NM Ln5 6 PDE Compatibility Mode

8/12/2019 NM Ln5 6 PDE Compatibility Mode

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Accuracy/consistency

The discretised equations are notthereal ones

x

u

real

e sc eme oes notso ve t e reaequations !

1/25/2012 Numerical Methods 5

approximatex

u

Important Properties of Numerical Schemes

Convergence

numerical scheme solution isconvergent if it comes closer and

closer to the analytical solution of the real ODE/PDE when the

ConsistencyConvergence

Stability

me s ep ecreases;

LaxTheorem: 2 conditions needed for convergence

Consistency

A scheme isconsistent if it gives a correct approximationof the ODE/PDE as the time/space step is decreased

verified using Taylor Series expansion

Stability

A scheme is stable if any initially finite perturbation

remains bounded as time grows

Verification: Matrix method, Fourier method, Domain of

dependence

1/25/2012 I.Popescu: Numerical Methods 6


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Accuracy/consistencyTo reduce the truncation error :

Decrease both t and x


Stability

. .reasonable range)

If the truncation error is small:

the discretised equation is consistent

x

taCr


with the real one.

Accuracy and suitability

Consistency of schemes for PDEs

nnnUCrCrUU

110

Ua

U


Stability

xt

uInitial profile of u

Exact solution, (Cr=1)

Cr=0.5, dx=100m

Cr=0.5, dx=50m

umerical diffusion


x

Numerical diffusion

causes amplitude error


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7

What you want to solve :

uu

Accuracy/consistency ConsistencyConvergence

Stability

What the scheme sees :

xt

3

2

2

uuuu

Numerical

diffusion


...32 xxxt

Truncation errorNumerical

dispersion


Stability of schemes for PDEs

nnnUCrCrUU

110

Ua

U


Stability

xt

uInitial profile of u

Exact solution, (Cr=1)

Cr=0.5, dx=100m

Cr=0.5, dx=50m

umerical diffusion


x

Numerical diffusion

causes amplitude error


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Accuracy and suitability Stability of schemes for PDEs

Because of numerical diffusion we try to use a better scheme,

like for instance Preismann scheme with psi=0.5;

a dis ersion e uation


Stability

0

x

Ua

t

U3

3

x

Uk

x

Ua

t

U

Dispersion equationu

Initial

profile

Analytical profile,

Advected

downstream

Numerical dis ersion


x

Computational

result

Due to derivatives estimation

Sharper profiles and oscilations

Create phase errors umerical dispersion

Numerical diffusion : profile smearing

1.2 Initial

Analytical

Accuracy/consistency ConsistencyConvergenceStability

0.4

0.6

0.8

1

u

Numerical


0

0.2

0 2 4 6 8 10 12 14 16 18 20

x


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Numerical dispersion : oscillations

1.2 Initial

Accuracy/consistency ConsistencyConvergence Stability

0.2

0.4

0.6

0.8

1

u

Numerical


-0.4

-0.2

0

0 2 4 6 8 10 12 14 16 18 20

x


Stability of explicit schemes Von Neumann stability method (Using Fourier

analysis)


Stability

Same principle: amplitude factor is less than 1



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Stability Stability of explicit schemes

Von Neumann stability method (Using Fourier analysis)

Same principle: amplitude factor is less than 1

xijtnnj eUU

00 xjixjtnitntnUU iir

n

j sincossincosexp0

0

n

j

n

j

n

j

n

j

NU

UCrCr

U

UA

1

1

1

i

1-1


xixijtnU

xjitnU

U

Un

j

n

j

exp

exp

1exp0

0

0

01

CrxixCrAN sincos1

Cr

Unit

circle

-i

-

Amplitude and phase portraits

Wave amplitude

Amplification factor =1

Any difference between the numerical phase speed

and true phase speed is the phase error

The graph that shows how the Fouriercomponents are amplified is called anamplitude portrait.


The graph that shows at what speed theFourier components travel is called aphaseportrait.


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Amplitude and phase portraits

x

2

M- The wave number - represents the

number of grid intervals needed to cover

0.75

0.8

0.85

0.9

0.95

1

1.05

1 10 100

A

1.2


2

MAArg

Cr

aAArgc N

NN

CrM

iM

CrAN

2

sin2

cos1

M

0

0.2

0.4

0.6

0.8

1 10 100

M

c/u

Phase and amplitude errors



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(B)Parabolic PDEs


Parabolic Equations Initial Value Problems

A 1D time-dependent parabolic eqn (b=c=0)

2u

au

Lx0

t

Com utational

T

xt t

)()0,( xfxu

)(),()(),0(

thtLutgtu

x

domain

0 L

With I.C

With B.C


x

u

0 L

t=0

f(x)

t

u

0 T

x=0

g(t)

t

u

0 T

x=L

h(t)


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2

2

x

ua

t

u

nnnnn 1

t

n

1nn

iu n

iu 1n

iu 1

1ni

u

Parabolic PDE:Solution Methods Explicit Methods2

iiiii uuuruu 11

x

1niu

i 1i1i

1n

I.C.: )(0

xifui B.C.:

tnun


)( tnhunN We can calculate the unknown values of from the known

values of starting from the initial condition

1niun

iu

0

iu

Parabolic PDE: Stability of the Explicit Method

The explicit method is unstable if the timestep is too large.

conditions is

Stability condition for derivative boundaryconditions is

210 r

a

xt

2

2


for )(fuu

n

uk

x

rk

2

10


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Parabolic Equation An example calculation

Solve

for2

2u

au

x 10

(Where u represents temperature)

with initial condition

t

0.15.0for)1(2

5.00for2)0,(

xx

xxxu

u

1


and boundary conditions

0),1(

0),0(

tu

tu

x10.5

Parabolic Equation An example calculation2

Solve

for

2u

au

x 10

(Where u represents temperature)

with initial condition

xt t

0.15.0for)1(2

5.00for2)0,(

xx

xxxu

u

1


and boundary conditions

0),1(

0),0(

tu

tu

1 21

222

2sinsinx

exp8),(

n xnn

tnntxu

Analytical (exact) solut ion

x1.


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Parabolic Example : Case 1 (r


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1Evolution of temperature distribution

Parabolic Example : Case 1 (r


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Evolution of temperature distribution

Parabolic Example : Case 3 (r>0.5)

Plot of the temperature distribution at 0 and 18time steps

0.5

0.6

0.7

0.8

0.9

1

temperature

t=0

t=0.1

0055.0

1.0

t

x

55.02

x

tar

analytic


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

x

UNSTABLE

solution is

meaningless

Parabolic Example :Evolution of maximum temperature (r=0.1)

0.9

1Evolution of maximum temperature

exact solution

r=0.1 solution

0.5

0.6

0.7

0.8

maximum

temperature

STABLE

solution


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.3

0.4

time


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0.9

1Evolution of m aximum temperature

exact solution

r=0.5 solution

Parabolic Example :

Evolution of maximum temperature (r=0.5)

0.5

0.6

0.7

0.8

maximumt

emperature

STABLE

solution but

not v.

accurate


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.3

0.4

time

Parabolic Example :Evolution of maximum temperature (r=0.55)

0.9

1Evolution of maximum temperature

exact solution

r=0.55 solution

0.5

0.6

0.7

0.8

maximumt

emperature UNSTABLE

solution is

meaningless


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.3

0.4

time


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Parabolic Example : Explicit Method Error

r M u % error

Comparison of temperature values at x=0.5 and t=0.1 (for N=11)

analytic - 0.3021 -

0.001 10000 0.3071 1.65

0.01 1000 0.3070 1.60

0.1 100 0.3056 1.16

0.5 20 0.3071 1.64

need to

increase N

to reduce


0.55 18 0.6336 109

0.6 16 7.2340 2294

error

furtherunstable

(C)Eliptic PDEs



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Elliptic Equations No time variable

),,(),(2

2

2

2

yxufyxux

u

y

u

y

x

Ly

Lx

0

0

- Poisson equation

- Laplace equation

),,(02

2

2

2

yxufx

u

y

u

0002

2

2

2

x

u

y

ufand

With different typs of B.C.:

Dirichlet : u is specified at the boundary


t

Neumann : erivative o u is speci ie at t e oun ary

Mixed(robin): both u and its derivative is specified atthe boundary

Example : Laplace equation

Equation is:

Lx0

2

2

2

2

x

u

y

u

y

1, jiu1j

j

1jjiu , jiu ,1jiu ,1

1, jiu

NLyy

M


xi 1i1iApproximate solution determined at all grid pointssimultaneously by solving single system of algebraicequations


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Elliptic PDE:Solution Methods Explicit Methods

y

j

1jjiu , jiu ,1jiu ,1

1, jiu

02

2

2

2

x

u

y

u

2

,1,,1

2

2 2

x

uuu

x

u jijiji

x

1, jiu

i 1i1i

1j


04 ,1,1,,1,1 jijijijiji uuuuu2

1,,1,

2

2 2

y

uuu

y

u jijiji

02

1,,1,

2

,1,,1

yx

jijijijijiji

Holds for all interior points of the domain

Finite difference methods

What you should remember



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What you should remember Numerical solutions to PDEs can be

obtained by discretising both space andtime.

Explicit numerical schemes for PDEs aresubject to stability constraints.

Implicit numerical schemes for PDEs arealways stable.

1/25/2012Numerical Methods

45

solution of non-linear PDEs.

The notions of consistency, stability andconvergence also hold for numericalschemes for PDEs.


First-order accurate schemes producenumerical diffusion; numerical profilesobtained are smoothed and may lead topeak underestimation. Numericaldiffusion leads to amplitude error.

Second-order accurate schemes producenumerical dispersion; numerical profilesexhibit artificial oscillations. Undesirable


behaviours (such as negativeconcentrations) may appear. Numericaldispersion causes phase error.


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What you should remember Decreasing t or x alone is not sufficient to

reduce the truncation error : tand xshouldbe reduced together.

T e MOC is a speci ic in o numerica met oused for advection modelling. Its maindrawback is that it is generally not conservative(some water, pollutant, or energy, may be lostartificially).

At least three points in space are needed to


solve a diffusion equation.

The design of 2-D and 3-D computational gridshould be carried out with care, long andnarrow grids should be avoided.

6 Finite volume methods

(FVM)



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FVM principle (1)FVMs are applicable to conservative equations, i.e. equations of the

form

problems)(1D0

x

F

t

U

problems)(3D0

problems)(2D0

z

H

y

G

x

F

t

U

y

G

x

F

t

U

F, G, H : Fluxes in x, y and z

U : Conserved variable

nnn

(5.1)


n

n

n

n

n

n

n

n

n

n

n

n

nnn

z

U

y

U

x

U

z

U

y

U

x

UUHH

z

U

y

U

x

U

z

U

y

U

x

UUGG

zyxzyxUFF

,,,,,,

,,,,,,

,,,,,,

Eqs. (5.1) express the conservation of U over any bounded volume of

space

(5.2) 0

zHyGxF

t

U

FVM principle (2)

: Divergence operator

By definition of the divergence, Eq. (5.2) can be rewritten as

0d.ddd nzHyGxFzyxU

t

n


F


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Application:

1) Discretise space into volumes

2) Compute the fluxes at the edges between the volumes

FVM principle (3)

e erm ne e c anges n v a a a ance equa on

(5.3) Vt

UU nn

outin1 FluxFlux


V

The conservation law together with piecewise constant data

having a single discontinuity is known asthe Riemann problem.

FVM application (1)

0

if

l

x

r

q xq

q x

husQus

hds

husQus

hds

h-discontinuity


Qds Qds


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husQus

hds

husQus

hds

h-discontinuity

FVM application (2)

ds ds

Three possible patterns:

zones of constant state (depth and velocity are

homogeneous over such zones),

a shock wave (information coming from upstream catches


and information downstream),

a rarefaction wave (information downstream travels faster

than the information upstream).

Transport equations

2/12/11

jjj

nj

nj FF

x

tCC0

x

CDuC

xt

C

FVM applications (3)

C

xj

Fj-1/2 Fj+1/2


0if2

0if2

1

1

1

1

1

2/1

uxx

CCDuC

uxx

CCDuC

F

jj

n

j

n

jn

j

jj

n

j

n

jn

j

j


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Applications in 1DDiscontinuous flows

PDE)(scalar0

x

U

t

U


When F [F] is a non-linear function of U [U], the solution

may become discontinuous

Ex. Burgers equation

PDEs)of(system0

xt

FU

form)ion(conservat0

2uu


invariant)Riemannais(d

dalong0

D

D

form)istic(character0

uut

x

t

u

x

uu

t

u

Applications in 1DBurgers Eq. (continued): formation of shocks from initially

smooth profiles


t

U

x

A

B

A

B

Shock (u travels faster

behind than ahead)


x

t0

t1

A B

dx/dt = u

u = Cst


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Applications in 1D (4)In finite volume methods: the flux is calculated from the

solution of a Riemann problem

FVM applications(6)

U

Initial

Final

U = Cst here


t

x

The characteristics are straight

lines in the phase space

Applications in 1DThe solution of the Riemann problem exists even though the

initial profile is discontinuous


Algorithm

1) At each interfacej-1/2, define the Riemann problem

2) Solve it => solutionUj-1/2

3) Compute the flux

njnj UU ,1

2/12/1 jj UFF


4) For each cell, carry out balance forU

2/12/11

jjj

nj

nj FF

x

tUU


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Multidimensional problemsWave splitting (scalar equations)

0

Uv

x

Uu

t

U

FVM applications(8)

M (xi1/2,yM)

ut

M(xi1/2 ut,yM)

Cell (i,j)Cell (i 1,j)

Interface (i 1/2,j)


vt

x

M(xi1/2ut,yMvt)

Multidimensional problems (2)

Wave splitting (systems of equations)


Decomposed into

0

yxt

byfollowed

00

xtxtUAUFU


=> Decompose the Riemann problem into 2 R.Ps: 1 along x

and 1 along y

00

ytyt

B


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Multidimensional problems (3)

Wave splitting (2)

FVM applications(10)

tpx )(

tp

y )1(

P(p, 1)

N(p) M

B

U(p)


tp

y )2,(

P(p, 2)A


Finite Volume Methods (FVMs) are well-suited for the solution of conservativePDEs. The weak solution of the PDE issought.

FVMs ensure mass conservation

automaticly and can handle shocks anddiscontinuities.

The solution of the advection PDE b the


Godunov-type FVMs involves thedefinition and the solution of a Riemannproblem.


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7. Finite Element Method

(FEM) - useful for problems withcomplicated geometries and

discontinuities, where analytical


solutions can not be obtained

What is it FEM?

The finite element method is a numerical methodfor solving problems of engineering andmathematical physics, useful for problems withcomp icate geometries an iscontinuities,where analytical solutions can not be obtained.



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Principle of the method -(3)Discretization

Model the domain by dividing it into an equivalent system ofsmaller domains or units (finite elements) interconnected atpoints common to two or more elements (nodes or nodal points)and/or boundary lines and/or surfaces.


Principle of the method -(4)

Discretization

eN

jjj

1

,

N1 and N2 are called Shape Functions or Interpolation Functions.

They express the shape of the assumed U.

For a linear representation of 1D elements:

N =1 N =0 at node 1


N1 =0 N2 =1 at node 2

N1 + N2 =1

1 2

N1

L1 2

N2

L

1 2

2

L


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MethodologyObtain a set of algebraic equations to solve for unknownnodal quantity (displacement).

Principle of the method -(5)

Secondary quantities (stresses and strains) are expressedin terms of nodal values of primary quantity

History

Hrennikoff [1941] - Lattice of 1D bars

McHenry [1943] - Model 3D solids

Courant [1943] - Variational form


Levy [1947, 1953] - Flexibility & Stiffness

Argryis and Kelsey [1954] - Energy Prin. for MatrixMethods

Turner, Clough, Martin and Topp [1956] - 2D elements

Clough [1960] - Term Finite Elements

Applications

Fluid Flow

Heat Transfer

Structural/Stress Analysis

Electro-Magnetic Fields

Soil MechanicsAcoustics



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Application -1D advection equation

0

x

Ua

t

U),(0.

0

yxwdxwx

yc

t

yL

0.0

*11

L

ji

in

ii

n

i

n

i dxNx

NUc

t

UU

0..0 0

*1*1

L L

ji

in

ijii

n

i

n

i dxNx

NUcdxNN

t

UU


11110

1

2

1

n

j

n

j

L

ij

in

i UUdxNx

NU

Application - 1D advection equation

0

x

Ua

t

U

111 1 nnL in N

110 2

jji ji x

j

n

jj

n

jj

n

jj dycybya

1

1

11

1

12/1

xa

j xx jj 2/12/12 12/1

xc

j


3 tc tcj 3 3 tc

nj

jn

j

jjn

j

j

j Ux

Uxx

Ux

tcd 1

2/12/12/1

1

2/1

636

2


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Modeling Considerations

Solve the tridiagonal system [A]Y=B Sy m m et r y -means correspondence in size, shape and position of U and

boundary conditions that are on opposite sides of a dividing line orplane;

Use of symmetry allows us to consider a reduced problem instead of the actual

problem.

The order of the total (global) stiffness matrix and the total number of equations can be

reduced.

Solution time is reduced!

Ba n d w i d t h- An envelope that begins with the firstnonzero component in each column of the [A] matrix


741

2

2PL

8

6

5

3

2

1

4

5

L


L L


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39

41

2

LP

5

3

2

1

4

L


L L

00000X0XX0XX

000000XX0XX0

0000000XXXXX

00000000X0XX

00X0XX000000

XX0XX0X000000XXXXXXX0000

00X0XX0XX000

000XX0XX0X00


X00X00000000

0X0XX0000000

X is a nonzero 2 x 2 block nb


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Bandwidth

nb = ndof( m + 1 )

ere:

nb is the semibandwidth

ndof is the number of degrees of freedom per node.

m is the maximum difference in node

numbers for any element.


Poor Shapes

b

h

b >> h

>>

very small cornerangles


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Poor Shapes

uadrilateral de eneratin

h1h2

h1 >> h2

into triangular shape

Quadrilateral approachingtriangular shape

Finite Element Program.



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START

Input Data

Zero [K] and {F}

Do JE=1,NELE

Compute element stiffness [k]

Assemble Global stiffness [K] and forces {F}

Apply B.C.s

Solve [K]{d}={f}

Compute element quantities

Output Results

END

INPUT

Control parameters

Number of Elements

Number of B.C.s

Geometry

x,y,z location of each node Element connectivity (which nodes are

associated with which elements)



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INPUT

Element Properties

Area

Moment of Inertia

Thickness

Location of Neutral Axis

Physical parameters Information


Programs

ALGOR

COSMOS/M

STARDYNE

IMAGES-3DMSC/NASTRAN


ADINA

NISA


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What you should remember Finite Element Methods (FEMs) seek a

weaksolution to the PDEs.

set of basis or shape functions. Thosecan be piecewise linear, or parabolic, etc.

The Galerkin technique uses a weightingof the solution by functions that are the


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NM Ln5 6 PDE Compatibility Mode

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