’s equations in metamaterials. The Maxwell’s equations in metamaterials are represented by integral-differential equations. Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain. The fully discrete numerical scheme is proved to be unconditionally stable. When polynomial of degree at most p is used for spatial approximation, our scheme is verified to converge at a rate of O(τ2+hp+1/2). Numerical results in both 2D and 3D are provided to validate our theoretical prediction.

The paper presents the size-dependant behaviors of the carbon nanotubes under electrostatic actuation using the modified couple stress theory and homotopy perturbation method. Due to the less accuracy of the classical elasticity theorems, the modified couple stress theory is applied in order to capture the size-dependant properties of the carbon nanotubes. Both of the static and dynamic behaviors under static DC and step DC voltages are discussed. The effects of various dimensions and boundary conditions on the deflection and pull-in voltages of the carbon nanotubes are to be investigated in detail via application of the homotopy perturbation method to solve the nonlinear governing equations semi-analytically.

In this paper, we introduce the Hamiltonian boundary value method (HBVM) to solve nonlinear Hamiltonian PDEs. We use the idea of Fourier pseudospectral method in spatial direction, which leads to the finite-dimensional Hamiltonian system. The HBVM, which can preserve the Hamiltonian effectively, is applied in time direction. Then the nonlinear Schrödinger (NLS) equation and the Korteweg-de Vries (KdV) equation are taken as examples to show the validity of the proposed method. Numerical results confirm that the proposed method can simulate the propagation and collision of different solitons well. Meanwhile the corresponding errors in Hamiltonian and other intrinsic invariants are presented to show the good preservation property of the proposed method during long-time numerical calculation.

The motion and rotation of an ellipsoidal particle inside square tubes and rectangular tubes with the confinement ratio R/a∈(1.0,4.0) are studied by the lattice Boltzmann method (LBM), where R and a are the radius of the tube and the semi-major axis length of the ellipsoid, respectively. The Reynolds numbers (Re) up to 50 are considered. For the prolate ellipsoid inside square and rectangular tubes, three typical stable motion modes which depend on R/a are identified, namely, the kayaking mode, the tumbling mode, and the log-rolling mode are identified for the prolate spheroid. The diagonal plane strongly attracts the particle in square tubes with 1.2≤R/a<3.0. To explore the mechanism, some constrained cases are simulated. It is found that the tumbling mode in the diagonal plane is stable because the fluid force acting on the particle tends to diminish the small displacement and will bring it back to the plane. Inside rectangular tubes the particle will migrate to a middle plane between short walls instead of the diagonal plane. Through the comparisons between the initial unstable equilibrium motion state and terminal stable mode, it is seems that the particle tend to adopt the mode with smaller kinetic energy.

In present work, we investigate numerical simulation of steady natural convection flow in the presence of weak magnetic Prandtl number and strong magnetic field by involving algebraic decay in mainstream velocity. Before passing to the numerical simulation, we formulate the set of boundary layer equations with the inclusion of the effects of algebraic decay velocity, aligned magnetic field and buoyant body force in the momentum equation. Later, finite difference method with primitive variable formulation is employed in the physical domain to compute the numerical solutions of the flow field. Graphical results for the velocity, temperature and transverse component of magnetic field as well as surface friction, rate of heat transfer and current density are presented and discussed. It is pertinent to mention that the simulation is performed for different values of algebraic decay parameter α, Prandtl number Pr, magnetic Prandtl number Pm and magnetic force parameter S.

Effects of viscous dissipation and radiation on MHD natural convection in oblique porous cavity with constant heat flux is studied numerically in the present article. The right inclined wall is maintained at a constant cold temperature Tc and the left inclined wall has a constant heat flux q with length S, while the remainder of the left wall is adiabatic. The horizontal walls are assumed to be adiabatic. The governing equations are obtained by applying the Darcy model and Boussinesq approximations. COMSOL's finite element method is used to solve the non-dimensional governing equations together with specified boundary conditions. The governing parameters of this study are Rayleigh number (Ra=10,100,200,250,500 and 1000), Hartmann number (0≤ Ha ≤20), inclination angle of the magnetic field (0° ≤ ω ≤ π/2), Radiation (0≤R≤15), the heater flux length (0.1≤H≤1) and inclination angle of the sloping wall (–π/3≤ϕ≤π/3). The results are considered for various values of the governing parameters in terms of streamlines, isotherms and averageNusselt number. It is found that the intensity of the streamlines and the isotherm patterns decrease with an increment in Hartmann number. The overall heat transfer is significantly increased with the increment of the viscous dissipation and the radiation parameters.

Let S: [0, 1]→[0, 1] be a chaotic map and let f* be a stationary density of the Frobenius-Perron operator PS: L1→L1 associated with S. We develop a numerical algorithm for approximating f*, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method.

A high-order numerical method for three-dimensional hydrodynamics is presented. The present method applies high-order compact schemes in space and a Runge-Kutta scheme in time to solve the Reynolds-averaged Navier-Stokes equations with the k-ε turbulence model in an orthogonal curvilinear coordinate system. In addition, a two-dimensional equation is derived from the depth-averaged momentum equations to predict the water level. The proposed method is first validated by its application to simulate flow in a 180° curved laboratory flume. It is found that the simulated results agree with measurements and are better than those from SIMPLEC algorithm. Then the method is applied to study three-dimensional hydrodynamics in a natural river, and the simulated results are in accordance with measurements.

In this article, natural convection of a temperature-sensitive magnetic fluid in a porous media is studied numerically by using lattice Boltzmann method. Results show that the heat transfer decreases when the ball numbers increase. When the magnetic field is increased, the heat transfer is enhanced; however the average wall Nusselt number increases at small ball numbers but decreases at large ball numbers due to the induced flow being more likely confined near the bottom walls with a high number of obstacles.

Multi-field coupling simulation method based on the physical principles is used to simulate the discharge characteristics of nanosecond pulsed plasma synthetic jet actuator. Considering the effect of the energy transferring for air, the flow characteristics of nanosecond pulsed plasma synthetic jet actuator are simulated. The elastic heating sources and ion joule heating sources are the two main sources of energy. Through the collisions, the energy of ions is transferred to the neutral gas quickly. The flow characteristics of a series of blast waves and the synthetic jet which erupt from the plasma synthetic jet (PSJ) actuator are simulated. The blast wave not only promotes outward, but also accelerates the gas mixing the inhaled gas from the outside cavity with the residual gas inside the cavity. The performances of PSJ actuator fluctuate in the first three incentive cycles and become stable after that.

In this work, we investigate wave propagation through a zero index meta-material (ZIM) waveguide embedded with triangular dielectric defects. We provide a theoretical guidance on how to achieve total reflection and total transmission (i.e., cloaking) by adjusting the defect sizes and/or permittivities of the defects. Our work provides a systematical way in manipulating wave propagation through ZIM in addition to the widely studied dielectric defects with cylindrical and rectangular geometries.

In the present paper a combined procedure for the quasi-dimensional modelling of heat transfer, combustion and knock phenomena in a “downsized” Spark Ignition two-cylinder turbocharged engine is presented. The procedure is extended to also include the effects consequent the Cyclic Variability. Heat transfer is modelled by means of a Finite Elements model. Combustion simulation is based on a fractal description of the flame front area. Cyclic Variability (CV) is characterized through the introduction of a random variation on a number of parameters controlling the rate of heat release (air/fuel ratio, initial flame kernel duration and radius, laminar flame speed, turbulence intensity). The intensity of the random variation is specified in order to realize a Coefficient Of Variation (COV) of the Indicated Mean Effective Pressure (IMEP) similar to the one measured during an experimental campaign. Moreover, the relative importance of the various concurring effects is established on the overall COV. A kinetic scheme is then solved within the unburned gas zone, characterized by different thermodynamic conditions occurring cycle-by-cycle. In this way, an optimal choice of the “knock-limited” spark advance is effected and compared with experimental data. Finally, the CV effects on the occurrence of individual knocking cycles are assessed and discussed.

In this paper, a new formulation is proposed to evaluate the origin intensity factors (OIFs) in the singular boundary method (SBM) for solving 3D potential problems with Dirichlet boundary condition. The SBM is a strong-form boundary discretization collocation technique and is mathematically simple, easy-to-program, and free of mesh. The crucial step in the implementation of the SBM is to determine the OIFs which isolate the singularities of the fundamental solutions. Traditionally, the inverse interpolation technique (IIT) is adopted to calculate the OIFs on Dirichlet boundary, which is time consuming for large-scale simulation. In recent years, the new methodology has been developed to efficiently calculate the OIFs on Neumann boundary, but the Dirichlet problem remains an open issue. This study employs the subtracting and adding-back technique based on the integration of the fundamental solution over the whole boundary to develop a new formulation of the OIFs on 3D Dirichlet boundary. Several problems with varied domain shapes and boundary conditions are carried out to validate the effectiveness and feasibility of the proposed scheme in comparison with the SBM based on inverse interpolation technique, the method of fundamental solutions, and the boundary element method.