PGD in computational mechanics
F. Chinesta, A. Leygue
EADS Chair, Ecole Centrale of Nantes
Reduced Bases Workshop
The past
The dream
Computational mechanics
1 – Models defined in high dimensional spaces;
2 – Separating the physical space;
3 – Parametric models: optimization, inverse methods and simulation in real time;
4 – Miscellaneous.
PGD & …
1 - Multidimensional models
Ψ x,t,q1,L ,qN( )
1
11
, , ,..,, , ,..,
( )( )
N
jj
NN
j
tt
t
d
d
Ψ Ψ x q q
x q qq
q
(3 1 3 )N D
Ψ q1,L ,qN( ) ≈ Qi
1 q1( )×L ×i=1
i=M
∑ QiN qN( )
JNNFM, 2006
1
* * *
( , ) ( ) ( ) ( ) ( )
( , ) ( ) ( ) ( ) ( )
n
j jj
u x y X x Y y x yR
u x y R x S y R x y
S
S
*( ) ( ) 0x y
u u u dx dy
M A M A
Successive enrichment:
R S
R⋅S* M (u(R,S,X j ,Yj )) −A( )x×y
dx dy=0•
x
⏐ →⏐ ⏐ S
S R
R* ⋅S M (u(R,S,X j ,Yj )) −A( )x×y
dx dy=0•
y
⏐ →⏐ ⏐ R
Variants:
- Newton linearization;
- Residual minimization in the case of non-symmetric operators;
- Other more efficient and optimal constructors are in progress.
Transient: INNFM 2007
Ψ t,q1,L ,qN( ) ≈ Ti t( )×Qi
1 q1( )×L ×i=1
i=M
∑ QiN qN( )
Non linear models: AR 2007
M Ψn−1( )Ψn =A
Extension of the radial approximation, Ladeveze 1985
Non linear models: CMS 2010
Newton and advanced fixed point M Ψn(s)( )Ψn(s+1) =A
BCF descriptions: MSMSE 2007
Quantum chemistry & Master Equation: IJMCE 2008, EJCM 2010
Complex flows:
Ψ t,x,q1,L ,qN( ) ≈ Ti t( )×Xi x( )×Qi
1 q1( )×L ×i=1
i=M
∑ QiN qN( )
ACME 2009, MCS 2010
2 - Separating the space …
Ψ x, y,z( ) ≈ Xi x( )×Yi y( )×Zi z( )
i=1
i=M
∑
CMAME 2008
Boundary conditions and separating non hexahedral domains
IJNME 2010
u(x, y, z,t)≈ F j ,1(x, y) ⋅F j ,2(z)
j=1
n
∑ ⋅F j ,3(t)
3D modeling in plate domains
CMAME, In press
3D solution with 2D complexity
1
( , , ) ( , ) ( )n
j jj i i
i
u x y z X x y Z z
Shells
xx
yy
zz
FEM solution:
PGD solution
610 dof
500 times steps, implicit
14s using matlab
∂T∂t
−k∂2T∂x2
=S
T (t, x,k)
2
2
TS
tk
x
T
,( )T t x
3 - Parametric modeling
3D
2D
allows the solution of a single problem instead the many solutions required by
usual strategies in:
Optimization & Inverse identification
( , , )T t x k
( , , )dT
t x kdk
is explicitly available
accounting for variability
( , , )g gT t x kk
Parametric solution computed off-line with all the sources of variability considered as extra-coordinates
On-line processing of such solution in real time and using light computing platforms: e.g. smartphones, …
Parametric modeling: MCS 2010, ACME 2010, JNNFM 2010
Process optimization: Composites Part A, 2011
Inverse identification: JNNFM 2011
Control & DDDAS: CMAME, MCS - Submitted
Accounting for variability: CMAME In press
Using smarthphones: CMAME In press, CMAME Submitted
u(x,t, p
1,L , p
Q) ≈ Xi (x) ⋅T(t)
i=1
n
∑ ⋅Pi1 p1( )L Pi
Q pQ( )
Shape optimization: ACME 2010
4 – Miscellaneous Error estimator: CMAME 2010
Coupling PGD & FEM: IJMCE 2011
Coupled models & multi-scale: IJNME 2010, IJMF 2010
Time multi-scale & parallel time integration: IJNME 2011, In press
u =u(τ )
u =u(τ ,t)
PGD based transient boundary element discretizations: EABE 2011
Convective stabilization:- Separating and then stabilizing- Stabilizing and then separating
Computational Mechanics, Submitted