0
Research Papers

Implementation and Assessment of a Residual-Based r-Adaptation Technique on Structured Meshes

[+] Author and Article Information
Aniruddha Choudhary

Department of Aerospace and
Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: anirudd@vt.edu

William C. Tyson, Christopher J. Roy

Department of Aerospace and
Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24061

1Corresponding author.

Manuscript received November 29, 2018; final manuscript received April 29, 2019; published online May 13, 2019. Assoc. Editor: Kevin Dowding.

J. Verif. Valid. Uncert 3(4), 041005 (May 13, 2019) (17 pages) Paper No: VVUQ-18-1036; doi: 10.1115/1.4043652 History: Received November 29, 2018; Revised April 29, 2019

In this study, an r-adaptation technique for mesh adaptation is employed for reducing the solution discretization error, which is the error introduced due to spatial and temporal discretization of the continuous governing equations in numerical simulations. In r-adaptation, mesh modification is achieved by relocating the mesh nodes from one region to another without introducing additional nodes. Truncation error (TE) or the discrete residual is the difference between the continuous and discrete form of the governing equations. Based upon the knowledge that the discrete residual acts as the source of the discretization error in the domain, this study uses discrete residual as the adaptation driver. The r-adaptation technique employed here uses structured meshes and is verified using a series of one-dimensional (1D) and two-dimensional (2D) benchmark problems for which exact solutions are readily available. These benchmark problems include 1D Burgers equation, quasi-1D nozzle flow, 2D compression/expansion turns, and 2D incompressible flow past a Karman–Trefftz airfoil. The effectiveness of the proposed technique is evident for these problems where approximately an order of magnitude reduction in discretization error (when compared with uniform mesh results) is achieved. For all problems, mesh modification is compared using different schemes from literature including an adaptive Poisson grid generator (APGG), a variational grid generator (VGG), a scheme based on a center of mass (COM) analogy, and a scheme based on deforming maps. In addition, several challenges in applying the proposed technique to real-world problems are outlined.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Eiseman, P. R. , 1987, “ Adaptive Grid Generation,” Comput. Methods Appl. Mech. Eng., 64(1–3), pp. 321–376.
Anderson, D. A. , 1987, “ Equidistribution Schemes, Poisson Generators, and Adaptive Grids,” Appl. Math. Comput., 24(3), pp. 211–227.
Hawken, D. , Gottlieb, J. , and Hansen, J. , 1991, “ Review of Some Adaptive Node-Movement Techniques in Finite-Element and Finite-Difference Solutions of Partial Differential Equations,” J. Comput. Phys., 95(2), pp. 254–302.
Baker, T. J. , 1997, “ Mesh Adaptation Strategies for Problems in Fluid Dynamics,” Finite Elem. Anal. Des., 25(3–4), pp. 243–273.
Dwight, R. P. , 2008, “ Heuristic a Posteriori Estimation of Error Due to Dissipation in Finite Volume Schemes and Application to Mesh Adaptation,” J. Comput. Phys., 227(5), pp. 2845–2863.
Ainsworth, M. , and Oden, J. , 2000, A Posteriori Error Estimation in Finite Element Analysis, Wiley Interscience, New York.
Roy, C. J. , 2009, “ Strategies for Driving Mesh Adaptation in CFD,” AIAA Paper No. 2009-1302.
Zhang, X. , Trepanier, J.-Y. , and Camarero, R. , 2000, “ A Posteriori Error Estimation for Finite-Volume Solutions of Hyperbolic Conservation Laws,” Comput. Methods Appl. Mech. Eng., 185(1), pp. 1–19.
Gu, X. , and Shih, T. , 2001, “ Differentiating Between Source and Location of Error for Solution-Adaptive Mesh Refinement,” AIAA Paper No. 2001–2660.
Zienkiewicz, O. C. , and Zhu, J. Z. , 1992, “ The Superconvergent Patch Recovery and a Posteriori Error Estimates—Part 2: Error Estimates and Adaptivity,” Int. J. Numer. Methods Eng., 33(7), pp. 1365–1382.
Laflin, K. R. , 1997, “ Solver-Independent r-Refinement Adaptation for Dynamic Numerical Simulations,” Ph.D. thesis, North Carolina State University, Raleigh, NC.
McRae, D. S. , 2000, “ r-Refinement Grid Adaptation Algorithms and Issues,” Comput. Methods Appl. Mech. Eng., 189(4), pp. 1161–1182.
Venditti, D. A. , and Darmofal, D. L. , 2000, “ Adjoint Error Estimation and Grid Adaptation for Functional Outputs: Application to Quasi-One-Dimensional Flow,” J. Comput. Phys., 164(1), pp. 204–227.
Venditti, D. A. , and Darmofal, D. L. , 2002, “ Grid Adaptation for Functional Outputs: Application to Two-Dimensional Inviscid Flows,” J. Comput. Phys., 176(1), pp. 40–69.
Fidkowski, K. J. , and Darmofal, D. L. , 2011, “ Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics,” AIAA J., 49(4), pp. 673–694.
Oberkampf, W. L. , and Roy, C. J. , 2010, Verification and Validation in Scientific Computing, Cambridge University Press, Cambridge, UK.
Borsboom, M. , 1998, “ Development of an Error-Minimizing Adaptive Grid Method,” Appl. Numer. Math., 26(1–2), pp. 13–21.
Sod, G. A. , 1978, “ A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws,” J. Comput. Phys., 27(1), pp. 1–31.
Klopfer, G. H. , and McRae, D. S. , 1983, “ Nonlinear Truncation Error Analysis of Finite Difference Schemes for the Euler Equations,” AIAA J., 21(4), pp. 487–494.
Fraysse, F. , Rubio, G. , de Vicente, J. , and Valero, E. , 2014, “ Quasi-a Priori Mesh Adaptation and Extrapolation to Higher Order Using τ-Estimation,” Aerosp. Sci. Technol., 38, pp. 76–87.
Tyson, W. C. , 2015, “ Application of r-Adaptation Techniques for Discretization Error Improvement in CFD,” Master's thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA. https://vtechworks.lib.vt.edu/handle/10919/78061
Phillips, T. S. , Derlaga, J. M. , Roy, C. J. , and Borggaard, J. , 2017, “ Error Transport Equation Boundary Conditions for the Euler and Navier–Stokes Equations,” J. Comput. Phys., 330, pp. 46–64.
Russell, R. D. , and Christiansen, J. , 1978, “ Adaptive Mesh Selection Strategies for Solving Boundary Value Problems,” SIAM J. Numer. Anal., 15(1), pp. 59–80.
Kautsky, J. , and Nichols, N. , 1980, “ Equidistributing Meshes With Constraints,” SIAM J. Sci. Stat. Comput., 1(4), pp. 499–511.
Anderson, D. A. , 1990, “ Grid Cell Volume Control With an Adaptive Grid Generator,” Appl. Math. Comput., 35(3), pp. 209–217.
Winslow, A. , 1966, “ Numerical Solution of the Quasi-Linear Poisson Equation,” J. Comput. Phys., 1(2), pp. 149–172.
Brackbill, J. , and Saltzman, J. , 1982, “ Adaptive Zoning for Singular Problems in Two Dimensions,” J. Comput. Phys., 46(3), pp. 342–368.
Brackbill, J. , 1993, “ An Adaptive Grid With Directional Control,” J. Comput. Phys., 108(1), pp. 38–50.
Huang, W. , and Russell, R. , 2011, Adaptive Moving Mesh Methods, Springer, New York.
Benson, R. , and McRae, D. , 1991, “ A Solution-Adaptive Mesh Algorithm for Dynamic/Static Refinement of Two and Three Dimensional Grids,” Third International Conference on Numerical Grid Generation in Computational Fluid Dynamics and Related Fields, Barcelona, Spain, June 3--7, pp. 185–199.
Liao, G. , and Anderson, D. , 1992, “ A New Approach to Grid Generation,” Appl. Anal., 44(3–4), pp. 285–298.
Moser, J. , 1965, “ On the Volume Elements on a Manifold,” Trans. Am. Math. Soc., 120(2), pp. 286–294.
Grisham, J. R. , Vijayakumar, N. , Liao, G. , Dennis, B. H. , and Lu, F. K. , 2015, “ A Comparison Between Local h-Refinement and a Novel r-Refinement Method,” AIAA Paper No. 2015-2040.
Choudhary, A. , 2014, “ Verification of Compressible and Incompressible Computational Fluid Dynamics Codes and Residual-Based Mesh Adaptation,” Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA. https://vtechworks.lib.vt.edu/handle/10919/51169
Roe, P. , 1997, “ Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,” J. Comput. Phys., 135(2), pp. 250–258.
van Leer, B. , 1979, “ Towards the Ultimate Conservative Difference Scheme—V: A Second-Order Sequel to Godunov's Method,” J. Comput. Phys., 32(1), pp. 101–136.
Derlaga, J. M. , Phillips, T. , and Roy, C. J. , 2013, “ SENSEI Computational Fluid Dynamics Code: A Case Study in Modern Fortran Software Development,” AIAA Paper No. 2013-2450.
van Leer, B. , 1982, “ Flux-Vector Splitting for the Euler Equations,” Eighth International Conference on Numerical Methods in Fluid Dynamics, Aachen, Germany, June 28–July 2, pp. 507–512.
Beam, R. M. , and Warming, R. , 1976, “ An Implicit Finite-Difference Algorithm for Hyperbolic Systems in Conservation-Law Form,” J. Comput. Phys., 22(1), pp. 87–110.
Banks, J. , Aslam, T. , and Rider, W. , 2008, “ On Sub-Linear Convergence for Linearly Degenerate Waves in Capturing Schemes,” J. Comput. Phys., 227(14), pp. 6985–7002.
Karamcheti, K. , 1966, Principles of Ideal-Fluid Aerodynamics, Krieger Publishing Company, Malabar, FL.
Vassberg, J. C. , and Jameson, A. , 2010, “ In Pursuit of Grid Convergence for Two-Dimensional Euler Solutions,” J. Aircr., 47(4), pp. 1152–1166.
Barth, T. , 1993, “ Recent Developments in High Order k-Exact Reconstruction on Unstructured Meshes,” AIAA Paper No. 93-0668.
Sharbatdar, M. , and Ollivier-Gooch, C. , 2018, “ Mesh Adaptation Using C1 Interpolation of the Solution in an Unstructured Finite Volume Solver,” Int. J. Numer. Methods Fluids, 86(10), pp. 637–654.
Jackson, C. W. , and Roy, C. J. , 2015, “ A Multi-Mesh CFD Technique for Adaptive Mesh Solutions,” AIAA Paper No. 2015-1958.

Figures

Grahic Jump Location
Fig. 1

Elemental step for 1D equidistribution of a function, W(x), using mesh node relocation

Grahic Jump Location
Fig. 2

Local cell cluster (reproduced with permission from K.R. Laflin [11])

Grahic Jump Location
Fig. 3

Exact solutions for 1D test cases. (a) 1D Burgers equation (Re = 128), and ((b)–(d)) quasi-1D nozzle flow.

Grahic Jump Location
Fig. 4

Adaptation for 1D Burgers equation with Re = 128 and N =65 using different adaptation drivers. (a) TE (enlarged view: −2 ≤ x 2), (b) DE (enlarged view: −2 ≤ x 2), and (c) node spacing on adapted meshes (enlarged view: −2 ≤ x 2).

Grahic Jump Location
Fig. 5

Adaptation for quasi-1D nozzle flow with N =65 nodes using TE as the adaptation driver. (a) Node spacing on the adapted mesh, (b) TE on uniform versus adapted mesh, and (c) DE (in %) obtained on uniform versus adapted mesh.

Grahic Jump Location
Fig. 6

Driver and mesh size comparisons for quasi-1D nozzle flow. (a) Node spacing on adapted meshes obtained using combined TE versus TE from individual governing equations, and (b) adapted meshes for different mesh sizes using combined TE as driver.

Grahic Jump Location
Fig. 7

Expansion fan case description. (a) Computational domain and boundary conditions, and (b) Mach contours for exact solution.

Grahic Jump Location
Fig. 8

Density DE at expansion fan root (N =65 × 65). (a) APGG, (b) VGG, (c) COM, (d) deformation schemes, and (e) uniform grid.

Grahic Jump Location
Fig. 9

DE distribution and grid size effects. (a) Density DE along x =0.5 slice (N =65 × 65). (b) Convergence of pressure DE.

Grahic Jump Location
Fig. 10

Oblique shock case description. (a) Computational domain and boundary conditions, and (b) Mach contours for exact solution.

Grahic Jump Location
Fig. 11

Adapted grids and pressure DE for oblique shock case (N =129 × 65). Full domain grid (left) and enlarged view of shock root (right): ((a) and (b)) uniform grid, ((c) and (d)) APGG scheme, and ((e) and (f)) COM scheme.

Grahic Jump Location
Fig. 12

Convergence of pressure DE with grid refinement

Grahic Jump Location
Fig. 13

Karman–Trefftz airfoil mapping

Grahic Jump Location
Fig. 14

Karman–Trefftz airfoil case description. (a) Computational domain and boundary conditions, and (b) Mach contours for exact solution.

Grahic Jump Location
Fig. 15

Adapted grids and u-velocity DE for Karman–Trefftz airfoil case (N =129 × 65). Leading edge (left) and trailing edge (right): ((a) and (b)) uniform grid, ((c) and (d)) APGG scheme, and ((e) and (f)) COM scheme.

Grahic Jump Location
Fig. 16

Convergence of pressure DE with grid refinement for Karman–Trefftz airfoil case

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In