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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
•Autonomous, forced (PSA,RFR)–Appropriate controller design.
• 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.
quan
tity
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:
–u = PN(p,q) + QF
F.P.I.
Princeton University
gPROMS:A General Purpose Package
gPROMS
gPROMS Model
Steady-state &
Dynamic Simulatio
n
Steady-state &
Dynamic Optimisatio
n
Parameter Estimatio
n Data Reconciliation
Nonlinear algebraic equation solvers
Differential algebraic equation solvers
Dynamic optimisation
solvers
Maximum likelihood estimation
solvers
Nonlinear programming
solvers
Princeton University
Mathematical solution methods in gPROMS• Combined symbolic, structural & numerical techniques
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
Princeton University
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
Princeton University
Bifurcation Diagrams
<u>
Around Hopf Around Turning Point
Princeton University
Eigenspectrum Around Hopf
Princeton University
Eigenvectors= 0.02
Princeton University
Arc-length continuation with gPROMS
),(y pyfdtd
System:
0
]*);([y
pyfDet
Solve (1) & (2)
p
y
),( pyf0 (1)Pseudo – arc length condition
0)()()()(1
011
01
Spp
Sppyy
Syy 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
xg
yg
yf
xf
1
Stability matrix
xpxf
),(
Jacobian of the ODE
DAEs :),,( pyxf
dtdx
),,( pyxg0)(* xyy ),( px
dtdx fODEs :
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
xxxDa
zx
zxPe
tx
wxxxxBDax
zx
zxPe
tx
22
212
222
21
22 ]
/1exp[)1(
Boundary Conditions:
0),0(11
1 xPez
tzx 0),0(22
2 xPez
tzx
0),1(1 z
tzx 0),1(2
z
tzx
(1)
(4)
(2)
(3)
Eqns (1)-(4): system of DAEs. Can also substitute to obtain system of ODEs.
Princeton University
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
Princeton University
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
Princeton University
+
)(yq)(yAq 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)
Stability Analysis without the Equations
Princeton University
•Isothermal operation•Modeling Equations (Nilchan & Pantelides)
Step 1 : Pressurisation
Step 2: Depressurisation
Rapid Pressure Swing Adsorption1-Bed 2-Step Periodic Adsorption Process
t=0 to T/2
Ci(z=0)=PfYf/(RTf)
P(z=0)=Pf
z=0
z=L
t= T/2 to T
P(z=0)=Pw
0)0( z
zCi
)(
)1(180
)(
3
2
2
1
2
2
iiiii
b
b
p
n
ii
ii
iib
it
qpmktq
dv
zP
CRTP
zCD
zvC
tq
tC
Mass balance in ads. bed
Darcy’s law
Rate of ads.
Princeton University
Rapid Pressure Swing Adsorption1-Bed 2-Step Periodic Adsorption Process
Production of oxygen enriched air
Zeolite 5A adsorbent (300m)
Bed 1m long, 5cm diameter
Short cycle
–1.5s pressurisation, 1.5s depressurisation
– T= 3s
Low feed pressure (Pf = 3 bar)
Periodic steady-state operation
–reached after several thousand cycles
q ,c (t=0) q , c (t=T/2)
q , c (t=T)
Must obtain:q , c (t=T) = q , c (t=0)
Princeton University
Typical RPSA simulation results(Nilchan and Pantelides, Adsorption, 4, 113-147, 1998)
20
25
30
35
40
45
50
0 50 100 150 200
c1(z=0.5) (mol/m3)
Time (s)
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Princeton University
PRM-gPROMS Spatial Profiles (t=T)
q1 mol/kg
0
0.1
0.2
0.3
0 0.2 0.4 0.6 0.8 1x
q2 mol/kg
0
0.1
0.2
0.3
0 0.2 0.4 0.6 0.8 1x
c1 mol/m3
0
10
20
30
0 0.2 0.4 0.6 0.8 1x
c2 mol/m3
0
30
60
90
0 0.2 0.4 0.6 0.8 1x
z z
z z
Princeton University
Leading Eigenvectors, =0.99484
c1
c2
q1
q2
0
0.04
0.08
0.12
0.16
0
0.0004
0.0008
0.0012
-0.15
-0.1
-0.05
09 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0
-6.00E-04
-4.00E-04
-2.00E-04
0.00E+00
c1
c2 q2
q1
Princeton University
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.
Princeton University
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.– Time-stepping or pseudo-timestepping.
• Jacobians NOT explicitly available. – Cannot perform systematic bifurcation/stability analysis studies.
Princeton University
Tracing Limit Cycles
continuation(I)
within gPROMS
continuation(II)
throughFORTRAN
F.P.I
R.P.Mthrough
FORTRAN
F.P.I
Getting systemJacobian
through an FPI
F.P.I
Tracing limit cycles
tracinglimit cycles
within gPROMS
Princeton University
Tracing Limit CyclesTracing limit cycles
),(y pyfdtd
SYSTEM:
Periodic Solutions: y(t+T)=y(t)
Period T not known beforehand
0),(
ypyf
dtdTdtd
0)()0y( Ty
( (0), ) 0G y p
( (0), ) (0) 0iG y p y a (0)( (0), ) 0idyG y p
dt