11 March 2009ICVT, Universitat Stuttgart
MAX−PLANCK−INSTITUT
TECHNISCHER SYSTEMEMAGDEBURG
DYNAMIK KOMPLEXER
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ERIC
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MA
GD
EBU
RG
Numerical nonlinear analysis in DIANA
Michael Krasnyk
Max Planck Institute for Dynamics of Complex Technical Systems, PSPD groupOtto-von-Guericke-University, IFAT
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Outline
1 Introduction
2 Steady-state point analysisLimit points continuationHigher co-dimension singularitiesSimulation models in DianaParameter continuation
3 Case Study I: Nonlinear analysis of CSTR
4 Case Study II: Nonlinear analysis of MCFC
5 Periodic solutions continuationAnalysis of periodic solutionsRecursive Projection Method
6 Case Study III: Periodic solutions in MSMPR Crystallizer
7 Summary
OvGU, IFAT Outline 2/24
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Introduction
Chemical Process Engineering
Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
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TM
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Introduction
Chemical Process Engineering Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
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Introduction
Chemical Process Engineering Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Introduction
Chemical Process Engineering Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Introduction
Chemical Process Engineering Dynamical Systems Analysis
cin
qin
c
T , c
t [s]
T [K]
q [l/h]
T [K]
State of the art
AUTO/MATCONT detection and continuation of bifurcation points and pe-riodic solutions in low-order ODE, no modeling tool
DIVA stationary and dynamic simulations of higher-order DAEfor engineering processes, outdated FORTRAN77 code
LOCA bifurcation analysis of large-scale CFD applications, lim-ited application problems
OvGU, IFAT Introduction 3/24
O
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TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Software tool Diana
Diana — Dynamic simulation and nonlinear analysis tool
developed at MPI Magdeburg
modularization, extensibility and object-oriented architecture
equation based models
numerical solvers based on free code
enhanced scripting and visualization
Objectives of the work
generation of C++ model code for Diana
symbolic differentiation of models (Maxima package)
parameter continuation of nonlinear problemshigher-order singularities of steady-state curvesefficient calculation of periodic solutions by reduction techniques
analysis of test models
OvGU, IFAT Introduction 4/24
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ERICKE UNIVERSITÄ
TM
AG
DE
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Software tool Diana
Diana — Dynamic simulation and nonlinear analysis tool
developed at MPI Magdeburg
modularization, extensibility and object-oriented architecture
equation based models
numerical solvers based on free code
enhanced scripting and visualization
Objectives of the work
generation of C++ model code for Diana
symbolic differentiation of models (Maxima package)
parameter continuation of nonlinear problemshigher-order singularities of steady-state curvesefficient calculation of periodic solutions by reduction techniques
analysis of test models
OvGU, IFAT Introduction 4/24
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Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
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Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
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Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Steady-state point continuation
Parameter continuation vs. dynamic simulation
Condition for steady-state points of an autonomous system x ∈ Rn, ν ∈ Rp is:
x = f (x , ν)!=0
x
0 t
xs
x(t, x0)
→
x
0 λ ∈ ν
x(t, x0)|t=∞
x(λ)
xs
λs
Stability is determined by eigenvalues of a linearized system at the steady-statepoint xs , νs :
VΛ =∂ f
∂ xV
OvGU, IFAT Steady-state point analysis 5/24
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Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S
L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
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Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
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Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
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AG
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Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
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Limit points analysis
Example: limit and hysteresis points (α, β, λ ∈ ν)
x
λ
α
S L
L
H
projection to
α-λ plane
α
λ
β
L L
H
P
OvGU, IFAT Steady-state point analysis/ Limit points continuation 6/24
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Lyapunov-Schmidt reduction 1
Reduction definition [Golubitsky and Schaeffer, 1985]
For a system x = f (x , ν) with f (xs , νs) = 0 and
L = fx(xs , νs) with dim ker L = 1
analysis of a limit points curve can be performed with a scalar equation g(z , ν)!
Lyapunov-Schmidt reduction defines a mapping φ : ker L → range L⊥
φ(v , ν) := (I − E)f (v + W (v , ν), ν)
The basis v0 ∈ ker L and v∗0 ∈ range L⊥ is defined by an adjoint system8><>:f (x , ν) = 0,
fx(x , ν)v0 − βv∗0 = 0, ||v0||2 = 1,
f Tx (x , ν)v∗0 − γv0 = 0, ||v∗0 ||2 = 1.
Reduced equation in the chosen basis {v0, v∗0 } is
g(z , λ) = 〈v∗0 , f (zv0 + W (zv0, λ), λ)〉, where z ∈ R and λ ∈ R ⊂ ν
1M. Golubitsky and D. G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, 1985.
OvGU, IFAT Steady-state point analysis/ Limit points continuation 7/24
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Lyapunov-Schmidt reduction 1
Reduction definition [Golubitsky and Schaeffer, 1985]
For a system x = f (x , ν) with f (xs , νs) = 0 and
L = fx(xs , νs) with dim ker L = 1
analysis of a limit points curve can be performed with a scalar equation g(z , ν)!Lyapunov-Schmidt reduction defines a mapping φ : ker L → range L⊥
φ(v , ν) := (I − E)f (v + W (v , ν), ν)
The basis v0 ∈ ker L and v∗0 ∈ range L⊥ is defined by an adjoint system8><>:f (x , ν) = 0,
fx(x , ν)v0 − βv∗0 = 0, ||v0||2 = 1,
f Tx (x , ν)v∗0 − γv0 = 0, ||v∗0 ||2 = 1.
Reduced equation in the chosen basis {v0, v∗0 } is
g(z , λ) = 〈v∗0 , f (zv0 + W (zv0, λ), λ)〉, where z ∈ R and λ ∈ R ⊂ ν
1M. Golubitsky and D. G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, 1985.
OvGU, IFAT Steady-state point analysis/ Limit points continuation 7/24
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Lyapunov-Schmidt reduction 1
Reduction definition [Golubitsky and Schaeffer, 1985]
For a system x = f (x , ν) with f (xs , νs) = 0 and
L = fx(xs , νs) with dim ker L = 1
analysis of a limit points curve can be performed with a scalar equation g(z , ν)!Lyapunov-Schmidt reduction defines a mapping φ : ker L → range L⊥
φ(v , ν) := (I − E)f (v + W (v , ν), ν)
The basis v0 ∈ ker L and v∗0 ∈ range L⊥ is defined by an adjoint system8><>:f (x , ν) = 0,
fx(x , ν)v0 − βv∗0 = 0, ||v0||2 = 1,
f Tx (x , ν)v∗0 − γv0 = 0, ||v∗0 ||2 = 1.
Reduced equation in the chosen basis {v0, v∗0 } is
g(z , λ) = 〈v∗0 , f (zv0 + W (zv0, λ), λ)〉, where z ∈ R and λ ∈ R ⊂ ν
1M. Golubitsky and D. G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, 1985.
OvGU, IFAT Steady-state point analysis/ Limit points continuation 7/24
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Higher co-dimension singularities
Classification of singular points with codim g 6 3
The classification theorem guarantees existence only the following possible singu-larities of g with codim g 6 3
codim g
0
1
2
3 z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ
gzzzz = 0gλ = 0
z3 ± zλ
gzzz = 0gzλ = 0
z2 ± λ3
gzz = 0 | d3g| = 0
z3 ± λ
gzzz = 0gλ = 0
z2 ± λ2
gzz = 0 | d2g| = 0
z2 ± λ
gzz = 0 gλ = 0
equilibrium
gz = 0
type
OvGU, IFAT Steady-state point analysis/ Higher co-dimension singularities 8/24
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Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
Types of correctors are used:
local parametrization
y(k+1)i = y
(k+1)i
pseudo-arclength parametrization
y (k+1) − y (k+1) ⊥ ~y (k+1)
x
ν
y (k) ~y(k)
t
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
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Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
In Diana two types of predictors are used:
tangential predictor ~y(k)
t
chord predictor ~y(k)
c
x
ν
y (k) ~y(k)
t
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
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AG
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Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
In Diana two types of predictors are used:
tangential predictor ~y(k)
t
chord predictor ~y(k)
c
x
ν
y (k) ~y(k)
t
y(ζ)
x
ν
y (k−1)
y (k)
~y(k)
c
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
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ON
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ERICKE UNIVERSITÄ
TM
AG
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Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
Types of correctors are used:
local parametrization
y(k+1)i = y
(k+1)i
pseudo-arclength parametrization
y (k+1) − y (k+1) ⊥ ~y (k+1)
x
ν
y (k) ~y(k)
t
y(ζ)
x
ν
y (k) y (k+1)
y (k+1)
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Parameter continuation
Parameter continuation solver
Predictor-corrector method computes a parametrized curve
y(ζ) := {x(ζ), ν(ζ)}
such thatF (y(ζ)) ≡ 0
with0 6 ζ 6 ζmax — arc-length parameter
Types of correctors are used:
local parametrization
y(k+1)i = y
(k+1)i
pseudo-arclength parametrization
y (k+1) − y (k+1) ⊥ ~y (k+1)
x
ν
y (k) ~y(k)
t
y(ζ)
x
ν
y (k) y (k+1)
y (k+1)
y(ζ)
OvGU, IFAT Steady-state point analysis/ Parameter continuation 9/24
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AG
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Outline
1 Introduction
2 Steady-state point analysisLimit points continuationHigher co-dimension singularitiesSimulation models in DianaParameter continuation
3 Case Study I: Nonlinear analysis of CSTR
4 Case Study II: Nonlinear analysis of MCFC
5 Periodic solutions continuationAnalysis of periodic solutionsRecursive Projection Method
6 Case Study III: Periodic solutions in MSMPR Crystallizer
7 Summary
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 10/24
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Model: Continuous Stirred Tank Reactor
Model equations [Zeyer et al., 1999]
cin
qin
c
T , c
The mass balances of the model read
cH2O2 = qin/V (cH2O2,in − cH2O2)− (r1 + r2 + r3)
cCH3CHO = qin/V (cCH3CHO,in − cCH3CHO) + (r1 − r2)
cCH3COOH = qin/V (cCH3COOH,in − cCH3COOH) + r2
ccat = qin/V (ccat,in − ccat)− (r4 − r5)
Reaction rates ri , i = 1, . . . , 5 are266664k1e
−E1/(RT )ccat cH2O2
k2e−E2/(RT )ccat cH2O2 cCH3CHO
k3e−E3/(RT )ccat cH2O2
k4e−E4/(RT )ccat
√cCH3CHO
k5e−E5/(RT )(cF ,ges − ccat)
377775The energy balance is
V ρcpT = ρcp qin(Tin − T ) + (UA)cool(Tcool − T ) + VP3
i=1 ri (−∆hR)i
VcoolρcpTcool = ρcp qcool(Tcool,in − Tcool) + (UA)cool(T − Tcool)
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 11/24
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Organizing center location
Unfolding results near a pitchfork point �
290 300 310
300
320
340
360
Tcool,in [K]
T[K
]
Steady-state continuation
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
300 310 320
1
2
3
Tcool,in [K]
c H2O
2,d
os
[mol/
l]
Hysteresis point continuation
290 295 300−2
0
2×10−8
Tcool,in [K]
gxx
Test function gxx (gx = 0)
300 310 320
−2000
−1000
0
1000
Tcool,in [K]
gqcool
Test function gλ (gx , gxx = 0)
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 12/24
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Organizing center location
Unfolding results near a pitchfork point �
290 300 310
300
320
340
360
Tcool,in [K]
T[K
]
Steady-state continuation
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
300 310 320
1
2
3
Tcool,in [K]
c H2O
2,d
os
[mol/
l]
Hysteresis point continuation
290 295 300−2
0
2×10−8
Tcool,in [K]
gxx
Test function gxx (gx = 0)
300 310 320
−2000
−1000
0
1000
Tcool,in [K]
gqcool
Test function gλ (gx , gxx = 0)
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 12/24
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ON
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ERICKE UNIVERSITÄ
TM
AG
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BURG
Organizing center location
Unfolding results near a pitchfork point �
290 300 310
300
320
340
360
Tcool,in [K]
T[K
]
Steady-state continuation
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
300 310 320
1
2
3
Tcool,in [K]
c H2O
2,d
os
[mol/
l]
Hysteresis point continuation
290 295 300−2
0
2×10−8
Tcool,in [K]
gxx
Test function gxx (gx = 0)
300 310 320
−2000
−1000
0
1000
Tcool,in [K]
gqcool
Test function gλ (gx , gxx = 0)
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 12/24
O
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Organizing center location
Unfolding results near a pitchfork point �
290 300 310
300
320
340
360
Tcool,in [K]
T[K
]
Steady-state continuation
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
300 310 320
1
2
3
Tcool,in [K]
c H2O
2,d
os
[mol/
l]
Hysteresis point continuation
290 295 300−2
0
2×10−8
Tcool,in [K]
gxx
Test function gxx (gx = 0)
300 310 320
−2000
−1000
0
1000
Tcool,in [K]
gqcool
Test function gλ (gx , gxx = 0)
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 12/24
O
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ON
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ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Organizing center location
Unfolding results near a pitchfork point �
290 300 310
300
320
340
360
Tcool,in [K]
T[K
]
Steady-state continuation
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
300 310 320
1
2
3
Tcool,in [K]
c H2O
2,d
os
[mol/
l]
Hysteresis point continuation
290 295 300−2
0
2×10−8
Tcool,in [K]
gxx
Test function gxx (gx = 0)
300 310 320
−2000
−1000
0
1000
Tcool,in [K]
gqcool
Test function gλ (gx , gxx = 0)
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 12/24
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ON
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TM
AG
DE
BURG
Organizing center location
Unfolding results near a pitchfork point �
290 300 310
300
320
340
360
Tcool,in [K]
T[K
]
Steady-state continuation
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
300 310 320
1
2
3
Tcool,in [K]
c H2O
2,d
os
[mol/
l]
Hysteresis point continuation
290 295 300−2
0
2×10−8
Tcool,in [K]
gxx
Test function gxx (gx = 0)
300 310 320
−2000
−1000
0
1000
Tcool,in [K]
gqcool
Test function gλ (gx , gxx = 0)
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 12/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Organizing center location
Unfolding results near a pitchfork point �
290 300 310
300
320
340
360
Tcool,in [K]
T[K
]
Steady-state continuation
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
290 295 30030
40
50
60
Tcool,in [K]
qcool
[l/h]
Limit point continuation
300 310 320
1
2
3
Tcool,in [K]
c H2O
2,d
os
[mol/
l]
Hysteresis point continuation
290 295 300−2
0
2×10−8
Tcool,in [K]
gxx
Test function gxx (gx = 0)
300 310 320
−2000
−1000
0
1000
Tcool,in [K]
gqcool
Test function gλ (gx , gxx = 0)
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 12/24
O
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Model: Continuous Stirred Tank Reactor
Unfolding results near a pitchfork point �
Pitchfork point neighborhood
0.58 0.6 0.62
310
315
320
cH2O2,dos [mol/l]
Tcool,in
[K]
Steady-state curveat the pitchfork point
1.8 1.9 2
312
314
316
qcool [l/h]T
[K]
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 13/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Model: Continuous Stirred Tank Reactor
Unfolding results near a pitchfork point �
Pitchfork point neighborhood
0.58 0.6 0.62
310
315
320
cH2O2,dos [mol/l]
Tcool,in
[K]
I
II
III
IV
1 2 3
305
310
315
320
qcool [l/h]
T[K
]
Domain I
1 2 3
310
320
330
qcool [l/h]
T[K
]
Domain II
1 2 3
310
320
330
qcool [l/h]
T[K
]Domain III
1 2 3
310
320
330
qcool [l/h]
T[K
]
Domain IV
OvGU, IFAT Case Study I: Nonlinear analysis of CSTR 13/24
O
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Outline
1 Introduction
2 Steady-state point analysisLimit points continuationHigher co-dimension singularitiesSimulation models in DianaParameter continuation
3 Case Study I: Nonlinear analysis of CSTR
4 Case Study II: Nonlinear analysis of MCFC
5 Periodic solutions continuationAnalysis of periodic solutionsRecursive Projection Method
6 Case Study III: Periodic solutions in MSMPR Crystallizer
7 Summary
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 14/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Model: Molten Carbonate Fuel Cell
Model equations [Mangold et al., 2004]
H
LyLz
ΦAEΦA
ΦCE
ΦC
anode
electrolyte
cathode
Itot
∆Φtot Energy balance and the corresponding bound-ary conditions are
∂Θ
∂τ=∂2Θ
∂η2 +`B − φtot´ i − Bi1Θ,
∂Θ
∂η
˛0,τ
= Bi2Θ(0, τ),∂Θ
∂η
˛1,τ
= −Bi2Θ(1, τ)
A correlation for i due to the Butler-Volmer reaction kinetics is
i = ψA exp
„γA Θ
1 + Θ
«(exp
−(1− βA)γeq φA
1 + Θ
!− KA
eq exp
βAγeq φA
1 + Θ
!)Ohm’s law for the electrolyte is
i = ψE exp
„γE Θ
1 + Θ
«“φA + φC − φtot
”, Itot =
1Z0
i dη
The simulation model is discretized with an equidistant grid and with 100 grid pointsand has 201 variables
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 15/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
••
codim-1 singularities
0 1 2 3 4×1040
1
2
Itot
Θ0
Steady-state point continuation
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
••
codim-1 singularities
0 1 2 3 4×1040
1
2
Itot
Θ0
Steady-state point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
•
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
•
•
codim-1 singularities
0 1 2 3 4×1040
1
2
Itot
Θ0
Steady-state point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
•
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
•
•
codim-1 singularities
0 1 2 3 4×1040
1
2
Itot
Θ0
Steady-state point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
•
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
•
•
codim-1 singularities
0 1 2 3 4×1040
1
2
Itot
Θ0
Steady-state point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
•
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
•
•
codim-1 singularities
0 1 2 3 4 5×1050
10
20
30
40
Itot
Θ0
Steady-state point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
•
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
•
•
codim-1 singularities
0 1 2 3 4×1040
1
2
Itot
Θ0
Steady-state point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
•
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
•
•
codim-1 singularities
0 1 2 3 4×1040
1
2
Itot
Θ0
Steady-state point continuation
z5 ± λ z4 ± zλ z3 ± λ2 z2 ± λ4
z4 ± λ z3 ± xλ z2 ± λ3
z3 ± λ z2 ± λ2
z2 ± λ
equilibrium
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Results of the MCFC analysis
Unfolding results near a winged cusp point
1 1.5 2 2.5 310
12
14
16
Bi2
γA
•
Pitchfork continuation
10 10.5 11 11.5 12 12.5
0.95
1
1.05
γA
γE
•
•
codim-1 singularities
0 1 2 3 4×1040
1
2
Itot
Θ0
Steady-state point continuation
0 0.25 0.5 0.75 10
5
10
15
η
Θ(η
)
Spatial profile, Itot = 37000
OvGU, IFAT Case Study II: Nonlinear analysis of MCFC 16/24
O
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Outline
1 Introduction
2 Steady-state point analysisLimit points continuationHigher co-dimension singularitiesSimulation models in DianaParameter continuation
3 Case Study I: Nonlinear analysis of CSTR
4 Case Study II: Nonlinear analysis of MCFC
5 Periodic solutions continuationAnalysis of periodic solutionsRecursive Projection Method
6 Case Study III: Periodic solutions in MSMPR Crystallizer
7 Summary
OvGU, IFAT Periodic solutions continuation 17/24
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
Task statement
Fixed point problem formulation
Find a fixed point {x∗ ∈ Rn, γ∗ ∈ Rm} of a mapping F with a constraint G
x (i+1) = F (x (i), γ), F : Rn × Rm → Rn ∈ C∞, n � 1,0 = G(x , γ), G : Rn × Rm → Rm ∈ C∞,
such that x∗ = F (x∗, γ∗) and G(x∗, γ∗) = 0.
Calculation of periodic solutions
For an IVP with ϕ(t, x0, λ), such that f (ϕ, ϕ, λ) ≡ 0
F (x , γ) := ϕ(x ,T , λ) — Poincare map,
G(x , γ) :=
s(x ,T , λ)n(x ,T , λ)
ffx (0) = x0 ∈ Rn
γ = {T , λ} ∈ R2
with the pseudo-arclength parameterisation n(x , γ) := 〈x − x , xt〉+ 〈γ − γ, γt〉 − σ
and the phase condition s(x ,T , λ) :=
Z T
0
〈ϕ(t, x , λ), ˙ϕ(t)〉 dtt
ϕϕ
OvGU, IFAT Periodic solutions continuation/ Analysis of periodic solutions 18/24
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ON
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TM
AG
DE
BURG
Task statement
Fixed point problem formulation
Find a fixed point {x∗ ∈ Rn, γ∗ ∈ Rm} of a mapping F with a constraint G
x (i+1) = F (x (i), γ), F : Rn × Rm → Rn ∈ C∞, n � 1,0 = G(x , γ), G : Rn × Rm → Rm ∈ C∞,
such that x∗ = F (x∗, γ∗) and G(x∗, γ∗) = 0.
Calculation of periodic solutions
For an IVP with ϕ(t, x0, λ), such that f (ϕ, ϕ, λ) ≡ 0
F (x , γ) := ϕ(x ,T , λ) — Poincare map,
G(x , γ) :=
s(x ,T , λ)n(x ,T , λ)
ffx (0) = x0 ∈ Rn
γ = {T , λ} ∈ R2
with the pseudo-arclength parameterisation n(x , γ) := 〈x − x , xt〉+ 〈γ − γ, γt〉 − σ
and the phase condition s(x ,T , λ) :=
Z T
0
〈ϕ(t, x , λ), ˙ϕ(t)〉 dtt
ϕϕ
OvGU, IFAT Periodic solutions continuation/ Analysis of periodic solutions 18/24
O
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
RPM: Idea of the method2
The stabilization procedure
Let M := F ∗x have eigenvalues {µk}n1 that for some δ > 0 are ordered as
|µ1| > · · · > |µnp | > 1− δ > |µnp+1| > · · · > |µn| and np � n
Then possible to define the maximal invariant subspace U of M belonging to {µk}np
1
with projectors P and Q := I − P that induce an orthogonal direct sum decompo-sition
Rn = U ⊕ U⊥ = PRn ⊕ QRn
A subspace decomposition of the original system leads to264 V Tq F
(i)x Vq − Iq 0 0
0 V Tp F
(i)x Vp − Ip V T
p F(i)γ
0 G(i)x Vp G
(i)γ
37524∆q(i)
∆p(i)
∆γ(i)
35 = −
264 V Tq (r (i) + F
(i)γ ∆γ(i))
V Tp (r (i) + F
(i)x Vq∆q(i))
G (i) + G(i)x Vq∆q(i)
375The resulting system has properties:
eigenvalues of the restricted to U⊥ matrix are |λ(V Tq F
(i)x Vq)| 6 1− δ
relatively small size of the bottom right part (np + m)× (np + m)
2G. M. Shroff and H. B. Keller. SIAM JNA, 30(4):1099–1120, Aug. 1993.
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 19/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
RPM: Idea of the method2
The stabilization procedure
Let M := F ∗x have eigenvalues {µk}n1 that for some δ > 0 are ordered as
|µ1| > · · · > |µnp | > 1− δ > |µnp+1| > · · · > |µn| and np � n
Then possible to define the maximal invariant subspace U of M belonging to {µk}np
1
with projectors P and Q := I − P that induce an orthogonal direct sum decompo-sition
Rn = U ⊕ U⊥ = PRn ⊕ QRn
A subspace decomposition of the original system leads to264 V Tq F
(i)x Vq − Iq 0 0
0 V Tp F
(i)x Vp − Ip V T
p F(i)γ
0 G(i)x Vp G
(i)γ
37524∆q(i)
∆p(i)
∆γ(i)
35 = −
264 V Tq (r (i) + F
(i)γ ∆γ(i))
V Tp (r (i) + F
(i)x Vq∆q(i))
G (i) + G(i)x Vq∆q(i)
375The resulting system has properties:
eigenvalues of the restricted to U⊥ matrix are |λ(V Tq F
(i)x Vq)| 6 1− δ
relatively small size of the bottom right part (np + m)× (np + m)
2G. M. Shroff and H. B. Keller. SIAM JNA, 30(4):1099–1120, Aug. 1993.
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 19/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
RPM: Idea of the method2
The stabilization procedure
Let M := F ∗x have eigenvalues {µk}n1 that for some δ > 0 are ordered as
|µ1| > · · · > |µnp | > 1− δ > |µnp+1| > · · · > |µn| and np � n
Then possible to define the maximal invariant subspace U of M belonging to {µk}np
1
with projectors P and Q := I − P that induce an orthogonal direct sum decompo-sition
Rn = U ⊕ U⊥ = PRn ⊕ QRn
A subspace decomposition of the original system leads to264 V Tq F
(i)x Vq − Iq 0 0
0 V Tp F
(i)x Vp − Ip V T
p F(i)γ
0 G(i)x Vp G
(i)γ
37524∆q(i)
∆p(i)
∆γ(i)
35 = −
264 V Tq (r (i) + F
(i)γ ∆γ(i))
V Tp (r (i) + F
(i)x Vq∆q(i))
G (i) + G(i)x Vq∆q(i)
375
The resulting system has properties:
eigenvalues of the restricted to U⊥ matrix are |λ(V Tq F
(i)x Vq)| 6 1− δ
relatively small size of the bottom right part (np + m)× (np + m)
2G. M. Shroff and H. B. Keller. SIAM JNA, 30(4):1099–1120, Aug. 1993.
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 19/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
RPM: Idea of the method2
The stabilization procedure
Let M := F ∗x have eigenvalues {µk}n1 that for some δ > 0 are ordered as
|µ1| > · · · > |µnp | > 1− δ > |µnp+1| > · · · > |µn| and np � n
Then possible to define the maximal invariant subspace U of M belonging to {µk}np
1
with projectors P and Q := I − P that induce an orthogonal direct sum decompo-sition
Rn = U ⊕ U⊥ = PRn ⊕ QRn
A subspace decomposition of the original system leads to264 V Tq F
(i)x Vq − Iq 0 0
0 V Tp F
(i)x Vp − Ip V T
p F(i)γ
0 G(i)x Vp G
(i)γ
37524∆q(i)
∆p(i)
∆γ(i)
35 = −
264 V Tq (r (i) + F
(i)γ ∆γ(i))
V Tp (r (i) + F
(i)x Vq∆q(i))
G (i) + G(i)x Vq∆q(i)
375The resulting system has properties:
eigenvalues of the restricted to U⊥ matrix are |λ(V Tq F
(i)x Vq)| 6 1− δ
relatively small size of the bottom right part (np + m)× (np + m)
2G. M. Shroff and H. B. Keller. SIAM JNA, 30(4):1099–1120, Aug. 1993.
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 19/24
O
O
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ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
RPM: Algorithm
Start
P subspacecalculation
Q part solution
P part solution
Convergence?
End
no
yes
Initial data
The input values are:
starting values x (0) and γ(0)
basis for the P part V(0)p = [v1, . . . , vnp+npe ]
Power subspace iteration approximates dominantnp + npe eigenvalues of F
(i)x (Arnoldi iteration)
Find ∆q with an iterative method (Picard, Jacobi,Gauss-Seidel iteration, GMRES)
Solve with a direct method one Newton step for ∆pand ∆γ
Results of the RPM algorithm:
the values x∗ and γ∗
np dominant eigenvalues µ and correspondingeigenvectors Vp of the mapping derivative F ∗x
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 20/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
RPM: Algorithm
Start
P subspacecalculation
Q part solution
P part solution
Convergence?
End
no
yes
P subspace calculation
The input values are:
starting values x (0) and γ(0)
basis for the P part V(0)p = [v1, . . . , vnp+npe ]
Power subspace iteration approximates dominantnp + npe eigenvalues of F
(i)x (Arnoldi iteration)
Find ∆q with an iterative method (Picard, Jacobi,Gauss-Seidel iteration, GMRES)
Solve with a direct method one Newton step for ∆pand ∆γ
Results of the RPM algorithm:
the values x∗ and γ∗
np dominant eigenvalues µ and correspondingeigenvectors Vp of the mapping derivative F ∗x
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 20/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
RPM: Algorithm
Start
P subspacecalculation
Q part solution
P part solution
Convergence?
End
no
yes
Q part solution (Picard iteration, ith step)
The input values are:
starting values x (0) and γ(0)
basis for the P part V(0)p = [v1, . . . , vnp+npe ]
Power subspace iteration approximates dominantnp + npe eigenvalues of F
(i)x (Arnoldi iteration)
Find ∆q with an iterative method (Picard, Jacobi,Gauss-Seidel iteration, GMRES)
Solve with a direct method one Newton step for ∆pand ∆γ
Results of the RPM algorithm:
the values x∗ and γ∗
np dominant eigenvalues µ and correspondingeigenvectors Vp of the mapping derivative F ∗x
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 20/24
O
O
TTV
ON
GU
ERICKE UNIVERSITÄ
TM
AG
DE
BURG
RPM: Algorithm
Start
P subspacecalculation
Q part solution
P part solution
Convergence?
End
no
yes
P part solution (Newton iteration, ith step)
The input values are:
starting values x (0) and γ(0)
basis for the P part V(0)p = [v1, . . . , vnp+npe ]
Power subspace iteration approximates dominantnp + npe eigenvalues of F
(i)x (Arnoldi iteration)
Find ∆q with an iterative method (Picard, Jacobi,Gauss-Seidel iteration, GMRES)
Solve with a direct method one Newton step for ∆pand ∆γ
Results of the RPM algorithm:
the values x∗ and γ∗
np dominant eigenvalues µ and correspondingeigenvectors Vp of the mapping derivative F ∗x
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 20/24
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RPM: Algorithm
Start
P subspacecalculation
Q part solution
P part solution
Convergence?
End
no
yes
Convergence check and results
The input values are:
starting values x (0) and γ(0)
basis for the P part V(0)p = [v1, . . . , vnp+npe ]
Power subspace iteration approximates dominantnp + npe eigenvalues of F
(i)x (Arnoldi iteration)
Find ∆q with an iterative method (Picard, Jacobi,Gauss-Seidel iteration, GMRES)
Solve with a direct method one Newton step for ∆pand ∆γ
Results of the RPM algorithm:
the values x∗ and γ∗
np dominant eigenvalues µ and correspondingeigenvectors Vp of the mapping derivative F ∗x
OvGU, IFAT Periodic solutions continuation/ Recursive Projection Method 20/24
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Outline
1 Introduction
2 Steady-state point analysisLimit points continuationHigher co-dimension singularitiesSimulation models in DianaParameter continuation
3 Case Study I: Nonlinear analysis of CSTR
4 Case Study II: Nonlinear analysis of MCFC
5 Periodic solutions continuationAnalysis of periodic solutionsRecursive Projection Method
6 Case Study III: Periodic solutions in MSMPR Crystallizer
7 Summary
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 21/24
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Model: MSMPR Crystallizer
Model equations [Pathath and Kienle, 2002]
Feed
q, cin
qq , c, F
q, c, Fhp
Productclassification
qf , c, Fhf
Finesdissolver
The population balance equation is
∂F
∂t= −∂(GF )
∂L− q
V(hf (L) + hp(L))F
with the boundary condition
F (0, t) =B(c, t)
G(c, t)=
kb(c(t)− csat)b
kg (c(t)− csat)g.
The recycle ratio of the fines dissolution andthe classified product removal are
hf = R1(1− h(L− Lf )), hp = 1 + R2h(L− Lp)The mass balance of solute is
MAdc
dt= (ρ−MAc)
„q
V+
1
ε
dε
dt
«+
qMAcin
V ε− qρ
V ε
„1 + kv
Z ∞
0
(hp − 1)FL3 dL
«,
where ε is the void fraction which is given by ε = 1− kv
Z ∞
0
FL3 dL
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 22/24
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Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1 III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
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Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1
III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
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Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1
III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
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Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1
III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
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Continuation results
Results of analysis (R2 = 0)
Hopf point continuation
5 10 15 20
0
50
100
b [-]
R1
III
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 0
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 75
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]R1 = 94
5 10 15 204.1
4.2
4.3
4.4
b [-]
c[m
ol/
l]
R1 = 98
OvGU, IFAT Case Study III: Periodic solutions in MSMPR Crystallizer 23/24
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Summary
Simulation system Diana
Lisp-module for the modeling tool ProMoT has been developed (transformationand symbolic differentiation of ProMoT models, C++ code generator)
C++ and Python interaction with CAPE-OPEN interfaces
solver classes for linear and nonlinear problems have been implemented
Numerical nonlinear analysis
implementation of continuation methods for steady-state, limit and Hopf points
generation of adjoint systems and reduced test functions for singular points
periodic solution continuation with simple bifurcations detection
recursive projection method has been applied to speed-up computation
OvGU, IFAT Summary 24/24
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References
Golubitsky, M. and Schaeffer, D. G. (1985).Singularities and groups in bifurcation theory. Vol. I, volume 1 of Applied Math-ematical Sciences.Springer-Verlag, New York.
Mangold, M., Krasnyk, M., and Sundmacher, K. (2004).Nonlinear analysis of current instabilities in high temperature fuel cells.Chemical Engineering Science, 59(22-23):4869–4877.
Pathath, P. K. and Kienle, A. (2002).A numerical bifurcation analysis of nonlinear oscillations in crystallization pro-cesses.Chemical Engineering Science, 57(10):4391–4399.
Zeyer, K. P., Mangold, M., Obertopp, T., and Gilles, E. D. (1999).The iron(III)-catalyzed oxidation of ethanol by hydrogen peroxide: a thermoki-netic oscillator.Journal of Physical Chemistry, 103A(28):5515–5522.
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