Parallel Adaptive Finite Element Methods for Problems in Natural Convection

Transcription

1 Parallel Adaptive Finite Element Methods for Problems in Natural Convection John W. Peterson CFDLab Univ. of Texas at Austin April 18, 2008

2 Outline 1 Physical Problem 2 Goals and Contributions 3 Governing Equations 4 Unsteady Simulations 5 Continuation Results 6 Parallel Adaptive Simulations 7 Conclusion and Future Work

3 Physical Problem 1 Physical Problem 2 Goals and Contributions 3 Governing Equations 4 Unsteady Simulations 5 Continuation Results 6 Parallel Adaptive Simulations 7 Conclusion and Future Work

4 Physical Problem Buoyancy basic concepts Buoyancy: Warmer fluids are slightly less dense than cold fluids. g Cold Plate Slightly Cooler Fluid Slightly Warmer Fluid Hot Plate

5 Physical Problem Buoyancy basic concepts Rayleigh s Number: a non-dimensional parameter which tells the relative importance of buoyant forces to dissipative effects Ra = Buoyant Force Dissipative Effects Rayleigh s famous calculation predicted Ra crit = 1708 for onset of fluid motion in an infinite horizontal layer.

6 Physical Problem Surface tension basic concepts Fluid surface tension decreases with temperature g Slightly Cooler Fluid Slightly Warmer Fluid

7 Physical Problem Surface tension basic concepts The same dissipative effects (viscosity, diffusion) retard fluid motion on the surface Marangoni s Number: a non-dimensional ratio of surface tension strength to dissipative effects Ma = Surface Tension Dissipative Effects

8 Physical Problem Aspect ratio Free T. Surface n=bit d L T. n=0 Side Walls T=Hot Bottom Wall Aspect Ratio Γ := L d

9 Parallel FEM for Natural Convection Physical Problem Hexagonal Cells In large (Γ 1) aspect ratio fluid layers, hexagonal convection cells form Be nard, 1900 Velarde et. al (Album of Fluid Motion)

10 Physical Problem Triangular Container Medale & Cerisier, 2002 Peterson, 2004 At moderate Ma, presence of hexagonal cells is independent of container shape Γ := A/d = 22

11 Physical Problem Convection far from onset Square Cells Γ At (c), Ma 781! Exhibits hysteresis Schatz et. al, 1999

12 Physical Problem Small aspect ratio domains In smaller aspect ratio domains (Γ 10), hexagons do not form The number of convection cells depends on the aspect ratio and other parameters E. L. Koschmieder and S. A. Prahl, Surface-tension-driven Bénard convection in small containers, J. Fluid Mech

13 Physical Problem Small aspect ratio domains Γ = 6.18, Ma = 96 Γ = 6.36, Ma = 93 Koschmieder, 1990

14 Goals and Contributions 1 Physical Problem 2 Goals and Contributions 3 Governing Equations 4 Unsteady Simulations 5 Continuation Results 6 Parallel Adaptive Simulations 7 Conclusion and Future Work

15 Goals and Contributions Goals Main Goal: Improve understanding of RBM convection in small aspect ratio containers. Hypothesis: In a square cross-section container of small aspect ratio Γ, there exist steady state cellular convection patterns which depend only on Γ.

16 Goals and Contributions Contributions Developed scaling (non-dimensionalization) for the RBM convection equations in which the aspect ratio, Γ, appears as an independent, continuous parameter. Developed a pseudo-arclength continuation algorithm for the RBM convection equations. Implemented generic continuation solver algorithm in LibMesh. Expanded catalog of spontaneous spatio-temporal pattern formation in RBM convection with high-fidelity numerical simulations.

17 Goals and Contributions Contributions Explicit second-order AB predictor implemented with continuation scheme. Performed linear stability analysis of solution branches; linked application code to state-of-the-art eigenanalaysis software package SLEPc. Performed exploratory AMR studies of combined Navier-slip thermocapillary convection. Several contributions to the LibMesh open-source library, and finite element analysis scientific community.

18 Governing Equations 1 Physical Problem 2 Goals and Contributions 3 Governing Equations 4 Unsteady Simulations 5 Continuation Results 6 Parallel Adaptive Simulations 7 Conclusion and Future Work

19 Governing Equations Standard Non-Dimensional Equations ( ) 1 u + (u )u σ Pr t = Ra Tĝ u = 0 T t + u T + u T 0 = 2 T

20 Governing Equations Boundary Conditions Bottom: u = 0 T = 0

21 Governing Equations Boundary Conditions Bottom: u = 0 T = 0 Sides: u = 0 T n = 0

22 Governing Equations Boundary Conditions Bottom: u = 0 T = 0 Sides: u = 0 T n = 0 Free surface: σ n = Ma T T n = Bi T

23 Governing Equations Variational Statement: Conservation of Momentum & Mass Ω Find u, p, T such that: [ 1 ( u Pr t ) + (u )u Ra Ω Ω ] v + ε(u): v p ( v) Tĝ v dx + Ma ( u)q dx = 0 dx + Ω s ( T v ) ds = 0

24 Governing Equations Variational Statement: Conservation of Energy Ω ( ) T t + u T + u T 0 w + ( T w) dx + Bi Tw ds = 0 Ω s hold v, q, w V Q W.

25 Governing Equations Numerical Method From the variational statement, we construct an LBB-stable Galerkin finite element discretization in space.

26 Governing Equations Numerical Method From the variational statement, we construct an LBB-stable Galerkin finite element discretization in space. We then perform a standard θ-method discretization in time (plus adaptive step-doubling timestep selection).

27 Governing Equations Numerical Method From the variational statement, we construct an LBB-stable Galerkin finite element discretization in space. We then perform a standard θ-method discretization in time (plus adaptive step-doubling timestep selection). This leads to a discrete system of nonlinear equations which is linearized and solved via Newton s method.

28 Governing Equations Library Description Tumor Angiogenesis Compressible RBM NS Petsc Laspack DD LibMesh STL Metis Library Structure Basic libraries are LibMesh s roots Application branches built off the library trunk MPI

29 Unsteady Simulations 1 Physical Problem 2 Goals and Contributions 3 Governing Equations 4 Unsteady Simulations 5 Continuation Results 6 Parallel Adaptive Simulations 7 Conclusion and Future Work

30 Unsteady Simulations Simulation Parameters Ma = 92 Ra = 30 (Ma/Ra 3.07, surf. ten. effects dominate) Pr = 880 (corresponds to a viscous silicon oil) Bi = 0.2 Values representative of those used by Koschmieder (1990) and Medale and Cerisier (2002) Direct quantitative comparisons with experiment not generally possible due to experimental uncertainty in material properties

31 Unsteady Simulations Adaptive Timestepping Adaptive step-doubling method with one-step θ scheme as base time discretization Steady state detection when the quantity u t un+1 u n t falls to approximately machine precision (10 13 to ) at t max. t max / t min = 10 2

32 Unsteady Simulations Representative Results, Γ = log t / t 0 log( U / t) Timestep 3

33 Unsteady Simulations Representative Results, Γ = log t / t 0 log( U / t) Timestep 3 1

34 Unsteady Simulations Representative Results, Γ = log t / t 0 log( U / t) Timestep 3 2a

35 Unsteady Simulations Representative Results, Γ = log t / t 0 log( U / t) Timestep 3 2b

36 Unsteady Simulations Representative Results, Γ = log t / t 0 log( U / t) Timestep 3 2c

37 Unsteady Simulations Representative Results, Γ = log t / t 0 log( U / t) Timestep 3 2d

38 Unsteady Simulations Representative Results, Γ = log t / t 0 log( U / t) Timestep 3 2e

39 Unsteady Simulations Representative Results, Γ = log t / t 0 log( U / t) Timestep 3 3

40 Unsteady Simulations Representative Results, Γ = 6 Movie: Time Evolution for Γ = 6

41 Unsteady Simulations Summary of Unsteady Results After collecting steady-state converged data for a range of aspect ratios, we plot it versus Nusselt number: Nu := ds Ω s T

42 Unsteady Simulations Summary of Unsteady Results After collecting steady-state converged data for a range of aspect ratios, we plot it versus Nusselt number: Nu := ds Ω s T The collected unsteady solution data does not provide the full story of the solution dependence on Γ.

43 Unsteady Simulations Summary of Unsteady Results After collecting steady-state converged data for a range of aspect ratios, we plot it versus Nusselt number: Nu := ds Ω s T The collected unsteady solution data does not provide the full story of the solution dependence on Γ. The branches of steady solutions appear to have discontinuous jumps near certain values of Γ.

44 Unsteady Simulations Summary of Unsteady Results Nu Cell 2 Cell 3 Cell 4 Cell Cell 6 Cell 7 Cell 8 Cell Aspect Ratio, Γ

45 Continuation Results 1 Physical Problem 2 Goals and Contributions 3 Governing Equations 4 Unsteady Simulations 5 Continuation Results 6 Parallel Adaptive Simulations 7 Conclusion and Future Work

46 Continuation Results Variation of Aspect Ratio d Γ=1 g L Γ=2 Γ=3 Aspect Ratio Γ := L/d We consider containers of constant d and increasing L

47 Continuation Results Scaled Equations Cartesian (x 1, x 2, x 3 ) coordinate system, gravity vector in the x 3 direction. Non-dimensionalize with respect to L for horizontal scales, d for vertical scales x 1 := x 1 L x 2 := x 2 L The gradient operator is transformed as 1 (,, Γ ) L x1 x2 x3 x 3 := x 3 d := 1 L Γ

48 Continuation Results Scaled Equations ( ) 1 u Pr t + (u Γ)u Γ σ Γ = Γ 3 Ra Tĝ Γ u = 0 T t + u Γ T + u Γ T 0 = Γ T 2

49 Continuation Results Variational Statement: Conservation of Momentum & Mass Ω Find u, p, T such that: [ ( ) 1 u Pr t + (u Γ)u Γ 3 Ra Ω Ω ] v + ε Γ (u): Γ v p ( Γ v) Tĝ v dx + Γ 2 Ma ( Γ u)q dx = 0 dx + Ω s ( T v ) ds = 0

50 Continuation Results Variational Statement: Conservation of Energy Ω ( ) T t + u Γ T + u Γ T 0 w + ( Γ T Γ w) dx + Γ 2 Bi Tw ds = 0 Ω s hold v, q, w V Q W.

51 Continuation Results Continuation Basics We write the (discretized) steady form of the variational statement as G(u, λ) = 0 The Jacobian G u may be singular depending on u, λ A standard continuation algorithm will fail near such points (Euler-Newton) Generic continuation in arbitrary parameters now available in LibMesh

52 Continuation Results Continuation Basics The pseudo-arclength continuation technique augments G with a constraint N(u, λ, s) = 0 s is the arclength parameter, s > 0 Must solve augmented Jacobian systems [ Gu G λ N t u N λ ] k [ ] k+1 δu = δλ [ ] k G N The augmented matrix is of full rank even when G u is singular

53 Continuation Results Continuation Algorithm Obtain u 0, λ 0 from unsteady solve Choose next parameter value: λ 1 = λ 0 + λ ( λ small) Solve steady equations for u 1, use u 0 as initial guess Compute tangents u/ s 1, λ/ s 1 via finite difference Compute initial guess ũ 2, λ 2 using the tangent for n = 2 to n steps do Solve augmented Jacobian system for new values u n, λ n Update tangents u/ s n, λ/ s n with add l linear solve Compute initial guess ũ n+1, λ n+1 end for

54 Continuation Results Continuation Algorithm Prediction & Correction Steps u λ

55 Continuation Results Continuation Algorithm Prediction & Correction Steps u λ

56 Continuation Results Continuation Algorithm Prediction & Correction Steps u λ

57 Continuation Results Continuation Algorithm Prediction & Correction Steps u λ

58 Continuation Results Continuation Algorithm Prediction & Correction Steps u λ

59 Continuation Results Arclength Stepsize Selection Nu Proper adaptive stepsize selection is required to prevent stepping past important branch features. Ma

60 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 1

61 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 2

62 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 3

63 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 4

64 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 5

65 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 6

66 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 7

67 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 8

68 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 9

69 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 10

70 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ 11

71 Continuation Results Representative Solution Branch Nu Aspect Ratio, Γ Koschmieder, 1990

72 Continuation Results Summary of Continuation Results Nu Aspect Ratio, Γ

73 Parallel Adaptive Simulations 1 Physical Problem 2 Goals and Contributions 3 Governing Equations 4 Unsteady Simulations 5 Continuation Results 6 Parallel Adaptive Simulations 7 Conclusion and Future Work

74 Parallel Adaptive Simulations Motivation Interested in controlling observed natural convection flow patterns and structure with non-invasive techniques such as surface treatments Shear-dependent slip, aka Navier slip, is a model for investigating these flow control ideas qualitatively and quantitatively

75 Parallel Adaptive Simulations Navier Slip BC Proposed by C. L. M. H. Navier, 1823 and independently by Maxwell, 1879 Amount of slip Rate of shear

76 Parallel Adaptive Simulations Navier Slip BC Proposed by C. L. M. H. Navier, 1823 and independently by Maxwell, 1879 Amount of slip Rate of shear Constant of proportionality is slip length, L

77 Parallel Adaptive Simulations Navier Slip BC Proposed by C. L. M. H. Navier, 1823 and independently by Maxwell, 1879 Amount of slip Rate of shear Constant of proportionality is slip length, L In the variational statement, we replace surface traction term (σ n) v = L 1 (u u wall ) v

78 Parallel Adaptive Simulations Navier Slip BC Proposed by C. L. M. H. Navier, 1823 and independently by Maxwell, 1879 Amount of slip Rate of shear Constant of proportionality is slip length, L In the variational statement, we replace surface traction term (σ n) v = L 1 (u u wall ) v Wall-normal and tangential slip lengths may be chosen independently

79 Parallel Adaptive Simulations Navier Slip Applications Moving contact line problem Flow near sharp corners/in small capillaries Electro-osmotically driven flow Hydrophobic channel flow Micro/nanoscale flows Surface polishing in semiconductor manufacturing

80 Parallel Adaptive Simulations Navier-slip in RBM Convection Performed a series of AMR simulations with unit Navier slip allowed on different quadrants of the container bottom.

81 Parallel Adaptive Simulations Navier-slip Quadrants and Reference Solution II III I IV Slip Quadrants Reference No-slip Solution

82 Parallel Adaptive Simulations Navier-slip Results Quadrant IV Quadrants I and III

83 Parallel Adaptive Simulations Navier-slip Results Quadrant III and IV Quadrants I, III, and IV

84 Parallel Adaptive Simulations Navier-slip Adapted Grids Temperature Contours Vel. Magnitude Contours

85 Parallel Adaptive Simulations Line Plots Through Slip Discontinuity x20x2 40x40x4 80x80x8 AMR(3) -4 u x 2

86 Parallel Adaptive Simulations Navier-slip Time Evolution Movies Movies: Top Surface Bottom Surface

87 Conclusion and Future Work 1 Physical Problem 2 Goals and Contributions 3 Governing Equations 4 Unsteady Simulations 5 Continuation Results 6 Parallel Adaptive Simulations 7 Conclusion and Future Work

88 Conclusion and Future Work Contributions Developed scaling (non-dimensionalization) for the RBM convection equations in which the aspect ratio, Γ, appears as an independent, continuous parameter. Developed a pseudo-arclength continuation algorithm for the RBM convection equations. Implemented generic continuation solver algorithm in LibMesh. Expanded catalog of spontaneous spatio-temporal pattern formation in RBM convection with high-fidelity numerical simulations.

89 Conclusion and Future Work Contributions Explicit second-order AB predictor implemented with continuation scheme. Performed linear stability analysis of solution branches; linked application code to state-of-the-art eigenanalaysis software package SLEPc. Performed exploratory AMR studies of combined Navier-slip thermocapillary convection. Several contributions to the LibMesh open-source library, and finite element analysis scientific community.

90 Conclusion and Future Work Future Work Additional parametric studies (Ma, Bi) including adaptive mesh refinement along solution branches Non-square (rectangular, circular, triangular... ) containers of small Γ Bifurcation tracking (via extended systems) Deforming free surface simulations Goal-oriented error indicators for driving AMR simulations to compute quantities of interest (Nu) Optimal shape design of containers to maximize heat transfer efficiency

91 Reference Slides Reference Material (Extra Slides)

92 Reference Slides Temperature Decomposition T T 0 T ~

93 Reference Slides Additional Remarks on the Scaling In the scaled variational statement, we obtained terms proportional to: Γ 3 Ra Γ 2 Ma Where Ra, Ma are defined in terms of the layer depth in the usual way Ra := ρ 0 β g T sd 3 αµ Ma := γ T T sd αµ

94 Reference Slides Additional Remarks on the Scaling In the scaled equations, the Γ 3 Ra term is constant with respect to d while the Γ 2 Ma term scales as d 1. Thus, the Ma term dominates for small d while the Ra term dominates for large d in both cases Ra term Ma term Classical d 3 d Scaled const d 1

95 Reference Slides Additional Remarks on the Scaling Alternatively, we can define analogous parameters Ra L := ρ 0 β g T sl 3 αµ Ma L := γ T T sl αµ in terms of the horizontal length scale L

96 Reference Slides Additional Remarks on the Scaling We then have Γ 3 Ra = Ra L Γ 2 Ma = = ( ) 2 L γ Ts d T d αµ ( ) L γ T T s L d αµ Let L be fixed = ΓMa L For small aspect ratio (Γ 0) layers, buoyant effects dominate Vice-versa for large aspect ratio layers

97 Reference Slides Surface Deflection General normal traction condition for curved surface σ n n = Kγ Kγ 0 Mean Curvature ( 1 K := + 1 ) R 1 R 2 Surface orientation n, t 1, t 2 is unknown; depends on solution {u, p, T}

98 Reference Slides Surface Deflection Surface deflections may be considered small if Kd Ca 1 Capillary number In the present work Ca := µα γ 0 d Ca

99 Reference Slides Surface Deflection Rahal et al. have recently (2007) employed interferometry to measure surface deflections The examined a related but slightly thicker (5 mm) fluid layer in a circular container of Γ := R/d = 6. Buoyant effects dominant (Ra/Ma 9) but thermocapillary flow still present. Rahal et al. observed convex cells (raised upwelling regions) on the order of a few ( 3) microns in height over distances of O(10) mm

100 Reference Slides Pseudo -Arclength Constraint We employ a linearized or pseudo arclength constraint as in Keller 1977, the LOCA library, and others N α (u, λ, s) := αi 2 (u u(s i )) t u s +(λ λ(s i )) λ si s (s s i ) si

101 Reference Slides Pseudo -Arclength Constraint We employ a linearized or pseudo arclength constraint as in Keller 1977, the LOCA library, and others N α (u, λ, s) := αi 2 (u u(s i )) t u s +(λ λ(s i )) λ si s (s s i ) si Solution and parameter tangents u si, λ s s si

102 Reference Slides Pseudo -Arclength Constraint We employ a linearized or pseudo arclength constraint as in Keller 1977, the LOCA library, and others N α (u, λ, s) := αi 2 (u u(s i )) t u s +(λ λ(s i )) λ si s (s s i ) si Solution and parameter tangents u si, λ s s si Scaling parameter α i ensures similar order-of-magnitude contributions from each constraint term

103 Reference Slides Schemes for Adaptively Selecting Arclength Stepsizes Ratio of successive tangents u λ si 1 s i+1 = s i u λ si

104 Reference Slides Schemes for Adaptively Selecting Arclength Stepsizes Most recent number of Newton iterations required [ ( ) ] 2 Nmax N i s i+1 = 1 + a s i N max 1

105 Reference Slides Generalized Eigenproblem The spatially-discretized equations can be written B u t = G(u, λ)

106 Reference Slides Generalized Eigenproblem The spatially-discretized equations can be written B u = G(u, λ) t B is a generalized mass matrix (linear operator)

107 Reference Slides Generalized Eigenproblem The spatially-discretized equations can be written B u = G(u, λ) t B is a generalized mass matrix (linear operator) u is a vector of FE coefficients

108 Reference Slides Generalized Eigenproblem The spatially-discretized equations can be written B u = G(u, λ) t B is a generalized mass matrix (linear operator) u is a vector of FE coefficients λ is a generalized parameter (Γ, Ma,... )

109 Reference Slides Generalized Eigenproblem Let u 0 be given such that 0 = G(u 0, λ)

110 Reference Slides Generalized Eigenproblem Let u 0 be given such that 0 = G(u 0, λ) Let η := u u 0

111 Reference Slides Generalized Eigenproblem Let u 0 be given such that 0 = G(u 0, λ) Let η := u u 0 Applying B to η we obtain B η = B u Bu 0

112 Reference Slides Generalized Eigenproblem Let u 0 be given such that 0 = G(u 0, λ) Let η := u u 0 Applying B to η we obtain B η = B u Bu 0 = G(u, λ) G(u 0, λ)

113 Reference Slides Generalized Eigenproblem Let u 0 be given such that 0 = G(u 0, λ) Let η := u u 0 Applying B to η we obtain B η = B u Bu 0 = G(u, λ) G(u 0, λ) = G(u, λ)

114 Reference Slides Generalized Eigenproblem Let u 0 be given such that 0 = G(u 0, λ) Let η := u u 0 Applying B to η we obtain B η = B u Bu 0 = G(u, λ) G(u 0, λ) = G(u, λ) = G(u 0 + η, λ)

115 Reference Slides Generalized Eigenproblem Let u 0 be given such that 0 = G(u 0, λ) Let η := u u 0 Applying B to η we obtain B η = B u Bu 0 = G(u, λ) G(u 0, λ) = G(u, λ) = G(u 0 + η, λ) = G(u 0, λ) + G u (u 0, λ)η + O( η 2 )

116 Reference Slides Generalized Eigenproblem Let u 0 be given such that 0 = G(u 0, λ) Let η := u u 0 Applying B to η we obtain B η = B u Bu 0 = G(u, λ) G(u 0, λ) = G(u, λ) = G(u 0 + η, λ) = G(u 0, λ) + G u (u 0, λ)η + O( η 2 ) G u (u 0, λ)η

117 Reference Slides Generalized Eigenproblem Letting A := G u (u 0, λ)

118 Reference Slides Generalized Eigenproblem Letting A := G u (u 0, λ) And inserting the ansatz η = exp(σt)x yields B η = Aη

119 Reference Slides Generalized Eigenproblem Letting A := G u (u 0, λ) And inserting the ansatz η = exp(σt)x yields B η = Aη σb exp(σt)x = A exp(σt)x

120 Reference Slides Generalized Eigenproblem Letting A := G u (u 0, λ) And inserting the ansatz η = exp(σt)x yields B η = Aη σb exp(σt)x = A exp(σt)x σbx = Ax

121 Reference Slides Generalized Eigenproblem Letting A := G u (u 0, λ) And inserting the ansatz η = exp(σt)x yields B η = Aη σb exp(σt)x = A exp(σt)x σbx = Ax A generalized eigenproblem for σ, x

122 Reference Slides Eigensolver Details Direct calculation of eigenvalues is O(n 3 ) To determine stability, we only need the sign of the rightmost eigenvalue The Arnoldi iteration is a relatively efficient technique for computing eigenvalues largest in magnitude (not necessarily rightmost)

123 Reference Slides Eigensolver Details Arnoldi is related to the power iteration Computes eigenvalues of the orthogonal projection of A onto the Krylov subspace Already implemented in the SLEPc library, a collection of eigensolvers built on top of PETSc

124 Reference Slides Spectral Transformations Arnoldi converges to the eigenvalues largest in magnitude Our generalized mass matrix B has time-independent constraint rows which effectively lead to eigenvalues We can use a spectral transformation such as shift-and-invert to avoid converging to large negative eigenvalues.

125 Reference Slides Spectral Transformations σbx = Ax Shift-and-invert transform: start with generalized eigenproblem

126 Reference Slides Spectral Transformations σbx = Ax (σ s) Bx = (A sb) x Choose real-valued shift s σ, subtract sbx from both sides

127 Reference Slides Spectral Transformations σbx = Ax (σ s) Bx = (A sb) x (σ s) (A sb) 1 Bx = (A sb) 1 (A sb) x Left multiply both sides by (A sb) 1

128 Reference Slides Spectral Transformations σbx = Ax (σ s) Bx = (A sb) x (σ s) (A sb) 1 Bx = (A sb) 1 (A sb) x (σ s) (A sb) 1 Bx = x Left multiply both sides by (A sb) 1

129 Reference Slides Spectral Transformations σbx = Ax (σ s) Bx = (A sb) x (σ s) (A sb) 1 Bx = (A sb) 1 (A sb) x (σ s) (A sb) 1 Bx = x (A sb) 1 Bx = (σ s) 1 x Multiply both sides by (σ s) 1

130 Reference Slides Spectral Transformations σbx = Ax (σ s) Bx = (A sb) x (σ s) (A sb) 1 Bx = (A sb) 1 (A sb) x (σ s) (A sb) 1 Bx = x (A sb) 1 Bx = (σ s) 1 x Tx = γx We are left with a standard eigenproblem, where T := (A sb) 1 B, γ := (σ s) 1

131 Reference Slides Spectral Transformations Shift-and-invert maps eigenvalues far from σ to zero (small magnitude) Arnoldi does not converge to these small-magnitude eigenvalues Can also be used to converge quickly to a suspected eigenvalue σ s

132 Reference Slides Solution of Bordered Linear Systems The continuation algorithm requires solution of bordered linear systems [ ] [ ] [ ] Gu G λ δu r = δλ ρ N t u N λ

133 Reference Slides Solution of Bordered Linear Systems The continuation algorithm requires solution of bordered linear systems [ ] [ ] [ ] Gu G λ δu r = δλ ρ N t u N λ δu, δλ Newton updates or new tangent directions

134 Reference Slides Solution of Bordered Linear Systems The continuation algorithm requires solution of bordered linear systems [ ] [ ] [ ] Gu G λ δu r = δλ ρ N t u N λ δu, δλ Newton updates or new tangent directions Right-hand sides: [ ] r = ρ [ ] G N or [ 0 N s ]

135 Reference Slides Solution of Bordered Linear Systems Can solve the augmented system efficiently via iterative methods e.g. Barragy and Carey, 1988 Here we employ the decomposition approach described by Keller, 1977

136 Reference Slides Solution of Bordered Linear Systems Can solve the augmented system efficiently via iterative methods e.g. Barragy and Carey, 1988 (+) Single solve step Here we employ the decomposition approach described by Keller, 1977 ( ) Multi-solve step required

137 Reference Slides Solution of Bordered Linear Systems Can solve the augmented system efficiently via iterative methods e.g. Barragy and Carey, 1988 (+) Single solve step (+) Augmented matrix has full rank Here we employ the decomposition approach described by Keller, 1977 ( ) Multi-solve step required ( ) Solves may involve nearly singular G u

138 Reference Slides Solution of Bordered Linear Systems Can solve the augmented system efficiently via iterative methods e.g. Barragy and Carey, 1988 (+) Single solve step (+) Augmented matrix has full rank ( ) Changes the system matrix sparsity pattern Here we employ the decomposition approach described by Keller, 1977 ( ) Multi-solve step required ( ) Solves may involve nearly singular G u (+) System matrix sparsity unchanged

139 Reference Slides Solution of Bordered Linear Systems Can solve the augmented system efficiently via iterative methods e.g. Barragy and Carey, 1988 (+) Single solve step (+) Augmented matrix has full rank ( ) Changes the system matrix sparsity pattern ( ) Requires modification of existing code Here we employ the decomposition approach described by Keller, 1977 ( ) Multi-solve step required ( ) Solves may involve nearly singular G u (+) System matrix sparsity unchanged (+) Requires minimal modifications to existing code

140 Reference Slides Solution of Bordered Linear Systems Solution Algorithm [ Gu G λ N t u N λ ] [ ] δu δλ 1 Declare auxiliary vectors y, z = [ ] r ρ

141 Reference Slides Solution of Bordered Linear Systems Solution Algorithm [ Gu G λ N t u N λ ] [ ] δu δλ 1 Declare auxiliary vectors y, z 2 Solve #1: G u y = G λ for y = [ ] r ρ

142 Reference Slides Solution of Bordered Linear Systems Solution Algorithm [ Gu G λ N t u N λ ] [ ] δu δλ 1 Declare auxiliary vectors y, z 2 Solve #1: G u y = G λ for y 3 Solve #2: G u z = r for z = [ ] r ρ

143 Reference Slides Solution of Bordered Linear Systems Solution Algorithm [ Gu G λ N t u N λ ] [ ] δu δλ 1 Declare auxiliary vectors y, z 2 Solve #1: G u y = G λ for y 3 Solve #2: G u z = r for z 4 Set: = [ ] r ρ δλ = ρ Nt uz N λ N t uy

144 Reference Slides Solution of Bordered Linear Systems Solution Algorithm [ Gu G λ N t u N λ ] [ ] δu δλ 1 Declare auxiliary vectors y, z 2 Solve #1: G u y = G λ for y 3 Solve #2: G u z = r for z 4 Set: 5 Set: = [ ] r ρ δλ = ρ Nt uz N λ N t uy δu = z (δλ)y

145 Reference Slides The Stick-Navier Slip Singularity log 10 u Slope=1/ x20x2 40x40x x80x8 AMR(3) AMR(4) log 10 r