Space-time goal-oriented reduced basis approximation for linear wave equation

(1)

& Advanced Materials

Space-time goal-oriented reduced basis

approximation for linear wave equation

Khac Chi Hoang*¹,

Pierre Kerfriden¹, Stéphane Bordas^{1 2}

1 Institute of Mechanics and Advanced Materials, School of Engineering, Cardiff University, UK

2 Faculté des Sciences, de la Technologie et de la Communication, University of Luxembourg, LU

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1. Motivation

 Linear Wave Equation

 Finite Element Approximation

 Reduced Basis Approximation

 (Standard) POD-Greedy Sampling Strategy

2. Proposed Algorithm

 Goal-oriented (GO) POD-Greedy Sampling Strategy

3. Numerical Example

 3D Dental Implant Model Problem

4. Conclusion

Outline

(3)

1. Motivation

2. Proposed Algorithm

3. Numerical Example

4. Conclusion

Outline

(4)

• Consider a spatial domain , Lipschitz continuous boundary

• Dynamic linear elasticity equilibrium (assume Rayleigh damping):

• Constitutive relations (homogeneous, isotropic materials)

• Initial conditions: and

• Boundary conditions: on ; surface traction on

Problem statement: strong form

  

d



2

( )

e e e

ij i i

i j

u u

x b  t



 t

 

  

 

   

( ) ( )

e e

e k k

ij ijkl

l l

u u

C x t x

 



         



          

( , 0) 0

ie

u x  ( , 0) 0

ei

u x

t

 



e

0 u 

i



_D



_{ij j}^e

n ˆ  t

_i



_N

(5)

• In space setting:

• Or, equivalently:

• Space-time quantity of interest:

Problem statement: weak form

N

2

( ) ( ) ( )

( ) , ,

e e e

i i i k

i i ijkl

j l

i ke e

ijkl i i i i

j l

u u v u

v v C

t t x x

t

v u

C b v v v Y

x x





 



  



  

  

    

 

   



 

     

 

  

   ^t ^

2

, , ; ( , ; ) ( ), ,

e e

u u

m v c v a u v f v v Y

t t

  

       

   

         

     

  

     

0 o 0

( )

^T

( , ; ) ( , )

^T

( ( , ; ))

e e e

s



u x t



x t dxdt u x t



dt

  



   ^

 ^u

^e

^{( )}  ^s

^e

^{( )?}

    

(6)

• “Method of Lines”:

spatial discretize (FE) + temporal discretize (Newmark)

 Discretize the time span into

 Solve following elliptic systems

 FE quantity of interest:

Finite element discretization

[0, ] T [ , t t

^k ^k^¹

] , 0  k  K  1

( K  1)

 ^u

^k^¹

^{( )} ^ ^{, ;} ^v

^

 ^ ^{( ),} ^v ^{ } ^v ^Y ^,

^

^ ^{, 1} ^ ^k ^ ^K ^ ¹

  



^u^k ¹^{( ), ;}^v



¹₂ ^{m u}⁽ ^k ¹^{( ), ; )}^v ₂¹_t^{c u}⁽ ^k ¹^{( ), ; )}^v ₄¹^{a u}⁽ ^k ¹^{( ), ; )}^v

  t      

      

 



1 1 1

2 2

1 1 1

( ) ( ( ), ; ) ( ( ), ; ) ( ( ), ; )

2 4

2 1

( ( ), ; ) ( ( ), ; ) ( ) ( ; ) 2

k k k

k k eq k

v m u v c u v a u v

t t

m u v a u v g t f v

t

     

    

  

   

 

  





0 ( , )0

( , ) 0, u t 0

u t

t

  

 



1 1

0

( ) ( ( , ; ))

k

K t

k t

s



u x t



dt

 



   ^

Trapezoidal rule

 ^u ^{( )}  ^s ^{( )?}

    

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• Introduce ; and

nested Lagrangian RO spaces

 Galerkin projection:

 Solve the following elliptic systems

 RO quantity of interest:

Reduced-order approximation: Galerkin projection

span{ ,1 },1

max

N n

Y    n  N  N  N

*

{

1

,

2

, ,

_N

},1

max

S             N  N

1 1

0

( ) ( ( , ; ))

k

K t

N t N

k

s



u x t



dt

 



   ^

 u

N^k^¹

^{( ), ;}  v



  ^{( ),} v   v Y

N

^,



 D ^{, 1}  k  K  ¹

 

1

( , ) ( , ) , , 1

N

k k

N N n n n N

n

u  t u  t   Y k K



     

 ^u

_N

^{( )}  ^s

_N

^{( )?}

    

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• Dual Weighted Residual (DWR) method

 Solve additionally a dual/adjoint problem

 Remove the snapshots cause small error / Keep the ones cause large error

• Build optimal goal-oriented basis functions based on all POD snapshots

 Use adjoint technique to build optimally basis functions based on all POD snapshots

• We want to build optimally goal-oriented basis functions without

computing/storing all the snapshots?

Approaches to build goal-oriented basis functions?

[Meyer et al. 2003] [Grepl et al. 2005]

[Bangerth et al. 2001] [Bangerth et al. 2010]

[Bui et al. 2007] [Willcox et al. 2005]

RB + Greedy sampling strategy

[Rozza, Huynh, Patera 2008]

(9)

1) Where: given a set of snapshots the POD space is defined as:

or, written as:

2) Projection error:

3) Residual

Standard POD-Greedy algorithm

(a) Set (b) Set (c) While (d)

(e) (f) (g) (h)

(i) end.

st 0

YN 

*st 0

   N  Nmax

 

st st st

proj( * , ),0^k

e  t k K

  



st st POD( st, )

N M N

Y _ Y



 M

N  N  M

 

train

*st arg max _u( )

  

  

 

st st st

* * *

S  S





(j)

st 2

1 st 2

1

( ; , ) ( )

( , )

K

k k Y

u K

N k Y

k

v t

u t



 



 





WM



1 _max



POD { , , },

M M

W     M

max1

{ }_{k k}^M_

max

1 max

2

span{ , , } max 1 1

arg min 1 inf

k M

M M

M k m mk

V k m

W v

M

    

    

 

 

 



_ 





proj( ) ( ) proj ( )

N

k k k

e   u   Y u 



¹



( ; , )v  t^k  ( )v  u_N^k^ ( ), ; v 

  

1  k  K 1

[Haasdonk et al. 2008] [Hoang et al. 2013]

(10)

1. Motivation

2. Proposed Algorithm

3. Numerical Example

4. Conclusion

Outline

(11)

Goal-oriented POD-Greedy algorithm

(e) (f) (g) (h)

(i) end.

st 0

YN 

*st 0

   N  Nmax

 

st st st

proj( * , ),0^k

e  t k K

  



st st POD( st, )

N M N

Y _ Y



 M

N  N  M

 

train

*st arg max _u( )

  

  

 

st st st

* * *

S  S





(j)

st 2

1 st 2

1

( ; , ) ( )

( , )

K

k k Y

u K

N k Y

k

v t

u t



 



 





(e) (f) (g) (h)

(i) end.

go 0

YN 

go

0

*   N  Nmax



^go



go go

proj( * , ),0^k

e  t k K

  



go go POD( go, )

N M N

Y _ Y



 M

N  N  M

 

train

go

* arg max _s( )

  

  

 

go go go

* * *

S  S





(j)

st s

go 2

2t

( )

( ) )

) ( (

N N s N

s s

s

 

  

NEW

(12)

1. Motivation

2. Proposed Algorithm

3. Numerical Example

4. Conclusion

Outline

(13)

3D Dental implant model problem

Example; in practice, use Dirac Delta loading with

Duhamel’s principle.

(14)

• Material properties

• Explicit bi/linear forms:

3D Dental implant model problem (cont.)

5

1

( , )

r r i i r

m w v  w v

 



 

3

5

1 3 1, 3

( , ; )

r

i r k i k

ijkl ijkl

r r j l j l

v w v w

a w v C C

x x x x

 

 

 

   

 

   

  

3

5

2 1 3 1, 3

( , ; )

r

i r k i k

r ijkl ijkl

r r j l j l

v w v w

c w v C C

x x x x

   

 

 

   

 

   

  

l

( )

f v v





 o o

( ) 1

| |

v v

 







(15)

• FE dof:

• Time integration: , ,

• Parameter domain:

• Training sample set:

Numerical results

26343





1 1 0

3

s

T  

^

   t 2 10 s

^⁶

T 500

K  t 



6 6 6 5 2

[1 10 ,25 10 ]Pa [5 10 , 5 10 ]

^ ^ ^P^

       



train

900 n 

(16)

• Offline stages of Std vs. GO POD-Greedy algorithms

Numerical results (cont.)

(17)

True error comparison (validation)

1

st

st 0

1

0

( , ) ( , ) ( )

( , )

k k

K t

N Y

k t

u K t

t Y k

u t u t dt

e

u t dt

 

















 

1

1 go

go 0

1

0

( , ) ( , ) ( )

( , )

k k

K t

N Y

k t

u K t

t Y k

u t u t dt

e

u t dt

 

















 

st ( ) st( )

( ) ( )

s N

s s

e s

 

 

go

go ( ) ( )

( ) ( )

s N

s s

e s

 

 

(18)

GO algorithm: error estimate

st go

off 2 ( ) ( )

ˆ ( )

( )

N N

s

s s

e s

 

 

go go

onl 2 ( ) ( )

ˆ ( )

( )

N N

s

s s

e s

 

 

st go

off 2

go

( ) ( )

( ) ( ) ( )

N N

s

N

s s

 

 

 

 



go go

onl 2

go

( ) ( )

( ) ( ) ( )

N N

s

N

s s

 

 

 

 



(19)

• Computational time:

• All calculations were performed on a desktop Intel(R) Core(TM) i7-3930K CPU

@3.20GHz 3.20GHz, RAM 32GB , 64-bit Operating System.

• The work focuses on real-time context with many online computations, the offline stage is done once and expensive: the total computational time is approximately 2 weeks (including all FEM solutions/outputs and RB true errors of the standard and goal-oriented algorithms) on this computer.

Computational time

(20)

1. Motivation

2. Proposed Algorithm

3. Numerical Example

4. Conclusion

Outline

(21)

• A new algorithm is proposed based on a novel idea: build optimally

goal-oriented basis functions without computing/storing all the

snapshots.

• The proposed algorithm still needs the standard POD-Greedy algorithm

as it provides an error estimate for the quantity of interest.

• Simple, easy to implement; same computational cost.

• Applicable to any regular output functional; it cooperates and improve

further the standard algorithm.

Conclusions

(22)

Thank you for your attention!

Khac Chi Hoang, Pierre Kerfriden, Stephane PA Bordas, Space-time

goal-oriented reduced basis approximation for linear wave equation,

arXiv:1305.3528

Questions?

Space-time goal-oriented reduced basis approximation for linear wave equation