Quasi-1D Finite Element Method for Isothermal Flow of Ionized Gas Through a Nozzle

CIS 888.11V/EE894R/ME894V A Case Study in Computational Science & Engineering

Quasi-1D Finite Element Methodfor Isothermal Flow of Ionized Gas

Through a Nozzle

Robert Lee

[email protected]


Comparison of finite differencesversus finite elements

• In FD, one approximates the derivatives by finite differences.

• In FEM, one approximates the unknown variable by basis functions

• On similar grids, the matrix equation produced by FEM and FD may actually be the same


Comparison of finite differencesversus finite elements (cont’d 2)

• FEM is ideally suited for unstructured grids while FD is applied to structured grids

structured unstructured


Method of Weighted Residuals

The method of weighted residuals is now the most popular way to obtain an equation for the application of FEM.

( ) ( )Lu x f xL is the operator (in our case diff. eq. of interest, and u is the unknown variable. f(x) is the forcing function.

Let us define an approximation of u given by .u Then

( ) ( ) ( )R x Lu x f x 1

( ) ( )N

j jj

u x x

0 1x


Method of Weighted Residuals (cont’d 2)

We must try to force R(x) = 0. One possible way calledpoint matching is

( ) 0 1,2, ,iR x i N

( ) ( ) 0 1,2, ,i iLu x f x i N

For 4, a typical plot of ( ) looks likeN R x

0 11x 3x2x 4x


Method of Weighted Residuals (cont’d 3)

Point matching is equivalent to the following:1

0

( ) ( ) 0 1,2, ,iR x x x dx i N

We must consider other functions.1

0

( ) ( ) 0 1,2, ,iR x v x dx i N

( ) is called the weighting or testing function.iv x


Finite Element Method (FEM)

• The basis functions j(x)are generated by simple functions defined piecewise (element by element) over the FEM grid.

• The basis and weighting must be smooth enough such that their derivatives in the weight residual equation exists (assume nth order derivatives), i.e.,

FEM provides a systematic and very general way ofgenerating the basis functions (usually polynomialapproximations. The criteria are:

2nj

nx

2n

in

v

x

is element of interest


FEM (cont’d 2)

• The basis functions are chosen in such a way that the coefficients defining the unknown quantity are precisely the value of the unknown quantity at the nodes.

1 1

( ) ( )N N

n nj j j j

j j

x x

1 1

( ) ( )N N

n nj j j j

j j

u u x u u x


FEM (cont’d 3)

Let us consider a 1-D grid

1h 4h3h2hElement 1 32 4

Node 1 2 3 4 5

Coordinate 1x 5x4x3x2x

1( )

0j i

i jx

i j

1( )

0 at other nodesj

j

x xx


FEM (cont’d 4)

1 2 3 4 5

Assuming that we want to use the same functions for both the basis and weighting functions, the simplest function is the linear function shown below

2 ( )x

1 2 3 4 5

1

2d

dx

1

1h

2

1h


Application to Conservation of Mass

1( ) 0uA

t A x

After applying FD to / t and linearization,

2(2 ) 0n n n nA d

u A uA u At dx

1n n 1n nu u u

n n n nu u u u

a x b

2(2 ) ( ) 0

bn n n n

i

a

A du A uA u A v x dx

t dx

Let

1,2, ,i N


Application to Conservation of Mass (cont’d 2)

We evaluate the above expression element by element

1

2(2 ) 0

m

Mn n n n

i im

A du A uA u A dx

t dx

Let ( ) ( )i iv x x

M is the number of elements

Apply integration by parts,1 1

1

m m

m

m

m m

x xx

xx x

dg dff dx fg g dx

dx dx

1,2, ,i N



leads to1

1

2(2 )

m

m

xMn n n n i

im x

A du A uA u A dx

t dx

1(2 ) | 0m

m

xn n n ni xu A uA u A

Note: Most of the endpoint contributions cancel outif A is continuous since and u are continuous.Consider two elements

1x 2x 3x

1,2, ,i N



After plugging in approximation of and u,

1

1 1 1 1

2m

m

xM N N Nn i

j j i j j k km j j kx

dAu A dx

t dx

(2 ) | 0n n n n bi au A uA u A

1,2, ,i N

1

1 1 1 1 1

2m

m

xM N N N Nn n n il l p p k k j j

m l p k jx

du A u A dx

dx


Single Element Evaluation

Consider integral for one elementm

mm 1m

and is nonzero only for , 1j jd dx j m m

1

1

( ) mm

m m

x xx

x x

11

( ) mm

m m

x xx

x x


Single Element Evaluation (cont’d 2)

1 1 1 12m

m

x m m mn i

j j i j j k kj m j m k mx

dAu A dx

t dx

, 1i m m

1 1 1 1 1

2 0m

m

x m m m mn n n il l p p k k j j

l m p m k m j mx

du A u A dx

dx

Let us consider the integral over a single elementAnd ignore the contribution from the boundary


Mapping from Global to Local System

mx 1mx

m

Global node number

Local node number

m m+1

1 2

Additional notation:

1( ) ( )mm x x 1 2( ) ( )m

m x x for mx

1 1 2 1 1 2n n n n

m m m m

1 1 2 1 1 2n n n n

m m m mu u u u u u u u


Element Evaluation in Local System1 2 2 2

1 1 1

2m

m

x mm m m n m i

j j i j j k kj j kx

dAu A dx

t dx

1,2i

1 12 2 2 2

1 1 1 1

2m m

m m

x xm mn m m n m n mi i

j k k j l l p pj k l px x

d du A dx A u dx

dx dx

Reordering terms,12 2

1 1

2m

m

x mm m n m m i

j j i k k jj kx

dAu A dx

t dx

1 2 2 2 2

1 1 1 1

2 0m

m

x mn m n m n m m il l p p k k j j

l p k jx

du A u A dx

dx

1,2i


Application to Conservation of Momentum

210

Tu u A

t A x x

After linearization, FD in time, and applying the method of weighted residuals, the expression for a single element is

12

m

n n ni

A ATu u dx

t x

2 212( ) ( ) 0

m

n n n n ni u A u A u u A dx

x

1,2, ,i N


Application to Conservation of Momentum (cont’d 2)

Applying integration by parts, grouping terms, and disregarding B.C.’s,

212 2

( )( )

m

n n i iA Tit

d d Au u A dx

dx dx

2

m

m

n n n iAit

in n ni T

du A u dx

dx

d Adu A dx

dx dx

1,2, ,i N


Application to Conservation of Momentum (cont’d 3)

Substituting basis function representation for unknowns and converting to local system,

12 2 221

21 1 1

( )( )

2

m

m

x m mn m m m n m m mi i

j k k i j k k j jj k kx

d d AA Tu u A dx

t dx dx

1

1

2 2 2 2

1 1 1 1

22 2 2

1 1 1

m

m

m

m

x mn m m m n m n m mi

j k k i j k k l l jj k k lx

mx min m n m n mi

k k l l k kk l kx

dAu u A dx

t dx

d Ad Tu A dx

dx dx

1,2i


Local Matrix Build

m m mk F

11 12

21 22

m mm

m m

k kk

k k

Subscripts associated withindices i j in equations.

(11) (12)

(21) (22)

m mij ijm

ij m mij ij

k kk

k k

( )For m pqijk

1: Conservation of massp

2: Conservation of momentump 1: mass density variable jq

2: velocity variable jq u


Local Matrix Build (cont’d 2)

1 2(11)

1

2m

m

x mm m m n m m iij j i k k j

kx

A dk u A dx

t dx

For example,

1 2(12)

1

m

m

x mm n m m iij k k j

kx

dk A dx

dx

1

2

m

m

m

jmj

ju


Local Matrix Build (cont’d 3)

1

2

m

m

m

FF

F

(1)

(2)

mm i

i mi

FF

F

( )For , m p

iF

1: Conservation of mass equationp 2 : Conservation of momentum equationp

For example,1 2 2

(1)

1 1

2m

m

x mm n m n m i

i l l p pl px

dF A u dx

dx


Transferring Local Matrix to Global Matrix

1 32 4

1 2 3 4 5GlobalNode #

LocalNode #

1 1 1 1 2222

For element3

3 3 3 311 33 12 34 21 43 22 44k k k k k k k k

3 3 3 31 3 2 4 1 3 2 4F F F F


Transferring Local Matrix to Global Matrix (cont’d 2)

1 111 12

1 1 2 221 22 11 12

2 2 3 321 22 11 12

43 3 4 1221 22 11

4 421 22

0 0 0

0 0

00

0 0

0 0 0

k k

k k k k

k k k kk

k k k

k k

1

2

3

4

5

11

1 22 1

2 32 1

3 42 1

42

F

F F

F F

F F

F


Evaluation of Area A

Assume A varies linearly within element m,2

1

m mj j

j

A A

1 is the area at local node 1mA

2 is the area at local node 2mA

m mi mi

i

d A d dAA

dx dx dx

2 2

1 1

mmjm m m mi

j j i jj j

ddA A

dx dx


Numerical Integration: Gauss Quadrature

Gauss quadrature of order N is given by

1

11

( ) ( )N

i i Ni

f x dx w f x R

i 22

are zeros of Legendre polynomials ( )

2w

(1 )

i N

i N i

x P x

x P x

42 1 2

2

2 !

(2 1) 2 !

N N

N

N d fR

dxN N


Numerical Integration: Gauss Quadrature (cont’d 2)

Note: Gauss quadrature is especially well suited for polynomialfunctions since for large enough N, RN=0. A polynomialof order 2N-1 can be exactly integrated by a quadrature of order N. For our case, we need coordinate transformation,

11 12

1

2

m m m m

m m

x x x x x

x xdx d

1 11 1

11

( ) ( ( )) ( ( ))2 2

m

m

x Nm m m m

i iix

x x x xf x dx f x d w f x


Numerical Integration: Gauss Quadrature (cont’d 2)

The highest order polynomial of interest for our problem is of order 4. We need quadrature of order 3,

1 3 2

1 3 2

0.7745966692414830 0.0

0.5555555555555560 0.8888888888888890w w w

Date post:	05-Jan-2016
Category:	Documents
Upload:	samuru
View:	24 times
Download:	2 times

Quasi-1D Finite Element Method for Isothermal Flow of Ionized Gas Through a Nozzle

Documents