High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

In this paper, we demonstrate the detailed numerical studies of three classical two dimensional detonation waves by solving the two dimensional reactive Euler equations with species with the fifth order WENO-Z finite difference scheme (Borges et al. in J. Comput. Phys. 227:3101?C3211, 2008) with various grid resolutions. To reduce the computational cost and to avoid wave reflection from the artificial computational boundary of a truncated physical domain, we derive an efficient and easily implemented one dimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the two dimensional unsteady reactive Euler equation when one of the directions of domain is periodical and inflow/outflow in the other direction. The numerical comparison among characteristic, free stream, extrapolation and PML boundary conditions are conducted for the detonation wave simulations. The accuracy and efficiency of four mentioned boundary conditions are verified against the reference solutions which are obtained from using a large computational domain. Numerical schemes for solving the system of hyperbolic conversation laws with a single-mode sinusoidal perturbed ZND analytical solution as initial conditions are presented. Regular rectangular combustion cell, pockets of unburned gas and bubbles and spikes are generated and resolved in the simulations. It is shown that large amplitude of perturbation wave generates more fine scale structures within the detonation waves and the number of cell structures depends on the wave number of sinusoidal perturbation.     

Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. This paper develops the PML concept for time-harmonic elastodynamics in Cartesian coordinates, utilising insights obtained with electromagnetics PMLs, and presents a novel displacement-based, symmetric finite-element implementation of the PML for time-harmonic plane-strain or three-dimensional motion. The PML concept is illustrated through the example of a one-dimensional rod on elastic foundation and through the anti-plane motion of a two-dimensional continuum. The concept is explored in detail through analytical and numerical results from a PML model of the semi-infinite rod on elastic foundation, and through numerical results for the anti-plane motion of a semi-infinite layer on a rigid base. Numerical results are presented for the classical soil–structure interaction problems of a rigid strip-footing on a (i) half-plane, (ii) layer on a half-plane, and (iii) layer on a rigid base. The analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by PML models even with small bounded domains.     

On the existence and convergence of the solution of PML equations

In this article we study the mesh termination method in computational scattering theory known as the method of Perfectly Matched Layer (PML). This method is based on the idea of surrounding the scatterer and its immediate vicinity with a fictitious absorbing non-reflecting layer to damp the echoes coming from the mesh termination surface. The method can be formulated equivalently as a complex stretching of the exterior domain. The article is devoted to the existence and convergence questions of the solutions of the resulting equations. We show that with a special choice of the fictitious absorbing coefficient, the PML equations are solvable for all wave numbers, and as the PML layer is made thicker, the PML solution converge exponentially towards the actual scattering solution. The proofs are based on boundary integral methods and a new type of near-field version of the radiation condition, called here the double surface radiation condition. Partly supported by the Finnish Academy, project 37692.     

基于正交轨迹的共形PML数值仿真方法

从材料的观点看,完全匹配层是各向异性的介质.场在完全匹配层遵从麦克斯韦方程.通过设计磁导率和介电常数张量参数来实现任意形状的共形完全匹配层.为匹配层动态稳定,可应用二维正交轨迹网格和FDTD方法实现了数值共形完全匹配层仿真,数值仿真结果表明基于正交轨迹网格的共形完全匹配层为一个共形FDTD或FEM方法提供了一个有效地吸收边界条件.     

Application of shifted Jacobi pseudospectral method for solving (in)finite-horizon min–max optimal control problems with uncertainty

The difficulty of solving the min–max optimal control problems (M-MOCPs) with uncertainty using generalised Euler–Lagrange equations is caused by the combination of split boundary conditions, nonlinear differential equations and the manner in which the final time is treated. In this investigation, the shifted Jacobi pseudospectral method (SJPM) as a numerical technique for solving two-point boundary value problems (TPBVPs) in M-MOCPs for several boundary states is proposed. At first, a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented. Then, by applying the generalised Euler–Lagrange equations and expanding the required approximate solutions as elements of shifted Jacobi polynomials, finding a solution of TPBVPs in nonlinear M-MOCPs with uncertainty is reduced to the solution of a system of algebraic equations. Moreover, the Jacobi polynomials are particularly useful for boundary value problems in unbounded domain, which allow us to solve infinite- as well as finite and free final time problems by domain truncation method. Some numerical examples are given to demonstrate the accuracy and efficiency of the proposed method. A comparative study between the proposed method and other existing methods shows that the SJPM is simple and accurate.     

A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations

In this work we consider Runge–Kutta discontinuous Galerkin methods for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics.     

An efficient parallel algorithm with application to computational fluid dynamics

When solving time-dependent partial differential equations on parallel computers using the nonoverlapping domain decomposition method, one often needs numerical boundary conditions on the boundaries between subdomains. These numerical boundary conditions can significantly affect the stability and accuracy of the final algorithm.In this paper, a stability and accuracy analysis of the existing methods for generating numerical boundary conditions will be presented, and a new approach based on explicit predictors and implicit correctors will be used to solve convection-diffusion equations on parallel computers, with application to aerospace engineering for the solution of Euler equations in computational fluid dynamics simulations. Both theoretical analyses and numerical results demonstrate significant improvement in stability and accuracy by using the new approach.     

Unified matrix-exponential FDTD formulations for modeling electrically and magnetically dispersive materials

Unified matrix-exponential finite difference time domain (ME-FDTD) formulations are presented for modeling linear multi-term electrically and magnetically dispersive materials. In the proposed formulations, Maxwell?s curl equations and the related dispersive constitutive relations are cast into a set of first-order differential matrix system and the field?s update equations can be extracted directly from the matrix-exponential approximation. The formulations have the advantage of simplicity as it allows modeling different linear dispersive materials in a systematic manner and also can be easily incorporated with the perfectly matched layer (PML) absorbing boundary conditions (ABCs) to model open region problems. Apart from its simplicity, it has been shown that the proposed formulations necessitate less storage requirements as compared with the well-know auxiliary differential equation FDTD (ADE-FDTD) scheme while maintaining the same accuracy performance.     

A down-stream boundary procedure for the euler equations

The steady slate problem for flow in a channel is considered. Down-stream boundary conditions are derived for the Euler equations by using Fourier-expansions of the solution. Not data are required at the down-stream boundary or at infinity, the condition used is simply the requirement of a bounded solution. A difference approximation is applied to the steady state equations, and Newton's method is used for solving the resulting non-linear system of equations.     

On the Accuracy and Stability of the Perfectly Matched Layer in Transient Waveguides

Energy transmitted along a waveguide decays less rapidly than in an unbounded medium. In this paper we study the efficiency of a PML in a time-dependent waveguide governed by the scalar wave equation. A straight forward application of a Neumann boundary condition can degrade accuracy in computations. To ensure accuracy, we propose extensions of the boundary condition to an auxiliary variable in the PML. We also present analysis proving stability of the constant coefficient PML, and energy estimates for the variable coefficients case. In the discrete setting, the modified boundary conditions are crucial in deriving discrete energy estimates analogous to the continuous energy estimates. Numerical stability and convergence of our numerical method follows. Finally we give a number of numerical examples, illustrating the stability of the layer and the high order accuracy of our proposed boundary conditions.     

FDTD方法吸收边界条件的研究及应用

用时域有限差分法(FDTD)求解电磁散射问题中,吸收边界条件的设置起着关键性作用.通过时间和空间上的递推算法对时域有限差分法中的两种吸收边界条件:Mur吸收边界条件和完全匹配层(PML)的吸收效果进行了比较和分析.同时,引入参数对PML的差分方程进行了优化,避免了将电磁场分裂为两个分量进行计算,进而降低了计算内存开销.实验结果证明PML具有更优越的吸收性能.最后,在FDTD算法中应用PML吸收层对一圆柱形导体的雷达散射截面积(RCS)进行数值仿真,验证了FDTD算法在计算雷达散射截面积(RCS)上的有效性.     

A Boundary Condition Capturing Method for Multiphase Incompressible Flow

In [6], the Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid compressible Euler equations. In [11], related techniques were used to develop a boundary condition capturing approach for the variable coefficient Poisson equation on domains with an embedded interface. In this paper, these new numerical techniques are extended to treat multiphase incompressible flow including the effects of viscosity, surface tension and gravity. While the most notable finite difference techniques for multiphase incompressible flow involve numerical smearing of the equations near the interface, see, e.g., [19, 17, 1], this new approach treats the interface in a sharp fashion.     

Weak imposition of Dirichlet boundary conditions in fluid mechanics

Weakly enforced Dirichlet boundary conditions are compared with strongly enforced conditions for boundary layer solutions of the advection-diffusion equation and incompressible Navier-Stokes equations. It is found that weakly enforced conditions are effective and superior to strongly enforced conditions. The numerical tests involve low-order finite elements and a quadratic NURBS basis utilized in the Isogeometric Analysis approach. The convergence of the mean velocity profile for a turbulent channel flow suggests that weak no-slip conditions behave very much like a wall function model, although the design of the boundary condition is based purely on numerical, rather than physical or empirical, conditions.     

A two-point difference scheme for computing steady-state solutions to the conservative one-dimensional Euler equations

An implicit finite-difference method is presented for obtaining steady-state solutions to the time-dependent, conservative Euler equations for flows containing shocks. The method uses a two-point central-difference scheme for the flux derivatives with dissipation added at supersonic points via the retarded density concept. Application of the method to 1-dimensional nozzle flow equations for various combinations of subsonic and supersonic boundary conditions show the method to be very efficient. Residuals are typically reduced to machine zero in approximately 35 time steps for 50 mesh points. For 1-dimensional Euler calculations, it is shown that the scheme offers two advantages over the more widely-used three-point schemes. The first is in regard to application of boundary conditions, and the second relates to the fact that the two-point algorithm is well-conditioned for large time steps.     

von Neumann stability analysis of first-order accurate discretization schemes for one-dimensional (1D) and two-dimensional (2D) fluid flow equations

In literature, the von Neumann stability analysis of simplified model equations, such as the wave equation, is typically used to determine stability conditions for the non-linear partial differential fluid flow equations (Navier–Stokes and Euler). However, practical experience suggests that such simplistic stability conditions are grossly inadequate for computations involving the system of coupled flow equations. The goal of this paper is to determine stability conditions for the full system of fluid flow equations – the Euler equations are examined, as any conditions derived for the Euler equations will apply to the Navier–Stokes (NS) equations in the limit of convection-dominated flows. A von Neumann stability analysis is conducted for the one-dimensional (1D) and two-dimensional (2D) Euler equations. The system of equations is discretized on a staggered grid using finite-difference discretization techniques; the use of a staggered grid allows equivalence to finite-volume discretization. By combining the different discretization techniques, ten solution schemes are formulated – eight solution schemes are considered for the 1D Euler equations, and two schemes for the 2D Euler equations. For each scheme, error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum-allowable CFL (Courant–Friedrichs–Lewy) number as a function of Mach number. The predictions are verified for selected schemes using the Riemann problem at incompressible and compressible Mach numbers. Very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, which offer a viable alternative to complicated mathematical expressions often reported in published literature, and should benefit everyday CFD (Computational Fluid Dynamics) users. The stability regions are used to discuss the effect of time integration (explicit vs. implicit), density bias in continuity equation and momentum convection term linearization on stability. A comparison of the predicted stability limits for 1D and 2D Euler equations with commonly-used stability conditions arising from the wave equation shows that the stability thresholds for the Euler equations lie well below those predicted by the wave equation analysis; in addition, the 2D Euler stability limits are more restrictive as compared to 1D Euler limits. Since the present analysis accounts for the full system of fluid flow (Euler) equations, the derived stability conditions can be used by CFD practitioners to estimate a timestep or CFL number to guide the stability of their computations.     

Numerical simulation of liquid sloshing phenomena in partially filled containers

In the simulation of the dynamic load excited by sloshing in a partially filled tank, appropriate boundary conditions need imposing to calculate the impact pressure. Traditionally, a thin artificial buffer zone is adopted near the tank ceiling and a linear combination of free surface dynamic and rigid wall boundary conditions are imposed inside the buffer zone. This investigation demonstrates that no special treatment is needed to describe the free surface, because a two-fluid approach based on a level set method is used to solve the Reynolds-averaged Navier-Stokes (RANS) equations in both water and air regions and the interface is treated as a variation of the fluid properties. All the boundary conditions adopted are those usually accepted in solutions of Navier-Stokes or Euler equations. Sloshing in a rectangular tank excited by a horizontal harmonic motion is assessed numerically at different filling levels and excitation frequencies. The dependency of numerical solution on grid resolution, time step size and the interface thickness are investigated. Further, numerical tests are conducted for a rectangular tank with both 45° and 60° chamfered ceiling corners subject to a harmonic rolling motion. The comparison of computed results with experimental data show the developed numerical method is capable of the simulation of dynamic pressure loads exerted on the tank walls and ceiling excited by fluid sloshing.     

On the mass and momentum transport in the Navier–Stokes slip layer

The Navier–Stokes slip boundary conditions are considered as conditions following from the mass and momentum balances within a thin, shell-like moving boundary layer. A problem of consistency between different models, that describes the internal and external friction in a viscous fluid, is stated within the framework of a proper form of the layer momentum balance. Appropriate constitutive equations for friction forces are formulated. The common features of the Navier, Stokes, Reynolds, and Maxwell concepts of a boundary slip layer are revalorized and discussed. Different mobility mechanisms connected with the transpiration phenomena, important for flows in micro- and nanochannels, are classified as a part of equations for the external friction.     

Towards development of unsteady near-wall interface boundary conditions for turbulence modeling

In near-wall turbulence modeling it is required to resolve a thin layer nearby the solid boundary, which is characterized by high gradients of the solution. An accurate enough resolution of such a layer can take most computational time. The situation even becomes worse for unsteady problems. To avoid time-consuming computations, a new approach is developed, which is based on a non-overlapping domain decomposition. The boundary condition of Robin type at the interface boundary is achieved via transfer of the boundary condition from the wall. For the first time interface boundary conditions of Robin type are derived for a model nonstationary equation which simulates the key terms of the unsteady boundary layer equations. In the case of stationary solutions the approach is automatically reduced to the technique earlier developed for the steady problems. The considered test cases demonstrate that unsteady effects can be significant for near-wall domain decomposition. In particular, they can be important in the case of the wall-function-based approach.     

Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains

The numerical simulation of coupled nonlinear Schrödinger equations on unbounded domains is considered in this paper. By using the operator splitting technique, the original problem is decomposed into linear and nonlinear subproblems in a small time step. The linear subproblem turns out to be two decoupled linear Schrödinger equations on unbounded domains, where artificial boundaries are introduced to truncate the unbounded physical domains into finite ones. Local absorbing boundary conditions are imposed on the artificial boundaries. On the other hand, the coupled nonlinear subproblem is an ODE system, which can be solved exactly. To demonstrate the effectiveness of our method, some comparisons in terms of accuracy and computational cost are made between the PML approach and our method in numerical examples.     

Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains

The numerical simulation of coupled nonlinear Schrödinger equations on unbounded domains is considered in this paper. By using the operator splitting technique, the original problem is decomposed into linear and nonlinear subproblems in a small time step. The linear subproblem turns out to be two decoupled linear Schrödinger equations on unbounded domains, where artificial boundaries are introduced to truncate the unbounded physical domains into finite ones. Local absorbing boundary conditions are imposed on the artificial boundaries. On the other hand, the coupled nonlinear subproblem is an ODE system, which can be solved exactly. To demonstrate the effectiveness of our method, some comparisons in terms of accuracy and computational cost are made between the PML approach and our method in numerical examples.