FINITE ELEMENT FORMULATION FOR CONVECTIVE-DIFFUSIVE PROBLEMS WITH SHARP GRADIENTS USING FINITE CALCULUS

Aleix Valls TomasInternational Center for Numerical Methods in Engineering

(CIMNE) Modulo C1. Despacho C2. Universidad Politécnica de

Cataluña. Campus Norte UPC, 08034 Barcelona, Spain

aleix@cimne.upc.edu

Introduction

A finite element method (FEM) for convective-diffusive problems presenting sharp gradients of the solution both in the interior of the domain and in boundary layers.The Finite Incremental Calculus (FIC) method is based in the solution by the Galerkin FEM of a modified set of governing equations, which includes characteristic length distances

Numerical Oscillations FIC Stabilization

Example: 1-D case (source less)Steady-state convection-diffusive problem

Taylor expansion second order

d2

FIC Governing equations

dd1

A C B

qA qB

( )

( )

2 231

1 12

2 232

2 22

2

2

A CC C

B CC C

dq d d qq q d d

dx dx

dq d d qq q d d

dx dx

= - + - O

= + + - O

- =0A Bq q

2

2 02

dq hd qdx dx

- =

1 2h d d= -

FIC Governing equations

Steady-state convection-diffusive problem

dqdx

ff= -

dq u k

dx

ff ffæ öæ ö æ ö÷÷ ç ÷ç ç- - - =÷÷ ÷çç ç ÷÷ ÷ç ç ÷çè ø è øè ø1

02

d d d d d d du k h u kdx dx dx dx dx dx dx

2

22hd qdx

- = FIC

( )1

02

r r in- ×Ñ = Wh

FIC Governing equations

Multidimensional case (with source term)Steady-state convection-diffusive problem

Governing equation

Boundary Conditions

Where h is characteristic length vector.

FIC Stabilization terms

0p on fff - = G

10

2pn qn q r onf× Ñ + - × = GD h n

:r Qff= - Ñ + Ñ Ñ +t tu D

Finite element discretization

A finite element interpolation of the unknown:

Application of the Galerkin FE method to Governing equations gives, after integrating by parts term

The last integral has been expressed as a sum of the elements contributions to allow for interelement discontinuities in the term

Note that the residual terms have disappeared from the Neumann boundary . This is due to the consistency between the FIC terms.

ˆt rÑh

ˆi iNff f= å;

( ) ( )1ˆ ˆ 0

2 eq

p ti i n i i

e

N rd N q d N N rdfW G W

W- Ñ + G+ Ñ + Ñ W=åò ò òt tn D h h

r̂Ñ

Finite element discretization

Integrating by parts the diffusive terms

In matrix form

( )1ˆ ˆ ˆ 02

e

eq

p ti i i n i i i

e

N N d N q d N Qd N N rdffW G W W

é ùÑ + Ñ Ñ W+ G- W- Ñ + Ñ W =ë û åò ò ò òt tu D h h

Ka = f

( ) ( )

( ) ( )

12

12

e

e

e t t tij i j i j i j

t ti i j

K N N N N N N d

N N N d

W

W

é ù= Ñ + Ñ + Ñ + Ñ Ñ W-ê úë û

- Ñ + Ñ Ñ Ñ W

ò

ò

t

t

u D hu h u

h h D

( )12e

eq

e t t pi i i i i nf N N N Qd N q d

WG

é ù= + Ñ + Ñ W- Gê úë ûò òh h

Finite element discretization

( )12e

e tij i j i jK N N N N d

W

é ù= Ñ + Ñ + Ñ Wê úë ûò tu D hu

12e

e ti i if N N Qd

W

é ù= + Ñ Wê úë ûò h

Simplifications: h constant over the element Linear elements Q constant

12

tG = + =D D D D hu

Computation of the characteristic length vector

For the sake of preciseness the method is explained for 2D problems. FIC balance equation in the principal curvature axes of the solution For simplicity we consider the 2D sourceless case (Q = 0) with an isotropic diffusion defined by a constant diffusion parameter k.

,x h

Computation of the characteristic length vector

The FIC balance equation is:

As and are the principal curvature axes of the solution then

2 2 2 2

2 2 2 2

2 2

2 2

2

02

hu u k u u k

hu u k

x

h

x h x h

x h

ff ff ff ffx h x x hx h x h

ff ffh x h x h

æ ö é æ öù¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶÷ ÷ç çê ú- - + + - - - + +÷ ÷ç ç÷ ÷÷ ÷ç çê ú¶ ¶ ¶ ¶ ¶è ø è ø¶ ¶ ¶ ¶ë ûé æ öù¶ ¶ ¶ ¶ ¶ ÷çê ú- - - + + =÷ç ÷÷çê ú¶ ¶ ¶ è ø¶ ¶ë û

2 2

0ff

x h h x¶ ¶

= =¶ ¶ ¶ ¶

x h

Computation of the characteristic length vector

Introducing previous simplification we can rewrite FIC equation as (linear elements):

In matrix form

2 2

2 2 02 2

u h u hu u k kxx h h

x hff ffx h x h

æ ö æ ö¶ ¶ ¶ ¶÷ç ÷ç- - + + + + =÷ ÷ç ç÷ ÷ç÷ç è ø¶ ¶ ¶ ¶è ø

( ) 0t tff¢ ¢ ¢ ¢ ¢- Ñ + Ñ Ñ =u D + D

002

00

2

u hk

u hk

xx

h h

é ùê úé ù ê úê ú ¢= = ê úê ú ê úê úë û ê úë û

D D

Computation of the characteristic length vector

The velocities along the principal curvature axes can be obtained by projecting the Cartesian velocities into the principal curvature axes

The characteristic length distances are defined as

where and are typical element dimensions along the principal curvature axes, respectively and and are the corresponding stabilization parameters.

cos sin; ,

sin cos

u u

u vx

h

a a

a a

é ùì ü ì üï ï ï ïï ï ï ïê ú¢= = = =í ý í ýê úï ï ï ï-ï ï ï ïî þ î þê úë ûu Tu T u

h l h lx x x h h ha a= =

lx lhxa ha

Computation of the characteristic length vector

Stabilization parameters:computed by considering the solution of two uncoupled 1D problems along the principal curvature axes.

Element dimensions

1coth ,

2

1coth ,

2

u l

k

u l

k

x x

x xx

h h

h h

x

hh

a g gg

a g gg

= - =

= - =

( )1 3

max , ,1 4

ti j i

j Tril d u i j

j Quadx h

= ¸ ®ìïï= = = íï = ¸ ®ïî

Orthotropic Matrix Diffusion

The next step is to transform the problem to global axes x, y

( ) 0t tGff- Ñ + Ñ Ñ =u D

G = +D D D

t ¢=D T D T

002

00

2

u hk

u hk

xx

h h

é ùê úé ù ê úê ú ¢= = ê úê ú ê úê úë û ê úë û

D DFIC governing equations introduce orthotropic diffusion matrix

About the FIC method

Remark 1:Clearly, if the principal curvature direction is parallel to the velocity direction, then

Where . Note that the method coincides with the standard SUPG approach in this case.

020 0

u hx x

x

é ùê ú

= ê úê úê úë û

D = D

u ux x= u

About the FIC method

Remark 2:The global balance diffusion matrix can be also computed from the expression of vector h in global axes as

12

x

y

hwith

h

ì üï ïï ï ¢= = =í ýï ïï ïî þ

t tD h u h T h

General iterative scheme

Step 0 (SUPG step). At each integration point choose , i.e. the gradient direction is taken coincident with the velocity direction. Compute .The expression of the balancing diffusion matrix coincides now precisely with the SUPG form.Solve for .

Verify that the solution is stable. This can be performed by verifying that there are not under or overshoots in the numerical results with respect to the expected physical values. If the SUPG solution is unstable, then implement the following iterative scheme.

For each iteration:Step 1 Compute at the element center. . Then compute and Solve for .Step 2 Estimate the convergence of the process. We have chosen the following convergence norm.

where N is the total number of nodes in the mesh and is the maximum prescribed value at the Dirichlet boundary. In above steps the left upper indices denote the iteration number.If condition is not satisfied, start a new iteration and repeat steps 1 and 2 until convergence. Indexes 0 and 1 are replaced now by i and i + 1, respectively.

0x = u

0 1 0, ,h ¢D D

0f

1 0x f= Ñ1f

( )12

21

1

1max

Ni i

j jjN

ff f ef

+

=

é ùê ú= - £ê úë ûå

maxf

1 1 1, ,h ¢D D

Examples

Example 1

( ) ( )

( ) ( )

( )

8

0,1 0,1

, 0,1

10

, 1

0P

x y

k

Q x y

f

f

-

W= ´

G = ¶W

=

=

=

=

u( ) ( )

P

k Q on

in f

ff

ff

×Ñ - Ñ × Ñ = W

= G

u

Example 1

SUPG FIC

Example 1Cut y=0.5

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

X

Ph

i

SUPG FIC

Example 2

( ) ( )

P

k Q on

in f

ff

ff

×Ñ - Ñ × Ñ = W

= G

u

( ) ( )

( )( )

( )

( )

8

0,1 0,1

0,1 0 0.5,

0, 1 0.5 1

10

, 1

0P

xx y

x

k

Q x y

f

f

-

W= ´

G = ¶W

£ £ìïï= íï - < £ïî=

=

=

u

Example 2

SUPG FIC

Example 2