Effects of the Jacobian Evaluation on Direct Solutions of the Euler Equations

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1 Middle East Technical University Aerospace Engineering Department Effects of the Jacobian Evaluation on Direct Solutions of the Euler Equations by Ömer Onur Supervisor: Assoc. Prof. Dr. Sinan Eyi

2 Outline Introduction Objectives Flow Analysis Accuracy Analysis & Results Performance Analysis & Results Conclusion Recommendations 2

3 Introduction Steady flow computations can be realized either by iterative means,, or using direct methods. Although iterative solvers make an unsteady flow analysis with an advance in time,, with direct methods a steady flow analysis is possible. Low memory requirements made the iterative methods popular untill the development of recent advanced computers. Solving the whole domain at once makes direct solvers stable, and faster convergence is possible due to small number of iterations. In CFD applications like design optimization,, and flutter analysis,, direct methods are preferable. 3

4 Introduction (Cont.) Direct solvers require the calculation of Jacobian matrix. Derivation of analytical Jacobians becomes more difficult as the discretization of governing equations become more complex. The best alternative is to compute the Jacobians numerically as accurate as possible. 4

5 Objectives To develop a direct flow solver code To compare the accuracy of numerical and analytical Jacobians used in direct flow solvers To investigate their effects on the performance (convergence and CPU time) of direct flow solvers To improve the efficiency of Jacobian matrix solution 5

6 Flow Analysis Flow code is a 2-D planar/axisymmetric Euler solver: 1st/2nd order finite-volume discretizations Steger-Warming, Van Leer, Roe upwind flux splitting schemes Newton's method both numerical and analytical Jacobian matrices UMFPACK sparse matrix solver package Different geometries,, grid sizes and BCs are considered. 6

7 Flow Analysis (Cont.) Steady, 2-D planar/axisymmetric Euler equations in generalized coordinates: F(W ˆ ˆ ) G(W ˆ ˆ ) + + σ H(W ˆ ˆ ) = 0 ξ η where Ŵ = J 1 ρ ρu ρv ρet ˆF = J 1 ρu ρuu + ξ x p ρvu + ξ y p ( ρet + p)u Ĝ = J 1 ρv ρuv + ηx p ρvv + ηy p ( ρet + p)v Ĥ = J 1 1 y ρv ρuv 2 ρv ( ρet + p)v 7

8 Flow Analysis (Cont.) Newton s method: R(W ˆ ˆ ) = 0 (Discretized Residual) n ˆR ˆ n ˆ n W R(W ) Ŵ = Wˆ = Wˆ + Wˆ n+ 1 n n [Jacobian Matrix] 8

9 Flow Analysis (Cont.) Analytical Jacobians: Derivation by hand / symbolic manipulators for simple flow models Difficult derivation for complex flow models Numerical Jacobians: Rˆ R(Wˆ ˆ i i + ε e j ) R(W) i = Ŵ ε j Good choice of finite-difference perturbation magnitude ε Usage of higher computer precision 9

10 Accuracy Analysis Error analysis: Condition error [loss of numerical precision] Truncation error [neglected terms in the Taylor series] Total error 2 2 ER f ( ξ ) ε E TOTAL( ε ) = E C( ε ) + E T ( ε ) = + 2 ε x 2 ε<< E C >> & ε >> E T >> ε opt E TOTAL min 10

11 Accuracy Analysis (Cont( Cont.) Optimum perturbation magnitude analysis: E ( ε ) 2 E 1 2 TOTAL R f( ξ ) = + = 0 ε ε 2 2 x 2 ε = 2 OPT 2 ER f( ξ ) 2 x E R can be taken as ε M or simply found by subtraction of double and single precision calculations of flux values. Second derivative of flux can be calculated using a forward/backward finite difference method with double precision or taken as equal to 1. Optimum perturbation magnitude is also found by trialerror 11

12 Accuracy Results Error in numerical flux Jacobians are analyzed for 15º ramp geometry with 33x25 grid M=2.0 free-stream flow inlet-outlet, symmetry and wall BCs Jacobians are calculated using already converged solution 1st order S-W flux splitting Different perturbation magnitude values are investigated for both single and double precision. Optimum perturbation magnitude analysis is performed for single precision. 12

13 Accuracy Results (Cont.) Effect of Control on Total Errors for Residual Jacobians [Single Precision, Forward Differencing] TotalError between ResidualJacobians ( R/ W) w/o Cont. (Max. E) Effect of Control on Total Errors for Residual Jacobians [Single Precision,Backward Differencing] ε =7 wcont.(max.e) w/o Cont. (Avg. E) w Cont. (Avg. E) TotalError between ResidualJacobians ( R/ W) ε = w/o Cont. (Max. E) wcont.(max.e) w/o Cont. (Avg. E) w Cont. (Avg. E) Perturbation Magnitude (ε) Perturbation Magnitude (ε)

14 Accuracy Results (Cont.) Change of Total Maximum Error with Perturbation Magnitude [Single Precision, Forward Differencing] TotalMax. Error between Jacobians F + / W Change of Total Average Error with Perturbation Magnitude [Single Precision, Forward Differencing] ε =7 F - / W G + / W G - / W R pl / W Total Avg. Error between Jacobians F + / W ε =7 F - / W G + / W G - / W R pl / W Perturbation Magnitude (ε) Perturbation Magnitude (ε)

15 Accuracy Results (Cont.) Effect of Axisymmetry on Total Errors for Residual Jacobians [Single Precision, Forward Differencing] Optimum Perturbation Magnitude (ε opt ) Analysis for Single precision TotalError between ResidualJacobians ( R/ W) 10 4 Planar (Avg. E) 10 3 Axis ym. (Avg. E) Planar (Max. E) Axisym. (Max. E) ε =7 Trial-Error Procedure Optimization Method Flux ε opt (max. error) ε opt (avg. error) ε opt (avg.) ε opt ( ε M ) F F G G For single precision; ε opt 7x Perturbation Magnitude (ε)

16 Accuracy Results (Cont.) Effect of Control on Total Errors for Residual Jacobians [Double Precision, Forward Differencing] Effect of Control on Total Errors for Residual Jacobians [Double Precision, Backward Differencing] TotalError between ResidualJacobians ( R/ W) 10 3 w/o Cont. (Max. E) wcont.(max.e) 10 1 w/o Cont. (Avg. E) w Cont. (Avg. E) ε = TotalError between ResidualJacobians ( R/ W) 10 3 w/o Cont. (Max. E) wcont.(max.e) 10 1 w/o Cont. (Avg. E) w Cont. (Avg. E) ε = Perturbation Magnitude (ε) Perturbation Magnitude (ε)

17 Accuracy Results (Cont.) Change of Total Maximum Error with Perturbation Magnitude [Double Precision, Forward Differencing] TotalMax. Error between Jacobians F + / W Change of Total Average Error with Perturbation Magnitude [Double Precision, Forward Differencing] ε = F - / W G + / W G - / W R pl / W TotalAvg. Error between Jacobians F + / W ε = F - / W G + / W G - / W R pl / W Perturbation Magnitude (ε) Perturbation Magnitude (ε)

18 Accuracy Results (Cont.) Effect of Axisymmetry on Total Errors for Residual Jacobians [Double Precision, Forward Differencing] Optimum Perturbation Magnitude (ε opt ) Analysis for Double precision TotalError between ResidualJacobians ( R/ W) 10 0 Planar (Avg. E) 10-1 Axis ym. (Avg. E) Planar (Max. E) Axisym. (Max. E) ε = Trial-Error Procedure Optimization Method Flux ε opt (max. error) ε opt (avg. error) ε opt F F G G For double precision; ε opt 4x Perturbation Magnitude (ε)

19 Performance Analysis Jacobian Matrix structure: a huge square matrix even for a simple geometry composed of either analytical or numerical Jacobians most of the elements are zero Matrix solver: Direct full matrix solvers have very high storage and memory costs. Storage of only non-zero elements improve the efficiency greatly. Sparse Matrix Solvers 19

20 Performance Analysis (Cont.) Matrix Solution Strategies: Frozen Jacobian [using same Jacobian matrix in subsequent iterations] Good initial guess [a time-like term addition to the Jacobian matrix diagonal] 0 n ˆ R(W ˆ ) 1 R n 0 n n 2 [] I + Wˆ = R(W ˆ ) t = t t Wˆ ˆ n R(W ) 2 Δt>> Original Newton s method 20

21 Performance Results Test Case Results: Test Case Results: Convergence and CPU time performance of the direct solver is analyzed for 15º ramp geometry with 33x25 grid M=2.0 free-stream flow inlet-outlet, symmetry and wall BCs Jacobians are calculated using 1st order S-W flux splitting Calculations are realized with forward and backward differencing in single and double precision. Effect of perturbation magnitude ε is investigated Effect of Jacobian freezing and time-like diagonal term addition are analyzed. 21

22 Performance Results (Cont.) 33x25 Grid for Ramp geometry 34x26 cells 4 flow variables total 3536 variables Jacobian matrix has million elements Solver Storage CPU time Full Matrix LU 240 MB 26 h Sparse Matrix 1.2 MB 20 s 22

23 Performance Results (Cont Cont.) Mach Contour for Ramp geometry [2-D planar, Double Precision, 1st order S-W] Mach Contour for Ramp geometry [2-D axisymmetric, Double Precision, 1st order S-W]

24 Performance Results (Cont.) Effect of t onconvergencehistory [2-D planar, Double Precision] Effect of t onconvergencehistory [2-D axisymmetric, Double Precision] 10 0 AnalyticalJac. No frz AnalyticalJac. No frz. Max. Density Residual (R 1 ) Max. Density Res idual (R 1 ) No t t rm =1.5 t rm =5. Full t #ofiterations t CPU Time (s) No t NaN t rm = t rm = Full t No t t rm =1.5 t rm =5. Full t #ofiterations t CPU Time (s) No t 9.79 t rm = t rm = Full t

25 Performance Results (Cont.) 10 1 EffectOfFreezingonConvergenceHistory [2-D planar, Double Precision] 10 1 Effect Of Freezing on Convergence History [2-D axisymmetric, Double Precision] 10-1 Analytical J ac. t rm = Analytical Jac. No t Max. Density Res idual (R 1 ) Max. Density Residual (R 1 ) No frz. R frz = R frz =1 R frz = No frz. R frz = R frz =1 R frz = #ofiterations Tolerance CPU Time (s) No freeze R frz = 1x R frz = 1x R frz = 1x #ofiterations Tolerance CPU Time (s) No freeze 9.79 R frz = 1x R frz = 1x 8.15 R frz = 1x

26 Performance Results (Cont.) 10 0 Convergence History [2-D planar, Double Precision, Forward Differencing] t rm =1.5 R frz = Convergence History [2-D a xis ymmetric, Double P re cis ion, Forward Diffe rencing] t rm =0. R frz = 10-3 Max. Density Residual (R 1 ) Max. Density Residual (R 1 ) An. J a c. ε =4 ε = ε = #ofiterations Jacobian CPU Time (s) Analytic ε = 4x ε = 4x ε = 4x An. J ac. ε =4 ε = ε = #ofiterations Jacobian CPU Time (s) Analytic 8.25 ε = 4x ε = 4x ε = 4x

27 Performance Results (Cont.) Different Flux Splitting Schemes: Different Flux Splitting Schemes: Convergence and CPU time performance of the direct solver is analyzed for the test case using 1st order Van Leer/Roe flux splitting Jacobians are calculated using 1st order Van Leer/Roe flux splitting numerically 1st order S-W flux splitting analytically Optimum perturbation magnitude ε opt is used in the calculations. Calculations are realized with forward and backward differencing in single and double precision. Benefits of using the same flux calculation scheme for both Jacobian and residual calculation are analyzed. 27

28 Performance Results (Cont Cont.) Mach Contour for Ramp geometry [2-D planar, Double Precision, 1st order VL] Mach Contour for Ramp geometry [2-D axisymmetric, Double Precision, 1st order VL]

29 Performance Results (Cont Cont.) Mach Contour for Ramp geometry [2-D planar, Double Precision, 1st order Roe] Mach Contour for Ramp geometry [2-D axisymmetric, Double Precision, 1st order Roe]

30 Performance Results (Cont.) 10 0 Convergence History [2-D planar, Double Precision, S-W An. Jac.] t rm =1.5 R frz = Convergence History [2-D planar, Double Precision, Num. Jac.] t rm =1.5(SW) t rm =2.(VL) t rm =3.(Roe) R frz = ε = Max. Density Residual (R 1 ) Max. Density Residual (R 1 ) Steger-Warming Van Leer Roe #ofiterations FS CPU Time (s) SW VL Roe Steger-Warming Van Leer Roe #ofiterations FS CPU Time (s) SW VL Roe

31 Performance Results (Cont.) 10 0 Convergence History [2-D axisymmetric, Double Precision, S-W An. Jac.] t rm =0. R frz = Convergence History [2-D axisymmetric, Double Precision, Num. Jac.] t rm =0.(SW,VL) t rm =1.5(Roe) R frz = ε = Max. Density Residual (R 1 ) Max. Density Residual (R 1 ) Steger-Warming Van Leer Roe #ofiterations FS CPU Time (s) SW 8.25 VL Roe Steger-Warming Van Leer Roe #ofiterations FS CPU Time (s) SW 9.72 VL Roe

32 Performance Results (Cont.) Higher-Order Discretizations: Higher-Order Discretizations: Convergence and CPU time performance of the direct solver is analyzed for the test case using 2nd order S-W/Van Leer/Roe flux splitting Van Albada limiter Jacobians are calculated using 2nd order S-W/Van Leer/Roe flux splitting numerically Optimum perturbation magnitude ε opt is used in the calculations. Calculations are realized with forward differencing in double precision. 32

33 Ömer Onur - AE500 Presentation Performance Results (Cont Cont.) Mach Contour for Ramp geometry [2-D planar, Double Precision, 2nd order S-W] Mach Contour for Ramp geometry [2-D axisymmetric, Double Precision, 2nd order S-W]

34 Ömer Onur - AE500 Presentation Performance Results (Cont Cont.) Mach Contour for Ramp geometry [2-D planar, Double Precision, 2nd order VL] Mach Contour for Ramp geometry [2-D axisymmetric, Double Precision, 2nd order VL]

35 Ömer Onur - AE500 Presentation Performance Results (Cont Cont.) Mach Contour for Ramp geometry [2-D planar, Double Precision, 2nd order Roe] Mach Contour for Ramp geometry [2-D axisymmetric, Double Precision, 2nd order Roe]

36 Performance Results (Cont.) Convergence History [2-D planar, Double Precision, Num. Jac., Van Albada] Full t No Freezing ε = Convergence History [2-D axisymmetric, Double Precision, Num. Jac., Van Albada] Full t No Freezing ε = Max. Density Residual (R 1 ) Max. Density Residual (R 1 ) Steger-Warming Van Leer Roe #ofiterations FS CPU Time (s) SW VL Roe Steger-Warming Van Leer Roe #ofiterations FS SW VL Roe CPU Time (s) Limit cycle Limit cycle Limit cycle 36

37 Performance Results (Cont.) Different Geometry and Flow Conditions: Different Geometry and Flow Conditions: Convergence and CPU time performance of the direct solver is analyzed for bump geometry with 65x17 grid M=2.0 and M=0.5 free-stream flows inlet-outlet, symmetry and wall BCs 2nd order Roe flux splitting with Van Albada limiter Jacobians are calculated using 2nd order Roe flux splitting numerically Optimum perturbation magnitude ε opt is used in the calculations. Calculations are realized with forward differencing in double precision. 37

38 Performance Results (Cont.) 65x17GridforBumpgeometry 38

39 Performance Results (Cont.) 39

40 Performance Results (Cont.) Convergence History [2-D planar, Double Precision, Num. Jac., Van Albada] Full t No Freezing ε = Convergence History [2-D axisymmetric, Double Precision, Num. Jac., Van Albada] Full t No Freezing ε = Max. Density Res idual (R 1 ) Max. Density Res idual (R 1 ) Roe Roe #ofiterations #ofiterations 40

41 Performance Results (Cont.) 41

42 Performance Results (Cont.) Convergence History [Subsonic 2-D planar, Double Precision, Num. Jac., Van Albada] Full t No Freezing ε = Convergence History [Subsonic 2-D axisymmetric, Double Precision, Num. Jac., Van Albada] Full t No Freezing ε = Max. Density Res idual (R 1 ) Max. Density Res idual (R 1 ) Roe Roe #ofiterations #ofiterations 42

43 Conclusion A direct flow solver code is developed. Accuracy of numerical Jacobians used in the solver is analyzed. A control mechanism is required in 1st order SW Jacobian calculation. Forward/backward differencing does not differ so much. Double precision improves accuracy significantly. Optimum perturbation magnitude is found by an optimization method. 43

44 Conclusion (Cont( Cont.) The effects of the accuracy of numerical Jacobians on the performance (convergence and CPU time) of direct flow solvers is investigated. Double precision improves convergence limits significantly. Optimum perturbation magnitude gives the same convergence with the analytical method. Calculation of fluxes with perturbation only for related cells reduced the execution time greatly. 44

45 Conclusion (Cont( Cont.) The efficiency of Jacobian matrix solution is improved by some strategies. A sparse matrix solver having low storage and memory requiremets is used. A time-like term addition to matrix diagonal stabilizes the solver in case of poor initial conditions. Removing this modification at the right time makes the convergence very fast. Jacobian matrix freezing at the right time decreases the execution time. 45

46 Conclusion (Cont( Cont.) Using the same flux calculation scheme for both Jacobian and residual calculation maintains faster convergence. According to the choice of the flux limiter, higherorder schemes may have convergence problems. Unsteady results like limit-cycle is observed in higherorder schemes. 46

47 Recommendations The control mechanism used in 1st order SW Jacobian calculation can be improved to give the best accuracy for all perturbation magnitude values. More advanced strategies can be considered to improve the Jacobian matrix solution. Higher-order schemes, and especially the choice and application of limiter can be analyzed deeply to obtain fully converged solution. 47

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