' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult - - PDF document
▶
SMART_READER_LITE
LIVE PREVIEW
' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult - - PDF document
Nov 25, 2023 154 likes •374 views
' $ r TU-Chemnitz-Zwick au r r r r r r r r r F akult at f ur Mathematik r r r r r r r r r r Ulrich R ude ' $ Higher o rder multilevel nite element metho ds based on extrap olation
SLIDE 1' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Higher
rder
multilevel nite element metho ds based
n
extrap
lation
quadrature Ulrich R ude Theo ry Institute
n
Numerical Quadrature August 22 { 24, 1994; Argonne, IL
Titel
0.1
SLIDE 2' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Problem Many physical p roblems can b e stated as va ria- tional minimizati
n
p roblems
f
the fo rm min E (u) fo r u 2 U where U is a function space and and E (u) involves integrals.
Problem
1.1
SLIDE 3' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Example 1 U = H 1 (0; 1) and E (u) = Z 1 d(x)
u
(x)
2
2f
(x)u(x) dx: This leads to the t w
p
int
b
unda
ry value p rob- lem (d(x)u ) = f in (0; 1); u(0) = u(1) = 0:
Examples
2.1
SLIDE 4' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Example 2 E (u) = Z 1
u
(x)
2
c
u 4 2
u
2 ! dx: This leads to the t w
p
int
BVP u 00 + c(u 3
u)
= in (0; 1); u(0) = 1 + ; u(1) = 1
:
Examples
2.2
SLIDE 5' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Example 3 U = H 1 ();
R
2 : E (u) = Z Z
(
ru ) 2
2f
u dxdy This b ecomes Laplace's equation u = f in
u
=
n
@
Examples
2.3
SLIDE 6' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Example 4 Equations
f
elasticit y U = H 1 ()
H
1 () E (u) = ~ E 1 +
Z
Z
"
@ u 1 @ x 1 @ u 1 @ x 1 + @ u 2 @ x 2 @ u 2 @ x 2 +
1
div
u divu + 1 2 ( @ u 1 @ x 2 + @ u 2 @ x 1 ) ( @ u 1 @ x 2 + @ u 2 @ x 1 ) # dxdy + Z
N
g 2;1 u 1 + g 2;2 u 2 ds; where ~ E is Y
ung's
elasticit y mo dulus
is
the P
isson
ratio g 2 = (g 2;1 ; g 2;2 ) T a re the surface tractions
Examples
2.4
SLIDE 7' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Example 4 cont'd : ? ? ? ? ? ? ? ? ? ? ? F
@
@ @ @ @ @ @ @ @ @ @ @ @ @ @
@
@ @ @ @ @ @ @ @ @ @ @ @ @ @
@
@ @ @ @ @ @ @ @ @ | {z }
D
E = 196 GP a
=
0:3 g 2;1 = g 2;2 = 8 > < > : F = 1000 N
n
the upp er pa rt
f
the b
unda
ry
therwise
Examples
2.5
SLIDE 8' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Classical Galerkin App roach
Select
a nite dimensional subspace U h
U
.
T
solve
min u h 2U h E (u h )
Evaluate
f
E (v h ) fo r v h 2 U h , e.g. E (v h ) = Z 1 d(x)v h v h
v
h f dx
Select
a nite element basis fo r U (small sup- p
rt)
F
rm
derivatives v h fo r the basis functions an- alytically
Compute
integrals numerically
Assemble
stiness matrix
Solve
(linea r) system.
Numerical
T echniques 3.1
SLIDE 9' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Direct Quadrature App roach
No
dal values (on mesh) u h = (u i ) i=0::: n ; u i = u(x i )
Find
dierentiation/quadrature rules fo r E h (u h )
E
(u) directly .
Solve
min u h 2R n E h (u h ) Example E (u) = Z 1 u (x)u (x) + u(x)f (x)dx E h (u h ) = h n X i=1
u
i
u
i1 h
2
+ u i f i + u i1 f i1 2 No rmal equations: u i1 + 2u i
u
i1 h 2 = f i
Numerical
T echniques 3.2
SLIDE 10' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Direct Quadrature App roach Problem Minimum do es not exist fo r E h (u h ) = 2h n=21 X j =0
u
2j +2
u
2j 2h
2
+ u 2j +1 f 2j +1
Numerical
T echniques 3.3
SLIDE 11' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Basic theo ry fo r linea r p roblem Theo rem: Given Hilb ert space V , jj
jj
2 =< ;
>
bilinea r fo rms a(; ), ~ a (; ), f 2 V
c
1 jjv jj 2
a(v
; v )
c
2 jjv jj 2 8v 2 V , W
V
~ a(v ; v )
a(v
; v ) 8v 2 V j ~ a(v ; v )
a(v
; v )j
jjv
jj 2 8v 2 W min u2V E (u) = min u2V a(u; u)
2f
(u) min ~ u2V ~ E ( ~ u) = min ~ u2V ~ a( ~ u; ~ u )
2f
( ~ u) min w 2W E (w ) = min w 2W a(w ; w )
2f
(w ) Then jju
~
ujj 2
c
jjw
jj 2 + jjw
ujj
2
Application:
W solution (FE) space, V auxilia ry la rger (FE) space fo r dening integration, a co r- rect, ~ a numerical bilinea r fo rm.
Numerical
T echniques 3.4
SLIDE 12' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Asymptotic expansions Dierentiati
n
b y basic dierences and integra- tion b y simple mid p
int
rule (and their 2D gen- eralization) lead to even asymptotic expansions
f
the fo rm E h (u) = E (u) + h 2 E 2 (u) + h 4 E 4 (u)
Reference:
1D case: Lyness 1968; 2D case: R. 1993, R. and Lyness 1994. Extrap
lation
f
functionals min
4
3 E h (u h )
1
3 E 2h (u h )
min
64
45 E h (u h )
20
45 E 2h (u h ) + 1 45 E 4h (u h )
Numerical
T echniques 3.5
SLIDE 13' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Application to Example 2 E (u) = Z 1
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Erro r fo r example 2 Metho ds
f
(fo rmal)
rder
2,4,6 L 2
Erro
r versus numb er
f
unkno wns
10 10
1
10
2
10
3
10
9
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
10
Erro r fo r 6th
rder
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
8
6
4
2
2 x 10
6
Numerical
T echniques 3.7
SLIDE 16' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Hiera rchical Basis
Hierarchical Displacements Hierarchical Function Original Function
Stiness matrix in hiera rchical basis K L;H l stiness matrix fo r coa rse mesh no des
nly
~ K l 1 fo r right hand sides f L;H l and ~ f l 1 , resp ectively . 4 3 K L;H l
1
3 ~ K l 1 = B @ K L;N l 1 4 3 K L;H l ;v m 4 3 K L;H l ;mv 4 3 K L;H l ;mm 1 C A
Hiera
rchical Basis and Multigrid 4.1
SLIDE 17' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Basic Theo ry Theo rem: The FE systems
f
algeb raic equation
4
3 K L;H l
1
3 ~ K l 1
u
H l =
4
3 f L;H l
1
3 ~ f l 1
and
K Q;H l u H l = f Q;H l have the same solution. Pro
fs:
V a riant 1: Elementa ry computation
f
the stiness matrices. T edious in 2D (M. Jung and R., 1994) V a riant 2: Implicati
n
f
asymptotic results fo r quadrature rules and their extrap
lation
(Lyness and R., unpublished).
Hiera
rchical Basis and Multigrid 4.2
SLIDE 18' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Multigrid Algo rithm Initial guess u N ;(k ;0) l 1. u N ;(k ;1) l = G V l (u N ;(k ;0) l ; K L;N l ; f L;N l ) 2. Coa rse{grid co rrection (a) d (k ) l 1 = 4 3 I l 1 l (f L;N l
K
L;N l u N ;(k ;1) l )
1
3 (f L;N l 1
K
L;N l 1 I l 1;inj l u N ;(k ;1) l ) (b) Solve K L;N l 1 w (k ) l 1 = d (k ) l 1 with
iterations
steps
f
a usual (l
1){
grid multigrid algo rithm (c) u N ;(k ;2) l = u N ;(k ;1) l + I l l 1 ~ w (k ) l 1 3. u N ;(k ;3) l = G N l (u N ;(k ;2) l ; K L;N l ; f L;N l ) Set u N ;(k +1;0) l = u N ;(k ;3) l
Hiera
rchical Basis and Multigrid 4.3
SLIDE 19' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Results fo r Example 4 Level l energy no rms
btained
from MG with Extrap. Quadratic Elts. 3 6.34388 6.34386 4 6.41933 6.41931 5 6.45374 6.45378 Advantages
f
extrap
lation
MG
Simple
stiness matrices
Ecient
solver
Easy
to implement in MG framew
rk
F
r
unstructured grids
Hiera
rchical Basis and Multigrid 4.4
SLIDE 20' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r Nonunifo rm Grid Example u = auf (1; 1) 2 u = cos (2 (x
y
))
sinh(2
(x + y + 2)) sinh 8
n
@ [1; 1] 2
Adaptive
renement
Non
unifo rm grids
No
extrap
lation
in algo rithm 1: Linea r elements
One
step
f
extrap
lation
in algo rithm 2: Quadratic elements
Non
unifo rm grids 5.1
SLIDE 21' ' $ $ TU-Chemnitz-Zwick au F akult
at
f ur Mathematik Ulrich R ude r r r r r r r r r r r r r r r r r r r r without extrap
