Discretization Error Reduction through r Adaptation in CFD

Transcription

1 Discretization Error Reduction through r Adaptation in CFD ASME Verification and Validation Symposium May 13 15, 2015 William C. Tyson, Christopher J. Roy, Joseph M. Derlaga, and Charles W. Jackson Aerospace and Ocean Engineering Department, Virginia Tech

2 Outline Introduction Adaptation Metrics & Weight Function Formulation Adaptation Schemes: Anderson s Adaptive Poisson Grid Generator Brackbill & Saltzman s Variational Grid Generator Center of Mass Deformation Map Method Test Cases: 2D Expansion Fan Karman Trefftz Airfoil Results Conclusions & Future Work 2

3 Introduction: Adaptation h Adaptation Cells subdivided to locally increase resolution Number of cells is not constant Unstructured p Adaptation Primarily used with FEM codes Local order of accuracy increased or decreased to resolve solution Structured/ unstructured r Adaptation Nodes in given mesh are relocated in the domain Number of cells remains constants Structured/ unstructured 3

4 Motivation What should be used to drive adaptation? Flow Features? Example: Pressure gradients across a shock wave Assuming error is being generated at flow feature; Not always the case This has also been shown to cause flow features to set up in incorrect locations Error estimate? Discretization error Difficult to estimate Has been shown to not get as good of results as other types of error Truncation Error! 4

5 Motivation Generalized Truncation Error Expression : The original differential equation: 0 The discretized equation: 0 Discretization Error : 0 1) Substitute numerical solution 2) Subtract = 0 3) If linear (or linearized) ε Roy,

6 ADAPTATION SCHEMES 6

7 Adaptive Poisson Grid Generator (Anderson 1987) Goal: Approximately equidistribute a given weight function across the domain Extends traditional elliptic grid generator to become adaptive Uses control functions, P and Q below, as a source term to control grid adaptivity Easily implemented by adding source term to original grid generation equations Anderson, D. Equidistribution Schemes, Poisson Grid Generators, and Adaptive Grids. Applied Mathematics and Computation, Vol. 24, 1987, pp

8 Variational Grid Generator (Brackbill & Saltzman 1982) Goal: Optimize grid quality and equidistribute weight function across domain Linearly combine functionals for smoothness, orthogonality, and weighted cell volume variation Derive Euler Lagrange equations to minimize the combination of the functionals λ λ : : : Brackbill, J and Saltzman, J. Adaptive Zoning for Singular Problem in Two Dimensions. Journal of Computational Physics, Vol. 46, 1982, pp

9 Center of Mass (Laflin 1997) Goal: Equidistribute a weight function across the domain No guarantee on mesh smoothness Equidistributes weight function by relocating nodes to the center of mass of the surrounding cells Laflin, K. R. Solver Independent r Refinement Adaptation for Dynamic Numerical Simulations. Doctoral Thesis, North Carolina State University,

10 Deformation Map Method (Liao et al. 1992) Goal: Specify cell volume variation exactly by prescribing the Jacobian of the transformation as the weight function Requires the solution of a Poisson equation for a potential function Gradient of potential function defines the mesh velocity in each direction Liao, G. and Su, J. Grid Generation via Deformation. Appl. Math. Lett. Vol. 5, No. 3, pp ,

11 Weight Function Formulation Since truncation error for governing equations is of different magnitudes, it is necessary to combine them in a consistent manner Weight function defined as average of truncation error from each equation normalized by its maximum value on the uniform/original mesh, #., 11

12 Adaptation Monitors Monitoring the adaptation process and knowing when to stop needs to be done in a manner consistent with the adaptation scheme being used Since most of the adaptation schemes used here enforce equidistribution to some degree, an area weighted truncation error metric and weight function equidistribution error are used to monitor adaptive convergence Area Weighted Truncation Error Metric Weight Function Equidistribution Error,,,, :, 12

13 TEST CASES 13

14 Test Cases 2D Supersonic Expansion Fan Geometry 12 downward turn Edge Case (Blue): Fan edges intersect bottom and left boundaries Corner Case (Black): Fan centered at bottom left corner of domain Inflow Conditions Mach kpa, 273 K M = 1.2 Computational Domain 65x65 Mesh 0 (x,y)

15 Test Cases Karman Trefftz Airfoil Geometry Airfoil surface determined by transforming a circle using Karman Trefftz airfoil transformation Inflow Conditions Mach 0.2 AOA = kpa, 347 K Computational Domain C grid 129x65 Mesh Figure: Wikimedia Commons 15

16 Test Cases Exact Solu ons Exact Trunca on Error These test cases were explicitly selected because they have known exact solutions Exact solutions were used to formulate exact truncation error for driving the adaptation Used exact truncation to cut down on computational time associated with using estimated truncation error Accurate truncation error estimates required a nearly converged solution Also, with using exact solutions, we are able to exactly evaluate the amount of discretization error improvement. 16

17 RESULTS 17

18 Results: Expansion Fan (Edge Case) Uniform Anderson 10.5 Brackbill & Saltzman 4.8 Exact Truncation Error: X mtm max, Center of Mass Deformation 18

19 Results: Expansion Fan (Edge Case) Uniform Anderson Brackbill & Saltzman Discretization Error: Pressure,, Center of Mass Deformation 19

20 Results: Expansion Fan (Corner Case) Uniform Anderson Brackbill & Saltzman Discretization Error: Density,, Center of Mass Deformation 20

21 Results: Expansion Fan (Corner Case) Uniform Anderson Brackbill & Saltzman Discretization Error: Density,, Center of Mass Deformation 21

22 Results: KT Airfoil Uniform Discretization Error: U velocity ,, Adapted 22

23 Summary Summary Several r adaptation techniques have been applied to a 2D expansion fan and a Karman Trefftz airfoil Truncation error has been used to drive adaptation Up to 6x improvement in discretization error achieved on adapted grids 23

24 Future Work Future Work Examine the efficiency of the entire adaptation/numerical solution process Test adaptation schemes using estimated truncation error Current adaptation methods were primarily developed for single block grids Some works needs to be done if this type of adaptation is to be applied to multi block grids 24

25 Questions Thank you! 25