Moving Kriging reconstruction for high-order finite volume computation of compressible flows. (English) Zbl 1297.76110
Summary: This paper describes the development of a high-order finite volume method for the solution of compressible viscous flows on unstructured meshes. The novelty of this approach is based on the use of moving Kriging shape functions for the computation of the derivatives in the numerical flux reconstruction step at the cell faces. For each cell, the successive derivatives of the flow variables are deduced from the interpolation function constructed from a compact stencil support for both Gaussian and quartic spline correlation models. A particular attention is paid for the study of the influence of the correlation parameter onto the accuracy of the numerical scheme. The effect of the size of the moving Kriging stencil is also investigated. Robustness and convergence properties are studied for various inviscid and viscous flows. Results reveal that the moving Kriging shape function can be considered as an interesting alternative for the development of high-order methodology for complex geometries.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
finite-volume method; compressible flows; unstructured mesh; high-order reconstruction; moving Kriging interpolationSoftware:
DACEReferences:
| [1] | Wang, Z., High-order methods for the Euler and Navier-Stokes equations on unstructured grids, Prog. Aerosp. Sci., 43, 1-4 (2007) |
| [2] | Luo, H.; Baum, J.; Lohner, R., On the computation of steady-state compressible flows using a discontinuous Galerkin method, Int. J. Numer. Methods Engrg., 73, 597-623 (2008) · Zbl 1159.76023 |
| [3] | Krivodonova, L.; Berger, M., High-order accurate implementation of solid wall boundary conditions in curved geometries, J. Comput. Phys., 211, 492-512 (2006) · Zbl 1138.76403 |
| [5] | Godfrey, A.; Mitchell, C.; Walters, R., Practical aspects of spatially high-order accurate methods, AIAA J., 31, 1634-1642 (1993) · Zbl 0783.76070 |
| [7] | Delanaye, M.; Geuzaine, P.; Essers, J., The quadratic reconstruction finite volume scheme: an attractive sequel to linear reconstruction, used on unstructured adaptive meshes, Lect. Notes Phys., 490, 617-622 (1997) |
| [8] | Ollivier-Gooch, C.; Altena, C. V., A high-order accurate unstructured mesh finite-volume scheme for the advection-diffusion equation, J. Comput. Phys., 181, 729-752 (2002) · Zbl 1178.76251 |
| [10] | Nejat, A.; Ollivier-Gooch, C., A high-order accurate unstructured finite volume Newton-Krylov algorithm for inviscid compressible flows, J. Comput. Phys., 227, 2582-2609 (2008) · Zbl 1388.76183 |
| [11] | Ollivier-Gooch, C.; Nejat, A.; Michalak, K., Obtaining and verifying high-order unstructured finite volume solution to the Euler equations, AIAA J., 47, 2105-2120 (2009) |
| [12] | Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 114, 45-58 (1994) · Zbl 0822.65062 |
| [13] | Ollivier-Gooch, C., Quasi-ENO schemes for unstructured meshes based on unlimited data-dependent least-squares reconstruction, J. Comput. Phys., 133, 6-17 (1997) · Zbl 0899.76282 |
| [14] | Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids, J. Comput. Phys., 144, 194-212 (1998) · Zbl 1392.76048 |
| [15] | Hu, C.; Shu, C., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 97-127 (1999) · Zbl 0926.65090 |
| [16] | Shu, C.-W., High-order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD, Int. J. Comput. Fluid Dynam., 17, 107-118 (2003) · Zbl 1034.76044 |
| [17] | Dumbser, M.; Kaser, M.; Titarev, V.; Toro, E., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226, 204-243 (2007) · Zbl 1124.65074 |
| [18] | Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Math. Comput., 87, 141-158 (1981) · Zbl 0469.41005 |
| [19] | Liu, W.; Li, S.; Belytschko, T., Moving least-squares reproducing kernel methods. I. Methodology and convergence, Comput. Methods Appl. Mech. Engrg., 143, 113-154 (1997) · Zbl 0883.65088 |
| [20] | Cueto-Felgueroso, L.; Colominas, I.; Fe, J.; Navarrina, F.; Casteleiro, M., High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. application to shallow water dynamics, Int. J. Numer. Methods Engrg., 65, 295-331 (2006) · Zbl 1111.76032 |
| [21] | Nogueira, X.; Colominas, I.; Cueto-Felgueroso, L.; Khelladi, S., On the simulation of wave propagation with a higher-order finite volume scheme based on reproducing kernel methods, Comput. Methods Appl. Mech. Engrg., 199, 1471-1490 (2010) · Zbl 1231.76181 |
| [22] | Nogueira, X.; Colominas, I.; Cueto-Felgueroso, L.; Khelladi, S.; Navarrina, F.; Casteleiro, M., Resolution of computational aeroacoustics problems on unstructured grids with a higher-order finite volume scheme, J. Comput. Appl. Math., 234, 2089-2097 (2010) · Zbl 1402.76084 |
| [23] | Cueto-Felgueroso, L.; Colominas, I.; Nogueira, X.; Navarrina, F.; Casteleiro, M., Finite-volume solvers and moving least-squares approximations for the compressible Navier-Stokes equations on unstructured grids, Comput. Methods Appl. Mech. Engrg., 196, 4712-4736 (2007) · Zbl 1173.76358 |
| [24] | Nogueira, X.; Cueto-Felgueroso, L.; Colominas, I.; Gomez, H.; Navarrina, F.; Casteleiro, M., On the accuracy of finite volume and discontinuous Galerkin discretizations for compressible flow on unstructured grids, Int. J. Numer. Methods Engrg., 78, 1553-1584 (2009) · Zbl 1171.76426 |
| [25] | Most, T.; Bucher, C., A moving least squares weighting function for the element-free Galerkin method which almost fulfills essential boundary conditions, Struct. Engrg. Mech., 21, 315-332 (2005) |
| [26] | Most, T.; Bucher, C., New concepts for moving least squares: an interpolating non-singular weighting function and weighted nodal least squares, Engrg. Anal. Bound. Elem., 32, 461-470 (2008) · Zbl 1244.74228 |
| [28] | Sukumar, N.; Wright, R., Overview and construction of meshfree basis functions: from moving least squares to entropy approximants, Int. J. Numer. Methods Engrg., 70, 181-205 (2007) · Zbl 1194.65149 |
| [29] | Dai, K.; Liu, G.; Lim, K.; Gu, Y., Comparison between the radial point interpolation and the Kriging interpolation used in meshfree methods, Comput. Mech., 32, 60-70 (2003) · Zbl 1035.74059 |
| [30] | Hardy, R., Theory and applications of the multiquadrics-biharmonic method (20 years of discovery 1968-1988), Comput. Math. Appl., 19, 163-208 (1990) · Zbl 0692.65003 |
| [31] | Wang, J.; Liu, G., A point interpolation meshless method based on radial basis functions, Int. J. Numer. Methods Fluids, 54, 1623-1648 (2002) · Zbl 1098.74741 |
| [32] | Arroyo, M.; Ortiz, M., Local maximum entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, Int. J. Numer. Methods Engrg., 65, 2167-2202 (2006) · Zbl 1146.74048 |
| [33] | Stein, L., Interpolation of Spatial Data: Some Theory for Kriging (1999), Springer · Zbl 0924.62100 |
| [34] | Simpson, T.; Mauery, T.; Korte, J.; Mistree, F., Kriging models for global approximation in simulation-based multidisciplinary design optimization, AIAA J., 39, 2233-2241 (2001) |
| [35] | Sakataa, S.; Ashidaa, F.; Zakob, M., Structural optimization using Kriging approximation, Comput. Methods Appl. Mech. Engrg., 192, 923-939 (2003) · Zbl 1025.74024 |
| [36] | Gu, L., Moving Kriging interpolation and element-free Galerkin method, Int. J. Numer. Methods Engrg., 56, 1-11 (2003) · Zbl 1062.74652 |
| [37] | Li, H.; Wang, Q.; Lam, K., Computational mechanics, Comput. Methods Appl. Mech. Engrg., 193, 2599-2619 (2004) · Zbl 1067.74598 |
| [38] | Tongsuk, P.; Kanok-Nukulchai, W., Further investigation of elementfree Galerkin method using moving Kriging interpolation, Int. J. Comput. Methods, 1, 1-21 (2004) · Zbl 1179.74182 |
| [40] | Bui, T.; Nguyen, T.; Nguyen-Dang, H., A moving Kriging interpolation-based meshless method for numerical simulation of Kirchhoff plate problems, Int. J. Numer. Methods Engrg., 77, 1371-1395 (2009) · Zbl 1156.74391 |
| [41] | Wong, F.; Kanok-Nukulchai, W., Kriging-based finite element method: element-by-element Kriging interpolation, Civil Engrg. Dimension, 11, 1, 15-22 (2009) · Zbl 1264.74275 |
| [42] | Wong, F.; Syamsoeyadi, H., Kriging-based timoshenko beam element for static and free vibration analyses, Civil Engrg. Dimension, 13, 42-49 (2011) |
| [43] | Bui, T.; Nguyen, M., Eigenvalue analysis of thin plate with complicated shapes by a novel mesh-free method, Int. J. Appl. Mech., 3, 21-46 (2011) |
| [44] | Shaw, A.; Bendapudi, S.; Roy, D., A Kriging-based error-reproducing and interpolating kernel method for improved mesh-free approximations, Int. J. Numer. Methods Engrg., 73, 1434-1467 (2008) · Zbl 1262.74034 |
| [45] | Batten, P.; Clarke, N.; Lambert, C.; Causon, D., On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. Sci. Comput., 18, 1553-1570 (1997) · Zbl 0992.65088 |
| [46] | Batten, P.; Leschziner, M.; Goldberg, U., Average-state jacobians and implicit methods for compressible viscous and turbulent flows, J. Comput. Phys., 137, 38-78 (1997) · Zbl 0901.76043 |
| [47] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439 (1988) · Zbl 0653.65072 |
| [48] | Jawahar, P.; Kamath, H., A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids, J. Comput. Phys., 164, 165-203 (2000) · Zbl 0992.76063 |
| [49] | Khelladi, S.; Nogueira, X.; Bakir, F.; Colominas, I., Toward a higher-order unsteady finite volume solver based on reproducing kernel particle method, Comput. Methods Appl. Mech. Engrg., 200, 2348-2362 (2011) · Zbl 1230.76033 |
| [50] | Huerta, A.; Vidal, Y.; Villon, P., Pseudo-divergence-free element free galerkin method for incompressible fluid flow, computer methods in applied mechanics and engineering, Comput. Methods Appl. Mech. Engrg., 2004, 1119-1136 (2004) · Zbl 1060.76626 |
| [51] | Chenoweth, S.; Soria, J.; Ooi, A., A singularity-avoiding moving-least-squares scheme for two-dimensional unstructured meshes, J. Comput. Phys., 228, 5592-5619 (2009) · Zbl 1280.76036 |
| [52] | Nogueira, X.; Cueto-Felgueroso, L.; Colominas, I.; Navarrina, F.; Casteleiro, M., A new shock-capturing technique based on moving least squares for higher-order numerical schemes on unstructured grids, Comput. Methods Appl. Mech. Engrg., 199, 2544-2548 (2010) · Zbl 1231.76218 |
| [53] | Bassi, F.; Rebay, S., High-order accurate discontinuous finite element solution of the 2d Euler equations, J. Comput. Phys., 138, 251-285 (1997) · Zbl 0902.76056 |
| [55] | Dolejsi, V.; Feistauer, M., A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow, J. Comput. Phys., 198, 727-746 (2004) · Zbl 1116.76386 |
| [59] | Delanaye, M.; Essers, J., Polynomial reconstruction finite volume scheme for compressible flows on unstructured adaptive grids, AIAA J., 35, 631-639 (1997) · Zbl 0903.76067 |
| [60] | Venkatakrishnan, V.; Mavriplis, D., Implicit method for the computation of unsteady flows on unstructured grids, J. Comput. Phys., 127, 380-397 (1996) · Zbl 0862.76054 |
| [61] | Berland, J.; Bogey, C.; Bailly, C., Low-dissipation and low-dispersion fourth-order Runge-Kutta algorithm, Comput. Fluids, 35, 10, 1459-1463 (2006) · Zbl 1177.76252 |
| [62] | Tam, C.; Webb, J., Dispersion-relation-preserving finite difference schemes for computational aeroacoustics, J. Comput. Phys., 107, 262-281 (1993) · Zbl 0790.76057 |
| [65] | Breuer, M., Large eddy simulation of the subcritical flow past a circular cylinder: numerical and modeling aspects, Int. J. Numer. Methods Fluids, 28, 1281-1302 (1998) · Zbl 0933.76041 |
| [66] | Son, J.; Hanratty, T., Velocity gradients at the wall for flow around a cylinder at Reynolds numbers from \(5 \times 10^3\) to \(10^5\), J. Fluid Mech., 35, 353-368 (1969) |
| [67] | Kravchenko, A. G.; Moin, P., Numerical studies of flow over a circular cylinder at \(Re_d = 3900\), Phys. Fluids, 12, 2, 403-417 (2000) · Zbl 1149.76441 |
| [68] | Nogueira, X.; Cueto-Felgueroso, L.; Colominas, I.; Gómez, H., Implicit large eddy simulation of non-wall-bounded turbulent flows based on the multiscale properties of a high-order finite volume method, Comput. Methods Appl. Mech. Engrg., 199, 9-12, 615-624 (2010) · Zbl 1227.76021 |
| [69] | Khelladi, S.; Nogueira, X.; Bakir, F.; Colominas, I., Toward a higher order unsteady finite volume solver based on reproducing kernel methods, Comput. Methods Appl. Mech. Engrg., 200, 29-32, 2348-2362 (2011) · Zbl 1230.76033 |
| [70] | Visbal, M. R.; Rizzeta, D. P., Large-eddy simulation on curvilinear grids using compact differencing and filtering schemes, J. Fluids Engrg., 124, 836-847 (2002) |
| [71] | Spyropoulos, E. T.; Blaisdell, G. A., Evaluation of the dynamic model for simulations of compressible decaying isotropic turbulence, AIAA J., 34, 990-998 (1996) · Zbl 0926.76055 |
| [72] | Sarkar, S.; Erlebacher, G.; Hussaini, M. Y.; Kreiss, H. O., The analysis and modelling of dilatational terms in compressible turbulence, J. Fluid Mech., 227, 473-493 (1991) · Zbl 0721.76037 |
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