AIAA-2009-4140.pdf

American Institute of Aeronautics and Astronautics

1

Performances of Numerical and Analytical Jacobians in Flow and Sensitivity Analysis

A. A. Ezertas1 and S. Eyi.2 Middle East Technical University , Ankara,,06531, Turkey

The effects of flux Jacobian evaluation on flow and sensitivity analysis are studied. A cell centered finite volume method with various upwinding schemes is used. A Newton’s method is applied for flow solution, and the resulting sparse matrix is solved by LU factorization. Flux Jacobians are evaluated both numerically and analytically. The sources of the error in numerical Jacobian calculation are studied. The optimum finite difference perturbation magnitude that minimizes the error is searched. The effects of error numerical Jacobians on the convergence of flow solver are studied. The sensitivities of the flow variables are evaluated by direct-differentiation method. The Jacobian matrix which is constructed in the flow solution is also used in sensitivity calculation. The influence of errors in numerical Jacobians on the accuracy of sensitivities is analyzed. Results showed that, the error in Jacobians significantly affects the convergence of flow analysis and accuracy of sensitivities. Approximately the same optimum perturbation magnitude enables the most accurate numerical flux Jacobian and sensitivity calculations.

I. Introduction Recent advances in computer technology and solution algorithms allow efficient solution of very large linear

systems of equations. These advances have been motivating researchers to develop implicit algorithms to solve the flow equations since usage of implicit methods is more beneficial compared to the explicit ones. Implicit flow solvers are more stable and the residual can be reduced to very low values within a small number of iterations. The equations of different disciplines can be strongly coupled with flow equations in an implicit algorithm. Another advantage of implicit methods is that, sensitivities can be calculated very efficiently in design optimization.

Providing quadratic convergence, Newton’s method is a widely used implicit technique for solving non-linear flow equations. Newton’s method requires the calculation of Jacobian matrices whose entries are the derivatives of discretized residual vector with respect to the flow variable vector. Although the size of this matrix can be very large, it is sparse in most of the flow problems. There are two methods available for calculating the Jacobian matrix: analytical and numerical methods. Analytical evaluation is more accurate but the differentiation procedure needs effort and it is time consuming1,2Moreover for each different flux discretization scheme, Jacobians are needed to be recalculated. Numerical evaluation can be performed easily by finite differencing the residual vector. Although this method is simple and independent from the complexity of the scheme, numerical evaluation has a lack of accuracy. Errors in numerical Jacobian calculations may hamper the convergence of Newton’s method. In numerical Jacobian calculations, error strongly depends on the finite difference perturbation magnitude. In small magnitudes of perturbation, condition error is dominant and in large magnitudes, truncation error becomes dominant. Hence the error becomes large for both small and large perturbation magnitudes. In order to evaluate the most accurate numerical Jacobian, the optimum perturbation magnitude should be used. As shown in Ref. 1, the optimum perturbation magnitude for numerical Jacobian calculation can be accurately estimated with a simple formula.

In gradient-based aerodynamic design optimization, the derivatives of objective function with respect to the design variables are needed. These derivatives are also called sensitivities. Sensitivities can be calculated by finite difference or analytical methods. Although the finite difference method is easy to use, analytical method has more advantages. In analytical method, sensitivities can be calculated more accurately and in a shorter CPU time. In analytical methods, due to the implicit relation between flow and design variables, the evaluation of sensitivities is not straightforward. There are two methods to calculate sensitivities analytically: direct-differentiation and by

1 Graduate Research Assistant, Aerospace Engineering Department. 2 Assoc. Professor, Aerospace Engineering Department

19th AIAA Computational Fluid Dynamics22 - 25 June 2009, San Antonio, Texas

AIAA 2009-4140

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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adjoint methods. In direct differentiation method, the discretized residual equations are differentiated with respect to design variables, and the resulting equations are solved for flow variable sensitivities. In adjoint method, the discretized residual equations are introduced as constraint functions, and the system of equations is solved for adjoint variables. Both methods require the construction and the solution of Jacobian matrix. Therefore, using Newton’s method for flow analysis has advantageous in the calculation of analytical sensitivities. Jacobian matrix which is constructed for flow analyses can also be used for sensitivity analyses. Sensitivities can be calculated very efficiently by solving the LU decomposed Jacobian matrix with different right hand sides. In the solution of flow equations with Newton’s method the flux Jacobian is used as a driver to the iterative procedure. Even tough the accuracy of flux Jacobian affects the convergence behavior of Newton’s method; it does not affect the accuracy of converged solution. However, in calculation of flow variable sensitivities, the flux Jacobian is the part of sensitivity equation. Therefore the accuracy of the sensitivities strongly depends on the accuracy of the flux jacobian3.

This study mainly composes of two parts. In the first part, the accuracy of the numerically evaluated flux Jacobian and its effect on the convergence of flow solution are studied. In the second part, the effects of the accuracy of numerical flux Jacobians on the accuracy of the sensitivities are analyzed. Sensitivities are calculated by direct differentiation method. The accuracy study is performed by comparing the results obtained from the numerically and analytically evaluated Jacobians. To perform the study, cell centered finite volume code is developed. The fluxes are computed by Steger Warming4, Van-Leer5,AUSM6 and Roe7 upwinding schemes. The analytical Jacobians are evaluated by differentiating Steger-Warming, AUSM, Van Leer fluxes for both first and second order discretizations. The sparse flux Jacobian matrix is LU factorized and solution is executed by using UMFPACK sparse matrix solver. The boundary conditions are implemented implicitly. The study is carried out for flows over the 10 percent circular arc, Ni bump8 geometry and the NACA0012 airfoil geometry.

II. Flow Model The steady state 2-D Euler equations in generalized coordinates can be written in non-dimensional form as

below:

ˆˆ ˆ ˆ( ) ( )

0F W G W∂ ∂∂ξ ∂η

+ = (1)

W is the conserved flow variable vector and F and G are the flux vectors in ξ, and η directions:

1ˆ

t

ρ

ρuW J

ρv

ρe

−

=

1ˆ x

y

t

ρU

ρuU pF J

ρvU p

( e p)U

ξξ

ρ

−

+ = + +

1ˆ x

y

t

ρV

ρuV η pG J

ρvV η p

( e p)Vρ

−

+ = + +

(2)

where ρ is the density, u and v are the components of the velocity vector, p is the pressure, et is the total energy per unit volume, U and V are contravariant velocity components. In Equation (2), J is the coordinate transformation Jacobian, ξ, and η are the curvilinear coordinates, and xξ , yξ , xη , yη are the transformation metrics.

The discretized form of the steady 2-D Euler equations given in Eq(1) can be written as:

ˆˆ

0F Gξ ηδ δξ η

+ =∆ ∆

(3)

For a cell centered finite volume method Eq (3) can then be written as:

1/ 2, 1/ 2, , 1/ 2 , 1/ 2ˆ ˆˆ ˆ( ) ( ) 0i j i j i j i jF F G G+ − + −− + − = (4)

The inviscid fluxes of the Euler equations represent the convective phenomena. The upwind flux splitting schemes are used for the spatial discretization of the flux vectors. In this study the splitted fluxes, F+ ,F- ,G+ and G-, are calculated by Steger Warming, Van Leer, AUSM and Roe method.


3

1/2, 1/2, 1/2,

, 1/2 , 1/2 , 1/2

ˆ ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ ˆˆ ˆ( ) ( )

L Ri j i j i j

L Ri j i j i j

F F W F W

G G W G W

+ −± ± ±

+ −± ± ±

= +

= +

(5)

where the i±1/2 and j±1/2 denotes a cell interface. The fluxes are calculated at the cell interfaces by using the flow variables interpolated from the cell center. The simple scheme with first order accuracy in space is obtained by assuming the values at the cell faces are equal to the values at the nearest cell centers.

1/2 1/2 1ˆ ˆ ˆ ˆ,L R

i i i iW W W W+ + += = (6)

Higher order accuracy in space can be obtained by using MUSCL (Monotonic Upstream-Centered Scheme Conservation Law) 9 scheme interpolation whose formulation is given below

( )[ ]{ }

( )[ ]{ }1/2

1/2 1 1

1ˆ ˆ (1 ) (1 )4

1ˆ ˆ (1 ) (1 )4

Li i i

Ri i i

W W r

W W r

φ κ κ

φ κ κ

+

+ + +

= + − ∇ + + ∆

= − + ∇ + − ∆ (7)

In Eq(7) r equals to:

ii

i

r∆

=∇

(8)

where forward∆ and backward ∇ operators are defined as:

1 1ˆ ˆ ˆ ˆ,i i i i i iW W W W+ −∆ = − ∇ = − (9)

MUSCL scheme requires the limiter functionφ , in order to prevent oscillations and spurious solutions in regions of

high gradients. The limiter function reduces the slopes used in the interpolation of flow variables to the cell faces. The parameter κ defines the order of the accuracy of the interpolation. For κ = -1, purely one sided upwind interpolation; for κ = 0, linear interpolation between one upstream and one downstream point, for κ =1/3 three point interpolation; are obtained with second order accuracy. For κ = 1 the upwind influence is lost and face values are calculated by the arithmetic mean of the neighboring cells.

Using the Van Albada’s limiter For κ = 0, and the Koren’ s limiter for κ =1/3 the resulting interpolation formulation is given below:

1/2 1/2

1/2 1 1/2

ˆ ˆ

ˆ ˆ

L Li i i

R Ri i i

W W

W W

δ

δ+ +

+ + +

= +

= − (10)

For κ = 0 2 2

2 2

( ) ( )

2ia b b a

a bδ + ∈ + + ∈

=+ + ∈

(11)

For κ = 1/3 2 2

2 2

(2 ) ( 2 )

3

a b b a

a b abδ + ∈ + + ∈

=+ − + ∈

(12)


4

where 1 1

,

,L i L i

R i R i

a b

a b+ +

= ∆ = ∇= ∇ = ∆

(13)

∈ is a small number which is used to prevent the activation of the limiter in the smooth regions of the flow domain. ∈ is defined as 0.008 for Van Albada’s limiter and as the square root of the cell area in 2-D problems for Koren’s limiter.

III. Solution Method The system of non-linear discretized governing equations can be written in the form:

ˆ ˆ( ) 0R W = (14)

where R is the residual vector and is defined as

ˆˆ ˆ ˆ( ) ( )ˆ ˆ( )

F W G WR W

∂ ∂∂ξ ∂η

= + (15)

Expanding ˆ( )R W in a Taylor series about (n)th iteration and discarding high order (or nonlinear) terms yields :

1ˆ

ˆ ˆ ˆ ˆ ˆ( ) ( )ˆ

n

n n nRR W R W W

W

∂∂

+ = + ∆

(16)

where ˆ

ˆR

W

∂∂

is the Jacobian matrix. Solving above equation for 1ˆ ˆ( )nR W+ = 0 formulates Newton’s Method as:

ˆ

ˆ ˆ( )ˆ

n

n nRW R W

W

∂∂

∆ = −

(17)

The new values of flow variable vector W at the (n+1)th iteration can be calculated as:

1ˆ ˆ ˆn n nW W W+ = + ∆ (18)

IV. Flux Jacobian In the solution of Euler equations with Newton’s method, the evaluation of the flux Jacobian matrix is needed.

The entries of Jacobian matrix are the derivatives of the residual vector with respect to the flow variables vector. In the calculation of these derivatives a finite difference method or analytical derivation method can be used, and the resulting matrices are called numerical or analytical Jacobians, respectively.

A. Analytical Jacobian Derivation Substituting Eq (5) into Eq (4) , the discretized residual vector can be calculated as:


5

, 1 1 1 1, , , ,

2 2 2 2

1 1 1 1, , , ,

2 2 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ( ) ( ) ( ) ( )

L R L Ri j

i j i j i j i j

L R L R

i j i j i j i j

R F W F W F W F W

G W G W G W G W

+ − + −

+ + − −

+ − + −

+ + − −

= + − +

+ + − +

(19)

Taking the derivatives of residual ,î jR with respect to a flow variable ,

ˆk lW , the residual Jacobian defined as:

1/2, 1/2, 1/2, 1/2,

, 1/2 , 1/2 , 1/

,1/2, 1/2, 1/2, 1/2,

, , , , ,

, 1/2 , 1/2 , 1/2

, ,

ˆ ˆ ˆ ˆˆˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆˆ ˆ ˆ

ˆ ˆ

i j i j i j i j

i j i j i j

L R L Ri j

i j i j i j i j

k l k l k l k l k l

L R

i j i j i j

k l k l

W W W WRA A A A

W W W W W

W W WB B B

W W

+ + − −

+ + −

+ − + −+ + − −

+ − ++ + −

∂ ∂ ∂ ∂∂= + − −

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂+ + −

∂ ∂2 , 1/2

, 1/2

, ,

ˆˆ

ˆ î j

L R

i j

k l k l

WB

W W

−−−

∂−

∂ ∂

(20)

where, ˆ ˆ ˆ ˆ, , ,

ˆ ˆ ˆ ˆL R L R

G GF FA A B B

W W W W

+ −+ −+ − + −∂ ∂∂ ∂= = = =

∂ ∂ ∂ ∂

Analytical derivation of Residual Jacobian needs two sets of derivatives. The first set composes of the derivatives of splitted fluxes with respect to the interpolated flow variables at cell faces, which are 1/2, 1/2, , 1/2 , 1/2

ˆ ˆ ˆ ˆ, , ,i j i j i j i jA A B B± ± ± ±+ − + − . The analytical relations of these derivatives are function of flux schemes but

they are independent of order of spatial discretization. In this study, to evaluate these derivatives the Steger-Warming, Van Leer and AUSM flux schemes are differentiated. The second set consists of the derivatives of interpolated flow variables at cell faces with respect to the flow variables at cell centers. The analytical relation of these derivatives varies according to order of spatial discretization but they are independent of flux schemes. As shown in Eq(6), for the first order discretization right ( ˆ RW ) and left ( ˆ LW ) flow variables are equal to the values at the cell center of the two cells that are just located at the right and left of the corresponding cell face. Therefore, in first order discretization k and l values in Eq(20) changes from i-1 to i+1 and j-1 to j+1, respectively.

1 , 1 1,, ,2 2

1 1, 1 ,, ,2 2

1 , 1 , 1, ,2 2

1 , 1 1 ,, ,2 2

ˆ ˆ ˆ ˆ,

ˆ ˆ ˆ ˆ,

ˆ ˆ ˆ ˆ,

ˆ ˆ ˆ ˆ,

L Ri j i ji j i j

L Ri j i ji j i j

L Ri j i ji j i j

L Ri j i ji j i j

W W W W

W W W W

W W W W

W W W W

++ +

−− −

++ +

−− −

= =

= =

= =

= =

(21)

Jacobian matrices in first order discretization can be evaluated as:

,1 1 1 1, , , ,2 2 2 2

,

ˆˆ ˆ ˆ ˆ

î j

i j i j i j i ji j

RA A B B

W+ − + −+ − + −

∂= − + −

∂ (22)

, , , ,1 1 1 1, , , ,2 2 2 2

1, , 1 1, , 1

ˆ ˆ ˆ ˆˆ ˆˆ ˆ, , ,

ˆ ˆ ˆ î j i j i j i j

i j i j i j i ji j i j i j i j

R R R RA B A B

W W W W− − + ++ + − −

+ + − −

∂ ∂ ∂ ∂= = = − = −

∂ ∂ ∂ ∂ (23)


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where 1/2, 1/2, , 1/2 , 1/21/2, 1/2, , 1/2 , 1/2

1/2, 1/2, , 1/2 , 1/2

ˆ ˆ ˆ ˆ, , ,ˆ ˆ ˆ ˆ

i j i j i j i ji j i j i j i jL R L R

i j i j i j i j

F F G GA A B B

W W W W

+ − + −+ − + −∂ ∂ ∂ ∂

= = = =∂ ∂ ∂ ∂

∓ ∓ ∓ ∓

∓ ∓ ∓ ∓

∓ ∓ ∓ ∓

For the second order discretization, the flow variables at the cell faces are calculated from interpolation of the flow variables at the center of the 4 neighboring cell using MUSCL method. Therefore, in second order discretization k and l values in Eq(20) changes from i-2 to i+2 and j-2 to j+2, respectively. Thanks to the continuous limiter functions used the MUSCL scheme is differentiable of flow variables. Hence analytical Jacobians are evaluated without any difficulty for high order schemes..

( )( )

( )( )

1 1 12 2 2

1 1 12 2 2

1 12 2

1 12 2

1, ,, , ,

, 1,, , ,

2, 1, , 1,, ,

1, , 1, 2,, ,

ˆ ˆ ˆ ˆ, ,

ˆ ˆ ˆ ˆ, ,

ˆ ˆ ˆ ˆ, , ,

ˆ ˆ ˆ ˆ, , ,

L R L R L Ri j i ji j i j i j

L R L R L Ri j i ji j i j i j

L R L Ri j i j i j i ji j i j

L R L Ri j i j i j i ji j i j

W W W W

W W W W

W W W W

W W W W

δ

δ

δ δ

δ δ

−− − −

++ + +

− − +− −

− + ++ +

=

=

=

=

(24)

Jacobian matrices in second order discretization can be evaluated as:

1/2, 1/2, 1/2, 1/2,

, 1/2 , 1/2 , 1/2 ,

,, , 1, 1,

, , , , ,

, , , 1 , 1

, , ,


ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ

ˆ ˆ ˆ

i j i j i j i j

i j i j i j i

L R L Ri j

i j i j i j i j

i j i j i j i j i j

L R L

i j i j i j i j

i j i j i j

W W W WRA A A A

W W W W W

W W W WB B B B

W W W

+ + − −

+ + −

+ − + −− −

+ − + −− −

∂ ∂ ∂ ∂∂= + − −

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂+ + − −

∂ ∂ ∂1/2

,ˆ

j

R

i jW

−

∂

(25)

1/2, 1/2, 1/2, 1/2,,

, , 1, 1,

1, 1, 1, 1, 1,


ˆ ˆ ˆ ˆ î j i j i j i j

L R L Ri j

i j i j i j i j

i j i j i j i j i j

W W W WRA A A A

W W W W W

+ + − −+ − + −− −

∂ ∂ ∂ ∂∂= + − −

∂ ∂ ∂ ∂ ∂∓ ∓ ∓ ∓ ∓

(26)

, 1/2 , 1/2 , 1/2 , 1/2,

, , , 1 , 1

, 1 , 1 , 1 , 1 , 1


ˆ ˆ ˆ ˆ î j i j i j i j

L R L Ri j

i j i j i j i j

i j i j i j i j i j

W W W WRB B B B

W W W W W

+ + − −+ − + −− −

∂ ∂ ∂ ∂∂= + − −

∂ ∂ ∂ ∂ ∂∓ ∓ ∓ ∓ ∓

(27)

1/2, 1/2,

1/2, 1/2,

, 1/2

,1/2, 1/2,

2, 2, 2,

,1/2, 1/2,

2, 2, 2,

,, 1/2 ,

, 2 , 2

ˆ ˆˆˆ ˆ

ˆ ˆ ˆ

ˆ ˆˆˆ ˆ

ˆ ˆ ˆ

ˆˆˆ ˆ

ˆ ˆ

i j i j

i j i j

i j

L Ri j

i j i j

i j i j i j

L Ri j

i j i j

i j i j i j

Li j

i j i j

i j i j

W WRA A

W W W

W WRA A

W W W

WRA A

W W

+ +

− −

+

+ −+ +

+ + +

+ −− −

− − −

+ −+

+ +

∂ ∂∂= +

∂ ∂ ∂

∂ ∂∂= − −

∂ ∂ ∂

∂∂= +

∂ ∂, 1/2

, 1/2 , 1/2

1/2

, 2

,, 1/2 , 1/2

, 2 , 2 , 2

ˆ

ˆ

ˆ ˆˆˆ ˆ

ˆ ˆ ˆ

i j

i j i j

R

i j

L Ri j

i j i j

i j i j i j

W

W

W WRA A

W W W

+

− −

++

+ −− −

− − −

∂

∂

∂ ∂∂= − −

∂ ∂ ∂

(28)

The main advantage of the analytical method is that the residual Jacobian can be calculated accurately. The order of error in the analytical method can be as small as the round-off error. Although the analytical method requires code development, run time of an analytical code is short. However, as the complexity of the discretized residual equations increases, the derivation of the analytical Jacobian becomes more complicated.


7

B. Numerical Jacobian Calculation Another alternative for Jacobian evaluation is to compute the Jacobian numerically. Using a small finite-difference perturbation magnitude ε, the numerical Jacobian can be calculated by the forward-difference method as follows10

ˆ ˆ ˆ ˆ ˆ ˆ ˆ( )ˆ ˆ

1, 1, ( )

m m m n m

n n

R R W R (W e ε) R (W)

εW W

where m mmax and n mmax nbound

∂ ∆ + −= =

∂ ∆

= = + (29)

where, ˆmR is the mth component of the residual vector and the ˆnW is the nth component of the flow variable vector,

en is the nth unit vector. The value of the nth component of the unit vector en is one, and all other components are zero. The size of the residual vector is defined by mmax, which equals to 4 times the number of interior cells in 2D flow problems. The size of the flow variable vector is larger as much as the number of boundary cells, nbound. In the numerical method, Jacobian evaluation does not require the large coding effort as needed in the analytical method. The same residual discretization is used for both the original and perturbed flow variables. This reuse of the same code is one of the important advantages of the numerical approach. Moreover, for cases in which the analytical derivation is difficult, numerical Jacobian can be obtained without any difficulty.

The inaccuracy and long computation time are the two main disadvantages of the numerical Jacobian evaluation. The error in numerical Jacobian is function of the finite-difference perturbation magnitude. The accuracy of the numerical jacobians can be improved with the usage of an optimum perturbation magnitude that minimizes the total error in the finite difference evaluation. The main reason that causes long computation time is the necessity of the residual vector calculation with each perturbed flow variable in the whole domain. For a given cell, the residual is only a function of flow variables in that cell and the neighboring cells according to the discretization used. In order to reduce computation time, the perturbed residual is computed only with flow variables in these cells. For first-order discretization, in addition to the cell in which the flow variable is perturbed, four neighboring cells are used. Considering four flow variables in each cell, 20 perturbed residual vector evaluations are required for the given cell. In second-order discretization, using eight neighboring cells in addition to the given cell, 36 perturbed residual vector evaluations are required. Although the speed and the accuracy of the analytical method may not be reached, the numerical Jacobian evaluation method may become faster and more accurate with some precautions.

C. Accuracy of Numerical Jacobians

Error Analysis In numerical Jacobian calculation, mainly two types of errors occur. These are truncation and condition errors12.

While truncation error is due to neglected terms in the Taylor’s series expansion, condition error is caused by loss of computer precision. In Eq(30), the truncation error due to the neglected terms in the Taylor series expansion can be written as:

2

2

ˆ ( )( )

ˆ 2m

Trun

n

RE

W

ζ εε ∂=

∂ (30)

where ζ=[ ˆnW , ˆ

nW ε+ ].

Because of computer precision, the exact values of the mth components of the vector ( )ˆ ˆmR W and their computed

value ˆ( )mR W� can be different due to round-off error ( )ˆmEr W :

ˆ ˆ ˆ ˆ( ) ( ) ( )

ˆ ˆ ˆ ˆ( ) ( ) ( )

m m m

m m m

R W R W Er W

R W R W Er Wε ε ε

= +

+ = + + +

�

� (31)


8

By using the computed function, ˆ( )mR W� , Eq(29) can be written as:

ˆ ˆ ˆ( )ˆ

ˆ ˆ ˆ ˆ ˆ( )ˆ

ˆ ˆ( ) ( )( )

ˆ ˆ

m m n m

n

m m n m m n m

n

m mm

n n

R W R (W e ) R (W)

W

R W R (W e ) R (W) Er (W e ) Er (W)

W

R W R WEc

W W

εε

ε εε ε

ε

∆ + −=

∆

∆ + − + −= +

∆

∆ ∆= +

∆ ∆

� � �

�

�

(32)

where Ec (ε) is the condition error. Considering a bound of round-off error { } ˆ ˆ max ( ) , ( )RE Er W Er W ε= + the

maximum of the condition error can be approximated as:

2

( ) REEc ε

ε= (33)

This bound of round-off error can be considered the precision error, which depends on the computer processor and compiler. For the computations of normalized variables where the magnitude of the computed values are around one, the precision error equals to machine epsilonMΣ . A reasonable estimate of MΣ can be given as follows:

1

2M mΣ = such that 1 1M+ Σ > (34)

where m is the number of possible highest bits in the binary representation of the mantissa. The machine epsilon

MΣ values of the compiler-computer configuration are found according to Eq(34). In this study, for single precision 71.19 10M

−Σ ≅ × , and for double precision 162.2 10M−Σ ≅ × values are reached.

Optimal Perturbation Magnitude Analysis The error in numerical Jacobian is highly dependent on perturbation magnitude, ε. For the small values of

perturbation magnitude the condition error grows up and dominates the total error. On the other hand as the magnitude of perturbation gets larger, the truncation error becomes dominantly larger. Hence, there should be an optimal value for the perturbation magnitude that minimizes the total error in numerical Jacobian. The error in the numerical Jacobian matrix equals to the sum of the errors in each matrix element. To minimize the error, each finite difference computations can be performed with their own optimum perturbation magnitudes. However this approach to minimize error will be highly impractical due to its cost. Alternative way is to find an optimum perturbation magnitude which minimizes the global total error arising from the finite differencing of each element. This can be achieved by evaluating the magnitude which satisfies the least squares minimization of the error. The formulation for the square of the total numerical Jacobian error is given below:

( )2

24( 1)2

21 1

2 ( )( )

2m

neighmmaxR m

TOTALm nn nn

E RE

w

ζ εεε

+

= =

∂= + ∂ ∑ ∑ (35)

In the Eq(35) the outer summation loops are constructed for the whole domain, excluding the ghost boundary cells where the residuals are not computed. The number of neighboring cells which are related to the residual vector is represented by neigh; it has a value of 4 in 1st order discretization and 8 in 2nd order discretization. The least squares minimization is performed by differentiation of the Eq(35) with respect to perturbation magnitude.


9

( )

( )

2 2 24( 1)

2 2 21 1

2 2 24(

2 21

2 2( ) ( ) ( ) 12

2 2

2 2( ) ( ) ( )2

2 2

m m

m m

neighmmaxR RTOTAL m m

m nn nn nn

neighR RTOTAL m m

nn nn nn

E EE R R

w w

E EE R R

w w

ε ζ ζεε ε ε

ε ζ ζε εε ε ε ε

+

= =

=

∂ ∂ ∂ = + − + ∂ ∂ ∂

∂ ∂ ∂ = − + − ∂ ∂ ∂

∑ ∑

( )

1)

1

22 2224( 1)

2 21 1

4( ) ( )2

4m

mmax

m

neighmmaxRTOTAL m

m nn nn

EE R

w

ε ζ εε ε ε

+

=

+

= =

∂ ∂ = − − ∂ ∂

∑ ∑

∑ ∑

(36)

The optimum ε which minimizes the total error is found by equating Eq (37) to zero

( )

( )

2

22 24( 1) 4( 1)max max 2

2 21 1 1 1

( )0

( ) 40

4

OPT

m

TOTAL

neigh neighm mm

Rm nn m nnnn

E

RE

w

ε ε

εε

ζεε

=

+ +

= = = =

∂=

∂

∂− = ∂

∑ ∑ ∑ ∑

(37)

The optimum perturbation magnitude can be evaluated as:

( ) ( )

max 2

1

224( 1)max

21 1

4( 1)

2( )

m

m

Rm

OPTneighm

m

m nn nn

neigh E

R

w

εζ

=

+

= =

+=

∂ ∂

∑

∑ ∑

(38)

Using the definition of the L2 norm, above equation can be simplified into the following form:

2

2

2

2

4( 1)2

( )

R

OPT

neigh E

R

w

εζ

+=

∂ ∂

(39)

Substituting above optimum perturbation magnitude into the Eq(35) the minimum error is evaluated as follows:

22 4( 1)max

2 221 1

2

( )( )( ) 2 4( 1) 2

m

neighmm

TOTAL OPT R Rm nn nn

RRE neigh E E

w w

ζζε+

= =

∂∂= + + ∂ ∂ ∑ ∑ (40)

Derived equations show that to estimate the numerical Jacobian with minimum error requires the calculation of second derivatives of residual vector with respect to flow variables and the round-off error. The round-off error in single precision can be estimated as the difference in residual vectors that are calculated with single and double precisions whereas the estimation of the round-off error in higher order precision calculations may not be easy. The evaluation of second derivatives can be employed by finite difference relation. Although the evaluated values may not be very accurate they may be useful for estimating the order of magnitudes.


10

Approximating Eq(39) for an arbitrary norm of Lp gives:

2

2

4( 1)2

( )

R pOPT

p

neigh E

R

w

εζ

+=

∂ ∂

(41)

The solution of CFD problems are commonly performed with normalized variables. Therefore assumption of the round-off errors to be equal to the machine epsilon may be reasonable. Calculation of second derivatives may not be easy. Therefore in most of the practical calculation, the order of second derivatives can be assumed as unit. These approximations will result in a simple formula which can be used to estimate optimum perturbation magnitude.

2 4( 1)

( ) 2 4( 1)

OPT M

TOTAL OPT M

neigh

E neigh

ε

ε

= + Σ

= + Σ (42)

The flux Jacobians are evaluated using wide range of perturbation magnitudes and the optimum value obtained

from trial error procedure was compared with corresponding results of Eq (42).

D. Structure of Jacobian Matrix and Solution Strategies The most of the entries in the Jacobian matrix equal to zero since the discretized residual equations only depend

on flow variables in neighboring cells. The Jacobian is a square matrix whose dimensions equal to the total number of flow variables. In the first order spatial discretization, five stencils are required and Jacobian is a block diagonal matrix that is made up of five 4x4 blocks. In second order discretization, nine stencils are required and the matrix is made up of nine 4x4 blocks. As a result whole matrix entries, except those block bands and boundary entries equal to zero. Although there is no need to compute and store the zero elements of the Jacobian matrix, full matrix solvers require the whole matrix to be constructed which is computationally expensive. The sparse matrix solvers that store only the non-zero elements in the matrix should be used to overcome the high cost of whole matrix storage.

In this study, the UMFPACK (Unsymmetric-pattern MultiFrontal PACKage) sparse matrix solver package11 is used in order to solve the linear system of equations. In this method, the full matrix is converted into sparse storage mode and then factorized using a sequence of small dense frontal matrices by LU factorization. The usage of a sparse matrix solver increases the efficiency of the flow solver significantly.

Newton’s method requires a good initial guess for convergence. This is a considerable drawback. In this study, flow variables are initialized with their free-stream values, although it may be a poor guess. Several ideas are available to modify Newton’s method to improve stability. For example, a time-like term can be added to the diagonal of the Jacobian matrix to make it more diagonally dominant. Increased diagonal dominance leads to a more stable linear solution. With the addition of a time-like term, the modified Newton’s method becomes:

[ ]ˆ1 ˆ ˆ( )ˆ

n

n nRI W R W

t W

∂∂

+ ∆ = − ∆

(43)

The original Newton’s method can be constructed as ∆t → ∞. In the modified Newton’s method, a small initial value ∆t0 is chosen and a new value of ∆t can be obtained using L2-norm of the residuals as

0

0 2

2

ˆ( )

ˆ( )

n

n

R Wt t

R W∆ = ∆ (44)

In this study, to improve the convergence from free-stream initial conditions, the modified Newton’s method is used. For all calculations the L2-norm is computed including all the residuals in the domain. The convergence of this


11

method is slow until ∆t gets very large. However, the modification in Newton’s method is useful in the early stages of iterations. Later, the solution becomes more accurate, and the diagonal term addition may not be needed. The withdrawal of the diagonal term from the matrix at the proper convergence level significantly reduces the number of iterations and CPU time. Initial and the withdrawal values of diagonal term that gives the best convergence performance are chosen by trial-error.

V. Effects of Flux Jacobian Evaluation on Flow Solution The accuracy of numerical Jacobian is studied in both internal and external flow computations. As an internal

flow application 10 percent circular arc, Ni bump geometry is studied. Computations are performed at three different incident Mach numbers: M∞ = 0.5, M∞ = 0.675 and M∞ = 2. Three different grid sizes are used. The medium sized grid has 65x17 nodes and it is shown in Figure-1. The characteristic boundary conditions based on Riemann invariants are used at the inflow and outflow boundaries4. Wall and symmetry boundary conditions are employed at the lower and upper sides of the boundaries. Figure 2 shows the Mach number contours of solution calculated for a free-stream Mach number of 0.675. In this solution, fluxes are calculated by using a second order AUSM scheme

As an external flow application, flow around NACA 0012 airfoil is studied at a free-stream Mach number of

0.85, and an angle of attack of 1degree. The grid sizes of 129x33, 193x49 and 225x65 were used. The 225x65 grid, which has 140 nodes on the airfoil is shown in Figure-3. The flow solution was performed for 0.85 Mach, 1degree angle of attack flow case.

In Figure-5 the pressure distribution obtained with solutions using analytical and numerical Jacobian evaluation is given. For numerical Jacobian the finite difference perturbation magnitude was taken as 3x10-8 which results from

Eq(39). It can be seen from the figure that using the optimal value suggested both Jacobian evaluation methods gave identical flow solution results.

One of the main objectives of this study is

evaluate the effects of perturbation magnitudes on the accuracy of numerical Jacobians. Hence, numerical Jacobians are calculated for a wide range of perturbation magnitudes. The deviation between the numerical and analytical Jacobian matrices is defined as error. Since the error itself is also a matrix, the norm values of error matrix are

Figure 1. Mach Contours for M∞= 0.675 .

Figure 2. 65x17 grid for Ni bump geometry.

Figure 4. Mach number distribution M∞= 0.85, αααα=1o

Figure 3. SurfaceMach number distribution.

M∞= 0.85, αααα=1o.


12

defined. Different types of norms are used. The change of the norm values of error matrix with respect to perturbation magnitude is plotted, and the results read from theses plots are compared with the results calculated by theoretical equations. The read perturbation magnitude that gives minimum error is compared with the optimum perturbation magnitude that is estimated with Eq.(39). The read norm values of minimum error is compared with the norm value of error at the optimum perturbation, that is estimated by neglecting the second term in Eq.(40). Effects of different discretization schemes, flow conditions, geometries and grid resolution on the error in numerical Jacobians are also studied. As stated earlier, the error in numerical Jacobian theoretically depends on two parameters: the norm values of second derivatives of the residuals with respect to the flow variables, and the round-off error. The evaluation of both of these terms may be impractical for general CFD applications. However, the evaluation of these terms may give an opportunity to validate the theoretical equations. In this study, for single precision computations, the round-off error is estimated as the differences between the residuals calculated by single and double precisions. The second derivatives can be calculated by a finite difference method. In order to find an appropriate perturbation magnitude for second derivatives, various perturbation magnitudes are tested. Results show that the norm values of second derivatives increases for perturbation magnitudes smaller than 10-8, but almost have no variation between 10-8 and 10-3. In the calculation of the norm values of second derivatives with double precision computations, the perturbation magnitude is set to 10-5.

Table-1. The estimated round-off error for single precision computation

Ni bump geometry; 65x17 grid , 0.675 inlet Mach #, 1st order AUSM fluxes

Round-off error Second derivatives

εεεεoptimum

( Eq.39 ) Minumum E T

( Eq.40 ) εεεεoptimum

(trial error) Minumum E T

(trial error)

L 1 4.75 x10-9 0.047 4.34 x10-4 0.885 3.96x10-4 0.927 L 2 7.76 x10-10 0.016 7.8 x10-4 1.3x10-2 3.1x10-4 8.9 x10-3 L∞ 2.44 x10-9 0.0202 - - 2.0x10-4 5.6 x10-4

Figure 6. Numerical Jacobian Error in Single Precision

(AUSM ,1st order,single, Mach#= 0.675)

Figure 5. 225x65 grid for Naca0012 airfoil.


13

The norm values of the round-off error for single precision and the second derivatives were tabulated in Table-1. The same table also presents the optimum perturbation magnitudes and the resulting errors from the usage of those values according to the Eq(39) and Eq(40).Figure-6 presents the variation of the error in numerical jacobian with perturbation magnitude, for the single precision 1st order flux calculation. To obtain the error values given in the vertical axis, the L1 and the L2 vector norms are used. Both optimum perturbation magnitudes and corresponding error calculated by theoretical relations also shown in Figure-6. Theoretical and trial and error results agree very well with each other. In the rest of the study, the numerical Jacobian error plots will be given for the double precision calculations and the error will be given in matrix norm definition.

The value of machine epsilon in single precision is approximately the square root of the value computed by double precision. Figure 7 shows that with the change of the computation precision from single to the double, the optimum values becomes the square of the values obtained for single precision. Since the change of the precision affects only the round-off error, and reduces the numerator of the Eq(39) by square, the location of the optimum and the total error changes accordingly. In Eq(39) the norm values of round-off and second derivatives are in division, hence the optimum perturbation magnitude does not vary with different norm definitions. However in Eq(40) the norms of round-off error and second derivatives are in multiplication. Hence, the total error varies with norm definition. The norms used in Figure 7 are L1, L∞ induced matrix norms and Frobenious entry-wise matrix norm, and they are defined below:

12

21

1 11 1 1 1

|| || max | | || || max | | || || | |m n m n

ij ij F ijj n j m

i j i j

A a A a A a∞≤ ≤ ≤ ≤= = = =

= = =

∑ ∑ ∑∑ (45)

The formulation of the second derivatives of flux residuals with respect to the flow variables can be obtained by differentiating the residual Jacobian as below:

( ) ( )

( )

1/2, 1/2, 1/2, 1/2,

, ,1/2, 1/2,,

1/2,

2 22 2 2, , 1/2, 1/2,

1/2, 1/2,2 2 2 22,, , ,

21/2,

2

ˆ ˆ

ˆ ˆ ˆ î j i j i j i j

k l k li j i jk l

i

i j

L L R Ri j i j i j i j

i j i jL R

k lk l k l k l

i j

L

W W W WR R F FA A

W W WW W W WW W

WF

W

+ + + +

+ +

−

−

+ −+ ++ −

+ +

+−

∂ ∂ ∂ ∂∂ ∂ ∂ ∂∂ = = + + + ∂ ∂ ∂∂ ∂ ∂ ∂∂ ∂

∂∂−

∂ ( )

( ) ( )

1/2, 1/2, 1/2, 1/2,

1/2,

, 1/2 , 1/2 , 1/2

, 1/2 , 1/2

2 221/2,

1/2, 1/2,2 2 2, ,, ,

22 2, 1/2 , 1/2

, 1/22 2 2, ,

j i j i j i j

i j

i j i j i j

i j i j

L L R Ri j

i j i jR

k l k lk l k l

L Li j i j

i jL R

k l k l

W W WFA A

W WW WW

W W WG GB

W WW W

− − −

−

+ + +

+ +

−−+ +

− −

+ −+ ++

+

∂ ∂ ∂∂− − −

∂ ∂∂ ∂∂

∂ ∂ ∂∂ ∂+ + +

∂ ∂∂ ∂

( ) ( )

, 1/2

, 1/2 , 1/2 , 1/2 , 1/2

, 1/2 , 1/2

2

, 1/2 2, ,

2 22 2, 1/2 , 1/2

, 1/2 , 1/22 2 2 2, ,, ,

i j

i j i j i j i j

i j i j

R R

i jk l k l

L L R Ri j i j

i j i jL R

k l k lk l k l

WB

W W

W W W WG GB B

W WW WW W

+

− − − −

− −

−+

+ −− −+ −

− −

∂+

∂ ∂

∂ ∂ ∂ ∂∂ ∂− − − −

∂ ∂∂ ∂∂ ∂

(46)

Figure 7. Effects of precison on the error in numerical Jacobian

Matrix. (AUSM ,1st order, Mach#= 0.675)


14

Figure 8. Effects of flux splitting method on error in Jacobian

Matrix ( AUSM, Ni bump,M#=0.675)

In first order flux discretization, the second derivatives of interpolated face variables with respect to the flow variables equal to zero. Hence in first order discretized fluxes, the magnitudes of the second order derivatives depend only on the flux evaluation method. The effects of the flux evaluation method on the magnitudes of the second derivatives are

investigated. The flow over bump geometry with inlet Mach number equals to 0.675 is solved using a grid size of 65x17 nodes with different first order methods. Each flux evaluation methods resulted with approximately the same sized second derivatives. The calculated L2 norm values of the second derivatives of the residual vectors are given in Table-2. The same table shows that the second derivatives do not vary with flux splitting method. Since the same flow problem is solved on the same grid the magnitudes of the flow variables and fluxes will approximately same even if the flux evaluation method varies. Hence the round off error is about the same in each flux calculation methods. The effect of flux evaluation method on the error in 1st order numerical discretization is presented in Figure-8. The same figure shows that the optimum perturbation magnitude does not vary with 1st order Steger-Warming, Van-Leer and AUSM flux evaluation methods.

In second order flux discretization, the second derivatives of interpolated face variables with respect to the flow variables are not zero. Hence these terms will affect the second derivatives of the residual vector. The formulation of the interpolated face variables is given in Eq(10). Differentiating this equation with respect to the flow variables twice gives :

2 2 2 2

1/2 1/2 1/2 1/2

2 2 2 2, , , ,

ˆ ˆ,

L L R Ri i i i

k l k l k l k l

W W

W W W W

δ δ∂ ∂ ∂ ∂= = −

∂ ∂ ∂ ∂∓ ∓ ∓ ∓ (47)

The second derivatives of the interpolated face variables directly depend on the second derivatives of the limiter functions. Limiters are known to stall the convergence of an iterative scheme, because of accidental switching in smooth flow regions20 In smooth flow regions difference between the values of flow variables at the neighboring cells is nearly zero. Therefore assuming that backward and forward differencing of flow variables (a and b in Eq 11 and Eq12) are equal to each other would be reasonable. The formulation for the second derivatives of the limiter function with this assumption will become very simple and it is given in Eq(47). This formulation presents that in smooth flow regions, as the values of ∈ get smaller the second derivatives of the limiter functions becomes larger.

2

2

ˆ

â b

a

W aa

δ

=

∂ =∈∂ +

(48)

Table-2. Effect of flux splitting method on second derivatives

Flux Evaluation

Method

22

21

( )n

n

R w

w=

∂ ∂

∑

SW 23.859 VL 25.186

AUSM 24.405


15

Figure 9. Effects of ∈∈∈∈ on the accuracy of the Jacobian

( Mach#= 0.675, Ni bump)

In the formulation of limiter functions, in order to prevent the activation of limiters a small number ∈ is used. There are different studies performed to define the values of ∈. 18,19. In this study, the effects of the magnitude of ∈ on the

accuracy of numerical Jacobians are studied. In Table 3, the root mean square of the second derivatives of the residual vector evaluated with different sizes of ∈ is tabulated. Consistent to Eqs (46),(47) and(48), the second derivatives of residual vector get larger as the size of the ∈ gets smaller. The variation of the L1 norm of the error in numerical Jacobian matrix with respect to the size of ∈ is given in Figure-9. For small ∈ values the second derivatives of interpolated face variables become larger compared to the second derivatives of splitted flux vectors. Hence for second order discretization, the size of the second derivatives of the residual vector is dominated by the second derivatives of the limiter functions for small ∈ values. The value of ∈ has no affect on the round-off error. Therefore, as the values of ∈ become smaller the magnitude of second derivatives increases while the round-off error stays constant. As a result, the small values of ∈ causes an increase in total error and a decrease in the optimum perturbation magnitude. The results shows that limiter functions calculated with small ∈ values significantly increase the magnitudes of the second derivative of residual vector and they can not be assumed as one in Eqs(39) and (40) . Hence the optimum perturbation magnitude deviates from the one proposed by Eq(42) significantly. For larger values of the ∈ the second derivatives of interpolated face variables become smaller compared to the second derivatives of splitted flux vectors. In this case, the size of the second derivatives of the residual vector is significantly affected by the second derivatives of the splitted flux vectors. Hence the usage of larger ∈ will not affect the accuracy of the Jacobian any more. Moreover the usage of very large ∈ values in limiter function will be improper. Since the variation in flow variable will become negligible compared to the large ∈ values, the definition of the interpolation with limiter function will be degraded. As a summary, the accuracy of the second order numerical Jacobian is bounded by that of the first order numerical Jacobian. On the other hand, the evaluation may become erroneous with the usage of small ∈ in the limiter function. In this study the most accurate evaluation for the second order numerical Jacobian is achieved with the usage of the ∈ values proposed in References[20] and[21]

Table-3. Effect of ∈ used in limiter

function on second derivatives

∈∈∈∈ used in limiter

22

21

1 ( )n

n

R w

n w=

∂ ∂

∑

∈ = 0.008 0.035 ∈ = 10-3 0.165 ∈ = 10-6 7.02 ∈ = 10-9 58.7 ∈ = 10-12 260


16

The effects of different geometries and

flow conditions on the accuracy of numerical Jacobian evaluation are studied. Figures 10 and 11 show the variation of the error with respect to perturbation magnitude for internal and external flow applications. In these figures, the effects of different flux splitting methods on the accuracy of numerical Jacobian evaluation for flows over bump and airfoil geometries are presented. The total error in the vertical axis is given in terms of the induced L1

matrix norm. In both results, the optimum perturbation magnitudes for first and second order discretizations are same since a proper ∈ value is used. Moreover the optimum perturbation magnitudes provided by these figures agree well with the proposed magnitude by Eq(42) . One of the important conclusions from these figures is that, although the effects of the flow condition and geometry on the magnitude of optimum perturbation are small the total error is significantly affected. The total error in the airfoil case is about two order of magnitudes larger compared to the bump case. To resolve the cause of this situation the Eq(39) and Eq(40) should be revised. In formulation which gives optimum perturbation magnitude the round off error and second derivatives are in division. In the formulation which gives the total error those two terms are in multiplication. Increasing the total error without changing the optimum perturbation magnitude may be caused by the proportional enlargement in both of round-off error and the second derivatives. In both flow cases, approximately the similar flow conditions are used. In both cases, transonic flow conditions cause a supersonic region on the geometry with a maximum mach number of 1.4. The main difference between the

two cases was the grids used in the solutions. In the problem of flow over Ni-bump, an H-grid whose inlet and outlet are located 1.75 times of chord length away from the leading and trailing edges. In flow over airfoil problem, a C-grid in which distance between far-field boundary and the airfoil geometry varies between 10 to 60 chord length is used. To be able to resolve the gradients, nodes were clustered near to airfoil geometry and were expanded at the far-field. The sizes of the cell faces at the far-field of C-grid are approximately 15 times larger than the ones used at the inlet and the outlet of H-grid. Due to the large cell faces it will be reasonable to expect large flux values near the far-field of the C-grid compared to the H-grid. This enlargement in the flux magnitudes may amplify the round-off error in residual computations. Since there is no practical way of round-off error estimation for double precision computations, the variation of round-off error according to the those two grids checked for single precision computation. Although the error results in Figures 10 and 11 were presented for double precision the single precision computation of the difference of round-off errors in two cases gives satisfactory clues. For single precision computation, the L2 norm of the round off error for c-grid is computed as approxiamtely4 x10-6 whereas it is

Figure 10. Effects of flux splitting method on error in Jacobian

Matrix ( Mach#= 0.675, Ni bump)

Figure 11. Effects of flux splitting method on error in Jacobian Matrix ( Mach#= 0.85,α=1o, Naca0012)


17

calculated as 1.2x108 for the H-grid with the same size of node numbers. The calculated single precision round-off errors with different norm definitions are tabulated in

The second derivatives of the residual at a given cell depends on the flow variable at that cell and 4 or 8 neighboring cells according to the order of discretization. Figure 12 and 13 present the contour plots of the averaged values of the second derivatives of the residual vector. Figure 13 shows that the second derivatives of the residual vector gets larger near boundaries and gets smaller near the wall. The comparison of the values of second derivatives for those two cases shows that in the regions closer to the wall geometries the second derivatives are in the same order of magnitudes. However in the far-field regions of the c-grid the second derivatives of the residual vector are approximately 2 orders of magnitude larger. Table-4, Figures 12 and 13 present that in flow solutions with larger grid cells, the round-off error and the second derivatives of the residual vector get larger compared to the ones that uses a grid with smaller sized cells.

The effects of the cell size on the second derivatives and on the total error in numerical Jacobian are shown by two alternative ways. Firstly, only the cells closer to the airfoil which has smaller sizes are used for the comparison with H-grid which has small sized cells. Secondly, the same flow problem is re-solved with a c-grid whose farfield is closer to the airfoil geometry with same number of nodes. The error plots of the numerical Jacobian for those cases are given in Figure-16 with the error plot of the H-grid case. The Figure 16 is drawn with the results obtained with 2nd order discretization. In the small region of the c-grid which is very close to the airfoil geometry, the magnitude of the total error is almost equal to the total error of the H-grid case since the cell size are almost equal. The results also show that by using a grid with closer far-field boundaries, uniformly distributed grid cells can be constructed and the total error may be reduced. However, as the far-field boundary gets closer to geometry the accuracy of the boundary conditions becomes more critical.

Table-4. The estimated round-off error for single precision computation

grid jmax 1imax 1

sin, ,

2 2

double glei j i j

i j

R R−−

= =

−∑ ∑ ( )2jmax 1imax 1

sin, ,

2 2

double glei j i j

i j

R R−−

= =

−∑ ∑

65x17 H-grid 7.41x10-9 1.3x10-8

129x33 H-grid 6.5x10-9 1.2x10-8

129x33 C-grid 1.4x10-7 4x10-6

Figure 12.Contours of second derivatives

Figure 13. Contours of second derivatives


18

After completing the search for the optimum perturbation magnitude for different flux evaluation methods and for different flow problems with different grid types, the variation of the error with grid resolution is studied. The analysis is performed both for the h-grid and c-grid cases. The results are presented in Figures 16 and 17. Calculations are performed using AUSM scheme for first and second order spatial discretizations. In second order discretization, limiter function of the Van Albada is used with ∈ value of 0.008. For the bump geometry an inlet Mach number of 0.675 and for the airfoil geometry the free stream Mach number of 0.85 and 1 degree of angle of attack is used. Like in the previous cases, for different grid resolutions, the optimum perturbation locations for bump and airfoil geometries coincides with each others, whereas the total error is larger in C-grid solution. To make the grid finer or coarser

does not change the magnitudes of the flow variables significantly. Therefore, the second derivatives and the round off error are almost constant as the grid resolution changes. As a result the optimum perturbation magnitude minimizing the total error is same for all coarse and fine grids. The optimum perturbation magnitudes presented by Figures 17 and 18 agree well with the magnitudes proposed by Eq(42).

As the last factor affecting the accuracy, the effect of the free-stream flow condition is analyzed. Figure-19 shows the change of errors with perturbation magnitude for three different flow conditions. The results presented are calculated for bump geometry by AUSM scheme with first and second order spatial discretizations. Results show that the optimum perturbation magnitude is not sensible to the free stream conditions of the flow problem chosen. The optimum magnitude did not varied significantly for the cases with different inlet mach numbers and its approximately equal to the magnitude proposed by Eq(42).

Figure 15. Alternative-2. The 129x33 Cgrid with closer farfield boundary

Figure 14. Alternative-1, The small cell sized region chosen for the comparison

Figure 16. Effects of thegrid size on the accuracy of Jacobian


19

Figure 17. Effects gridre solution on error in Jacobian Matrix

( AUSM, MACH#=0.675, Ni bump)

Figure 18. Effects grid resolution on error in Jacobian Matrix ( AUSM, MACH#=0.675, Naca0012)

Up to here, different factors which may affect the accuracy of the numerical Jacobian are analyzed. All the results show that usage of the finite difference perturbation magnitude prescribed by Eq(42) is satisfactory enough to construct accurate numerical Jacobian matrices compared to analytical ones. The Eq(42) was the simplification of the Eq(39) with the assumption of second derivatives equal to one and round-off error equals to machine epsilon. Although the assumed values for round-off error and second derivatives are not reasonable for most of the cases the proportion of these two terms does not change. Therefore, the assumption works well and it will be concluded that the optimum perturbation magnitude nearly equals to 3.0x10-8 for double precision and 6.9x10-4 for single precision computations. Analyzing the ways of accurate computation of numerical Jacobians, the effect of the accuracy of the Jacobian on the convergence of the flow solution and on the accuracy of the sensitivity analysis will be presented in remaining parts.

To make the flow solution with

Newton’s method more stable some modifications are applied on it. In earlier iterations, diagonal terms are added to Jacobian to have stable linear solution. Reaching a proper convergence level, diagonal terms are withdrawn to reduce the total number of iterations and CPU time. The number of iterations and overall CPU time depend on the magnitudes of the diagonally added terms. The best convergence behavior can be achieved with the usage of optimum values for the initial and the withdrawal values of these terms. The optimum initial and the withdrawal values vary for different flow solutions since the Jacobian matrix changes. In this study, the initial and withdrawal values for diagonally added terms are fixed at a reasonable values, rather than searching the

optimums for each cases. Different fixed values are used for the first and the second order discretizations since the change of order of discretization greatly changes the Jacobian matrix.

The effects of accuracy of the Jacobian matrix on the convergence behavior of the Newton’s method is studied before by 1,2 . In those studies, it was shown that usage of estimated (numerical) Jacobian rather than exact (analytical) Jacobian will degrade the convergence performance of the solver. In next section, the variation of convergence histories and the CPU time with the usage of analytical and numerical Jacobian are presented. Figure 19 and Figure 20 give the convergence histories for the flow solutions over bump and airfoil geometries. Calculations are performed with both first and second order spatial discretizations. Figures present the convergence performances of the analytically evaluated Jacobian and the numerically constructed Jacobians with different finite


20

difference perturbation magnitudes. The number of node points of grids for the bump and airfoil geometries are 65x17 and 129x33, respectively. For solution on bump geometry initial and withdrawal values of diagonally added terms are chosen as 3 and 5, respectively for the first order spatial discretization and in second order discretization chosen as 50 and 500. In airfoil case initial and withdrawal values of diagonally added terms are chosen as 3 and 5, respectively for the first order spatial discretization and in second order discretization chosen as 200 and 20000. Figures for convergence histories shows that the best convergence performances are given by the analytically derived Jacobian and the numerical one which is computed by the optimum perturbation magnitude. As the accuracy of the numerical Jacobian degrades the convergence performance also degrades.

Figure 19. Effects of flow condition on error in Jacobian Matrix

( AUSM, Ni bump)

Figure 20. Convergence History ( AUSM, Ni bump)

Figure 21 Convergence-history

( AUSM, Naca0012)


21

In Figure 22 the effect of the ∈ used in the limiter function on the convergence performance is given. To generate the results given in Figure 22, Each numerical Jacobian is calculated by its own optimum perturbation magnitude which can be read from Figure-9. Figure 23 presents the CPU time spent in an Newton iteration. Figure shows that, the most of the CPU time is spent for Jacobian matrix evaluation, when the large grid size is used eventhough 1st order discretization is utilized.

VI. Sensitivity Governing Equations The gradient based optimization method is widely used technique in aerodynamic shape optimization. The

objective is maximizing or minimizing the aerodynamic loads by using design variables. To be able to perform this, gradients of objective functions in design variable space is required. The gradients of aerodynamic loads with respect to design variables are known as objective function sensitivities. In general, aerodynamic loads are functions of state flow variables, geometrical variables and design variables .14

( ) ( ){ }j jC C W β , X β , β= (49)

Hence applying the chain rule sensitivities of the aerodynamic loads can be calculated as follows

. j j j j

k k k k

dC C C CW X

d W Xβ β β β∂ ∂ ∂ ∂ ∂= + + ∂ ∂ ∂ ∂ ∂

(50)

The second term in the right hand side is zero when design variables are independent from the geometry. The third term in the right hand side is zero when geometrical design variables are used since their effect on total derivative is already stated in the second term. In this study geometrical design variables are used. The aerodynamic geometry is modified by using Hicks Henne functions15. The weightings of the functions are chosen as design variables.

j j j

k k k

dC C CW X

d W Xβ β β∂ ∂ ∂ ∂= + ∂ ∂ ∂ ∂

(51)

Figure 23. CPU time spent in an iteration ( AUSM, Naca0012)

Figure 22. Effects of ∈∈∈∈ on convergence history ( AUSM, Ni bump)


22

where Cj, w, X , βk are aerodynamic loads, flow variables, grid coordinates and design variables, respectively The aerodynamic loads have explicit dependence on the flow variables and the coordinates of the geometry. Therefore the evaluations of ∂Cj /∂W and ∂Cj /∂X derivatives are simple. As an example, for Euler equations, the dependence of lift coefficient, CL, and drag coefficient, CD, on the flow variables and coordinates of the geometry can be illustrated as follows.

cos sin

sin cos

L y x

D y x

C C C

C C C

α αα α

= −

= + (52)

Cx and Cy are the force coefficients in x and y directions and can be evaluated as:

1 1

NE NE

x x j y y jj j

C C C C= =

= =∑ ∑ (53)

In above equation, NE denotes the number elements on the geometry.

( )( )

1

1

j j

j j

x j p j b b

y j p j b b

C C y y

C C x x

+

+

= −

= − (54)

Pressure coefficient can be calculated as:

21

2

j

wall jp

l

PC

Vρ∞ ∞

= (55)

The pressure on the geometry can be written in terms of flow variables as below:

( ) ( )( ) ( )2 2

12

j

j

wall wall j

wall j wallwall j

u vP E

ρ ργ ρ

ρ

+ = − −

(56)

In this study, aerodynamic geometry is perturbed by Hicks Henne functions and new coordinates of the grid is obtained by using Equation 34. The modification to geometry is applied such that the domain is perturbed mostly near the wall and effect of perturbation is diminished away from the wall. The grid points furthest to the wall are not affected by the perturbations applied to the wall.

max

max

( )1new old k k

j jX X f x

jβ −= + −

(57)

Hence the grid coordinates have an explicit dependence on the design variables, and the grid sensitivities can be evaluated simply as:

( )1max

max

−−=

∂∂

j

jjxf

Xk

kβ (58)

Equations from52 to 58 show some explicit relations such that Cj= Cj(X,w) and X =X(βk). Therefore the evaluation of ∂Cj /∂w, ∂Cj /∂X, ∂X/∂βk derivatives are straightforward. However, the evaluation of derivative ∂w/∂βk is not simple since there is no explicit relation between flow variables and design variables. There are two methodologies


23

to pass this step of derivation to reach final formulation that gives sensitivities of the objective functions. These methods are direct differentiation method and adjoint method. Adjoint method bypasses the requirement for evaluation of the flow sate sensitivities, ∂w/∂βk, by introducing Lagrange multipliers. The choice between those two methods is made according to the size of the objective functions and the design variables. If the number of design variables is larger than the number of objective functions, the adjoint method is computationally more desirable compared to the direct differentiation method for iterative solvers. However the usage of direct solvers in flow solution removes the computational difference between those two methods. The LU factors computed in flow solution can be reused in sensitivity analysis and both of these methods can be applied efficiently. In this study the direct

differentiation method is chosen for sensitivity analysis. Direct differentiation method is formulated by the differentiation of the discrete residual vector. At the steady state flow condition, residual vector, which is a function of flow variables and grid coordinates, equals to zero.

( ) ( )( ){ } { }0k kR W , Xβ β = (59)

The flow sate sensitivity, ∂w/∂βk is evaluated from direct differentiation of steady state flow governing equations as follows16,17

0k k

R w R X

w Xβ β ∂ ∂ ∂ ∂ + = ∂ ∂ ∂ ∂

(60)

1

k k

w R R X

W Xβ β

− ∂ ∂ ∂ ∂ = − ∂ ∂ ∂ ∂

(61)

Above equation shows that, the calculation of flow state sensitivities requires the evaluation of flux Jacobian matrix, ∂R /∂w. In previous sections, the flux Jacobian were evaluated analytically and numerically, and the accuracy of numerical Jacobians were analyzed. The main interest of this section is to investigate the effect of accuracy of the numerical flux Jacobian on the accuracy of flow state sensitivities. For this purpose, various numerical flux Jacobians with different finite difference perturbation magnitudes are used to obtain sensitivities. The error in sensitivities calculated with numerical flux Jacobians is defined as.

sensitivity

numeric analytick kjacobian jacobian

W WError

β β∂ ∂= −∂ ∂

(62)

The Newton’s method is efficiently used in evaluation of sensitivities since the jacobian matrix appearing in Equation (54) was already LU decomposed to solve Equation (14). Hence in Equation (54) each of flow state sensitivities are calculated in single iteration by simple backward and forward substitution.

Figure 24. Effects of flux splitting method on error in

sensitivity ( AUSM, Ni bump)


24

VII. Effects of Flux Jacobian Evaluation on Sensitivities The effect of the accuracy of the numerical Jacobian on the sensitivities calculated by direct differentiation

method is studied. As explained in previous section, flow state sensitivities are the dominant components of the objective function sensitivities, hence the accuracy of the flow state sensitivities are analyzed in detail. The error

analysis is performed by constructing a flow state sensitivity error vector by substituting vectors evaluated from numerical Jacobians from the analytical ones. The L1 norm of the error vector is presented in results.

The code solves the flow equations with Newton’s method for a given conditions. After calculating a converged solution analytical Jacobian is calculated by using final flow variables. The right hand side of the sensitivity governing equations is evaluated. Using the analytical Jacobian, analytical sensitivities are evaluated. A loop is constructed in which the finite difference perturbation magnitude is incremented. In the loop for the each perturbation magnitudes numerical Jacobians are calculated. After the numerical Jacobian calculation numerical sensitivity is evaluated. For the each perturbation magnitude in the loop, calculated numerical Jacobian and sensitivity is compared with the previously stored analytical Jacobian and the sensitivities, and error plots are drawn. The Hicks Henne functions are used to perturb the geometry. The function which has the maximum perturbation at 30% of the chord is chosen.

The effect of the flux splitting methods on the accuracy of the sensitivity analysis is studied. Figures 24 and 25 show the effect of the accuracy of different flux splitting schemes on the error in numerical sensitivities. Results are given for identical cases used in the Jacobian accuracy analysis. Comparisons of Figures 24 and 25 with Figures10 and 11 show that the locations of the optimum perturbation in numerical Jacobian and sensitivity evaluations are same. The location of the optimum perturbation can be estimated accurately by Eq(42).

The effect of the grid resolution on the accuracy of the sensitivity analysis is studied. Figures 26 and 27 present sensitivity accuracy analysis for bump and airfoil geometries, respectively. The same grids used in analysis of Jacobian accuracy are also used in the sensitivity analysis.

Figure 25. Effects of flux splitting method on error in sensitivity

(AUSM, Naca0012,M#=0.85) ( AUSM)

Figure 25. Effects grid resolution on error in sensitivity (AUSM, Ni bump,M# =0.675)


25

Results show that grid resolution does not affect the optimum perturbation magnitude whereas the magnitude of the error in sensitivity is increased by the grid resolution. When the number of the grid nodes increases, the size of the sensitivity error vector also increases. Therefore L1 vector norm gets larger with increasing vector size.

The effect of the free-stream flow condition on the accuracy of the numerical sensitivity vector is studied. Sensitivity equations are solved for flows over the bump geometry with different free-stream Mach numbers. Figure- 29 shows that the location of the optimum perturbation magnitude is not affected significantly by the flow conditions. But the error may change according to the flow conditions used.

VIII. Conclusion

The Euler flow equations are solved with the Newton’s method. The source of the error in numerical Jacobian evaluation is analyzed. The finite difference perturbation magnitude significantly affects the accuracy of the numerical Jacobian. Using the optimum finite difference perturbation magnitude the error in numerical Jacobian can be minimized. Some simple formulas are derived to evaluate the optimum perturbation magnitude. Evaluated values are compared with the ones given by trial and error method with a range of perturbation magnitudes. The optimum perturbation magnitude is function of the magnitudes of round-off error and the second derivatives of the residual vector. In first order flux discretization the magnitude of second derivatives are not very large. In second order discretization these derivatives may have large values. This may be due to the large values of the second derivatives of the limiter functions

in the smooth flow regions. The effect of the accuracy of the Jacobian matrix on the convergence of the flow solution is studied. The numerical Jacobian evaluated with the optimum perturbation magnitudes can show the same convergence performance that is obtained by the analytic Jacobian. The effect of the finite difference perturbation magnitude on the accuracy of the numerical sensitivities is analyzed. The same optimum perturbation magnitude enabled the most accurate numerical Jacobian and sensitivities for a wide range of flow problems.

Figure 29. Effects of of flow condition on error in sensitivity

( AUSM, Ni bump)

Figure27. Effects of grid resolution on error in sensitivity

( C-grid, AUSM, M# = 0.85)


26

IX. References

1 Onur, O. and Eyi, S., “Effects of the Jacobian Evaluation on Newton’s Solution of the Euler Equations”, International Journal for Numerical Methods in Fluids, Vol. 49, pp 211-231, 2005.

2 Orkwis,P.,D., and Venden, K.,J., ”On the Accuracy of Numerical Versus Analytical Jacobians”, AIAA Paper 94-0176, 1994

3 Rizk, M. H., “The Use of Finite-Differenced Jacobians for Solving the Euler Equations for Evaluating Sensitivity Derivative”, AIAA Paper 94-2213, 1994.

4 Steger, J. L., and Warming, R. F., “Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to Finite-Difference Methods”, Journal of Computational Physics, Vol. 40, 1981, pp. 263-293.

5 Van Leer, B., “Flux Vector Splitting for the Euler Equations”, ICASE Report 82-30, September 1982.

6 Liou, M.-S. “A sequel to AUSM: AUSM+”, Journal of Computational Physics, Vol 129 (1996), pp. 364–382

7 Roe, P. L., “Characteristics-Based Schemes for the Euler Equations”, Annual Review of Fluid Mechanics, Vol. 18, 1986, pp. 337-365.

8 Ni, R. H., “A Multiple-Grid Scheme for Solving the Euler Equations” AIAA Journal vol.20, no.11 pp.1565-1571, 1982.

9 Hirsch, C., Numerical Computation of Internal and External Flows, Vol. I-II, John Wiley & Sons, Chichester, 1990.

10 Dennis J.E., Schnabel R.B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, New Jersey, 1983.

11 Davis, T. A., UMFPACK Version 4.1 User Manual, University of Florida, Florida, 2003.

12 Gill, P.E., Murray W., and Wright M.H., Practical Optimization, Academic Press, London, 1992.

13 Kelley C.T., Iterative Methods for Linear and Nonlinear Equations, Society for Industrial and Applied Mathematics, Philadelphia, 1995.

14 Korivi V.M., Taylor A.C., Newman P.A., Hou G.J.W. and Jones H.E.., “An Approximately Factored Incremental Strategy For Calculating Consistent Discrete Aerodynamic Sensitivity Derivatives”, AIAA Paper 92-4746, 1992

15 Lee, K. and Eyi, S., “Aerodynamic Design Via Optimization” ,Journal of Aırcraft Vol. 29, No.6 November-December 1992, pp 1012-1029.

16 Taylor, A. C. III, Korivi, V. M., and Hou, G. W., “Sensitivity Analysis Applied to The Euler Equations : A Feasibility Study with Emphasis on Variation of Geometric Shape,” AIAA Paper 91-0173 1991

17 Taylor, A. C. III, , Hou, G. W., and Korivi, V. M., “ A Methodology for Determining Aerodynamic Sensitivity Derivatives with Respect to Variation of Geometric Shape” AIAA Paper 91-1101 1991

18 Radespiel, R., and Kroll, N., “Accurate Flux Vector Splitting for Shocks and Shear Layers”, Journal of Computational Physics 121,1995, pp 66-78.

19 Godlewski, E., and Raviart, P.-A., Numerical Approximation of Hyperbolic Systems of Conservation Laws ,Springer, New York,1996.

20 V. Venkatakrishnan, On the accuracy of limiters and convergence to steady state solutions,J.Comp. Phys., 118 (1995), pp. 120--130.

21 Van Albada, G.D., Van Leer, B., Roberts, W.W., “A Comparative Study of Computational Methods in Cosmic Gas Dynamics”, Astronomy and Astrophysics, Vol 108, 1982, pp. 76-84,

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