Pinhas Z. Bar-Yoseph

Computational Mechanics Lab.

Mechanical Engineering, Technion

23.3.2006 ISCM-20

Copyright by PZ Bar-Yoseph©

c vt

,

,

jt j

dt

dt

yf y

u f g

u F u F u u 0

1

1

1

0

The time interval

is partitioned into subintervals

where and belong to an orde

one spec

red partition

of time levels 0 .

Each time subinterval is represented by

0,

tral

,

ele

n n n

n n

N

t t

t t t T

I T

I t t

.

Within each spectral element, the dependent variables are

expressed in terms of -th order Lagrangian interpolants

through the Legendre-Gauss-Lobatto poi

ment

nts.

p

t

nInt

1nt

1nt

T T

, , ,

Discon

acts as a stabilizer operator

li

tinuous

m

Galerkin Method (DGM)

nn

tI

Jumpoperator

dt

dt

where

z 0

yw f y w y 0

y y y

v v x z

0

, , 0

0

dt t

dt

yf y

y y

0

Method of Weighed Residuals (MWR

,

)

n nT

dTR T t t

dt

r T t T

1

0

n

n

n

t

T t tt

T Rdt T r

0

0 0

0,dT

T t tdt

T t T

1

0

n

n

nT

tn n

tr t t

R

dTT T dt T T t T

dt

2

Space of piecewise polynomials of degree 0

with no continuity requirements

across interelement boundaries:

: P , np n n

I

T p

v L I v I I I

S

S

1

1 1 2 11,

1 1 1 22 6

,

e e e

e e

ne

n

Tt

T

K C M

C M

U

1

1

Galerkin + Linear elem t

enn n

n n

n n

T N T N T

where T T t

Galerkin Method (CGM)

0,

Con

tinu

ous

0n e

nT T t

F

1 3

2 3

1 A-Stable

n n n

AmplificationfactorA

T T A T

A

Galerkin Method (DGM)D

1 0,

0

iscont

0

in u

0

uo s

e e enT t

F K K

12

2 3

4 6

1

lim =0 Asymptotic annihilation

L -Stable

n n n

A

T T A T

A

A

Bar-Yoseph, Appl. Num. Math. 33, 435-445, 2000

DSM for Dynamic systems

Aharoni & Bar-Yoseph, Comp. Mech. 9, 359-374, 1992

Hamiltonian

Generalized displacement

Generalized momentum

H

q

p

0

0 00

0 00 0

ft

t

H Hdt

t t

t t

q p p Q qp q

p p q

q q p

nt steptimenth

, n 1 n 1p q

,n np q

,n np t q t

nt

t1nt

nt

1nt

Discontinuous element

Plat & Bar-Yoseph, 27th Israel Conf. Mech. Eng. 683-685, 1998

Nonlinear Spatio-Temporal Dynamics of a

Flexible Rod

Bar-Yoseph, Appl. Num. Math. 33, 435-445, 2000

Nave, Bar-Yoseph & Halevi, Dynamics. & Control. 9, 279-296, 1999

The unicycle system, presents an example of inherently unstable system which can be autonomously controlled and stabilized by a skilled rider

Wu -required to maintain the unicylce’s upright position

Tu -required to maintain lateral stability

-the friction torque is assumed to be dependent on the yew rate only

fM

The adaptive technique performed very well for all stiff systems that we have experienced with (convection, radiation and chemical reactions), and is competitive with the best Gear-type routines

dim, 1(1)

Linear Eq's. (scalar wave eq.)

Nonlinear Eq's.

jt j j n u f g

0 1

0 1

0, , ] , [ ]0, [

'

,0

'

, ,

0

u uc x t x x T

t xIC s

u x f x

BC s

u x t u x t

where

c

f x

x

DGM's milestones papers:

, " Triangular mesh methods for the neutron transprt equation",

Proc. Conf. Math. Models &

Comutational Techniques for Analysis of Nuclear Systems, Michigan

Reed & Hill

, 1973 .

Le , " On a finite element method for solving the neutron transprt equation",

Mathematical Aspects of Finite Elements

sa

in

int & Rav

PDE's,

iart

1974

u x

x

Classical artificia

Conti

l dif

nuous Gale

fusio

Dis

Exact solutio

continuous Ga

rk

le

in

n

n

rkin

ν

t

x

1G

2G

3G

3

0

u in G I

u u on

3 3 3

3 3

11

1

2

12

3

23

2

0w u d w u d w u d

u u d

uu

d

u

n ν n ν

n ν n ν

3 3

0G

w udG w u d

First Order

Upwind FD scheme

0p

x

t

e

G

t

x

x

tx

t

f

1n

n

1j j 1j

1p

ˆ ˆ 0e e e

t x

e e et x

G

u uw c d w u d w cu d

t x

DGM

10 0 1

indicates the time slab

indicates element number within the time slab

n n nt x i i j j jc c c

n

j

D D L R U L U R U

T T

T T0

T T0

,

ˆ ˆ,

ˆ ˆ,

e e

e et t

e ex x

e et x

G G

e ei t t

e ei x x

where

d dt x

d d

c d c d

N ND W D W

L W N L W N

R W N R W N

Space-Time Discontinuous Approximations

Bar-Yoseph & Elata, IJNME, 29, 1229-1245, 1990

Conventional el. "Gauss-Lagrange" el.

2

Space of piecewise polynomials of degree 0

with no continuity requirements

across interelement boundaries:

: P , p n e n e

G

u p

S v L G v G G G

S

int Basis functions are

at the integration points.

po wise orthogonal

0 0

2 1 2 1 2 2 1 1

1 2 1 2 2 2 1 1,

2 1 2 1 1 1 2 212 12

1 2 1 2 1 1 2 2

4 2 0 0 0 0 4 2

2 4 0 0 0 0 2 4,

0 0 0 0 012 12

0 0 0

ˆFor and bilinear element:

0

t t x x

i i

x tx t

x xx x

D D D D

L L L

= =

L

W W N

0 0

0 0 0

0 0 0 0

4 0 2 0 0 4 0 2

0 0 0 0 0 0 0 0,

2 0 4 0 0 2 0 412 12

0 0 0 0 0 0 0 0

i i

t tt t

R R R R

11 0

Von Neumann Analysis

exp 2 / , / , 1

n n j nj j

wavewavelengthnumber

where

i k k L x i

U U U

1 10 0 0 0

n nt x i i D D L R R U L U

110 0t x i i

Amplificationmatrix

A D D L R R L

Bar-Yoseph & Elata & Israeli, IJNME, 36, 679-694, 1993;

Golzman & Bar-Yoseph (Project)

Bar-Yoseph & Elata & Israeli, IJNME, 36, 679-694, 1993;

Golzman & Bar-Yoseph (Project)

Bar-Yoseph & Elata, IJNME, 29, 1229-1245, 1990

Adaptive

Refinements

: This problem can be solved by use of ,

where two elements are sufficient to reconstruct t

tilt

he e

ed el

xact

em

solution.

entsNote

Fischer & Bar-Yoseph, IJNME, 48, 1571-1582, 2000

Adaptive Level of Details

Technique for Meshing

Advanced CAD Visualization Methods

Morphing between Meshes at Different Times

DGM Elements are discontinuous.

CGM

Conforming elements.

Elemental Contributions

are to

GLO

assemb

BAL

generate

the set of equat

led

ions.

The size of the equations is equal to

the inside the corresponding element.

The element matrices can be inverted by

using symbolic manipolator once an

LOCAL

d

for all!

edofn

"Conforming" eleCGM ments

"Non-conforming"

elem

DGM

ents

The degree of the

approximating polynomial

can be easily changed

from one element to the other.

Space-Time Discontinuous Approximations

x

u t

DGM remains compact

with high-order polynomial basis

(essential for unstructured mesh computation)

•Gauss-Lobatto nodes are clustered near element boundaries and are chosen because of their interpolation and quadrature properties.• Mass lumping by nodal quadrature.• Exponential rate of convergence.•The increase in the due to the discontinuity at the interelement boundaries is balanced in high order elements.

Discontinuous SPECTRAL ELEMENTS

dofn

x

can

since refinement or unrefinement of the mesh

can be achieved without taking into account

the continuity restrictions

ty

adaptivity

pical of co

ea

nfo

strategie

rming FEM

sily ha

(needs

n

DGM

tra

d

n

sle

sition elements).

x

t

Parallel adaptive DGFE computation.

type of convergence

can be easily implemented,

since no continuity requirement

is imposed on the

test and trail space of

-

functions.

h p

a priori

11

x21

9 876

543

10

15141312

16

t

V

highly paralleliD zGM are eable.

The compact (nearest neighbor) scheme

minimizes interelement communication.

Mjinjjt 11, 0fu

iii fff

Flux Splitting

Mjh

N

i

N

i

jjt

te

ie

ie

e

11,;]][[

]][[]][[

,

21211

Vwu,w

f,wf,w

fuw

ii

0

Bar-Yoseph,Comput. Mech., 5, 145-160, 1989

Nonlinear Wave Eq.

0

020

2

202

2

22

2

2

1

11

,

Ec

Xu

cc

where

X

uc

t

u

Miles Rubin (2005)

uuAf uuAf

i i

i 0ff i i

i 0ff

Flux splitting for non homogeneous uf i

where :

The effective wave speed:

In a matrix form:

2

1

u

uu

),0(,2

2

Lxxt

0fu

1

22

20 1

11

1

u

uu

c

f

0

0

011

11

10

10

01

2

122

20

2

1

u

u

xuc

u

u

t

2

2

20

2

11

1

1

ucc

Traper & Bar-Yoseph (Project)

• The Jacobian matrix:

• The eigenvalues:

• The corresponding eigenvectors:

011

11

10 2

2

20

uc

uA

2

222

02

2

222

01

11

211

11

211

u

uuc

u

uuc

111

211

2

222

01 u

uuc

v

111

211

2

222

02 u

uuc

v

22222

22

2201

22

2201

2

222

0

1

211

1211

2

1

2

1

11

1

2

1

11

211

2

1

uuuu

uuucu

uu

ucu

u

uuc

ff

22222

22

2201

22

2201

2

222

0

2

211

1211

2

1

2

1

11

1

2

1

11

211

2

1

uuuu

uuucu

uu

ucu

u

uuc

ff

2222

22

220

2

220

2

222

0

1

211

1211

2

1

2

1

11

1

2

1

11

211

2

1

uuu

uuuc

u

uc

u

uuc

AA

2222

22

220

2

220

2

222

0

2

211

1211

2

1

2

1

11

1

2

1

11

211

2

1

uuu

uuuc

u

uc

u

uuc

AA

tx,

Displacement 2.0,1,1,5.0 00 cL

Traper & Bar-Yoseph (Project)

Velocity 2.0,1,1,5.0 00 cL

Strain 2.0,1,1,5.0 00 cL

• -Time for breakdown [Lax (1964)]:

1,, 00max2

2

xxucr KT

3.0396

2.0

12

2.00max

10

1

11

20

2

2

2

,2

0,

22

0

2

c

LT

Lu

cK

ucK

cr

x

u

crT

Velocity at t = 3 sec

1u

2.0,1,1,5.0 00 cLx

bilinear

biquadratic

Coarse (& Uniform) grid

Strain at t = 3 sec

2u

x

bilinear

biquadratic

2.0,1,1,5.0 00 cL

Explicit Vs. Implicit schemes

Bar-Yoseph et al., JCP, 119, 62-74, 1995

The discontinuous approximation

can capture shock waves

and other discontinuities

with accuracy.

Bar-Yoseph & Moses, IJNMHFF, 7, 215-235, 1997

c vt

t

c v

numerical flux

, 0

, 0

, ,

,

Runge-Kutta Discontinuous

,

Galerkin (RKDG ; LDGM)

0e e e

where

dd d d

dt

H

u F u F u u

u F u u

F u u F u F u u

uu u F u u n u F u u

Cockburn& Shu, JCP, 84, 90, 1989; Basi & Rebay, JCP, 131,267-279, 1997

c vt

c vt

, 0 (1)

0

Runge-Kutta Dis

(2)

, 0

continuous Galerkin (RKDG)

u F u F u u

D u

u F u F u D

D

(3)

(2)

0 (4)e e e

numericalflux

d d d

H

DD D u n D u

c

v

c

numerical flux

v v

numerical flux

(3)

, , 0 (5)

e e e

e e

edd d d

dt

d d

H

H

uu u F u n u F u

u F u D n u F u D

c v

c c D

c v

: When evaluating boundary integral of (5) along ,

the flux terms, , & , , are not uniquely defined

due to the discontinuous approximation.

H , H 1 2 ,

H 1 2 ,

eNote

u n F u F u D

F u u u n

F u D

v , . (6) F u D n

d

dt

UM K U 0

Semi-discrete method

This system of ODE's is integrated

with a Runge-Kutta method.