Frontal and multi-frontal solvers: Generalization to isogeometric finite element method

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Maciej Paszynski

Department of Computer Science

AGH University of Science and Technology, Krakow, Poland maciej.paszynski@agh.edu.pl

http://home.agh.edu.pl/paszynsk

http://www.ki.agh.edu.pl/en/staff/paszynski-maciej http://www.ki.agh.edu.pl/en/research-groups/a2s

Main collaborators Victor Calo (KAUST)

Leszek Demkowicz (ICES, UT) David Pardo (IKERBASQUE)

Frontal and multi-frontal solvers:

Generalization to isogeometric finite

element method

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INTRODUCTION

B-SPLINES BASED FINITE ELEMENT METHOD

Strong formulation

 d

dx A x   ^{d u}

d x







  B x   ^{d u}

d x  C x   ^u ^ ^{f x}  

u 0

 

^ ⁰

A 1

 

^{d u 1}

 

d x  u 1

 

^ ^

Weak formulation

Find u V 



u H ¹

 

0,1 ^{: u 0}

 

^ ⁰



^s.t.

b v,u

 

^ ^{l v}

 

^, ^v ^^V ^



^v ^H ¹

 

^0,1 ^{: v 0}

 

^ ⁰



b v,u

 

^ ^{A x}

 

^{d v}

d x d u

d x B x

 

^{v x}

 

^{d u}

d x C x

 

^{v x}

 

^{u x}

 



 

d x

0



1 ^ ^ ^{v 1}

 

^{u 1}

 

l v   ^ ^ ^{v 1}  

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Find u  V   u H

¹

  0,1 ^{: u 0}   ^ ⁰  ^s.t.

b v,u   ^ ^{l v}   ^, ^v ^ ^V ^  ^v ^H

¹

  ^0,1 ^{: v 0}   ^ ⁰ 

INTRODUCTION

B-SPLINES BASED FINITE ELEMENT METHOD

Using B-splines as basis functions

u

 

x ^ ^Ni,p

 

x

i



^di v

 

x ^ ^N^{j ,p}

 

^x

b N



_{j ,p}

 

x ^,Ni,p

 

x



^aⁱ ^ ^{l N}



^{j ,p}

^{ }

^x



i



^, ^j

contribution of b(N2,1;N3,1) Linear B-splines

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Find u  V   u H

¹

  0,1 ^{: u 0}   ^ ⁰  ^s.t.

b v,u   ^ ^{l v}   ^, ^v ^ ^V ^  ^v ^H

¹

  ^0,1 ^{: v 0}   ^ ⁰ 

INTRODUCTION

B-SPLINES BASED FINITE ELEMENT METHOD

Using B-splines as basis functions

u

 

x ^ ^Ni,p

 

x

i



^dⁱ ^v

 

^x ^ ^N^{j ,p}

 

^x

b N



_{j ,p}

 

x ^,Ni,p

 

x



^aⁱ ^ ^{l N}



^{j ,p}

^{ }

^x



i



^, ^j

contribution of b(N2,2;N3,2) Quadratic B-splines

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GRAPH GRAMMAR PRODUCTIONS AS ATOMIC TASKS

We assign indices to grammar productions in order to localize the places where the graph grammar productions were fired

The elimination tree obtained by executing the following sequence of productions (P1)-(P2)₁-(P2)₂-(P2)₃-(P2)₄-(P3)₁-(P3)₂-(P3)₃-(P3)₄-(P3)₅-(P3)₆

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SCHEDULER BASED ON GRAPH COLORING

Dependency relation for construction of the elimination tree (P1)D{(P2)₁,(P2)₂}

(P2)₁D{(P2)₃,(P2)₄} (P2)₃D{(P3)₁,(P3)₂} (P2)₄D{(P3)₃,(P3)₄} (P2)₂D{(P3)₅,(P3)₆} Alphabet:

A = {(P1) , (P2)₁, (P2)₂, (P2)₃ , (P2)₄ , (P3)₁ , (P3)₂ , (P3)₃ , (P3)₄ , (P3)₅ , (P3)₆}

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SCHEDULER BASED ON GRAPH COLORING

Dependency graph

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SCHEDULER BASED ON GRAPH COLORING

Dependency graph

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TRACE THEORY BASED SCHEDULER

(P1)-(P2)₁-(P2)₂-(P2)₃-(P2)₄- (P3)₁-(P3)₂-(P3)₃-(P3)₄-(P3)₅-(P3)₆

[(P1)][(P2)₁(P2)₂][(P2)₃(P2)₄(P3)₅(P3)₆][(P3)₁(P3)₂(P3)₃(P3)₄] Scheduling according to Foata Normal Form:

Thus, the execution of the solver consists of several steps, where independent tasks are executed in concurrent, interchanged with the synchronization barriers.

    

 



_k

 

_k



_i^k ^k_j

k j k i k

k i

n l n n l

l

Da a

l j

l i

k

Ia a l

j i k

A a

a a

a a a

a

n

1 1

2 1 2

2 2 2 1 1 1

2 1 1

,..., 1 ,...,

1 ,..., 1 ,

...

1 1

 















i<>j where I=AxA\D

Foata Normal Form

(alphabet)

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GRAMMAR BASED NUMERICAL INTEGRATION

using Gaussian quadrature the integration over the domain can be substituted by a weighted summation over Gauss points

b N



_{j ,p}

 

x ^{, N}i ,p

 

x



^

 A x

 

^{d N}^{j ,p}

 

^x

d x

d N_{i ,p}

 

x

d x B x

 

^Nj ,p

 

x ^{d N}^{i ,p}

 

^x

d x C x

 

^Nj ,p

 

x ^Ni ,p

 

x











d x

0



1

  N_{j ,p}

 

1 ^Ni ,p

 

1

l N 

_i,p

  x  ^ ^ ^N

^i,p

^{ } ¹

A x

 

^{d N}^i,p

 

^x

d x

d N_{j ,p}

 

x

d x B x

 

^{d N}^i,p

 

^x

d x N_{j ,p}

 

x ^^{C x}

 

^Ni,p

 

x ^Nj ,p

 

x











d x

0



1 ^

w_l A x

 

_l ^{d N}^i,p

 

^x^l

d x

d N_{j ,p}

 

x_l

d x B x

 

_l ^{d N}^i,p

 

^x^l

d x N_{j ,p}

 

x_l ^^{C x}

 

l ^Ni,p

 

x_l ^Nj ,p

 

x_l













l



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GRAMMAR BASED NUMERICAL INTEGRATION

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GRAMMAR BASED NUMERICAL INTEGRATION

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PROCESS OF THE ELIMINATION

EXPRESSED BY GRAPH GRAMMAR PRODUCTIONS

Generation of frontal matrices at leaves of the eliminaton tree expressed as the execution of graph grammar productions (A1)-(A)⁴-(AN)

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Graph grammar productions generating local frontal matrices for left boundary, interior and right boundary nodes for linear B-splines

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Graph grammar productions merging element frontal matrices at parent level

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Graph grammar production eliminating fully assembled row at parent level

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Graph grammar production for solution at root level

Graph grammar production for merging element frontal matrices at root level

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Graph grammar production for recursive backward substitution

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Expression of the solver execution by graph grammar productions

(A1)-(A)⁴-(AN) (generation of frontal matrices at leaves of the elimination trees) (A2)³ (merging contributions at father nodes)

(E2)³ (elimination of fully assembled nodes)

(A2) – (E2) (merging at parent node followed by elimination)

(Aroot) – (Eroot) (merging at root node followed by full forward elimination) (BS)⁴ (backward substitutions)

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SCHEDULER BASED ON GRAPH COLORING

Dependency relation for the solver algorithm {(A1),(A)₁}D(A2)₁

{(A)₂,(A)₃}D(A2)₂ {(A)₄,(AN)}D(A2)₃ (A2)₁D(E2)₁

(A2)₂D(E2)₂ (A2)₃D(E2)₃

{(E2)₁,(E2)₂}D(A2)₄ (A2)₄D(E2)₄

{(E2)₃(E2)₄}D(Aroot) (Aroot)D(Eroot)

(Eroot)D{(BS)₁,(BS)₂ (BS)₁D{(BS)₃,(BS)₄}

Alphabet:

A={(A1), (A)₁, (A)₂ , (A)₃, (A)₄ , (AN), (A2)₁, (A2)₂ , (A2)₃ , (E2)₁ , (E2)₂ , (E2)₃ , (A2)₄ , (E2)₄ , (Aroot) , (Eroot) , (BS)₁ , (BS)₂ , (BS)₃, (BS)₄}

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SCHEDULER BASED ON GRAPH COLORING

Dependency graph

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SCHEDULER BASED ON GRAPH COLORING

Dependency graph

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TRACE THEORY BASED SCHEDULER

Scheduling according to Foata Normal Form:

(A1)-(A)₁-(A)₂-(A)₃-(A)₄- (AN)-(A2)₁-(A2)₂- (A2)₃-(E2)₁-(E2)₂-(E2)₃- (A2)₄- (E2)₄- (Aroot)-(Eroot)-(BS)₁-(BS)₂-(BS)₃-(BS)₄

[(A1)(A)₁(A)₂(A)₃(A)₄(AN)][(A2)₁(A2)₂(A2)₃][(E2)₁(E2)₂(E2)₃] [(A2)₄][(E2)₄] [(Eroot)][(Aroot)][(Eroot)][(BS)₁(BS)₂][(BS)₃(BS)₄]

Thus, the execution of the solver consists of several steps, where independent tasks are executed in concurrent, interchanged with the synchronization barriers.

    

 



_k

 

_k



_i^k ^k_j

k j k i k

k i

n l n n l

l

Da a

l j

l i

k

Ia a l

j i k

A a

a a

a a a

a

n

1 1

2 1 2

2 2 2 1 1

1 2 1 1

,..., 1 ,...,

1 ,..., 1 ,

...

1 1

 















Foata Normal Form

(alphabet)

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GRAPH GRAMMAR PRODUCTIONS EXPRESSING THE SOLVER ALGORITHM

Linear B-splines

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GRAPH GRAMMAR PRODUCTIONS EXPRESSING THE SOLVER ALGORITHM

Quadratic B-splines

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NUMERICAL EXPERIMENTS

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1D NUMERICAL RESULTS LINEAR B-SPLINES

NVidia Tesla c2070, 6GB memory, 448 CUDA cores, each one with 1.15GHz clock

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1D NUMERICAL RESULTS QUADRATIC B-SPLINES

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1D NUMERICAL RESULTS CUBIC B-SPLINES

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1D NUMERICAL RESULTS QINTIC B-SPLINES

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COMPARISON WITH CPU MUMPS SOLVER

NVidia Tesla c2070, 6GB memory, 448 CUDA cores, each one with 1.15GHz clock Intel(R) Core(TM)2 Quad CPU Q9400 with 2.66GHz clock, 8GB of memory

Linear B-splines Quadratic B-splines

Cubic B-splines Qintic B-splines

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