Graph grammar
and algorithmic transformations
Lecture topics:
Review of 1D solver
2D Finite Element Method
Generalization of the multi-frontal solver algortihm
to 2D problems
GENERALIZATION TO 1D FINITE ELEMENT METHOD
Strong formulation
Find u  C
2
0 , l 
such that
  a ( x ) u ' ( x )  ' b ( x ) u ' ( x )  c ( x ) u ( x )  f ( x )
u (0)  0
a (l ) u ' (l )   u (l )  
Weak formulation
Find u  V such that
l
l
 au ' v ' bu ' v  cuv dx   u ( l ) v ( l )   f v dx   v ( l )
0
0

b u , v 
v  V  v  H
1

0 , l  : v ( 0 )  0 
l v 
GENERALIZATION TO 1D FINITE ELEMENT METHOD
Finite element method disretization
N
u 
a
i
N
 a b e , e   l e 
v  ej
ei
i
i 1

i
j
j
j  1,..., N
i 1
l
  ae i ' e j ' be i ' e j  ce i e j dx   e i ( l ) e j ( l )
b ei , e j 
0

l
  
l ej 
f e j dx   e j ( l )
0
  ae i ' e j ' be i ' e j  ce i e j dx   ae i ' e j ' be i ' e j  ce i e j dx   e i ( l ) e j ( l )
b ei , e j 
K1
 ae
K1
i
K2

' e j ' be i ' e j  ce i e j dx
 ae
K2
i

' e j ' be i ' e j  ce i e j dx   e i ( l ) e j ( l )
EIGHT FINITE ELEMENTS EXAMPLES
Element matrices:
Eight finite elements mesh
each matrix contains degrees of
freedom (unknowns) associated with
two vertex shape functions,
and two interior shape functions
with vertex shape functions
and two interior shape functions
on each element
EIGHT FINITE ELEMENTS EXAMPLES
Elimination of the two interior shape functions over each element
(meaning: elimination of rows with unknowns – called degrees of
freedom – associated with the interior shape functions)
EIGHT FINITE ELEMENTS EXAMPLES
Merging of the two Schur complements, for pairs of two adjacent elements
EIGHT FINITE ELEMENTS EXAMPLES
Elimination of the one common vertex shape functions,
in each pair of adjacent elements
EIGHT FINITE ELEMENTS EXAMPLES
Merging of the two Schur complements, for sets of four adjacent elements
EIGHT FINITE ELEMENTS EXAMPLES
Elimination of the one common vertex shape functions,
in each set of four adjacent elements
EIGHT FINITE ELEMENTS EXAMPLES
Merging of the resulting Schur complements – all elements are involved
EIGHT FINITE ELEMENTS EXAMPLES
Elimination of the one common vertex shape functions
EIGHT FINITE ELEMENTS EXAMPLES
Recursive backward substitutions
EIGHT FINITE ELEMENTS EXAMPLES
Recursive backward substitutions
EIGHT FINITE ELEMENTS EXAMPLES
Recursive backward substitutions
1D Hp Finite Element
1D hierarchical shape functions:
2D Hp Finite Element
4 vertices
4 mid-edge nodes
1mid-face nodes
The reference element shape functions are created as tensor products
of 1D hierarchical shape functions
2D Hp Finite Element
Vertex shape functions and second order edge and interior shape functions
1
e hp
3
e hp
5
e hp
2
e hp
4
e hp
8
e hp
7
9
e hp
e hp
14
e hp
16
e hp
17
e hp
2D Hp Finite Element
One bilinear shape function for each of 8 vertices
(order of approximation equal to 1 in each vertex)
2D Hp Finite Element
shape functions for each of 4 mid-edge nodes (various orders of approximation)
2D Hp Finite Element
face bubble shape functions for interior node
2D hp FINITE ELEMENT METHOD
15
u 

i
i
u hp e hp
i 1
We seek the solution u of some weak form of PDE as a linear
i
combination of shape functions e hp spread over finite element mesh
1
3
e hp
2
e hp
8
e hp
4
e hp
7
e hp
e hp
5
e hp
9
e hp
2D hp FINITE ELEMENT METHOD
1
3
e hp
2
e hp
5
e hp
e hp
4
e hp
7
e hp
9
e hp
8

e hp
i
hp
b e ,e
i
The coefficients u hp
(called „degrees of freedom” d.o.f.)
are obtained by solving
system of linear equations –
finite element disecretization
of a weak (variational) form of PDE
15

j
hp

i
j
  

i
j
 and l e 
u hp b e hp , e hp  l e hp
i

j  1,..., 15
j
i 1
where b e hp , e hp
j
hp
i
j
are some integrals of shape functions e hp , e hp
ORDERING ISSUES
Full forward elimination
O(15^3)
ORDERING ISSUES
Partial
forward elimination
O(6*9^2)
ORDERING ISSUES
Partial
forward elimination
O(6*9^2)
ORDERING ISSUES
Full forward elimination
of the interface problem matrix
O(3^3)
ORDERING ISSUES
Backward substitutions
ORDERING ISSUES
”Naive” ordering: (order of elimination of rows)
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15
One frontal matrix
Computational cost
O(15^3)
”Smart” ordering: (order of elimination of rows)
1,2,6,7,11,12; 4,5,9,10,14,15; 3,8,13
Two frontal matrices
Computational cost
O(6*9^2)+O(3^3)
Parallel multi-frontal solver
PARALLEL VERSION FOR FOUR ELEMENTS EXAMPLE
PARALLEL VERSION FOR FOUR ELEMENTS EXAMPLE
PARALLEL VERSION FOR FOUR ELEMENTS EXAMPLE
Mesh refinements
CONSTRUCTION OF THE REFINEMENT TREES
CONSTRUCTION OF THE REFINEMENT TREES
CONSTRUCTION OF THE REFINEMENT TREES
CONSTRUCTION OF THE REFINEMENT TREES
CONSTRUCTION OF THE ELIMINATION TREE
BASED ON THE HISTORY OF MESH REFINEMENTS
For any initial mesh, the elimination tree can be created based on
nested dissection algorithm.
CONSTRUCTION OF THE ELIMINATION TREE
BASED ON THE HISTORY OF MESH REFINEMENTS
The elimination tree created for the initial mesh is updated when the mesh is refined
(elimination tree is constructed dynamically, during mesh refinements)
CONSTRUCTION OF THE ELIMINATION TREE
BASED ON THE HISTORY OF MESH REFINEMENTS
The elimination tree created for the initial mesh is updated when the mesh is refined
(elimination tree is constructed dynamically, during mesh refinements)
Ordering
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• Local matrices for active elements – leaves of the elimination tree –
are created
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• Interior degrees of freedom are eliminated at every leaf
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• Schur complement contributions are merged at parent level
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• Fully assembled degrees of freedom are eliminated at parent nodes level
(degrees of freedom related to common edges shared by both son elements
can be eliminated now)
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• Contributions to Schur complement are merged again at the next level
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• Fully assembled degrees of freedom are eliminated
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• Finally, Schur complement contributions are merged at the tree root node
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• The common interface problem is solved at the tree root node
(The size of the common interface problem corresponds to the size of
crossection of the entire domain)
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• The solution obtained at root node is utilized at son nodes.
• The backward substitution is executed
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• The solution utilized at the second level nodes is utilizes at their sone nodes.
• The backward substitution is executed
ELIMINATIONS BASED ON REFINEMENT TREES
Level of
initial mesh elements
Level of
refinement trees
• The solution obtained at the third level nodes is utilized at leaf nodes.
• The backward substitution is executed on the level of leaf nodes
CLASSICAL MULTIFRONTAL SOLVER ALGORITHM
•
In case of mesh with multiple different adaptation levels, the multifrontal solver
algorithm scans the mesh elements and eliminates degrees of freedom starting
from leaves to the root of the adaptation tree, on each coarse mesh element
independently.
CLASSICAL MULTIFRONTAL SOLVER ALGORITHM
•
Interface size: green nodes
Interface size: blue nodes
CLASSICAL MULTIFRONTAL SOLVER ALGORITHM
•
Interface size: green nodes
Interface size: blue nodes
The matrix size is growing !!!
The matrix size is growing !!!
PROPOSED MODIFICATION TO
MULTIFRONTAL SOLVER ALGORITHM
•
Interface size: green nodes
Constant matrix size !!!
PROPOSED MODIFICATION TO
MULTIFRONTAL SOLVER ALGORITHM
•
Interface size: green nodes
Constant matrix size !!!