CFD I_2

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The equations governing the steady 2-D Newtonian flow are

or identically

But this is just a theoretical example in which the flow is assumed to havenull thickness

If we want to make a more adequate approach that takes into account the

third dimension we have to use the Shallow Water equations (SSWW)

governing equations

shallow waterscomputational fluid dynamics I

xfux

p

y

uv

x

uu

t

u

1

yfvy

p

y

vvx

vut

v

1

0, iiu ufuuu

p

t

1

0

y

v

x

u

2,1i

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governing equations


The assumptions to be made are

The distribution of the horizontal velocity along the vertical direction is assumed

to be uniform

An integration in height is carried out, and the horizontal velocity is taken as the

mean value of the horizontal velocities along the vertical direction

The main direction of the flow is the horizontal one, and only very small flows

take place on vertical planes

The acceleration in the vertical direction is negligible compared to gravity and a

hydrostatic distribution of the pressure is assumed

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Integrating the continuity equation along the z-axis

As the Leibniz rule to bring the derivatives into the integral sign gives

it is obtained

governing equations

shallow waters. continuity eq.computational fluid dynamics I

0 bh

h

h

h

hwhwdz

y

vdz

x

u

bb

0

z

w

y

v

x

u

h

hbH=h+hb

h

0 xh

hux

hhuudzxdzx

u bb

h

h

h

h bb

0 bbbh

h

bb

h

h

hwhwy

hhv

y

hhvvdz

yx

hhu

x

hhuudz

xbb

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w(h), (vertical component of the velocity on the surface) is given by

Substituting in the former equation

Noting that , and taking and renaming the main velocities as

the continuity equation is obtained as

0

th

t

hvdz

yudz

x

b

h

h

h

h bb

hvyh

hux

h

t

h

dt

dh

hw

0

t

hb

uudzH

u

h

hb

1vvdz

Hv

h

hb

1

0

y

vH

x

uH

t

h

governing equations

shallow waters. continuity eq.computational fluid dynamics I

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As the vertical acceleration is negligible, the third dynamic equation

can be written as

Integrating this equation in depth and assuming the atmospheric pressure

to be zero it is obtained

Deriving with respect tox and y

governing equations

shallow waters. dynamic eq.computational fluid dynamics I

01

zf

z

p

zfz

w

y

w

x

w

z

p

z

ww

y

wv

x

wu

t

w

2

2

2

2

2

21

dzz

pdzf

h

h

h

hz

bb

phphphhf bbz

x

p

x

hfz

1

y

p

y

hfz

1

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The first dynamic equation results into

Adding the continuity equation multiplied by u, it is obtained

this is

as

2

2

2

2

2

2

zu

yu

xu

xhff

zuw

yuv

xuu

tu zx

2

2

2

2

2

2

z

u

y

u

x

u

x

hff

z

w

y

v

x

uu

z

uw

y

uv

x

uu

t

uzx

2

2

2

2

2

22

z

u

y

u

x

u

x

hffz

uw

y

uv

x

u

t

uzx

governing equations


z

wuw

z

u

y

vuv

y

u

x

uu

t

u

z

uw

y

uv

x

u

t

u

2

2

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Integrating in depth the former expression

Taking into account that , it is obtained

Cancelling terms

xhhuxhhudzuxthhuudztb

b

h

h

h

h bb

222

dzHx

hffhwhuhwhu

y

hhvhu

y

hhvhuuvdz

y

h

hzxbb

h

h

bbb

bb

u

hvy

hhu

x

h

t

h

dt

dhhw

dzHx

hffhv

y

hhu

x

h

t

hhuhv

y

hhu

x

h

t

hhu

y

hhvhu

y

hhvhuuvdz

y

h

hzxb

bb

bbb

h

h

bbb

bb

u

dzHx

h

ffuvdzydzuxudzt

h

hzx

h

h

h

h

h

h bbbb

u

2

x

hhu

x

hhudzu

xt

hhuudz

t

bb

h

h

h

h bb

222

2

2

2

2

2

22

z

u

y

u

x

u

x

hff

z

uw

y

uv

x

u

t

uzx

governing equations


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Taking mean velocities it is obtained

The viscosity effects can be evaluated as

where v is the turbulent viscosity

Where are the shear stresses acting on the surface (due to the wind action)and on the bottom (due to the roughness of the channel)

= Wind drag coefficient

= Manning coefficient

= Wind velocity components

= Air density

dzHx

hff

y

uvH

x

Hu

t

uH h

hzx b

u

2

xxb

bs

h

hH

y

u

x

udz

2

2

2

2

u

xx bs ,

iaw

s

WWC

i 34

2

h

i

b H

uVgn

Hi

WC

n

iW

a

governing equations


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Developing the derivatives in the left hand side

Taking into account the continuity eq.

the former eq becames

governing equations


yHuvH

yvuv

yu

xHuH

xuu

tHuH

tu

yuvH

xHu

tuH

)(2 2

2

0

y

HvH

y

v

x

HuH

x

u

t

h

y

vH

x

uH

t

h

vHy

u

y

HvH

y

v

x

HuH

x

u

t

HuH

x

uuH

t

u

y

uvH

x

Hu

t

uH

)(

2

vH

y

uH

x

uuH

t

u

y

uvH

x

Hu

t

uH

0

2

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The derivatives of the depth with respect tox and y are

Carrying out the same operations for they dimension, and developing the

derivatives taking into account the last expression it is obtained

wherefc is the Coriolis factor

34

2

2

2

2

2

h

xaw

cH

uVgn

H

WWC

y

u

x

uvf

x

hg

y

uv

x

uu

t

u

34

2

2

2

2

2

h

yaw

cH

vVgn

H

WWC

y

v

x

v

ufy

h

gy

v

vx

v

ut

v

governing equations


xhxhhxHb

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The shallow water equations result into

with boundary conditions

impermeability , (no slip)

discharge

contour stresses ,

water level

governing equations


34

2

2

2

2

2

h

xaw

cH

uVgn

H

WWC

y

u

x

uvf

x

hg

y

uv

x

uu

t

u

34

2

2

2

2

2

h

yaw

cH

vVgn

H

WWC

y

v

x

vuf

y

hg

y

vv

x

vu

t

v

0 yvH

xuH

th

0Nu 0Tu

QdsHuN0NN

0TT

thth 0

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If in the N-S dynamic equation we substitute the non-linear velocities by a known

velocity field and the rest of the velocities by the a scalar unknown we arrive to

the convection diffusion equation that rules the transport of substances byconvective and diffusive actions.

The equations are

or in 1D

where is the quantity being transported, kis the diffusion coefficient, Ui is the

known velocity field, and Q are the external sources of the quantity. These are

also known as the Transport Equations

governing equations

convection-diffusion equationcomputational fluid dynamics I

f

zyxz

W

y

V

x

U

t

2

2

2

2

2

2

0 QkU jjjjt ,,,

0

Q

xk

xxU

t

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CFDI

finite elements in fluids

computational fluid dynamics I

3. Finite Elements in Fluids

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There is no analytical solution for most engineering problems such as

fluid flow

The determination of the velocity and pressure field is required in a

domain ofinfinite degrees of freedom

The Finite Element Method (developed about 1950 for structures)

substitutes the domain by another with a finite number of freedom

degrees, thus an approximation of the solution is obtained

Some important names in the finite element history are Courant, Turner,

Clough, Zienkiewicz, Brookes, Hughes,

Now it is used not only in structural mechanics but also in heat

conduction, seepage flow, electric and magnetic fields, and of course

in fluid dynamics


general issuescomputational fluid dynamics I

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sms.avi

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2D h zoom at the mine.avi

3D H(x,y), water depth colour (only 600 days).avi

largo modulos.avi

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X

Y

0 100 200

0

50

100

150

200

250

VEL

1.78125

1.6625

1.54375

1.425

1.30625

1.1875

1.06875

0.950002

0.831252

0.712501

0.593751

0.475001

0.356251

0.237501

0.11875

1.69975E-06

9.60324E-07

4.19696E-07

8.11084E-08

2D H (water level).avi

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The main way of solving continuum problems in the finite element method are the

following

The direct approach (matrix analysis), by using a direct physical reasoning to establish

the element properties. Requires very simple basic elements (bars, pipelines,)

Variational approach (e.g. Rayleigh-Ritz based method), in this method the stiffness

matrix is obtained as a result of the resolution of a variational problem

Weighted residual approach (e.g. Galerkin Method), as a result of weighting the

differential equations and integrating them in the domain



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Main steps of the finite element method

Subdivide the domain in a finite number of elements interconnected a the nodes,

where the unknowns (p, u) are going to be determined

It is assumed that the variation of the unknowns can be approximated by a simple

function

The approximation functions are defined in terms of the values of the field

variables at the nodes

When the equilibrium or variational equations has been obtained the new finite

unknowns are introduced into the equations

The system of equations is solved and the unknowns are determined at the nodes The approximat ion functions give the solution in the rest of the domain points

Following, the fem solution of the one simple 1-D problem is to be considered

on a 6-step basis



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In the traditional Rayleigh-Ritz methods the interpolating functions have to be

defined over the enti re domain and have to satisfy the boundary conditions.

Meanwhile in the FEM the interpolat ing trial funct ions are defined on a finite elementbasis, being more versatile when the shape is not simple enough

The limitation is that the FEM trial functions have to satisfy in addition some

convergence conditions (continuity and completeness and compatibility)



'...as the nature of the universe is the most perfect and the work of the

Creator is wiser, there's nothing that takes place in the universe inwhich the ratio of maximum and minimum does not appear. So there is

no doubt whatsoever that any effect of the universe can be explained

satisfactorily because of its final causes, through the help of the method

of maxima and minima, as can be by the very causes taking place

Leonhard Euler

(Basel,1707- Saint Petersburg,1783)

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When using a variational approach, the aim is to find the vector function of

unknowns, that makes a minimum or a maximum of the functional I (typically

the energy)

After the discretezation has been carried out in terms ofE smaller parts the

piecewise approximation is introduced so that

or in terms of the so called shape functions Ni

where are the values of the unknowns at the nodes


variational approachcomputational fluid dynamics I

dSx

gdVx

FI

,...,,...,

e

aprox

e

ee

NNe

2211

i

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Afterwards, the condition of extremezation ofIwith respect to i is imposed

Adding all those element contributions it is obtained

Assuming I to be a quadratic functional of the element equation results in



0

M

i

I

II

I

2

1

E

e i

e

i

II

1

0

eee

e

e

PKI

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After the assembling process it is obtained

where and

After applying the boundary conditions the system is solved for the nodal

unknowns i

Once i are known, we can obtain other variables as a post-processing value



PK

E

e

e

1

KK

E

e

e

1

PP

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Example. Find the velocity distribution of an inviscid fluid flowing trough a

varying cross section pipe shown in the figure

The governing equations are defined by finding the potential that minimizes the

energy integral equation

with the boundary condition u(x=0)=u0, where the cross section area is


variational approach, examplecomputational fluid dynamics I

dxdx

dAI

L

0

2

2

1

LxeAA

0

u0

A0

L

A1 A2

1 2 3

l(1) l(2)

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1st step. Discretization

Divide the continuum into two finite elements. The values of the potential

function at the three nodes will be the unknowns of the fem

2nd step. Select an interpolation model, easy but leading to convergence

The potential function will be taken as linear

and can be evaluated at each element as

where l(e) is the length of the e element

bxax

eeeee

l

xx 121



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3rd step. Derivation of stiffness matrices K(e) and load vectors P(e) by using

a variational principle

Deriving the interpolating function with respect to x it is obtained

where the cross sectional areas can be taken for the first and second element

as and

where the nodal unknowns are respectively and

eee

l

oe

eeeel

e

eel

e xl

Adx

lAdx

dx

dAI

2

2

2

1

2

12

12

2

1

2

2

0

2

12

0

2

eeTe

e

e

e

ee

e

eeeee

l

A

l

AI K

2

1

11

11

2

1

2

2

2

1

2112

2

1

2

2

2

10 AA

2

21 AA

2

11)(

3

22)(



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3rd step. (cont)

The minimal potential energy principle gives , if we take into account the

external inflow

where Q is the mass flow rate across section

therefore, if we derive the functional Ifor each basic element

or in matrix form

eTeeeTeeeee

e

eeeee QQ

l

AI QK

2

1

2

22211

12

2

1

2

2

0

i

I

02

2121221

1

12

2

1

2

2

111

)()()( eeee

eeeeeeee

e

e

Ql

AQQ

l

AI

022 2122212122

1

2

2

221

)()()( eee

e

eeeeeeee

e

e

QlAQQ

lAI

0

2

1

eeeeTeeeTe

ii

eI

QKQK

AuQ


variational approach, example


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4th step. Assembly of the stiffness and load vectors

Once we have obtained the matrices for all the basic elements as

we can assemble the system to obtainQK

11

111

11

l

AK

11

112

22

l

AK

0

111uA

Q

23

20

uAQ

22

00

3

2

1

2

2

2

2

2

2

2

2

1

1

1

1

1

1

1

1

0

0

0

uA

uA

l

A

l

Al

A

l

A

l

A

l

A

l

A

l

A



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5th step. Resolution of the system

As we need a reference value for the potentials (u3

is an unknown) we can set

equal to 0

Taking A(1) as 0.80 A0 and A(2) as 0.49 A0, and l

(1)= l(2)=L/2, the system of two

equations with two unknowns gives

6th step. Computation of the results

Once we have obtained the potentials, the velocities can be derived by using

the equivalence

which gives the velocities at elements 1 and 2 as

Lu01 651. Lu02 0271.

112

ldx

du

0

1

251 uu .

0

2

052 uu .



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In this method the FE equations can be directly obtained from the governing

equations (or equilibrium equations)

The discretization is made and the field variable is approximated as

where i are constants and Ni(x) are linearly independent functions chosen

such that the boundary conditions are satisfied

A quantity R known as the residual or error is defined as

The weighted function of the residual is taken as

where f(R)=0 when R=0


weighted residualscomputational fluid dynamics I

GF

n

i

ii xNx1

~

~~

FGR

0 dVRwfV

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There are several approaches to the weighted residuals method such as the

collocation method, the Least Squares method and the most commonly used of

all, the Galeking method

In the Galerkin method the weighting functions are chosen to be equal to the

trial functions and f(R) is taken as R

with i=1,2,,n

In the rest of the aspects the method is similar to the variational



0 dVRNV

i

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Example. Find the velocity distribution of an inviscid fluid flowing trough a

varying cross section tube shown in the figure

The governing equations are given by the continuity equation

with the boundary condition u(x=0)=u0, where the cross section area is



02

2

dx

d

LxeAA 1

u0

A1

L

A2 A3

1 2 3

l(1) l(2)

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1st step. Discretization

Divide the continuum into two finite elements. The values potential function in

the three nodes will be the unknowns of the fem

2nd step. Select an interpolation model, easy but leading to convergence

The potential function will be taken as linear

and can be evaluated at each element as

where l(e) is the length of element e

bxax

eeee

l

xx 121



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This can also be obtained through the shape functions which have to be 1 at its

node and zero at the others, that is

this is

(the same as obtained before)



xNxNx 2211

elxN 11

el

xN 21

l(e)

1

l(e)

eeel

x

l

x

l

xx 12121 1

fi i l i fl id

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3rd step. Derivation of stiffness matrices K(e) and load vectors P(e) by using

equilibrium. Obtaining of a weak form

The integral of the weighted residual is

integrating by parts

00 2

2

dxdxd

wel

i


weighted residuals


dxdxddv

2

2

dxdv

iwu idwdu

00000

00 2

2

dxdxdwdxddxdwdxldlwdwdxddxdwdxdxdw

il

i

e

eii

ll

i

l

i

ee

e

e

12

00 uwulwdx

dx

d

dx

dwi

e

i

li

e

fi it l t i fl id

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weighted residuals


This is

The elementary matrices result into

As

where

As the derivatives are

the elementary matrices result into

0 eee PK

122

121

02211

00 uwulwdx

dx

dN

dx

dN

dx

dwdxNN

dx

d

dx

dwi

e

i

li

li

ee

dx

dx

dN

dx

dN

dx

dN

dx

dNdx

dNdx

dNdx

dNdx

dN

dxdx

dN

dx

dN

dx

dNdx

dNe

2212

2111

21

2

1

K

2

1

u

ue

P

Ldx

dN 11 Ldx

dN 12

11

111

11

11

0

22

22

e

l

ee

eee

ldx

ll

lle

K

2

11)(

3

22)(

fi it l t i fl id

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4th step. Assembly of the stiffness and load vectors

Once we have obtained the matrices for all the basic elements as

we can assemble the system to obtainQK

11

1111

1

lK

11

1112

2

lK

0

01u

P

2

20

uP

2

0

3

2

1

22

2211

11

0

110

1111

011

u

u

ll

llll

ll


weighted residuals



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5th step. Resolution of the system

6th step. Computation of the results

As can be seen, the system of equations obtained by the weighted residuals

method is the same as in the variational method except for the absence of the

density (which can be removed as it is a constant), and the cross section areas.

The areas are not present in the second formulation as the system is solved invelocities and not in flow rates. To avoid this fact a two dimensional model

should be considered.


weighted residuals



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Finite elements = Piecewise approximation of the solution by dividing the

region into small pieces

This approximation is usually made in terms of a power series (polynomial) which

is easy to integrate and easy to be improved in accuracy by increasing the order,

fitting in this way the shape of the polynomial to that of the solution (see figure)

When the polynomial is of higher order (bigger than one) the midside and/or

interior nodes have to be used in addition to the corner nodes

Some other approximations such as Fourier series could also be used

Problems involving curved boundaries can be solved using isoparametricelements which are not straight-sided


discretization



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discretization



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The mesh can be improved by

Subdividing selected elements (h-refinement)

Increasing the order of the polynomial of selected elements (p-refinement) Moving node points (r-refinement)

Defining a new mesh

In higher order elements the midside and/or interior nodes have to be used in

addition to the corner nodes in order to match the number of nodal degrees offreedom with the number of constants

As it will be shown a different interpolation for the velocity and pressure

unknowns is required for fem in fluids

Basic elements to be considered

Triangular linear

Quadrilateral linear

Triangular linear (natural)

Triangular quadratic


discretization



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The FEM is an approximation that converges to the exact solution as the element

size is reduced if:

i. The field variable and its derivatives must have representation as the element

size reduces to zero

For example, second derivatives cannot be represented with linear functions

Then the elements are said to be complete

ii. The field variable and its derivatives should be continuous within the element (Cr

piecewise differentiable, where ris the maximum order of derivatives within the

integrand)

(The polynomials are inherently continuous and satisfy this requirement)

The field variable and its derivatives, up to the r-1-th, must be continuous at the

element boundaries

Then the elements are said to be compatible or conforming


discretization, convergence


dxdx

dr

r


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If we had for instance, flat penthouses as interpolating functions, the

interpolating surface would be discontinuous (would break and split up)

Still, there are many fem basic elements that not verifying the former properties

still provide meaningful solutions (such as the checker board pressure mode)


discretization, convergence


5 . 00 1 0 .0 0 1 5 .0 0 2 0 .0 0 2 5 .0 0 3 0 .0 0 3 5 .0 0 4 0 .0 0 4 5 .0 0 5 0 .0 0 5 5 .0 0

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

55.00

0 . 00 1 0 .0 0 2 0 .0 0 3 0 .0 0 4 0 .0 0 5 0 .0 0 6 0 .0 0 7 0 .0 0 8 0 .0 0 9 0 .0 0 1 0 0 .0 0

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00


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Let the basic linear triangular element connecting nodes 1, 2,and 3 be

The equation that gives the

surface (plane) is

(1)

that leads to the following

equations

yxyx 321 ,

131211yx

232212 yx

333213 yx

1

2

3

22 , yx

11, yx 33 , yxx

y


discretization, triangular linear b.e.



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The solution of the former system gives

(2)

where

substituting (2) in (1) and rearranging terms it is obtained

3322111 21

aaaA

33221122

1 bbb

A

33221132

1 ccc

A

132 yyb 321 yyb 231 xxc

312 xxc

123 xxc 213 yyb

23321 yxyxa

31132 yxyxa

12213 yxyxa




33

22

11

1

1

1

2

1

yx

yx

yx

A


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The interpolating function results

where

(3) The shape functions take the value of 1 at its node and cero at the rest

These expressions are complicated and depend onx andy

332211 ,,,, yxNyxNyxNyx

2332233211112

1

2

1xxyyyxyxyx

Aycxba

AyxN ,

3113311322222

1

2

1xxyyyxyxyx

Aycxba

AyxN ,

1221122133332

1

2

1xxyyyxyxyx

Aycxba

AyxN ,





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For anA element matrix equal to

The integrals are

As the integrand is a constant there is no need to integrate numerically

dxdy

y

N

y

N

x

N

x

NA

jiji

ij

e

A

discretization, triangular linear


dxdy

y

N

y

L

x

N

x

N

y

N

y

N

x

N

x

N

y

N

y

N

x

N

x

N

yN

yN

xN

xN

yN

yN

xN

xN

yN

yN

xN

xN

y

N

y

N

x

N

x

N

y

N

y

N

x

N

x

N

y

N

y

N

x

N

x

N

e

e

333323231313

323222221212

313121211111

A

e

dxdy

xxyysim

xxxxyyyyxxyy

xxxxyyyyxxxxyyyyxxyy

Ae

e

2

12

2

21

12312113

2

31

2

13

1223213231231332

2

23

2

32

24

A


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The basic element matrix results

That now can be assembled

in the stiffness matrix to yield

discretization, triangular linear


2

12

2

21

12312113

2

31

2

13

1223213231231332

2

23

2

32

4xxyysim

xxxxyyyyxxyy

xxxxyyyyxxxxyyyyxxyy

Ae

e A

6

2

9

10

9

8

7

6

5

4

3

2

1

10

9

8

7

6

5

4

3

2

1

f

f

f

ff

f

f

f

f

f


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The need of integrating the shape functions and their derivatives overthe domain leads to the use of the natural (local) coordinates, which

allows for an element based integration that simplifies the calculations

The natural triangular system of referenced is defined with the lineardependent coordinatesL1, L2, andL3

whereAi is the area defined by the point P and the opposite side

The shape functions for this triangular linear element are

A

AL 11 AAL 22

A

AL 33

1

3

2P

A2 A1

A3

ii LN 321 ,,i



1321 LLL


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The shape functions are in fact as seen in (3)

Or in matrix form

1

3

2P

A2 A1

A3



A

A

yx

yxyx

ALN ikk

jj

ii2

2

1

11

2

1

jkkjjkkjii xxyyyxyxyxAyxLyxN 21

,,

y

x

xxyyyxyx

xxyyyxyx

xxyyyxyx

AL

L

L 1

21

12211221

31133113

23322332

3

2

1

33

22

11

1

1

1

2

1

yx

yx

yx

A


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The derivatives ofL1,L2 andL3 being

1

3

2P

A2 A1

A3



Ayy

xL

2321

Ayy

xL

2132

A

xx

y

L

2

312

A

xx

y

L

2

231

Ayy

xL

2213

A

xx

y

L

2

123


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For natural coordinates in triangles the same procedure can be usedexcept for the fact that one of the three coordinates is lineardependant and can be dropped from the integration leading to a

change in the integration limits

where the jacobian determinant is

and the integral is

Axxyyxxyy

AyL

xL

yL

xLJ

21

41 231331322

1221

121

0

1

02112

1

0

1

0

11 1dLdLLLgdLdLyxfJdxdyyxf

LL

,,,

1321 LLL

discretization, triangular quadratic


121

0

1

021

1

2 dLdLLLgAdyxfL

,,

CFD I_2

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Transcript of CFD I_2