As a continuation of developing multiscale method for the transport phenomena, a unified gas kinetic scheme (UGKS) for multi-scale and multi-component plasma simulation is constructed. The current scheme is a direct modeling method, where the time evolution solutions from the Vlasov-BGK equations of electron and ion and the Maxwell equations are used to construct a scale-dependent plasma simulation model. The modeling scale used in the UGKS is the mesh size scale, which can be comparable to or much larger than the local mean free path. As a result, with the variation of modeling scales in space and time through the so-called cell's Knudsen number and normalized Larmor radius, the discretized governing equations can recover a wide range of plasma evolution from the Vlasov equation in the kinetic scale to different-type of magnetohydrodynamic (MHD) equations in the hydrodynamic scale. The UGKS provides a general evolution model, which goes to the Vlasov equation in the kinetic scale and many types of MHD equations in the hydrodynamic scale, such as the two fluids model, the Hall, the resistive, and the ideal MHD equations. All current existing governing equations become the subsets of the UGKS, and the UGKS bridges these distinguishable governing equations seamlessly. The construction of UGKS is based on the implementation of physical conservation laws and the un-splitting treatment of particle collision, acceleration, and transport in the construction of a scale-dependent numerical flux across a cell interface. At the same time, the discretized plasma evolution equations are coupled with the Maxwell equations for electro-magnetic fields, which also cover a scale-dependent transition between the Ampère's law and the Ohm's law for the calculation of electric field. The time step of UGKS is not limited by the relaxation time, the cyclotron period, and the speed of light in the ideal-MHD regime. Our scheme is able to give a physically accurate solution for plasma simulation with a wide range of Knudsen number and normalized Larmor radius. It can be used to study the phenomena from the Vlasov limit to the scale of plasma skin depth for the capturing of two-fluid effect, and the phenomena in the plasma transition regime with a modest Knudsen number and Larmor radius. The UGKS is validated by numerical test cases, such as the Landau damping and two stream instability in the kinetic regime, and the Brio-Wu shock tube problem, and the Orszag-Tang MHD turbulence problem in the hydrodynamic regime. The scheme is also used to study the geospace environment modeling (GEM), such as the challenging magnetic reconnection problem in the transition regime. At the same time, the magnetic reconnection mechanism of the Sweet-Parker model and the Hall effect model can be connected smoothly through the variation of Larmor radius in the UGKS simulations. Overall, the UGKS is a physically reliable multi-scale plasma simulation method, and it provides a powerful and unified approach for the study of plasma physics.

For 2D elastic-plastic flows with the hypo-elastic constitutive model and von Mises' yielding condition, the non-conservative character of the hypo-elastic constitutive model and the von Mises' yielding condition make the construction of the solution to the Riemann problem a challenging task. In this paper, we first analyze the wave structure of the Riemann problem and develop accordingly a Four-Rarefaction wave approximate Riemann Solver with Elastic waves (FRRSE). In the construction of FRRSE one needs to use an iterative method. A direct iteration procedure for four variables is complex and computationally expensive. In order to simplify the solution procedure we develop an iteration based on two nested iterations upon two variables, and our iteration method is simple in implementation and efficient. Based on FRRSE as a building block, we propose a 2nd-order cell-centered Lagrangian numerical scheme. Numerical results with smooth solutions show that the scheme is of second-order accuracy. Moreover, a number of numerical experiments with shock and rarefaction waves demonstrate the scheme is essentially non-oscillatory and appears to be convergent. For shock waves the present scheme has comparable accuracy to that of the scheme developed by Maire et al., while it is more accurate in resolving rarefaction waves.

In this work, a robust, consistent, and coherent approach, termed as Modified Ghost Method (MGM), is developed to deal with the multi-medium interaction with elastic-plastic solid. This approach is simple to implement and keeps the solvers intact, and can handle multi-medium problems which involve various media including gas, liquid and solid. The MGM is first validated by two-dimensional (2D) cases and then is applied to study the interaction between elastic-plastic solid structure and the underwater explosion. The development of the wave system is described and analyzed. Furthermore, two kinds of complex solid structure subjected to underwater explosion are simulated. Finally, a complex solid structure immersed in water subjected to underwater explosion is simulated and analyzed. The numerical experiments show the viability, effectiveness and versatility of the proposed method which is able to accurately predict the wave pattern at various interfaces.

Moving mesh methods provide an efficient way of solving partial differential equations for which large, localised variations in the solution necessitate locally dense spatial meshes. In one-dimension, meshes are typically specified using the arclength mesh density function. This choice is well-justified for piecewise polynomial interpolants, but it is only justified for spectral methods when model solutions include localised steep gradients. In this paper, one-dimensional mesh density functions are presented which are based on a spatially localised measure of the bandwidth of the approximated model solution. In considering bandwidth, these mesh density functions are well-justified for spectral methods, but are not strictly tied to the error properties of any particular spatial interpolant, and are hence widely applicable. The bandwidth mesh density functions are illustrated in two ways. First, by applying them to Chebyshev polynomial approximation of two test functions, and second, through use in periodic spectral and finite-difference moving mesh methods applied to a number of model problems in acoustics. These problems include a heterogeneous advection equation, the viscous Burgers’ equation, and the Korteweg-de Vries equation. Simulation results demonstrate solution convergence rates that are up to an order of magnitude faster using the bandwidth mesh density functions than uniform meshes, and around three times faster than those using the arclength mesh density function.

This paper presents the simulation of complex separation flows over a modern fighter model at high angle of attack by using an unstructured/hybrid grid based Detached Eddy Simulation (DES) solver with an adaptive dissipation second-order hybrid scheme. Simulation results, including the complex vortex structures, as well as vortex breakdown phenomenon and the overall aerodynamic performance, are analyzed and compared with experimental data and unsteady Reynolds-Averaged Navier-Stokes (URANS) results, which indicates that with the DES solver, clearer vortical flow structures are captured and more accurate aerodynamic coefficients are obtained. The unsteady properties of DES flow field are investigated in detail by correlation coefficient analysis, power spectral density (PSD) analysis and proper orthogonal decomposition (POD) analysis, which indicates that the spiral motion of the primary vortex on the leeward side of the aircraft model is highly nonlinear and dominates the flow field. Through the comparisons of flow topology and pressure distributions with URANS results, the reason why higher and more accurate lift can be obtained by DES is discussed. Overall, these results show the potential capability of present DES solver in industrial applications.

We analyze the geometrically consistent schemes proposed by E. Lu and Yang [6] for one-dimensional problem with finite range interaction. The existence of the reconstruction coefficients is proved, and optimal error estimate is derived under sharp stability condition. Numerical experiments are performed to confirm the theoretical results.

We describe a numerical model to simulate the non-linear elasto-plastic dynamics of compressible materials. The model is fully Eulerian and it is discretized on a fixed Cartesian mesh. The hyperelastic constitutive law considered is neohookean and the plasticity model is based on a multiplicative decomposition of the inverse deformation tensor. The model is thermodynamically consistent and it is shown to be stable in the sense that the norm of the deviatoric stress tensor beyond yield is non increasing. The multimaterial integration scheme is based on a simple numerical flux function that keeps the interfaces sharp. Numerical illustrations in one to three space dimensions of high-speed multimaterial impacts in air are presented.

This paper proposes a novel distance derivative weighted ENO (DDWENO) limiter based on fixed reconstruction stencil and applies it to the second- and high-order finite volume method on unstructured grids. We choose the standard deviation coefficients of the flow variables as the smooth indicators by using the k-exact reconstruction method, and obtain the limited derivatives of the flow variables by weighting all derivatives of each cell according to smoothness. Furthermore, an additional weighting coefficient related to distance is introduced to emphasize the contribution of the central cell in smooth regions. The developed limiter, combining the advantages of the slope limiters and WENO-type limiters, can achieve the similar effect of WENO schemes in the fixed stencil with high computational efficiency. The numerical cases demonstrate that the DDWENO limiter can preserve the numerical accuracy in smooth regions, and capture the shock waves clearly and steeply as well.

This paper presents topology optimization of capillary, the typical two-phase flow with immiscible fluids, where the level set method and diffuse-interface model are combined to implement the proposed method. The two-phase flow is described by the diffuse-interface model with essential no slip condition imposed on the wall, where the singularity at the contact line is regularized by the molecular diffusion at the interface between two immiscible fluids. The level set method is utilized to express the fluid and solid phases in the flows and the wall energy at the implicit fluid-solid interface. Based on the variational procedure for the total free energy of two-phase flow, the Cahn-Hilliard equations for the diffuse-interface model are modified for the two-phase flow with implicit boundary expressed by the level set method. Then the topology optimization problem for the two-phase flow is constructed for the cost functional with general formulation. The sensitivity analysis is implemented by using the continuous adjoint method. The level set function is evolved by solving the Hamilton-Jacobian equation, and numerical test is carried out for capillary to demonstrate the robustness of the proposed topology optimization method. It is straightforward to extend this proposed method into the other two-phase flows with two immiscible fluids.

We propose a multi-layer approach to simulate hyperpycnal and hypopycnal plumes in flows with free surface. The model allows to compute the vertical profile of the horizontal and the vertical components of the velocity of the fluid flow. The model can describe as well the vertical profile of the sediment concentration and the velocity components of each one of the sediment species that form the turbidity current. To do so, it takes into account the settling velocity of the particles and their interaction with the fluid. This allows to better describe the phenomena than a single layer approach. It is in better agreement with the physics of the problem and gives promising results. The numerical simulation is carried out by rewriting the multilayer approach in a compact formulation, which corresponds to a system with nonconservative products, and using path-conservative numerical scheme. Numerical results are presented in order to show the potential of the model.

Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable, and isotropic elastic solid, which is immersed in a homogeneous compressible air or fluid. The paper concerns the numerical solution for such an acoustic-elastic interaction problem in three dimensions. An exact transparent boundary condition (TBC) is developed to reduce the problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by using a PML equivalent TBC. An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.

For problems governed by a non-normal operator, the leading eigenvalue of the operator is of limited interest and a more relevant measure of the stability is obtained by considering the harmonic forcing causing the largest system response. Various methods for determining this so-called optimal forcing exist, but they all suffer from great computational expense and are hence not practical for large-scale problems. In the present paper a new method is presented, which is applicable to problems of arbitrary size. The method does not rely on time-stepping, but on the solution of linear systems, in which the inverse Laplacian acts as a preconditioner. By formulating the search for the optimal forcing as an eigenvalue problem based on the resolvent operator, repeated system solves amount to power iterations, in which the dominant eigenvalue is seen to correspond to the energy amplification in a system for a given frequency, and the eigenfunction to the corresponding forcing function. Implementation of the method requires only minor modifications of an existing time-stepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss the method, first, in the context of the linear Ginzburg-Landau equation and then, the two-dimensional lid-driven cavity flow governed by the Navier-Stokes equations. Most importantly, we demonstrate that for the lid-driven cavity, the optimal forcing can be computed using a factor of up to 500 times fewer operator evaluations than the standard method based on exponential time-stepping.