Upload
doantruc
View
218
Download
0
Embed Size (px)
Citation preview
16/10/2002 Seminar: Numerical Integration in more dimensions 2
Outline
The role of a mapping function inmultidimensional integration
Gauss approach in more dimensions and quadrature rules
Critical analysis of acceptability of a given quadrature rule
16/10/2002 Seminar: Numerical Integration in more dimensions 3
Problem definition
! We have and we want to compute I:ℜ→ℜ⊂Ω nf :
! We want to implement some numerical method, which ought to be (as usual) accurate and cheap (e.g. small number of operations)
∫Ω= xdfI
16/10/2002 Seminar: Numerical Integration in more dimensions 4
Covering
! Step 1: covering of the domain with replicas of a basic geometry
Ω
( ) ( )∑∫∫ ≅Ω
lel
dfdf xxxx
Error 1: covering error
el
16/10/2002 Seminar: Numerical Integration in more dimensions 5
Mapping function
! Step 2: introduction of a mapping function F" Example 1: squares
( ) ( ) ( ) ( )[ ] ( )
ll
l0
l0
l2
l0
l1l
Ay
x
y
xF
bx
vvvvv
+=
+
−−=
ˆˆ
ˆ|
ˆ
ˆ
x0
el
x10
y1
eFl
( )l2v ( )l
3vy
el( )l0v
( )l1v
16/10/2002 Seminar: Numerical Integration in more dimensions 6
Mapping function
x0
y
( )l1v
el
" example 2: triangles
( ) ( ) ( ) ( )[ ] ( )
ll
l0
l0
l2
l0
l1l
Ay
x
y
xF
bx
vvvvv
+=
+
−−=
ˆˆ
ˆ|
ˆ
ˆ
Fl
x10
y1
e
( )l2v
el( )l0v
16/10/2002 Seminar: Numerical Integration in more dimensions 7
Properties of the mapping function F
! Fl is affine: ( ) lll AF bxx += ˆˆlinear
constant
∑ ∑ =i i
iii 1x αα ,
F maps affine combinations
to affine combinations
Fl
Fl
e
e el
eltriangles are mapped to triangles, rectangles to parallelograms, etc.
…that is,
16/10/2002 Seminar: Numerical Integration in more dimensions 8
The role of the mapping function
( ) ( )∑∫∫ ≅Ω
lel
dfdf xxxx
( ) ( )( ) ( )
( ) ( )( )∫∫ ∫
⋅=
=∂=
ell
ee
ll
dFfA
dFFfdfl
ˆ
ˆ
ˆˆdet
ˆdetˆ
xx
xxxx
! Step 1:
! Step 2:
16/10/2002 Seminar: Numerical Integration in more dimensions 9
Quadrature
! Step 3: integration over the basic geometry
( )( ) ( ) ( )∑∫∫ ≅=i
iie
e l gwdgdFf xxxxx ˆˆˆˆˆˆ
ˆ
Error 2: quadrature error
16/10/2002 Seminar: Numerical Integration in more dimensions 10
Integration over a surface
! Suppose we have a function defined over a surface
! Thanks to the properties of the mapping function, we can use the same approach:
( ) ( ) ( ) ( )[ ] ( )ll
l0
l0
l2
l0
l1l A
y
x
y
xF bxvvvvv +=+
−−=
ˆ
ˆ
ˆ|
ˆ
ˆ
…simply v(l) given in a suitable reference system…
16/10/2002 Seminar: Numerical Integration in more dimensions 11
How can one reduce errors?
Covering error Quadrature error! More accurate formulas! Smaller volumes (where
necessary, depending on f)Ω
16/10/2002 Seminar: Numerical Integration in more dimensions 12
Open problem
Given a basic geometry, find the least amount of points and weights such that
is exact for all monomials of degree d and lower
" Let’s look at some examples…
( ) ( )xxx ˆˆˆˆ
∑∫ ≅i
ie
gwdg
16/10/2002 Seminar: Numerical Integration in more dimensions 13
d=1 in the square
3 equations # 1 point, 1 weight
Mathematical problem - Physical interpretation
1
x y
1
1
0
y
x0.5
0.5
==
==
==
∫
∫
∫
e11
e11
e1
yw21dydxy
xw21dydxx
w1dydx1
ˆ
ˆ
ˆ
ˆ
ˆ 1w1 =ˆ
16/10/2002 Seminar: Numerical Integration in more dimensions 14
d=1 in the triangle
3 equations # 1 point, 1 weight
Mathematical problem - Physical interpretation
1
x y
1
1
0
y
x
21w1 =ˆ
31
31
==
==
==
∫
∫
∫
e11
e11
e1
yw61dydxy
xw61dydxx
w21dydx1
ˆ
ˆ
ˆ
ˆ
ˆ
16/10/2002 Seminar: Numerical Integration in more dimensions 15
d=2 in the square
6 equations # 2 points, 2 weights
Mathematical problem - Physical interpretation
1
x y
x2 y2xy
x
y
1
1
0
∫
∫
∫
∫
∫
∫
+==
+==
+==
+==
+==
+==
e222111
e
222
211
2
e
222
211
2
e2211
e2211
e21
yxwyxw41dydxxy
ywyw31dydxy
xwxw31dydxx
ywyw21dydxy
xwxw21dydxx
ww1dydx1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆˆˆˆ
ˆˆ
ˆˆ
ˆˆ
ˆˆ
...no Mathematica solution…...no Mathematica solution…
16/10/2002 Seminar: Numerical Integration in more dimensions 16
d=3 (e.g. in the triangle)
10 equations # how many points?
3 points # too many equations?4 points # not enough
1
x y
x2 y2xy
x3 y3x2y xy2
16/10/2002 Seminar: Numerical Integration in more dimensions 17
Biquadratic polynomials for the square
! Let’s choose two monomials p(x) and q(y) and let them be of degree d at most.
! If we choose d=3
! 16 equations # 4 points, 4 weightsx3
x3y
x3y2
x3y3x2y3xy3y3
x2x1
x2yxyy
x2y2xy2y2
...no Mathematica solution (in a reasonable time)…...no Mathematica solution (in a reasonable time)…
16/10/2002 Seminar: Numerical Integration in more dimensions 18
Cross product Gauss
Gauss 1D in [0,1] Gauss 2D in [0,1]2
0 1
0
1
0 1
$ only for domains like [a,b]n
( ) ( ) ( ) ( )∑ ∫ ∑∑∫∫ ∫∑ ===i i j
jijiiii
ii xxgwwdyyxgwdyyxgwdxdyyxg ˆ,ˆ,ˆ,ˆ,
16/10/2002 Seminar: Numerical Integration in more dimensions 19
Higher degree formulas
! Many of them in the literature
" Example 1: degree 6 in the triangle with 12 points
" Example 2:degree 20 in the triangle with 79 points!
D.A.Dunuvant, HIGH DEGREE EFFICIENT SYMMETRICAL GAUSSIAN QUADRATURE RULES FOR THE TRIANGLE, International Journal for Numerical Methods in Engineering, vol. 21, 1129-1148 (1985)
A.H. Stroud & D. Secrest, GAUSSIAN QUADRATURE FORMULA, Prentice-Hall, 1966
x10
y
1
1
25
4
3
67 8
9
10 1112
16/10/2002 Seminar: Numerical Integration in more dimensions 20
Summary
Do the number of the unknows
correspond to the number of the
equations?
Does the non-linear system have
a solution?
Is the found solution
acceptable?
WE HAVE AN ACCEPTABLE QUADRATURE
METHOD!
Condition 1: Are all the points inside the element? Condition 2: Are all the weights wi positive?
ix
16/10/2002 Seminar: Numerical Integration in more dimensions 21
Condition 1: ∀ iexi ˆˆ ∈
4x
1x
2x3x
F
∂Ω( )4F x
( )3F x( )1F x
( )2F x
Ω
What is the value of if does not belong to Ω?( )( )4Ff x ( )4F x
16/10/2002 Seminar: Numerical Integration in more dimensions 22
Condition 2: wi ≥ 0, ∀ i
! To always have non-negative integrals for non-negative functions
0>∫ dxfb
a
( ) 07
1<∑
=i
ii xfw
Exact solution:
Approximation: if w2< 0
b
y
x0 a
w4 w6
f(x)
w5w3w2w1 w7
! Finite Elements Methods (FEM)
16/10/2002 Seminar: Numerical Integration in more dimensions 23
Why weights always ≥ 0 in FEM
! Stiffness matrix A is positive definite
! Positive definite property is needed for the iterative solvers of Krylov type (all fast iterative solvers)
! Negative weights might cause positive definite property to be lost
0022>>+∇=== ∫∑∑ uifuuuau jij
i ji
T …Auu
∑=i
iiuu ϕ ( ) xda ijijij ∫ +∇∇= ϕϕϕϕ[ ]n1 uu ,,…=u
16/10/2002 Seminar: Numerical Integration in more dimensions 24
Do we really get negative weights?
! Newton-Cotes approach in [-1,1]:
! Negative weights also in many formula using with Gauss approach
299376427368
299376260550
299376272400
29937648525
299376106300
2993761606711
141754540
1417510496
14175928
141755888
141759899
34
313211
AAAAAAn 654321
−−
−−