gproms_pres

8/7/2019 gproms_pres

1/24

Princeton University

A Stability/Bifurcation Framework

For 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


2/24


Motivation

A large number of existing scientific, large-scale legacy codesBased 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 performanceLocate 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 smallsubspace.

Even when Jacobians are not explicitly available (!)

parameter

bif.quanti

ty


3/24


Recursive Projection Method (RPM)

Recursively identifies subspace of

slow eigenmodes, P

Subspace Pof fewslow eigenmodes

Subspace

Q=I-P

Reconstruct solution:

u = p+q = PN(p,q)+QF

Pic

ard

itera

tions

Newton

iterations

Treats timstepping routine, as ablack-box

Timestepper evaluates un+1= F(un)Initial state un

Timestepping

Legacy Code

Convergence?

Final state uf

F(un)

YES

Picard

iteration

NO

Substitutes pure Picard iteration with

Newton method inPPicard iteration in Q = I-P

Reconstructs solution u from sum of

the projectors P and Q ontosubspace P and its orthogonalcomplement Q , respectively:u = PN(p,q) + QF

F.P.I.


4/24


gPROMS:A General Purpose Package

gPROMS

gPROMSModel

Steady-state &

Dynamic

Simulation

Steady-state &

Dynamic

Optimisation

ParameterEstimatio

n DataReconciliation

Nonlinearalgebraicequationsolvers

Differentialalgebraicequationsolvers

Dynamicoptimisation

solvers

Maximumlikelihoodestimation

solvers

Nonlinearprogramming

solvers


5/24


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 ICsautomatic block triangularisation


6/24


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

0.2,03.0,0.4 10 === aa

)( 012

32

avauvv

vuuuu

t

t

+=

+=


7/24


Bifurcation Diagrams

Around Hopf Around Turning Point


8/24


9/24


10/24


Arc-length continuation with gPROMS

),(y

pyfdt

d=System:

0=

]

*);([

y

pyfDet

Solve (1) & (2)

p

y

),( pyf=0 (1)

Pseudo arc length condition

0)()(

)()(

101

101 =

+

Spp

S

ppyy

S

yy T(2)

continuation(II)

through

FORTRAN

F.P.I

continuation

(I)within

gPROMS


11/24


System Jacobian

R.P.M.

through

FORTRAN

F.P.I

Getting system

Jacobian

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

dtdx =

),,( pyxg=0)(* xyy =),( px

dt

dxf=ODEs :

Cannot get correct

Jacobian from

augmented system

Obtain correct

Jacobian of leading

eigenspectrum


12/24


Tubular Reactor: A DAE system

Dimensionless equations:

]/1

exp[)1(2

21

1

2

1

21

11

x

xxDa

z

x

z

xPe

t

x

++

=

wxx

x

xBDaxz

x

z

x

Pet

x2

2

2

12

2

2

2

21

2

2

]/1exp[)1( +++

=

Boundary Conditions:

0),0(

111 =

=xPe

z

tzx0

),0(22

2 =

=xPe

z

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.


13/24


Bifurcation/Stability with RPM-gPROMS

0

0 . 2

0 . 4

0 . 6

0 . 8

1

0 . 1 0 . 1 1 0 . 1 2 0 . 1 3 0 . 1 4

D a

x1 Hopf pt.

Model solved as DAE system2 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


14/24


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

1

2

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


15/24


+

)(yq)(yAq kk

+

SYSTEM AROUND STEADY STATE

y(k)

)(yy kk =+1

Choose1

q with 11=q

For j =1 Until Convergence DO

(1) Compute and store jAq

(2) Compute and store jtqAqh tjjt ,...2,1,,, ==

(3)

==

j

ttjtjj

qhAqr1

,

(4)2/1

,1 , =+ jjjj rrh

(5) jjjj hrq ,11 / ++=

End For

q

LeadingSpectrum

-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


16/24


Isothermal operationModeling Equations (Nilchan & Pantelides)

Step 1 :

Pressurisation

Step 2:Depressurisation

Rapid Pressure Swing Adsorption

1-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

z

Ci

)(

)1(180

)(

3

2

2

1

2

2

iiiii

b

b

p

n

i

i

ii

iib

it

qpmktq

d

v

z

P

CRT

P

z

CD

z

vC

t

q

t

C

=

=

=

+

=

+

=

Mass balance in ads. bed

Darcys law

Rate of ads.


17/24


Rapid Pressure Swing Adsorption

1-Bed 2-Step Periodic Adsorption Process

Production of oxygen enriched air

Zeolite 5A adsorbent (300 m)

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)


18/24


19/24


PRM-gPROMS Spatial Profiles (t=T)

q1mol/kg

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1x

q2mol/kg

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8 1x

c1mol/m3

0

10

20

30

0 0.2 0.4 0.6 0.8 1x

c2mol/m3

0

30

60

90

0 0.2 0.4 0.6 0.8 1x

z z

z z


20/24


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


21/24


Conclusions

Can construct a RPM-based computational framework around large-scaletimestepping 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.


22/24


gPROMS

General purpose commercial package for modelling, optimization and control of processsystems.

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. JacobiansNOT explicitly available.

Cannot perform systematic bifurcation/stability analysis studies.


23/24


Tracing Limit Cycles

continuation

(I)

within

gPROMS

continuation(II)

through

FORTRAN

F.P.I

R.P.M

through

FORTRAN

F.P.I

Getting system

Jacobian

through an FPI

F.P.I

Tracing limit cycles

tracing

limit cycleswithin gPROMS


24/24


Tracing Limit CyclesTracing limit cycles

),(y pyfdt

d =SYSTEM:

Periodic Solutions: y(t+T)=y(t)

Period T not known beforehand

=

0

),(y

pyf

dt

dTdt

d

0)()0y( = Ty

( (0), ) 0G y p =

( (0), ) (0) 0iG y p y a =

(0)( (0), ) 0i

dyG y p

dt =

gproms_pres

Documents