FORCE schemes on unstructured meshes I: Conservative hyperbolic systems

This paper is about the construction of numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple space dimensions on structured and unstructured meshes. The work is a multi-dimensional extension of the one-dimensional FORCE flux and is closely related to the work of Nessyahu–Tadmor and Arminjon. The resulting basic flux is first-order accurate and monotone; it is then extended to arbitrary order of accuracy in space and time on unstructured meshes in the framework of finite volume and discontinuous Galerkin methods. The performance of the schemes is assessed on a suite of test problems for the multi-dimensional Euler and Magnetohydrodynamics equations on unstructured meshes.     

A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension

We describe a cell-centered Godunov scheme for Lagrangian gas dynamics on general unstructured meshes in arbitrary dimension. The construction of the scheme is based upon the definition of some geometric vectors which are defined on a moving mesh. The finite volume solver is node based and compatible with the mesh displacement. We also discuss boundary conditions. Numerical results on basic 3D tests problems show the efficiency of this approach. We also consider a quasi-incompressible test problem for which our nodal solver gives very good results if compared with other Godunov solvers. We briefly discuss the compatibility with ALE and/or AMR techniques at the end of this work. We detail the coefficients of the isoparametric element in the appendix.     

Efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes

This paper presents efficient second-order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows. For compressible unsteady flows, a time-dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables. As a result, a compact scheme on the unstructured meshes is developed. For incompressible steady flows, a continuous second-order gas-kinetic BGK type scheme is presented, for which the time-dependent gas distribution function with a continuous particle velocity is used on unstructured meshes. The efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells. Therefore, even with the stencil of a first-order scheme, a high resolution method is constructed. Numerical examples are presented which are compared with the benchmark solutions and the experimental measurements.     

Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes

We consider a non-linear finite volume (FV) scheme for stationary diffusion equation. We prove that the scheme is monotone, i.e. it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. The scheme is extended to regular star-shaped polygonal meshes and isotropic heterogeneous coefficients.     

Filtered density function simulator on unstructured meshes

A new computational filtered density function (FDF) methodology is developed for large eddy simulation (LES) of turbulent reacting flows. This methodology is based on a Lagrangian Monte Carlo (MC) FDF solver constructed on a domain portrayed by an unstructured mesh. The base filtered transport equations on this mesh are solved by a finite-volume (FV) method. The consistency of the hybrid FV-MC solver and the realizability of the simulated results are demonstrated via LES of a temporally developing mixing layer. The overall performance of the model is appraised by comparison with direct numerical simulation (DNS) data. The algorithmic implementation in the commercial software ANSYS-FLUENT facilitates future FDF-LES of turbulent combustion in complex configurations.     

Finite-volume lattice Boltzmann schemes in two and three dimensions

Simple and practical finite-volume schemes for the lattice Boltzmann equation are derived in two and three dimensions through the application of modern finite-volume methods. The schemes use a finite-volume vortex-type formulation based on quadrilateral elements in two dimensions and trilinear hexahedral elements in three dimensions. It is shown that the schemes are applicable to domains with irregular boundaries of arbitrary shape in two and three dimensions.     

Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes

In this paper we consider the very high order approximation of solutions of the Euler equations. We present a systematic generalization of the residual distribution method of [1] to very high order of accuracy, by extending the preliminary work discussed in [2] to systems and hybrid meshes. We present extensive numerical validation for the third and fourth order cases with Lagrange finite elements. In particular, we demonstrate that we both have a non-oscillatory behavior, even for very strong shocks and complex flow patterns, and the expected accuracy on smooth problems.     

Directional integration on unstructured meshes via supermesh construction

Unstructured meshes are in widespread use throughout computational physics, but calculating diagnostics of simulations on such meshes can be challenging. For example, in geophysical fluid dynamics, it is frequently desirable to compute directional integrals such as vertical integrals and zonal averages; however, it is difficult to compute these on meshes with no inherent spatial structure. This is widely regarded as an obstacle to the adoption of unstructured mesh numerical modelling in this field. In this paper, we describe an algorithm by which one can exactly compute such directional integrals on arbitrarily unstructured meshes. This is achieved via the solution of a problem of computational geometry, constructing the supermesh of two meshes. We demonstrate the utility of this approach by applying it to a classical geophysical fluid dynamics system: the thermally driven rotating annulus. This addresses an important objection to the more widespread use of unstructured mesh modelling.     

Development and comparison of different spatial numerical schemes for the radiative transfer equation resolution using three-dimensional unstructured meshes

In the present work four different spatial numerical schemes have been developed with the aim of reducing the false-scattering of the numerical solutions obtained with the discrete ordinates (DOM) and the finite volume (FVM) methods. These schemes have been designed specifically for unstructured meshes by means of the extrapolation of nodal values of intensity on the studied radiative direction. The schemes have been tested and compared in several 3D benchmark test cases using both structured orthogonal and unstructured grids.     

A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes

This paper proposes a new WENO procedure to compute multi-scale problems with embedded discontinuities, on non-uniform meshes.In a one-dimensional context, the WENO procedure is first defined on a five-points stencil and designed to be fifth-order accurate in regions of smoothness. To this end, we define a finite-volume discretization in which we consider the cell averages of the variable as the discrete unknowns. The reconstruction of their point-values is then ensured by a unique fifth-order polynomial. This optimum polynomial is considered as a symmetric and convex combination, by ideal weights, of four quadratic polynomials.The symmetric nature of the resulting interpolation has an important consequence: the choice of ideal weights has no influence on the accuracy of the discretization. This advantage enables to formulate the interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights.We adapt this procedure for the non-linear weights to maintain the theoretical convergence properties of the optimum reconstruction, whatever the problem considered.The resulting scheme is a fifth-order WENO method based on central interpolation and TVD Runge–Kutta time-integration. We call this scheme the CWENO5 scheme.Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In those experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems. Finally, the new algorithm is directly extended to bi-dimensional problems.     

虚拟现实中基于图像的三维空间漫游方法

最近几年基于图像的虚拟现实技术渐渐引起人们的兴趣。它是一种通过实景拍摄获取图像来表示虚拟场景的技术。提出了一种基于图像的三维空间漫游方法 ,它是基于视图变形技术的原理 ,在三维空间中选定三幅视图构成一组图像基 ,再由这组基插值中间任何姿态视图 ,最后这些视图以视频方式显示出来。它避免了 3D重建问题 ,人们可以从任何视点或视向来观赏对象。     

Finite-energy static solutions to chiral models in three space dimensions

Finite-energy static kink solutions in three space dimensions, carrying a topologically conserved charge, are exhibited for a new type of chiral pion model. The usual scaling instability problems are circumvented by an unconventional choice of action. This action has no free field sector, but small excitations about classical static solutions of exhibit causal propagation and are candidates for quantization. Another type of scaling stable chiral model is also discussed. It does have a free field sector but involves fourth-order derivatives in the action.Some global aspects of solutions to non-linear chiral theories are discussed in the appendix.     

String amplitudes in arbitrary dimensions

We calculate gravitational dressed tachyon correlators in non critical dimensions. The 2D gravity part of the theory is constrained to constant curvature. Then scaling dimensions of gravitational dressed vertex operators are equal to their bare conformal dimensions. Considering the model asd+2 dimensional critical string we calculate poles of generalized Shapiro-Virasoro amplitudes.     

Lorentz anomaly in arbitrary dimensions

It is shown that the Lorentz invariance is broken in gauge theories of chiral Weyl fermions in flat space-time via one-loop quantum corrections. Abelian gauge fields contribute to this anomaly in even dimensions larger than or equal to four and non-Abelian gauge fields do in even dimensions larger than or equal to six. The anomaly is proportional toD/2–1 power to the charge, whereD is a number of space-time dimensions.     

Mesh quality effects on the accuracy of CFD solutions on unstructured meshes

The order of accuracy and error magnitude of node- and cell-centered schemes are examined on representative unstructured meshes and flowfield solutions for computational fluid dynamics. Specifically, we investigate the properties of inviscid and viscous flux discretizations for isotropic and highly stretched meshes using the Method of Manufactured Solutions. Grid quality effects are studied by randomly perturbing the base meshes and cataloguing the error convergence as a function of grid size. For isotropic grids, node-centered approaches produce less error than cell-centered approaches. Moreover, a corrected node-centered scheme is shown to maintain third order accuracy for the inviscid terms on arbitrary triangular meshes. In contrast, for stretched meshes, cell-centered schemes are favored, with cell-centered prismatic approaches in particular showing the lowest levels of error. In three dimensions, simple flux integrations on non-planar control volume faces lead to first-order solution errors, while second-order accuracy is recovered by triangulation of the non-planar faces.     

A large-scale wave-current coupled module with wave diffraction effect on unstructured meshes

Based on the extended mild-slope equation,a large-scale wave module is developed.By combining the eikonal equation and the modified wave action equation,the wave model can account for diffraction in most situations such as in the lee of islands and breakwaters,and using unstructured meshes provides great flexibility for modelling the wave in the complex geomorphology of barriers and islands,also allowing for refinement of the grid resolution within computationally important domains.The numerical implementation of the module is based on the explicit second-order upwind finite-volume schemes in geographic space,the Flux-Corrected Transport(FCT)algorithm in frequency space and the implicit Crank-Nicolson method in directional space.The three-dimensional hydrodynamic module is then modified to couple with the wave model,where the wave readily provides the depth-dependent radiation stress and the wave-induced turbulence coefficient for the current fields,and the wave propagation takes into account the current-induced advection,refraction and diffraction of wave energy and the effect of water level.The applicability of the proposed model to calculate Snell’s Law,wave transformation over the breakwaters and the elliptic shoal,wave propagation over the rip current field and the undertow on a sloping beach is evaluated.Numerical results show that the present model makes better predictions of the near-shore wave propagation and complex three-dimensional(3D)near-shore circulation driven by the waves,considering analytical solutions and experimental values.     

The Ising model on random lattices in arbitrary dimensions

We study analytically the Ising model coupled to random lattices in dimension three and higher. The family of random lattices we use is generated by the large N limit of a colored tensor model generalizing the two-matrix model for Ising spins on random surfaces. We show that, in the continuum limit, the spin system does not exhibit a phase transition at finite temperature, in agreement with numerical investigations. Furthermore we outline a general method to study critical behavior in colored tensor models.     

New nonlinear conduction mechanisms in charge-density-wave condensates in two or three space dimensions

It is shown that in phase-pinned charge-density-wave condensates an external electric field induces thermally activated continuous creation of new domains, when the pinning potential is periodic in the phase π of the condensate wave function. The mechanism gives rise to a nonlinear dc conductivity at finite temperatures, which seems to account for the anomalous conductivity observed in NbSe₃.     

Radiating black hole solutions in arbitrary dimensions

We prove a theorem that characterizes a large family of non-static solutions to Einstein equations in N-dimensional space-time, representing, in general, spherically symmetric Type II fluid. It is shown that the best known Vaidya-based (radiating) black hole solutions to Einstein equations, in both four dimensions (4D) and higher dimensions (HD), are particular cases from this family. The spherically symmetric static black hole solutions for Type I fluid can also be retrieved. A brief discussion on the energy conditions, singularities and horizons is provided.