iMAM,CardiffUniversity O.Goury,P.Kerfriden,S.P.A.Bordas Dealingwithinterfacesinpartitionedmodelorderreductionforapplicationtononlinearproblems

(1)

Introduction Primal Partitioned POD method Dual Partitioned POD Conclusion

Dealing with interfaces in partitioned model

order reduction for application to nonlinear

problems

(2)

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 _{Primal Partitioned POD method}

Domain decomposition methods

System approximation

Results

3 Dual Partitioned POD

What was wrong with the primal method?

Continuity

Issues when used with MOR

4 Conclusion

(3)

Introduction

Primal Partitioned POD method Dual Partitioned POD Conclusion

Why model order reduction? A straigthforward solution?

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

(4)

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

4 Conclusion

(5)

Introduction

Non-linear expensive simulations

Problems depending on microscale phenomena =⇒

requires very fine mesh: expensive simulations

Surgical simulation: real-time brain surgery simulation

(6)

Projection-based model order reduction

We want to solve a parametrised mechanical problem:

F

_int

(

U(λ), λ)

|

{z

}

Non-linear

+

F

_ext

(λ) =

0 (1)

We are interested in the solution

U(λ) for many different values

of λ.

Projection-based model order reduction assumption:

Solutions

U(λ) for different parameters λ are contained in a

space of small dimension span((

C

_i

)

i∈

_J1,n

c

K

)

(7)

Introduction

Proper Orthogonal Decomposition (POD)

Look for

U as U = C α. Where does the basis C comes from?

Solve the full problem a certain number of times varying

the input parameter λ

You obtain a base of solutions (the snapshot):

(

U

₁

,

U

₂

, ...,

U

_n

S

) =

S

Extract the essence of this snapshot space (the basis

C)

by minimising the cost function:

J =

1 n

S

n

S

X

i=1

(k

U

_i

− CC

T

U

_i

k

2 )

(2)

Reduced system: min

α

F

_int

C α

+ F

_ext

In the Galerkin framework:

C

T

F

_int

C α

+ C

T

_F

(8)

Introduction

Proper Orthogonal Decomposition (POD)

Look for

U as U = C α. Where does the basis C comes from?

Solve the full problem a certain number of times varying

the input parameter λ

Extract the essence of this snapshot space (the basis

C)

by minimising the cost function:

J =

1 n

S

n

S

X

i=1

(k

U

_i

− CC

T

U

_i

k

2 )

(2)

Reduced system: min

α

F

_int

C α

+ F

_ext

In the Galerkin framework:

C

T

F

_int

C α

+ C

T

_F

ext

=

0

(9)

Introduction

Proper Orthogonal Decomposition (POD)

Look for

U as U = C α. Where does the basis C comes from?

Solve the full problem a certain number of times varying

the input parameter λ

You obtain a base of solutions (the snapshot):

(

U

₁

,

U

₂

, ...,

U

_n

S

) =

S

Extract the essence of this snapshot space (the basis

C)

by minimising the cost function:

J =

1 n

S

n

S

X

i=1

(k

U

_i

− CC

T

U

_i

k

2 )

(2)

Reduced system: min

α

F

_int

C α

+ F

_ext

In the Galerkin framework:

C

T

F

_int

C α

+ C

T

_F

(10)

Introduction

Proper Orthogonal Decomposition (POD)

Look for

U as U = C α. Where does the basis C comes from?

Solve the full problem a certain number of times varying

the input parameter λ

You obtain a base of solutions (the snapshot):

(

U

₁

,

U

₂

, ...,

U

_n

S

) =

S

Extract the essence of this snapshot space (the basis

C)

by minimising the cost function:

J =

1 n

S

n

S

X

i=1

(k

U

_i

− CC

T

U

_i

k

2 )

(2)

In the Galerkin framework:

C

F

_int

C α

+ C

F

_ext

=

0

(11)

Introduction

Proper Orthogonal Decomposition (POD)

Look for

U as U = C α. Where does the basis C comes from?

Solve the full problem a certain number of times varying

the input parameter λ

You obtain a base of solutions (the snapshot):

(

U

₁

,

U

₂

, ...,

U

_n

S

) =

S

Extract the essence of this snapshot space (the basis

C)

by minimising the cost function:

J =

1 n

S

n

S

X

i=1

(k

U

_i

− CC

T

U

_i

k

2 )

(2)

Reduced system: min

α

F

_int

C α

+ F

_ext

In the Galerkin framework:

C

T

F

_int

C α

+ C

T

_F

(12)

Proper Orthogonal Decomposition (POD)

Look for

U as U = C α. Where does the basis C comes from?

Solve the full problem a certain number of times varying

the input parameter λ

You obtain a base of solutions (the snapshot):

(

U

₁

,

U

₂

, ...,

U

_n

S

) =

S

Extract the essence of this snapshot space (the basis

C)

by minimising the cost function:

J =

1 n

S

n

S

X

i=1

(k

U

_i

− CC

T

U

_i

k

2 )

(2)

Reduced system: min

α

F

_int

C α

+ F

_ext

In the Galerkin framework:

C

T

F

_int

C α

+ C

T

_F

ext

=

0

(13)

Introduction

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

(14)

Example

(15)

Introduction

Snapshots

Load orientation of 15 degrees:

(16)

First 3 modes of the POD basis

Mode 1:

Mode 2:

Mode 3:

(17)

Introduction

Fracture not well captured

= Construction of reduced order model

POD

Solution at arbitrary angle using the reduced model

Compute particular realisations (snapshots) Reduced basis + + approximated by

(18)

What can we do?

Idea: juste divide up the domain and select regions that are

“reducible”

(19)

Introduction

Primal Partitioned POD method

Dual Partitioned POD Conclusion

Domain decomposition methods System approximation Results

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

(20)

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

4 Conclusion

(21)

Introduction

Partitioned reduced basis

≈

Construction of partitioned reduced order model

approximated by α1· α2· Partitioned POD β1·

+

=

+

=

β3· 2· β

Solution for arbitrary parameter using reduced model

Locally non correlated: Compute particular realisations

(22)

Is that good enough?

Speed-up actually poor

Equation “

C

T

F

_int

C α

+ C

T

_F

ext

=

0“ quicker to solve but

C

T

F

_int

C α

still expensive to evaluate

Need to do something more =⇒ system approximation

(23)

Introduction

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

(24)

Idea

Integrate only over some nodes of the domain

Reconstruct the operators using a second POD basis

(25)

Introduction

“Gappy“ technique

Originally used to reconstruct altered signals

F

_int

C α

approximated by

_F

^

int

C α

= D β

F

_int

C α

is evaluated exactly only on a few selected

nodes:

_F

\

int

C α

β

found through: min

(26)

Introduction

“Gappy“ technique

Originally used to reconstruct altered signals

F

_int

C α

approximated by

_F

^

int

C α

= D β

F

_int

C α

is evaluated exactly only on a few selected

nodes:

_F

\

int

C α

(27)

Introduction

“Gappy“ technique

Originally used to reconstruct altered signals

F

_int

C α

approximated by

_F

^

int

C α

= D β

F

_int

C α

is evaluated exactly only on a few selected

nodes:

_F

\

int

C α

β

found through: min

(28)

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

4 Conclusion

(29)

Introduction

(30)

0

50

100

150

200

250

10

−4

10

−3

10

−2

10

−1

runtime

re

la

ti

v

e

er

ro

r

Partitioned POD + SA

Partitioned POD

Full Scale Inexact

(31)

Introduction

(32)

0

100

200

300

400

500

600

700

10

−4

10

−3

10

−2

10

−1

runtime (seconds)

re

la

tiv

e

er

ro

r

Partitioned POD + SA

Partitioned POD

Full Scale Inexact

(33)

Introduction

Domain decomposition methods System approximation Results 1 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5

Ratio of total number of elements over number of controlled elements

S

p

ee

d

u

p

(34)

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

4 Conclusion

(35)

Introduction Primal Partitioned POD method

Dual Partitioned POD

Conclusion

What was wrong with the primal method? Continuity

Issues when used with MOR

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

(36)

Continuity

The displacement at the interface could not be reduced

which restrained the potential speedup:

(37)

Conclusion

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

(38)

Continuity

The continuity is now enforced with the use of Lagrange

multipliers

We can now include the interface degrees of freedom in

the reduction

(39)

Conclusion

(40)

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

4 Conclusion

(41)

Conclusion

Continuity at the interface?

The continuity is forced at each node of the interface even

though each subdomain only has few degrees of freedom

Leads to a badly defined problem: too many continuity

conditions

Lagrange

Multipliers

(42)

Need to make sure the basis is good!

Possible discontinuities:

(43)

Introduction Primal Partitioned POD method Dual Partitioned POD

Conclusion

Outline

1 _Introduction

Why model order reduction?

A straigthforward solution?

2 Primal Partitioned POD method

Domain decomposition methods

System approximation

Results

3 _{Dual Partitioned POD}

What was wrong with the primal method?

Continuity

Issues when used with MOR

(44)

Conclusion

The primal partitioned POD can handle localised high

non-linearities and lead to a good speed-up while keeping

a reasonable accuracy.

The dual partitioned POD has a higher potential since it

can extend the domain of reducibility to the interfaces

between subdomains.

(45)

Introduction Primal Partitioned POD method Dual Partitioned POD

Conclusion