Case-Mode Analysis of Finite Element Spatial DiscretizationsAlexander E. Maslowski, Dr. Marvin L. Adams

Texas A&M University

Abstract

We analyze the following problem:

• One dimensional, steady state, discrete-ordinates.

• Continuous Finite Element (CFE) with discontinuous weight function.

Case-mode analysis shows:

• CFE methods (CFEMs) yield the correct dispersion relation.

• CFEMs yield Padè approximations of the exact single-cell attenuation factor for each mode. The Padè approximations have numerator and denominator of the same polynomial order.

• CFE s can not be truly robust because their single-cell attenuation factors approach +1 as cell become thicker.

We present results showing that the analysis correctly predicts the single-cell attenuation factor of each mode.

Analytic TransportWe consider the following transport equation:

The homogeneous solution is spanned by the following modes:

The exact single-cell attenuation factor (SCAF) for a given mode is:

We aim at deriving similar properties for the CFE approximation:

We restrict our study to the CFEMs that are symmetric and use discontinuous weight functions mapped to each cell.

1

1

2 2

Mm

m m n nm

d cx w x Q x

dx

/

1

, , 1M

x nm m m

m nn

wx e

/m x

exactm

x xg e

x

1

,1

, cell J

m m kj kjj

x b x x k

CFEM Analysis: Without ScatteringWe seek the CFEM SCAF, f, such that:

From the CFE equations the attenuation factor is:

Since our CFEMs are symmetric, they treat right and left moving particles identically. We can show:

Thus, no CFEM in this family is robust – each has SCAF ±1

1/ 2 1/ 2m k m kx f x

, 1 1 1

k k k kJ J m k

m m

x xL M L M

11 1

,1

J

J mk J mk iJ ii

f L M L M

20 1 2

20 1 2

11lim

a a af f f

f a a a

CFEM Analysis: With ScatteringWe seek a separable solution and SCAF, g, defined by:

We modify the basis function to define it in terms of the unknown attenuation factor:

Using the CFE equations we obtain:

It follows that the following eigen-problem is formed:

If the same procedure is done for the pure-absorber case, we can conclude that f and g are the same functions with the directional cosine replaced by the relaxation length. We can also use this equation to show:

,m kj m kja z , , 1 , ,1 , 1 1m k J m k k J kg z g z

1 , 1

, 1,

, 1.

kjkj

k k J

b x jb x

b x g b x j

1

M

m mm

w a

11 1

1 12

M M

m mk mk n nm n

cz w z w aI L M L M

1 1 1

k kz z

xL M L M

m mm

a

To test the theory, individual Case-modes must be isolated. We found a Case-mode orthogonality relation for SN. We use it to obtain scalar-flux modes from the individual angular fluxes:

Separating the Modes

Numerical Experiment: Two Region Slab S4

inc= 40

(normal)

inc= 20

(grazing) 15

cells

5 cells

4 cells

10 cells

x = 0.200 cm

x = 1.500 cm

x = 1.333 cm

x = 0.564 cm

Same Material Properties

Total Scalar Flux

0

5

10

15

20

25

30

0 5 10 15 20

Position (cm)

Sca

lar

Flu

x (P

artic

les/

cm2 -s

)

DiamondQuadraticHigh LevelExact

inc= 40

(normal)

inc= 20

(grazing) x = 0.667 cm t = 3.0 cm2

c = 0.9

x = 2 cm t = 2.0 cm2 c = 0.9999

15 cells

5 cells

Total Scalar Flux

0

5

10

15

20

25

30

0 5 10 15 20

Position (cm)

Sca

lar

Flu

x (P

artic

les/

cm2 -s

)

DiamondQuadraticHigh LevelExact

'

''1

l lM

m m l l ll lm m m

w C

'

' '' 2

' '

'

Nm n

m mNmn n m

n mNm n m n

m mm n m

w

w

w

Case-Mode Analysis of Finite Element Spatial Discretizations Alexander E. Maslowski, Dr. Marvin L. Adams

Texas A&M University

Conclusions

We found the CFEM have the following properties:

•The single-cell attenuation factor is a Pade’ rational polynomial approximation of exp(-), where is the traversal distance in mean-free paths for the pure absorber case, and the cell width in relaxation lengths for the scattering case.

•CFEMs obtain the correct angular shape function for each mode: 1/().

•The rational polynomial approximation is not robust: it approaches +1 in the thick-cell limit.

The analysis describes the behavior of all methods of the defined CFE family. It applies to all slab problems regardless of mesh spacings.

We expect future studies to focus on using this analysis, or the insight gained from it, to improve solutions of multidimensional problems.

Positive Asymptotic Mode Negative Asymptotic Mode Positive Transient Mode Positive Transient Mode

Positive Asymptotic Mode Negative Asymptotic Mode Positive Transient Mode Positive Transient Mode

Case 2: Resolved Boundary Layers

Case 1: Unresolved Boundary Layers