Princeton University A Stability/Bifurcation Framework For Process Design C. Theodoropoulos 1 , N. Bozinis 2 , C. Siettos 1 , C.C. Pantelides 2 and I.G. Kevrekidis 1 1 Department of Chemical Engineering, Princeton University, Princeton, NJ 08544 2 Centre for Process System Engineering, Imperial College, London, SW7 2BY, UK
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Princeton University A Stability/Bifurcation Framework For Process Design C. Theodoropoulos 1, N. Bozinis 2, C. Siettos 1, C.C. Pantelides 2 and I.G. Kevrekidis.
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Princeton University
A Stability/Bifurcation FrameworkFor Process Design
C. Theodoropoulos1, N. Bozinis2, C. Siettos1,
C.C. Pantelides2 and I.G. Kevrekidis1
1Department of Chemical Engineering,Princeton University, Princeton, NJ 08544
2 Centre for Process System Engineering, Imperial College, London, SW7 2BY, UK
Princeton University
Motivation• A large number of existing scientific, large-scale legacy codes
–Based on transient (timestepping) schemes. • Enable legacy codes perform tasks such as bifurcation/stability analysis
–Efficiently locate multiple steady states and assess the stability of solution branches.–Identify the parametric window of operating conditions for optimal performance–Locate periodic solutions
• RPM: method of choice to build around existing time-stepping codes.–Identifies the low-dimensional unstable subspace of a few “slow” eigenvalues–Stabilizes (and speeds-up) convergence of time-steppers even onto unstable steady-states.–Efficient bifurcation analysis by computing only the few eigenvalues of the small subspace.
•Even when Jacobians are not explicitly available (!)
parameter
bif.
qua
n tit
y
Princeton University
Recursive Projection Method (RPM)
• Recursively identifies subspace of slow eigenmodes, P
Subspace P of few slow eigenmodes
Subspace Q =I-P
Reconstruct solution:u = p+q = PN(p,q)+QF
Pica
rdite
ratio
ns Newtoniterations
• Treats timstepping routine, as a “black-box”
– Timestepper evaluates un+1= F(un)Initial state un
TimesteppingLegacy Code
Convergence?
Final state uf
F(un)
YES
Picard iteration
NO
• Substitutes pure Picard iteration with
–Newton method in P–Picard iteration in Q = I-P
• Reconstructs solution u from sum of the projectors P and Q onto subspace P and its orthogonal complement Q, respectively:
symbolic differentiation for partial derivatives automatic identification of problem sparsity structural analysis algorithms
• Advanced features: exploitation of sparsity at all levels support for mixed analytical/numerical partial derivatives handling of symmetric/asymmetric discontinuities at all levels
• Component-based architecture for numerical solvers open interface for external solver components hierarchical solver architectures
• mix-and-match
• external solvers can be introduced at any level of the hierarchy
•well-posedness
•DAE index analysis
•consistency of DAE IC’s
•automatic block triangularisation
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FitzHugh-Nagumo: An PDE-based Model
• Reaction-diffusion model in one dimension
• Employed to study issues of pattern formation
in reacting systems – e.g. Beloushov-Zhabotinski
reaction
– u “activator”, v “inhibitor”
– Parameters:
– no-flux boundary conditions
– , time-scale ratio, continuation parameter
• Variation of produces turning points
and Hopf bifurcations
0.2,03.0,0.4δ 10 aa
)(εδ 012
32
avauvv
vuuuu
t
t
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Bifurcation Diagrams
<u>
Around Hopf Around Turning Point
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Eigenspectrum Around Hopf
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Eigenvectors
= 0.02
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Arc-length continuation with gPROMS
),(y
pyfdt
dSystem:
0
]
*);([
y
pyfDet
Solve (1) & (2)
p
y
),( pyf0 (1)
Pseudo – arc length condition
0)()(
)()(
101
101
SppS
ppyy
S
yy T
(2)
continuation(II)
throughFORTRAN
F.P.Icontinuation
(I)within
gPROMS
Princeton University
System Jacobian
R.P.M.through
FORTRAN
F.P.I
Getting systemJacobian
through an FPI
F.P.IContinuation
within gPROMS
x
g
y
g
y
f
x
f
1
Stability matrix
x
pxf
),(
Jacobian of the ODE
DAEs :),,( pyxf
dt
dx
),,( pyxg0)(* xyy ),( px
dt
dxfODEs :
Cannot get “correct”Jacobian from augmented system
Obtain “correct”Jacobian of leading eigenspectrum
Princeton University
Tubular Reactor: A DAE system
Dimensionless equations:
]/1
exp[)1(2
21
121
21
11
x
xxDa
z
x
z
xPe
t
x
wxx
xxBDax
z
x
z
xPe
t
x2
2
212
222
21
22 ]
/1exp[)1(
Boundary Conditions:
0),0(
111
xPez
tzx0
),0(22
2
xPez
tzx
0),1(1
z
tzx0
),1(2
z
tzx
(1)
(4)
(2)
(3)
Eqns (1)-(4): system of DAEs. Can also substitute to obtain system of ODEs.
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Bifurcation/Stability with RPM-gPROMS
0
0.2
0.4
0.6
0.8
1
0.1 0.11 0.12 0.13 0.14
Da
x1
Hopf pt.
•Model solved as DAE system•2 algebraic equations @ each boundary
•101-node FD discretization
•2 unknowns (x1,x2) per node
•State variables: 99 (x 2) unknowns at inner nodes•Perform RPM-gPROMs at 99-space to obtain correct Jacobian
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Eigenspectrum
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1 1.5
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1 1.5
Da=0.110021
Da=0.1217380.00E+00
1.00E-02
2.00E-02
3.00E-02
4.00E-02
5.00E-02
6.00E-02
0 20 40 60 80 100 120
0.00E+00
5.00E-03
1.00E-02
1.50E-02
2.00E-02
2.50E-02
3.00E-02
3.50E-02
0 20 40 60 80 100 120
Re
Im
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+
)(yq)(y
Aq kk
SYSTEM AROUND STEADY STATE
y(k)
)(yy kk 1
C h o o s e 1q w i t h 11 q
F o r j = 1 U n t i l C o n v e r g e n c e D O
( 1 ) C o m p u t e a n d s t o r e jAq
( 2 ) C o m p u t e a n d s t o r e jtqAqh tjjt ,...2,1,,,
( 3 )
j
ttjtjj qhAqr
1,
( 4 ) 2/1
,1 , jjjj rrh
( 5 ) jjjj hrq ,11 /
E n d F o r
ε q
LeadingSpectrum
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Matrix-free ARNOLDI
Large-scale eigenvalue calculations(Arnoldi using system Jacobian):R.B. Lechouq & A.G. Salinger, Int. J. Numer. Meth.(2001)
Conclusions• Can construct a RPM-based computational framework around large-scale
timestepping legacy codes to enable them converge to unstable steady states and efficiently perform bifurcation/stability analysis tasks.
– gPROMS was employed as a really good simulation tool– communication with wrapper routines through F.P.I.
• Both for PDE and DAE-based systems. • Have “brought to light” features of gPROMS for continuation around turning
points and information on the Jacobian and/or stability matrix at steady states of systems.
• Employed matrix-free Arnoldi algorithms to perform stability analysis of steady state solutions without having either the Jacobian or even the equations!
• Used the RPM-based superstructure to speed-up convergence and perform stability analysis of an almost singular periodically-forced system
• Have enabled gPROMS to trace autonomous limit cycles• Newton-Picard computational superstructure for autonomous limit cycles.
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gPROMS
• General purpose commercial package for modelling, optimization and control of process systems.
• Allows the direct mathematical description of distributed unit operations• Operating procedures can be modelled
– Each comprising of a number of steps
• In sequence, in parallel, iteratively or conditionally.• Complex processes: combination of distributed and lumped unit operations
– Systems of integral, partial differential, ordinary differential and algebraic equations (IPDAEs).
– gPROMS solves using method of lines family of numerical methods. • Reduces IPDAES to systems of DAEs.