Chris Lacor

On the Stability and Accuracy of the Spectral Difference Method

In this article, it is shown that under certain conditions, the spectral difference (SD) method is independent of the position of the solution points. This greatly simplifies the design of such schemes, and it also offers the possibility of a significant increase in the efficiency of the method. Furthermore, an accuracy and stability study, based on wave propagation analysis, is presented for several 1D and 2D SD schemes. It was found that higher than second-order 1D SD schemes using the Chebyshev–Gauss–Lobatto nodes as the flux points have a weak instability. New flux points were identified which produce accurate and stable SD schemes. In addition, a weak instability was also found in 2D third- and fourth-order SD schemes on triangular grids. Several numerical tests were performed to verify the analysis.

An accuracy and stability study of the 2D spectral volume method

In this article, the accuracy and the stability of 2D spectral volume schemes are studied by means of an analysis of the wave propagation properties. It is shown that several SV partitions suffer from a weak instability. Stable schemes with lower dispersion and diffusion errors are proposed. Numerical tests show an important improvement in the accuracy of the fourth-order scheme.

Dispersion and dissipation properties of the 1D spectral volume method and application to a p-multigrid algorithm

In this article, the wave propagation properties of the 1D spectral volume method are studied through analysis of the Fourier footprint of the schemes. A p-multigrid algorithm for the spectral volume method is implemented. Restriction and prolongation operators are discussed and the efficiency of the smoothing operators is analyzed. The results are verified for simple 1D advection problems and for a quasi-1D Euler flow. It is shown that a significant decrease in computational effort is possible with the p-multigrid algorithm.

Convection algorithms based on a diagonalization procedure for the multidimensional Euler equations

Convection algorithms for the multidimensional Euler equations are presented based on characteristic propagation properties, with the aim of following closely the physical transfer of information. As opposed to the one dimensional case, information in two or three dimensional Euler flows is propagated in infinitely many directions, each one corresponding to an arbitrary wave front normal. It is shown that a complete decoupling (diagonalization) of the Euler equations can be obtained by an appropriate choice of wave front propagation directions. Actually,two directions are sufficient, one related to the pressure gradient and another related to the local strain tensor. This new formulation allows the definition of numerical schemes with upwind properties depending only on the local flow properties and not on the mesh geometry.

Efficient Large-eddy Simulations of Low Mach Number Flows Using Preconditioning and Multigrid

In the present paper, an implicit time accurate approach combined with multigrid, preconditioning and residual smoothing is used for the large-eddy simulation (LES) of low Mach number flow. In general, due to the restriction imposed on the time step by the physics of the flow, the advantage of an implicit method over an explicit one for LES is not obvious. It is shown that for the test cases considered in this paper, the present approach allows an efficiency gain of a factor 4–7 compared to the use of a purely explicit approach. The efficiency varies according to the test case, grid clustering, physical time step and requested residual drop. Numerical difficulties are catalogued and mitigatory procedures are introduced. Several problems with available experimental and DNS data are employed to verify the efficiency of the method.

Short Note: On the connection between the spectral volume and the spectral difference method

In the last few years, two new high-order accurate methods for unstructured grids have been developed, namely the spectral volume (SV) and the spectral difference (SD) method. Both methods are related to the well-known discontinuous Galerkin (DG) method, see for instance [1], in the sense that they also use piecewise continuous polynomials as solution approximation space. The development of the SV method was mainly reported in publications by Wang [2], Wang et al. [3–5,8], Liu et al. [6] and Sun et al. [7]. Further contributions were made by Chen [12,13], Van den Abeele et al. [14] and Van den Abeele and Lacor [15]. The development of the SD method was reported more recently in publications by Liu et al. [9] and Wang et al. [10]. In 1D, the SD method is identical to the multi-domain spectral method proposed by Kopriva [11]. The SV method is strongly related to the finite volume (FV) method. As with the FV method, the SV solution variables are averaged values over control volumes (CVs) and the residuals corresponding to these solution variables can be written as the sum of the fluxes through the CV faces. The SV method differs from the FV method in the choice of the stencils used to approximate the fluxes through these CV faces. By partitioning each cell, or spectral volume (SV), into CVs in a similar way, a unique stencil, valid for all cells in the mesh, can be obtained. This is the major advantage of the SV method over high-order accurate unstructured FV methods, where these stencils depend on the local mesh geometry. At faces between two SVs, a Riemann solver is used to compute a unique flux from the two solutions in the neighbouring SVs. Similarly, the SD method is strongly related to the finite difference (FD) method. For both the FD and the SD method, the solution variables are pointwise values, for which the residuals can be written as a function of the flux derivatives in the solution points. With the SD method the computational domain is subdivided in

Robust parameter design optimization using Kriging, RBF and RBFNN with gradient-based and evolutionary optimization techniques

Abstract The dual response surface methodology is one of the most commonly used approaches in robust parameter design to simultaneously optimize the mean value and keep the variance minimum. The commonly used meta-model is the quadratic polynomial regression. For highly nonlinear input/output relationship, the accuracy of the fitted model is limited. Many researchers recommended to use more complicated surrogate models. In this study, three surrogate models will replace the second order polynomial regression, namely, ordinary Kriging, radial basis function approximation (RBF) and radial basis function artificial neural network (RBFNN). The results show that the three surrogate model present superior accuracy in comparison with the quadratic polynomial regression. The mean squared error (MSE) approach is widely used to link the mean and variance in one cost function. In this study, a new approach has been proposed using multi-objective optimization. The new approach has two main advantages over the classical method. First, the conflicting nature of the two objectives can be efficiently handled. Second, the decision maker will have a set of Pareto-front design points to select from.

Acoustic Velocity Formulation for Sources in Arbitrary Motion

This paper deals with the acoustic velocity field simulation generated by interaction of flow with moving bodies. Starting from the Ffowcs Williams and Hawkings equation, an analytical formulation of the acoustic velocity is derived for sources in arbitrary motion. This makes the imposition of the boundary condition on a (rigid) scattering surface much more straightforward, as, if the traditional pressure formulation is used, then the pressure gradient must be calculated. Computational results for a pulsating sphere, dipole source, and a propeller case with subsonic tips verify this formulation.

Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier-Stokes equations

The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower-upper symmetric Gauss-Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier-Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a p-multigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier-Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge-Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5-10 compared with a well tuned E-RK scheme.

Optimization of time integration schemes coupled to spatial discretization for use in CAA applications

In this paper errors arising from spatial and temporal discretization are discussed. A new formulation to optimize time integration schemes is proposed. The new optimization also takes the errors coming from the spatial discretization into account, which leads to a minimization of the total errors. It is applicable to central and upwind type spatial discretizations. A six stage Runge-Kutta scheme is optimized for central and upwind spatial discretizations. Its efficiency is demonstrated on a one dimensional convection equation and on a Linearized Euler problem. The results indicate an important improvement.