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A GENERAL EFFECTIVE PROCEDURE FOR
COMBINING COLLOCATION AND DOMAIN
DECOMPOSITION METHODS
Ismael Herrera* and Robert Yates**Ismael Herrera* and Robert Yates***UNAM and **Multisistemas de Computo*UNAM and **Multisistemas de Computo
MEXICOMEXICO
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THE PROBLEM
• The main technical difficulty stems from the fact that the standard collocation method (orthogonal spline collocation: OSC) yields non-symmetric matrices, even for formally symmetric differential operators.
Combining collocation and DDM presents difficulties that must be overcome
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SOLUTION OF THE PROBLEM
• In recent years new collocation methods have been introduced which yield symmetric matrices when the differential operators are formally symmetric . Generically they are known as TH-collocation.
• TH-collocation combines orthogonal collocation with a special kind of Finite Element Method: FEM-OF.
New collocation methods
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STRUCTURE OF THIS TALK
This talk is divided into two parts:
1. Finite Element Method with Optimal
Functions (FEM-OF).
2. TH-collocation
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NOTATIONS
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PIECEWISE DEFINED FUNCTIONS
1 ,..., Eu u u
1,..., E
Σ
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THE BOUNDARY VALUE PROBLEM WITH PRESCRIBED JUMPS (BVPJ)
2 1,*
"Boundary Value Problem with Prescribed Jumps (BVPJ)" :
Given f,g, j D find u D such that
Pu f, Bu g and Ju = j
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GREEN´S FORMULAS IN
DISCONTINUOUS FUNCTIONS(GREEN-HERRERA FORMULAS,1985)
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*
" ´
* *
Green s Formulas for the BVPJ"
Green - Herrera formulas have the general form :
P - B - J = Q - C K
" ´
* *
Green s Formulas for the BVP"
Green's formulas have the general form :
P - B = Q - C
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* ,
, , * , ,
, , * , ,
, , * , ,
,
w u u w u w
Boundary operators
u w n u w u w on
Jump and average operators
- u w n u w u w on
u w u w n and u w u w n
Pu w w ud
L L D
D =B C
D =J K
J D K D
L
, * , *
, , , * , * ,
, , , * , * ,
x Q u w u wdx
Bu w u w dx C u w u w dx
Ju w u w dx K u w u w dx
L
B C
J K
A GENERAL GREEN-HERRERA FORMULA FOR
OPERATORS WITH CONTINUOUS COEFFICIENTS
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WEAK FORMULATIONS OF THE BVPJ
"
"
"
* * *
Starting with a Green - Herrera formula
P - B - J = Q C K
Weak formulation in terms of the data" :
P - B - J u f g j
Weak formulation in terms of the complement
*
, ,
*
* * *
ary information" :
Q* -C - K u f g j
"Classification of the information"
Data of the BVPJ : Pu, Bu, Ju
Complementary information : Q u C u K u
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FINITE ELEMENT METHOD
with OPTIMAL FUNCTIONS
A target of information is defined. This is denoted by “S*u”.
FEM-OF are procedures for gathering such information.
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CONJUGATE DECOMPOSITIONS
J J
*
1).- A pair of decompositions are introduced :
K = S + R and J = S + R
2).- S u is the 'sought information'
3).- When the 'sought information' is given, the equation
J R P - B - R u = f - g - j
defines well - posed 'local' problems.
* 0
S R S J R J
P
J P R P
REMARKS.- Here and in what follows :
A).- j j j with j S u and j R u
B).- The function u is 'defined' by
P B R u f g j & S u
C).- Homogeneous boundary conditions w
ill be assumed
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OPTIMAL FUNCTIONS
1
2
0
0
JB J P B R
T Q C R
Optimal Base Functions
O D P B R N N N
Optimal Test Functions
O w D Q C R w N N N
v v
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THE STEKLOV-POINCARÉ APPROACH
ˆ ˆ
ˆ, ,
* *B
J J P S B
Let u O , then S u S u if and only if
- S u w = S u w - j ,w , w O
THE TREFFTZ-HERRERA APPROACH
1
*
ˆ ˆ
ˆ,
* *
T
Let u D , then S u S u if and only if
- S u w = f - j,w , w O
THE PETROV-GALERKIN APPROACH
ˆ ˆ
ˆ ˆ, , , ,
* *B
*J J P S T
Let u O , then S u S u if and only if
- S u w - S u w f j w = S u w - j ,w , w O
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ESSENTIAL FEATURES OFFEM-OF METHODS
B T
B T
The linear spaces of 'optimal functions', O and O ,
are replaced by finite dimensional spaces, O and O ,
whose members are approximate 'optimal' base and
test functions.
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THREE VERSIONS OF FEM-OF
1
, ,
, ,
B
P B
* * *P T
POINCARÉ - STEKLOV FEM - OF : Seek for u O such that
P - B - J u w = f - Pu ,w - P - B - J u w , w O
TREFFTZ - HERRERA FEM - OF : Seek for u D such that
Q - C - K u w = f - Pu ,w - P - B - J u w , w O
PETROV GALERKIN FEM - O
, ,
,
B
* * *
P T
F : Seek for u O such that
P - B - J u w Q - C - K u w =
f - Pu ,w - P - B - J u w , w O
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EXAMPLE
SECOND ORDER ELLIPTIC
*
* ,
,
,
* ,
n n n
n n
u a u bu cu and w a w b w cw
w u u w u w
u w a u w w u buw
u w a u w u a w b w
u w a w b w u w
L L
L L D
D
J
K na u
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* ,
* ,
n n
n
The 'average' on the internal boundary
u w a w b w u, on
u w w a u, on , on
S
R
A POSSIBLE CHOICE OF THE ‘SOUGHT INFORMATION’
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CONJUGATE DECOMPOSITIONS
:
, ,
, , ,
, , ,
n nJ J n
J J
Define
u w a u w and u w u a w b w
Then
u w u w u w
w u w u w u
S R
J S R
K S R
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THE SYMMETRIC POSITIVE CASE
, ,* * *
B T
The 'optimal' base and test functions are the same.
The bilinear form
P - B - J w Q - C - K w
is symmetrical and positive definite on O O .
v v
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TH-COLLOCATION
• This is obtained by locally applying orthogonal collocation to construct the approximate optimal functions.
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SECOND ORDER ELLIPTIC EQUATIONS
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*
0, 1,...,
0,
0,
0, 1,...,
0,
0,
i
B
i
T
a b c
w a w b w cw
Optimal base functions
in , i E
O on
on
Optimal test functions
w in , i E
w O w on
w on
v v v v
v
v v
v
L
L *
L
L
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CONSTRUCTION OF THE OPTIMAL FUNCTIONS
• An optimal function is uniquely defined when its ‘trace’ is given on Σ.
• Piecewise polynomials, up to a certain degree, are chosen for the traces on the internal boundary Σ.
• Then the well-posed local problems are solved by orthogonal collocation.
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Support of an ‘Optimal Function’
( , )i jx y
CONSTRUCTION BY ORTHOGONAL COLLOCATION
Cubic-Cubic: Four Collocation Points
1 1
, 0,1
0 0 0
1 1 0
2 0 1
* *
,
( ) ( )
( ) ( )
( ) ( )
, ;
ij
ij
ij
ij
ij ij i j
i j
i j
i j
ij
w b x y C H x H y
b H x H y
b H x H y
b H x H y
x y Gaussian points in = 1,..,4
* *( , )x yCollocation at each
ij
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COMPARISON WITH ‘OSC’
• Steklov-Poincaré FEM-OF yields the same solution as
OSC. However, now the system-matrix is positive definite
for differential systems that are symmetric and positive.
• Trefftz-Herrera FEM-OF yields the same order of
accuracy as OSC, although its solution is not necessarily
the same. The system-matrix is positive definite for
differential systems that are symmetric and positive.
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Support of an ‘Optimal Function’
( , )i jx y
CONSTRUCTION BY ORTHOGONAL COLLOCATIONLinear-Quadratic (One collocation point)
* *( , )x yCollocation at each
ij
1 1
* *
,
1,...,4
,
ij i j ij i i j j
ij
w l x l y C l x l x l y l y
x y Gaussian point in
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THE BILINEAR FORM
,
* * * , , ,
0
B T
P B J w
a w b w c w dx
Q C K w w O O
It is positive definite when b and c 0.
v
v v v
v v
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TH-COLLOCATION FOR
ELASTOSTATIC PROBLEMS OF ANISOTROPIC
MATERIALS AND ITS PARALLELIZATION
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*:
0, 1,...,
0,
0,
0, 1,...,
0,
0,
i
B
i
B
C
Optimal base functions
in , i E
O on
on
Optimal test functions
w in , i E
w O w on
w on
L L
L
L
v v v
v
v v
v
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CONSTRUCTION OF THE OPTIMAL FUNCTIONS
• The displacement fields are chosen to be piecewise polynomials, up to a certain degree, on the internal boundary, Σ.
• Then the well-posed local problems are solved by orthogonal collocation.
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THE BILINEAR FORM
,
: :
* * * , , ,
B T
P B J w
w C dx
Q C K w w O O
It is positive definite.
v
v
v v
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ISOTROPIC MATERIALS
,
2 :
* * * , , ,
B T
P B J w
w w dx
Q C K w w O O
It is positive definite.
v
v v
v v
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*
0, 1,...,
0,
0,
0, 1,...,
0,
0,
i
B
i
B
Optimal base functions
in , i E
O on
on
Optimal test functions
w in , i E
w O w on
w on
L L
L
L
v v v v
v
v v
v
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CONCLUSIONS
For any linear differential equation or system of such equations, TH-collocation supplies a new and more effective manner of using orthogonal collocation in combination with DDM. It has attractive features such as:
1. Better structured matrices,
2. The approximating polynomials on the internal boundary and in the element interiors can be chosen independently,
3. The number of collocation points can be reduced.
