Riemann solvers with primitive parameter vectors for two-dimensional compressible flows of a real gas

An analysis of averaging procedures is presented for an approximate Riemann solver for the equations governing the compressible flow of a real gas. This study extends earlier work for the Euler equations with ideal gases.     

Solution properties and approximate Riemann solvers for two-layer shallow flow models

The 4 × 4 system of governing equations for two-layer shallow flow models is known to exhibit particular behaviours such as loss of hyperbolicity under certain flow configurations. An eigenvalue analysis of the conservation part of the equations shows that the loss of hyperbolicity is due only to the reaction exerted by each fluid onto the other at the interface between the fluids. Three Riemann solvers derived from the HLL formalism are presented. In the first solver, the pressure-induced terms are accounted for by the source term; in the second solver, they are incorporated into the fluxes; the third solver uses the same formulation as the first, except that the mass and momentum balance for the bottom layer are replaced with the balance equations for the system formed by the two layers as a whole. Numerical results using the three solvers are presented for (1) static conditions such as two fluids of identical densities at rest above each other, (2) dam-break flows involving the collapse of a body of light fluid over a uniform layer of a denser fluid, and (3) Liska and Wendroff’s ill-posed test cases [24] involving two-layer flows over a topographic bump. The three solvers produce quasi-undistinguishable results for the dam-break flows, and produce sharp solutions over the full range of density ratio, from 0 to 1. However, only the third solver allows a strict preservation of static configurations. Moreover, a method is proposed to assess the convergence of the numerical solutions in the configurations for which no analytical solution can be obtained.     

Parallel simulation of three-dimensional complex flows: Application to two-phase compressible flows and turbulent wakes

In this paper, we present parallel simulations of three-dimensional complex flows obtained on an ORIGIN 3800 computer and on homogeneous and heterogeneous (processors of different speeds and RAM) computational grids. The solver under consideration, which is representative of modern numerics used in industrial computational fluid dynamics (CFD) software, is based on a mixed element-volume method on unstructured tedrahedrisations. The parallelisation strategy combines mesh partitioning techniques, a message-passing programming model and an additive Schwarz algorithm. The parallelisation performances are analysed on a two-phase compressible flow and a turbulent flow past a square cylinder.     

Numerical and computational efficiency of solvers for two-phase problems

We consider two-phase flow problems, modelled by the Cahn–Hilliard equation. In this work, the nonlinear fourth-order equation is decomposed into a system of two coupled second-order equations for the concentration and the chemical potential.We analyse solution methods based on an approximate two-by-two block factorization of the Jacobian of the nonlinear discrete problem. We propose a preconditioning technique that reduces the problem of solving the non-symmetric discrete Cahn–Hilliard system to a problem of solving systems with symmetric positive definite matrices where off-the-shelf multilevel and multigrid algorithms are directly applicable. The resulting solution methods exhibit optimal convergence and computational complexity properties and are suitable for parallel implementation.We illustrate the efficiency of the proposed methods by various numerical experiments, including parallel results for large scale three dimensional problems.     

A robust shock-capturing scheme based on rotated Riemann solvers

This paper presents a robust finite volume shock-capturing scheme based on the rotated approximate Riemann solver. A general framework for constructing the rotated Riemann solver is described and a rotated Roe scheme is discussed in detail. It is found that the robustness of the rotated shock-capturing scheme is closely related to the way in which the direction of upwind differencing is determined. When the upwind direction is determined by the velocity-difference vector, the rotated Roe scheme demonstrates a robust shock-capturing capability and the shock instabilities or carbuncle phenomena can be eliminated completely. The dissipation property associated with the linear field of the rotated flux-difference splitting scheme is analyzed, and several test cases are presented to validate the proposed scheme.     

An immersed-boundary method for compressible viscous flows

This paper combines a state-of-the-art method for solving the preconditioned compressible Navier-Stokes equations accurately and efficiently for a wide range of the Mach number with an immersed-boundary approach which allows one to use Cartesian grids for arbitrarily complex geometries. The method is validated versus well documented test problems for a wide range of the Reynolds and Mach numbers. The numerical results demonstrate the efficiency and versatility of the proposed approach as well as its accuracy, from incompressible to supersonic flow conditions, for moderate values of the Reynolds number. Further improvements, obtained via local grid refinement or non-linear wall functions, can render the proposed approach a formidable tool for studying complex three-dimensional flows of industrial interest.     

Exact and approximate rhythm matching algorithms

An interesting problem in music information retrieval is to classify songs according to rhythms. A rhythm is represented by a sequence of “Quick” (Q) and “Slow” (S) symbols, which correspond to the (relative) duration of notes, such that S?=?2Q. Christodoulakis et?al. presented an efficient algorithm that can be used to classify musical sequences according to rhythms. In this article, the above algorithm is implemented, along with a naive brute force algorithm to solve the same problem. The theoretical time complexity bounds are analyzed with the actual running times achieved by the experiments, and the results of the two algorithms are compared. Furthermore, new efficient algorithms are presented that take temporal errors into account. This, the approximate pattern matching version, could not be handled by the algorithms previously presented. The running times of two algorithmic variants are analyzed and compared and examples of their implementation are shown.     

Exact and approximate state estimation for nonlinear dynamic systems

This paper presents a modified approach to solve state estimation problems of nonlinear dynamic systems involving noise free, uncorrelated and correlated state and measurement noise processes. The basic approach makes use of the matrix minimum principle together with the Kolmogorov and Kushner's equations to minimize the error-variance, taken to be the estimation criterion. The filtering equations obtained for nonlinear systems with white noise process are exact, but for non-white noise processes the results obtained are approximate.

For systems with polynomial or product types non-linearities, the proposed algorithms can be evaluated without the need of approximation under the assumption that the estimator errors are Gaussian. Such an assumption is significantly different from the most commonly used assumption that the state is Gaussian. Simulation results obtained from the proposed filtering algorithms are compared to various other approximate nonlinear filters. The results indicate the superiority of the proposed filter over those of other filters investigated.     

Exact and approximate equilibria for optimal group network formation

We consider a process called the Group Network Formation Game, which represents the scenario when strategic agents are building a network together. In our game, agents can have extremely varied connectivity requirements, and attempt to satisfy those requirements by purchasing links in the network. We show a variety of results about equilibrium properties in such games, including the fact that the price of stability is 1 when all nodes in the network are owned by players, and that doubling the number of players creates an equilibrium as good as the optimum centralized solution. For the general case, we show the existence of a 2-approximate Nash equilibrium that is as good as the centralized optimum solution, as well as how to compute good approximate equilibria in polynomial time. Our results essentially imply that for a variety of connectivity requirements, giving agents more freedom can paradoxically result in more efficient outcomes.     

Computation of subcritical compressible flows

A finite element analysis and iterative solution of steady plane inviscid compressible flows is developed. Specific attention is directed to subcritical flows in which the nonlinear governing equation is elliptic, and to slightly supercritical mixed flows. The underlying variational theory for this nonlinear flow problem and a corresponding finite element formulation are presented. A Newton-Raphson iteration is introduced within this approximate analysis to provide efficient solution of subcritical flows and also slightly supercritical flows in which the nonlinear flow equation is of mixed type. The performance of the algorithm is studied with particular reference to the effect of a two-parameter scheme in which incident Mach number $\text{M}{}_{\mathrm{x}}$ and specific heat ratio γ may be varied. By thus regularizing the operator during the early iteration history, more efficient computations can be realized in the mildly-transonic flow regime of higher incident Mach numbers.     

The PCICE-FEM scheme for highly compressible axisymmetric flows

The recently developed PCICE-FEM scheme (Journal of Computational Physics, vol. 198, 659, 2004) is extended to two-dimensional axisymmetric geometries. The main discretization problem for nodal-based axisymmetric formulations lies in deriving a closed form as the radial coordinates approach zero along the axis of symmetry. This problem is addressed by employing the finite element piecewise linear approximations to both the flow variables and (separately) to the nodal values of the radial coordinates. The resulting formulation is an elegant treatment of the axisymmetric coordinate system with out noticeable loss of spatial accuracy and little additional cost in computational effort. An overview of the PCICE algorithm for the axisymmetric governing equations will be followed by a detailed axisymmetric finite element formulation for the PCICE-FEM scheme. The ability of the PCICE-FEM scheme to accurately and efficiently simulate highly compressible axisymmetric flows is demonstrated.     

High-order residual-based compact schemes for compressible inviscid flows

A residual-based compact scheme, previously developed to compute d-dimensional inviscid compressible flows with third-order accuracy on a 3^d-point stencil [Lerat A, Corre C. Residual-based compact schemes for multidimensional hyperbolic system of conservation laws. Comput Fluids 2002;31:639–61], is generalized to larger stencils allowing to reach a very high order of accuracy. Compactness is retained since for instance a seventh-order accurate dissipative approximation can be achieved on a 5^d-point stencil, without requiring the linear system solutions associated with usual compact schemes. The high-order generalization is also performed for unsteady flows. Applications to model problems and unsteady inviscid flows are presented.     

Exact and approximate flexible aggregate similarity search

Aggregate similarity search, also known as aggregate nearest-neighbor (Ann) query, finds many useful applications in spatial and multimedia databases. Given a group Q of M query objects, it retrieves from a database the objects most similar to Q, where the similarity is an aggregation (e.g., \({{\mathrm{sum}}}\), \(\max \)) of the distances between each retrieved object p and all the objects in Q. In this paper, we propose an added flexibility to the query definition, where the similarity is an aggregation over the distances between p and any subset of \(\phi M\) objects in Q for some support \(0< \phi \le 1\). We call this new definition flexible aggregate similarity search and accordingly refer to a query as a flexible aggregate nearest-neighbor ( Fann ) query. We present algorithms for answering Fann queries exactly and approximately. Our approximation algorithms are especially appealing, which are simple, highly efficient, and work well in both low and high dimensions. They also return near-optimal answers with guaranteed constant-factor approximations in any dimensions. Extensive experiments on large real and synthetic datasets from 2 to 74 dimensions have demonstrated their superior efficiency and high quality.     

Exact and approximate construction of offset polygons

The Minkowski sum of two sets $A,B\in {\mathbb{R}}^2$, denoted $A\oplus B$, is defined as $\{a+b\mid a\in A,b\in B\}$. We describe an efficient and robust implementation of the construction of the Minkowski sum of a polygon in ${\mathbb{R}}^2$ with a disc, an operation known as offsetting the polygon. Our software package includes a procedure for computing the exact offset of a straight-edge polygon, based on the arrangement of conic arcs computed using exact algebraic number-types. We also present a conservative approximation algorithm for offset computation that uses only rational arithmetic and decreases the running times by an order of magnitude in some cases, while having a guarantee on the quality of the result. The package will be included in the next public release of the Computational Geometry Algorithms Library, Cgal Version 3.3. It also integrates well with other Cgal packages; in particular, it is possible to perform regularized Boolean set-operations on the polygons the offset procedures generate.     

Exact and approximate solutions of the perspective-three-pointproblem

Model-based pose estimation techniques that match image and model triangles require large numbers of matching operations in real-world applications. The authors show that by using approximations to perspective, 2D lookup tables can be built for each of the triangles of the models. An approximation called `weak perspective' has been applied previously to this problem; the authors consider two other perspective approximations: paraperspective and orthoperspective. These approximations produce lower errors for off-center image features than weak perspective     

The extension of incompressible flow solvers to the weakly compressible regime

Numerical simulation schemes for incompressible flows such as the Simple scheme are extended to weakly compressible fluid flow. A single time scale, multiple space scale asymptotic analysis is used to gain insight into the limit behavior of the compressible flow equations as the Mach number vanishes. Motivated by these results, multiple pressure variables (MPV) are introduced into the numerical framework. These account separately for thermodynamic effects, acoustic wave propagation and the balance of forces. Discretized analogues of the averaging and large scale differencing procedures known from multiple scales asymptotics allow accurate capturing of various physical phenomena that are operative on very different length scales. The MPV approach combines the explicit numerical computation of global compression from the boundary and the long wavelength acoustics on coarse grids with an implicit pressure or pressure correction equation that formally converges to the corresponding incompressible one when the Mach number tends to zero.     

Preconditioned methods for simulations of low speed compressible flows

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid compressible flow at low speeds is formulated. The preconditioned system of equations are hyperbolic in time and remain well-conditioned in the incompressible limit. The preconditioning formulation is easily generalized to multicomponent/multiphase mixtures. When applying conservative methods to multicomponent flows with sharp fluid interfaces, nonphysical solution behavior is observed. This stimulated the authors to develop an alternative solution method based on the nonconservative form of the equations which does not generate the aforementioned nonphysical behavior. Before the results of the application of the nonconservative method to multicomponent flow problems is reported, the accuracy of the method on single component flows will be demonstrated. In this report a series of steady and unsteady inviscid flow problems are simulated using the nonconservative method and a well-known conservative scheme. It is demonstrated that the nonconservative method is both accurate and robust for smooth low speed flows, in comparison to its conservative counterpart.     

A Newton-like finite element scheme for compressible gas flows

Semi-implicit and Newton-like finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers, which is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. Numerical evidence for unconditional stability is presented. It is shown that the proposed approach offers higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense.