ACTA AERONAUTICAET ASTRONAUTICA SINICA >
Adaptive finite volume method with Walsh basis functions
Received date: 2022-05-15
Revised date: 2022-07-01
Accepted date: 2022-07-27
Online published: 2022-08-08
Supported by
National Natural Science Foundation of China(U2141254)
The Finite Volume Method with Walsh Basis Functions （FVM-WBF method） is a novel numerical method with the ability to capture discontinuity inside grids. While globally increasing the number of Walsh basis functions can effectively improve the numerical resolution， it also leads to a large increase in computational costs. To balance the resolution and the computational efficiency of the FVM-WBF method， an adaptive finite volume method with Walsh basis functions is proposed according to the numerical properties of the Walsh basis functions. The proposed method dynamically adjusts the number of Walsh basis functions in the grid based on the features of the flow field. Sufficient basis functions are employed only in the local region where the flow field structure changes dramatically， so as to avoid the explosive growth of the computation caused by the global increase of the basis functions. Several cases are selected to test the adaptive FVM-WBF method in comparison with the original FVM-WBF method， including two-dimensional Double Mach reflection problem， Rayleigh-Taylor instability problem and the flow over NACA0012 airfoil. Theoretical analysis and numerical results show that the adaptive FVM-WBF method has the inherent capability of convenient dynamic adaptation， and a balance between high resolution and high efficiency in the numerical simulations has been achieved by intelligently adapting the number of Walsh basis functions in the flow field.
Jiong REN , Gang WANG , Guodong HU , Xiaolu SHI . Adaptive finite volume method with Walsh basis functions[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2023 , 44(8) : 127444 -127444 . DOI: 10.7527/S1000-6893.2022.27444
| 1 | 阎超. 航空CFD四十年的成就与困境［J］. 航空学报， 2022， 43（10）： 526490. |
| YAN C. Achievements and predicaments of CFD in aeronautics in past forty years［J］. Acta Aeronautica et Astronautica Sinica， 2022， 43（10）： 526490 （in Chinese）. | |
| 2 | BALAN A， PARK M A， WOOD S， et al. Verification of anisotropic mesh adaptation for complex aerospace applications［C］∥ AIAA Scitech 2020 Forum. Reston： AIAA， 2020. |
| 3 | 张扬， 张来平， 赫新， 等. 基于自适应混合网格的脱体涡模拟［J］. 航空学报， 2016， 37（12）： 3605-3614. |
| ZHANG Y， ZHANG L P， HE X， et al. Detached eddy simulation based on adaptive hybrid grids［J］. Acta Aeronautica et Astronautica Sinica， 2016， 37（12）： 3605-3614 （in Chinese）. | |
| 4 | LISEIKIN V D. The construction of structured adaptive grids—A review［J］. Computational Mathematics and Mathematical Physics， 1996， 36（1）： 1-32. |
| 5 | PAN J H， WANG Q， ZHANG Y S， et al. High-order compact finite volume methods on unstructured grids with adaptive mesh refinement for solving inviscid and viscous flows［J］. Chinese Journal of Aeronautics， 2018， 31（9）： 1829-1841. |
| 6 | HAMFELDT B F， SALVADOR T. Higher-order adaptive finite difference methods for fully nonlinear elliptic equations［J］. Journal of Scientific Computing， 2018， 75（3）： 1282-1306. |
| 7 | 唐静， 崔鹏程， 贾洪印， 等. 非结构混合网格鲁棒自适应技术［J］. 航空学报， 2019， 40（10）： 122894. |
| TANG J， CUI P C， JIA H Y， et al. Robust adaptation techniques for unstructured hybrid mesh［J］. Acta Aeronautica et Astronautica Sinica， 2019， 40（10）： 122894 （in Chinese）. | |
| 8 | BIGONI C， HESTHAVEN J S. Adaptive WENO methods based on radial basis function reconstruction［J］. Journal of Scientific Computing， 2017， 72（3）： 986-1020. |
| 9 | 唐志共， 陈浩， 毕林， 等. 自适应笛卡尔网格超声速黏性流动数值模拟［J］. 航空学报， 2018， 39（5）： 121697. |
| TANG Z G， CHEN H， BI L， et al. Numerical simulation of supersonic viscous flow based on adaptive Cartesian grid［J］. Acta Aeronautica et Astronautica Sinica， 2018， 39（5）： 121697 （in Chinese）. | |
| 10 | 陈浩， 袁先旭， 王田天， 等. 国家数值风洞（NNW）工程中的黏性自适应笛卡尔网格方法研究进展［J］. 航空学报， 2021， 42（9）： 625732. |
| CHEN H， YUAN X X， WANG T T， et al. Advances in viscous adaptive Cartesian grid methodology of NNW Project［J］. Acta Aeronautica et Astronautica Sinica， 2021， 42（9）： 625732 （in Chinese）. | |
| 11 | 陈浩， 华如豪， 袁先旭， 等. 基于自适应笛卡尔网格的飞翼布局流动模拟［J］. 航空学报， 2022， 43（8）： 125674. |
| CHEN H， HUA R H， YUAN X X， et al. Simulation of flow around fly-wing configuration based on adaptive Cartesian grid［J］. Acta Aeronautica et Astronautica Sinica， 2022， 43（8）： 125674 （in Chinese）. | |
| 12 | CHO M， JUN S. r-Adaptive mesh generation for shell finite element analysis［J］. Journal of Computational Physics， 2004， 199（1）： 291-316. |
| 13 | AMEUR F BEN， BALIS J， VANDENHOECK R， et al. R-adaptive algorithms for supersonic flows with high-order flux reconstruction methods［J］. Computer Physics Communications， 2022， 276： 108373. |
| 14 | LI W Z， LUO H， PANDARE A， et al. A p-adaptive discontinuous Galerkin method for compressible flows using charm++［C］∥ AIAA Scitech 2020 Forum. Reston： AIAA， 2020. |
| 15 | BASSI F， BOTTI L， COLOMBO A， et al. A p-adaptive matrix-free discontinuous Galerkin method for the implicit LES of incompressible transitional flows［J］. Flow， Turbulence and Combustion， 2020， 105（2）： 437-470. |
| 16 | XIA M T， SHAO S H， CHOU T. A frequency-dependent p-adaptive technique for spectral methods［J］. Journal of Computational Physics， 2021， 446： 110627. |
| 17 | PANOURGIAS K， EKATERINARIS J A. Three-dimensional discontinuous Galerkin h/p adaptive numerical solutions for compressible flows［C］∥ 53rd AIAA Aerospace Sciences Meeting. Reston： AIAA， 2015. |
| 18 | NTOUKAS G， MANZANERO J， RUBIO G， et al. An entropy-stable p-adaptive nodal discontinuous Galerkin for the coupled Navier-Stokes/Cahn-Hilliard system［J］. Journal of Computational Physics， 2022， 458： 111093. |
| 19 | 任炯， 王刚. 一种在网格内部捕捉间断的Walsh函数有限体积方法［J］. 力学学报， 2021， 53（3）： 773-788. |
| REN J， WANG G. A finite volume method with Walsh basis functions to capture discontinuity inside grid［J］. Chinese Journal of Theoretical and Applied Mechanics， 2021， 53（3）： 773-788 （in Chinese）. | |
| 20 | REN J， WANG G. A Walsh-function-based finite volume method to capture discontinuity inside grid cell［C］∥ AIAA Aviation 2020 Forum. Reston： AIAA， 2020. |
| 21 | GNOFFO P A. Global series solutions of nonlinear differential equations with shocks using Walsh functions［J］. Journal of Computational Physics， 2014， 258： 650-688. |
| 22 | WALSH J L. A closed set of normal orthogonal functions［J］. American Journal of Mathematics， 1923， 45 （1）： 5-24. |
| 23 | REN J， WANG G， MA B P. Multidimensional extension and application of entropy-consistent scheme for navier-stokes equations on unstructured grids［C］∥ 23rd AIAA Computational Fluid Dynamics Conference. Reston： AIAA， 2017. |
| 24 | ISMAIL F， ROE P L. Affordable， entropy-consistent Euler flux functions II： Entropy production at shocks［J］. Journal of Computational Physics， 2009， 228（15）： 5410-5436. |
| 25 | REN J， WANG G， MA M S. A group of CFL-dependent flux-limiters to control the numerical dissipation in Multi-stage unsteady calculation［J］. Journal of Scientific Computing， 2019， 81（1）： 186-216. |
| 26 | GOTTLIEB S， KETCHESON D I， SHU C W. High order strong stability preserving time discretizations［J］. Journal of Scientific Computing， 2009， 38（3）： 251-289. |
| 27 | SHI J， ZHANG Y T， SHU C W. Resolution of high order WENO schemes for complicated flow structures［J］. Journal of Computational Physics， 2003， 186（2）： 690-696. |
| 28 | LIU Y L， ZHANG W W， ZHENG X B. An accuracy preserving limiter for the high-order discontinuous Galerkin method on unstructured grids［J］. Computers & Fluids， 2019， 192： 104253. |
/
| 〈 |
|
〉 |
CJA