Modeling and Numerical Analysis of Partial
Differential Algebraic Systems for the Simulation
of Networks
Caren Tischendorf
Humboldt-Universität zu Berlin
Elgersburg Workshop 2014
March 06, 2014
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Outline
1 What are PDAEs?
2 Applications
3 Modeling
4 Challenges
5 Numerical Simulation
6 Purely Imaginary Eigenvalues
C. Tischendorf
Modeling and Simulation with PDAEs
Page 1/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
What are PDAEs?
Differential-Algebraic Equation
f (u 0 (t), u(t), t) = 0
with
∂
∂y f
(y , u, t) being singular.
Example:
u10 (t) = g1 (u1 (t), u2 (t), t)
0 = g2 (u1 (t), u2 (t).t)
C. Tischendorf
Modeling and Simulation with PDAEs
Page 2/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Partial Differential-Algebraic Equations (PDAEs)
ODE
DAE
+ constraints
PDE
C. Tischendorf
PDAE
Modeling and Simulation with PDAEs
Page 3/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Partial Differential-Algebraic Equations (PDAEs)
ODE
DAE
+ constraints
PDE
PDAE
PDAE example:
ut (x , t) = uxx (x , t) + v (x , t)
0 = g(u(x , t), v (x , t), x , t)
C. Tischendorf
Modeling and Simulation with PDAEs
Page 3/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Why PDAEs?
Why are we interested in PDAEs?
C. Tischendorf
Modeling and Simulation with PDAEs
Page 4/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Applications
water networks
electronic circuits
gas networks
C. Tischendorf
Modeling and Simulation with PDAEs
blood circuit
Page 5/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Flow Networks
network as a graph:
C. Tischendorf
Modeling and Simulation with PDAEs
Page 6/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Flow Networks
network as a graph:
C. Tischendorf
Modeling and Simulation with PDAEs
Page 6/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Flow Networks
network as a graph:
q4
q9
q6
q8
q7
q3
q5
q2
q1
The currents qi flow along the branches.
C. Tischendorf
Modeling and Simulation with PDAEs
Page 6/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Flow Networks
p5
p7
network as a graph:
q4
q9
p6
p4
q3
q6
q5
p1
q1
q8
p8
q7
p3
q2
p2
The currents qi flow along the branches.
At the nodes we have pressures/potentials pj .
C. Tischendorf
Modeling and Simulation with PDAEs
Page 6/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Flow Balance at Each Node
p5
p7
p6
−q1 − q5 = 0
q4
q9
p4
q3
q6
q5
p1
C. Tischendorf
q1
q8
p8
q7
p3
q2
p2
q1 + q2 = 0
..
.
−q3 − q4 + q6 − q7 = 0
..
.
Modeling and Simulation with PDAEs
q8 = 0
Page 7/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Flow Balance at Each Node
p5
p7
p6
−q1 − q5 = qs1
q4
q9
p4
q3
q6
q5
p1
q1
q8
p8
q7
p3
q2
p2
q1 + q2 = qs1
..
.
−q3 − q4 + q6 − q7 = qs4
..
.
q8 = qs8
if sources and/or sinks exist
C. Tischendorf
Modeling and Simulation with PDAEs
Page 7/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Flow Balance at Each Node
p5
p7
p6
−q1 − q5 = qs1
q4
q9
p4
q3
q6
q5
p1
q1
q8
p8
q7
p3
q2
p2
q1 + q2 = qs1
..
.
−q3 − q4 + q6 − q7 = qs4
..
.
q8 = qs8
if sources and/or sinks exist
Aq=qs
C. Tischendorf
Modeling and Simulation with PDAEs
Page 7/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Flow Balance Equations for Networks
AT q=qs
The matrix A
• describes the topology of the network
• maps branches to nodes
• has only entries -1, 0 and +1
The vector q consists of all branch currents. It contains
• the electric currents i for electric/electronic networks
• the water flows m for water networks
• the gas flows q for gas networks
C. Tischendorf
Modeling and Simulation with PDAEs
Page 8/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Model Extension
So far, we implicitly assumed that the flow ql at the left node of a branch
equals the flow qr at the right node of the same branch.
q
branch:
nl
C. Tischendorf
nr
Modeling and Simulation with PDAEs
Page 9/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Model Extension
So far, we implicitly assumed that the flow ql at the left node of a branch
equals the flow qr at the right node of the same branch.
q
branch:
nl
nr
Such a modeling is only sufficient if one can neglect the time delay of the
flow along a branch.
C. Tischendorf
Modeling and Simulation with PDAEs
Page 9/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Model Extension
So far, we implicitly assumed that the flow ql at the left node of a branch
equals the flow qr at the right node of the same branch.
q
branch:
nl
nr
Such a modeling is only sufficient if one can neglect the time delay of the
flow along a branch.
In general, we need both, ql and qr as network variables.
ql
qr
branch:
nl
C. Tischendorf
Modeling and Simulation with PDAEs
nr
Page 9/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
General Flow Balance Equations for Networks
Al ql + Ar qr =qs
The matrix AL
• maps left branch ends to nodes
• has only the entries -1 and 0
Die Matrix AR
• maps right branch ends to nodes
• has only the entries +1 and 0
A = Al +Ar
C. Tischendorf
Modeling and Simulation with PDAEs
Page 10/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Potential and Pressure Differences
w=AT p
The vector w reflects the potential/pressure differences. It contains
• the electric voltages u for electric networks
• water pressure differences h for water networks
• gas pressure differences ∆p for gas networks
The vector p consists of pressures/potentials. It contains
• the electrostatic potentials e for electric networks
• water pressures p for water networks
• gas pressures p for gas networks
C. Tischendorf
Modeling and Simulation with PDAEs
Page 11/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
F(q,p,w,u,t)=0
for electric networks, e.g.
• Ohmic resistances: w=Rq
C. Tischendorf
Modeling and Simulation with PDAEs
Page 12/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
F(q,p,w,u,t)=0
for electric networks, e.g.
• Ohmic resistances: w=Rq
d
• capacitances: q=C dt
w
C. Tischendorf
Modeling and Simulation with PDAEs
Page 12/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
F(q,p,w,u,t)=0
for electric networks, e.g.
• Ohmic resistances: w=Rq
d
• capacitances: q=C dt
w
• diodes: q=q0 exp ( ww − 1)
0
C. Tischendorf
Modeling and Simulation with PDAEs
Page 12/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
F(q,p,w,u,t)=0
for electric networks, e.g.
• Ohmic resistances: w=Rq
d
• capacitances: q=C dt
w
• diodes: q=q0 exp ( ww − 1)
0
• transistors: q=
R
Γ
[(Jn + Jp ) · ν χ1 − ε ∂t grad ψ · ν χ2 ] dσ
Jn =qe (Dn grad n−µn n grad ψ)
Jp =qe (−Dp grad p−µp p grad ψ)
αψ+β grad ψ·ν=s(p)
C. Tischendorf
on Γ
1
−∂t n+ q div Jn =R(n,p,Jn ,Jp )
e
1
∂t p+ q div Jp =−R(n,p,Jn ,Jp )
e
div (ε grad ψ)=qe (n−p−N)
Modeling and Simulation with PDAEs
Page 12/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Example Circuit Equations
Circuits with capacitors, inductors, resistors, voltage and current sources
AC qC + AL qL + AR qR + AV qV + AI qI = 0
>
>
>
>
wC = A>
C p, wL = AL p, wR = AR p, wV = AV p, wI = AI p
qC =
C. Tischendorf
d
d
fC (wC ), wL = fL (qL ), qR = fR (wR ), wV = vset , qI = iset
dt
dt
Modeling and Simulation with PDAEs
Page 13/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Example Circuit Equations
Circuits with capacitors, inductors, resistors, voltage and current sources
MNA:
AC qC + AL qL + AR qR + AV qV + AI qI = 0
>
>
>
>
wC = A>
C p, wL = AL p, wR = AR p, wV = AV p, wI = AI p
qC =
C. Tischendorf
d
d
fC (wC ), wL = fL (qL ), qR = fR (wR ), wV = vset , qI = iset
dt
dt
Modeling and Simulation with PDAEs
Page 13/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Example Circuit Equations
Circuits with capacitors, inductors, resistors, voltage and current sources
MNA:
AC
d
fC (wC ) + AL qL + AR fR (wR ) + AV qV + AI iset = 0
dt
>
>
>
>
wC = A>
C p, wL = AL p, wR = AR p, wV = AV p, wI = AI p
qC =
C. Tischendorf
d
d
fC (wC ), wL = fL (qL ), qR = fR (wR ), wV = vset , qI = iset
dt
dt
Modeling and Simulation with PDAEs
Page 13/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Example Circuit Equations
Circuits with capacitors, inductors, resistors, voltage and current sources
MNA:
AC
d
fC (wC ) + AL qL + AR fR (wR ) + AV qV + AI iset = 0
dt
>
>
>
>
wC = A>
C p, wL = AL p, wR = AR p, wV = AV p, wI = AI p
qC =
C. Tischendorf
d
d
fC (wC ), wL = fL (qL ), qR = fR (wR ), wV = vset , qI = iset
dt
dt
Modeling and Simulation with PDAEs
Page 13/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Example Circuit Equations
Circuits with capacitors, inductors, resistors, voltage and current sources
MNA:
AC
d
>
fC (A>
C p) + AL qL + AR fR (AR p) + AV qV + AI iset = 0
dt
>
>
>
>
wC = A>
C p, wL = AL p, wR = AR p, wV = AV p, wI = AI p
qC =
C. Tischendorf
d
d
>
>
fC (A>
fL (qL ), qR = fR (A>
C p), AL p =
R p), AV p = vset , qI = iset
dt
dt
Modeling and Simulation with PDAEs
Page 13/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Example Circuit Equations
Circuits with capacitors, inductors, resistors, voltage and current sources
MNA:
d
>
fC (A>
C p) + AL qL + AR fR (AR p) + AV qV + AI iset = 0
dt
d
fL (qL )
A>
Lp =
dt
A>
V p = vset
AC
>
>
>
wC = A>
C p, wL = AL p, wR = AR p, wV = vset , wI = AI p
d
>
qC = fC (A>
C p), qR = fR (AR p), qI = iset
dt
C. Tischendorf
Modeling and Simulation with PDAEs
Page 13/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
F(qL ,qR ,p,w,u,t)=0
for water networks, e.g.
• reservoirs: p = p set
d
• tanks: C dt
p= qs
C. Tischendorf
Modeling and Simulation with PDAEs
Page 14/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
F(qL ,qR ,p,w,u,t)=0
for water networks, e.g.
• reservoirs: p = p set
d
• tanks: C dt
p= qs
• pipes:
∂t p + a∂x q = 0
∂t q + b∂x p + c|q|q + d = 0
p(nl ) = pnl ,
C. Tischendorf
p(nr ) = pnr ,
q(nl ) = ql ,
Modeling and Simulation with PDAEs
q(nr ) = qr
Page 14/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Example Water Network Equations
water network with pipes and reservoirs
APl ql + APr qr = qset
pR = pset
∂t p + a∂x q = 0
∂t q + b∂x p + c|q|q + d = 0
p(nl ) = pnl ,
C. Tischendorf
p(nr ) = pnr ,
q(nl ) = ql ,
Modeling and Simulation with PDAEs
q(nr ) = qr
Page 15/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
F(qL ,qR ,p,w,u,t)=0
for gas networks, e.g.
• valves: q=qL =qR = g(w, t)
C. Tischendorf
Modeling and Simulation with PDAEs
Page 16/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
F(qL ,qR ,p,w,u,t)=0
for gas networks, e.g.
• valves: q=qL =qR = g(w, t)
• pipes:
∂t % + ∂x q
∂t q + ∂x p + ∂x (%v 2 ) + gp∂x h
p
p(nL ) = pnL ,
C. Tischendorf
p(nR ) = pnR ,
= 0
λ(q)
%v |v |
2D
= γ(T )z(p, T )%
= −
q(nL ) = qL ,
Modeling and Simulation with PDAEs
q(nR ) = qR
Page 16/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
for blood circuits, e.g. systemic circuit
Systemic Circuit
Pulmonary Circuit
[Danielsen, Ottesen 2004]
C. Tischendorf
Modeling and Simulation with PDAEs
Page 17/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Modeling of Network Elements
for blood circuits, e.g. heart chambers
[Danielsen, Ottesen 2004]
C. Tischendorf
Modeling and Simulation with PDAEs
Page 18/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Model Equations for Networks
1
flow balance at each node: AL qL + AR qR = 0
2
potential/pressure differences: w=AT p
3
element models: F(qL ,qR ,p,w,u,t)=0
Depending on the element models the resulting system represents a
system of
• linear/nonlinear equations
• differential algebraic equations (DAEs)
• partial differential algebraic equations (PDAEs)
C. Tischendorf
Modeling and Simulation with PDAEs
Page 19/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Challenges for (P)DAEs
• existence of solutions
• uniqueness of solutions
• stability
• numerical methods (feasible, convergent and stable)
• solver for the discretized PDAE
• implementation of numerical methods (monolithic or co-simulation)
• model order reduction
• parameter optimization
• uncertainty quantification
• control
C. Tischendorf
Modeling and Simulation with PDAEs
Page 20/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Weak Formulation of PDAEs
generalized evolution equation approach
0
A∗ [Du(t)] + B(u(t), t) = r (t)
f.a.a. t ∈ I
Du(t0 ) = z0 ∈ Z
• real Banach spaces V , Z , evolution triple Z ⊆ H ⊆ Z ∗
• D : V → Z surjective, A = T D
• B : V × I → V∗
solution space
0
W21 (I, V , Z , H) = u ∈ L2 (I, V ) ∃ [Du(t)] ∈ L2 (I, Z ∗ )
C. Tischendorf
Modeling and Simulation with PDAEs
Page 21/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Unique Solvability I
0
A∗ [Du(t)] + B(u(t), t) = r (t)
f.a.a. t ∈ I
Du(t0 ) = z0 ∈ Z
is uniquely solvable [T. 2004, Matthes 2012] on W21 (I, V , Z , H) if
• B : V × I → V ∗ strongly monotone, satisfies growth condition
• r ∈ L2 (I; V ∗ )
Drawback:
Coupled systems often do not lead to a strongly monotone operator B
C. Tischendorf
Modeling and Simulation with PDAEs
Page 22/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Unique Solvability II
d
Pm(Px (t)) + F (x (t), t) + PM(x (t), u(t), t) = 0,
dt
A(u(t)) − R(x (t)) = 0
t ∈ I,
has a unique solution [Matthes 12] if
• P ∈ Rn×n is orthoprojector and Q := I − P
• the map Px 7→ Pm(Px ) is strongly monotone (on Im P) and ∈ C 1
• F : Rn × I → Rn is continuous and Lipschitz continuous w.r.t. x
• The map φ : Im Q → Im Q with w 7→ QF (z + w , t) is bijective and
φ−1 (z, t) is continuous and Lipschitz continuous w.r.t. z ∈ Im P, t ∈ I
• A : V → V ∗ is strongly monotone and hemicontinuous on V
• M : Rn × V → Rn is continuous and Lipschitz continuous w.r.t. x , u
• R : Rn → V ∗ is Lipschitz continuous
• (x (t0 ), u(t0 )) ∈ {(x , u) ∈ Rn × V | QF (x , t0 ) = 0, A(u) − R(x ) = 0}
C. Tischendorf
Modeling and Simulation with PDAEs
Page 23/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Discretization of PDAEs
ODE
DAE
spatial discretization
parabolic PDE
C. Tischendorf
PDAE
Modeling and Simulation with PDAEs
Page 24/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Simple DAE Examples
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + u2 (t) + g(t)
C. Tischendorf
⇒
u10 (t) = −u1 (t) − g(t) + h(t)
u2 (t) = −u1 (t) − g(t)
Modeling and Simulation with PDAEs
Page 25/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Simple DAE Examples
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + u2 (t) + g(t)
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + g(t)
C. Tischendorf
⇒
⇒
u10 (t) = −u1 (t) − g(t) + h(t)
u2 (t) = −u1 (t) − g(t)
u2 (t) = −g 0 (t) − h(t)
u1 (t) = −g(t)
Modeling and Simulation with PDAEs
Page 25/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Simple DAE Examples
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + u2 (t) + g(t)
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + g(t)
⇒
⇒
u20 (t) = u3 (t)
u10 (t)
= u2 (t) + h(t)
0 = u1 (t) + g(t)
C. Tischendorf
u10 (t) = −u1 (t) − g(t) + h(t)
u2 (t) = −u1 (t) − g(t)
u2 (t) = −g 0 (t) − h(t)
u1 (t) = −g(t)
u3 (t) = −g 00 (t) − h0 (t)
⇒
u2 (t) = −g 0 (t) − h(t)
u1 (t) = −g(t)
Modeling and Simulation with PDAEs
Page 25/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Simple DAE Examples
Index 1
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + u2 (t) + g(t)
⇒
u10 (t) = −u1 (t) − g(t) + h(t)
u2 (t) = −u1 (t) − g(t)
Index 2
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + g(t)
⇒
u2 (t) = −g 0 (t) − h(t)
u1 (t) = −g(t)
Index 3
u20 (t) = u3 (t)
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + g(t)
C. Tischendorf
u3 (t) = −g 00 (t) − h0 (t)
⇒
u2 (t) = −g 0 (t) − h(t)
u1 (t) = −g(t)
Modeling and Simulation with PDAEs
Page 25/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Perturbation Index
u20 (t) = u3 (t)
u10 (t) = u2 (t) + h(t)
0 = u1 (t) + g(t)
C. Tischendorf
Modeling and Simulation with PDAEs
Page 26/ 42
What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Perturbation Index
u3 (t) = n2 ε cos(nt) − h0 (t)
u20 (t) = u3 (t)
u10 (t)
= u2 (t) + h(t)
⇒
0 = u1 (t) + ε cos(nt)
ku1 − uδ1 k = ε,
u2 (t) = nε sin(nt) − h(t)
u1 (t) = −ε cos(nt)
ku2 − uδ2 k = nε,
ku3 − uδ3 k = n2 ε
ill-posed problem !
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Modeling and Simulation with PDAEs
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What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Perturbation Index
u3 (t) = n2 ε cos(nt) − h0 (t)
u20 (t) = u3 (t)
u10 (t)
= u2 (t) + h(t)
⇒
0 = u1 (t) + ε cos(nt)
ku1 − uδ1 k = ε,
u2 (t) = nε sin(nt) − h(t)
u1 (t) = −ε cos(nt)
ku2 − uδ2 k = nε,
ku3 − uδ3 k = n2 ε
ill-posed problem !
ku − uδ k ≤ c (kδk + kδ 0 k + kδ 00 k)
perturbation index 3
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Modeling and Simulation with PDAEs
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What are PDAEs?
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Modeling
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PDAE Discretization
Approach:
1
spatial discretization
2
time discretization
Goal: Find space discretizations yielding low index DAEs!
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Modeling and Simulation with PDAEs
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What are PDAEs?
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Spatial Discretization for Water Networks
water networks with reservoirs and pipes:
APl ql + APr qr = qset
1
−∂t A>
Pl p + Da (qr − ql ) = 0
h
1
1
>
∂t qr + Db (A>
Db A >
Pr p + APl p) + c|qr |qr + d =
R pset
h
h
is a DAE system of index 1 if the direction of pipes is chosen as follows:
1
Each reservoir has only pipes directing away from the reservoir.
2
Each other node of the network has at least one pipe directing to
this node.
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Modeling and Simulation with PDAEs
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What are PDAEs?
Applications
Modeling
Challenges
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Problems when Solving (P)DAEs
• The algebraic constraints demand implicit or half-implicit methods.
• The solution does not depend continuously on the input data (for
higher index DAEs).
• Initial values have to fulfil (hidden) constraints.
• Numerical methods as implicit multistep or one-step methods may
fail.
• Simulation results may depend on the (P)DAE formulation.
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Modeling and Simulation with PDAEs
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Trouble when Simulating DAEs
index 2 example [Gear, Petzold 86]
x10 + ηtx20 + (1 + η)x2 = q1
x1 + ηtx2 = q2
unique solution:
x2
= q1 − q20
x1
= q2 − ηtx2
numerical solution:
The methods fail for parameter values η < −0.5.
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Modeling and Simulation with PDAEs
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Reformulation of the DAE
equivalent formulation
x10 + (ηtx )02 + x2 = q1
x1 + ηtx2 = q2
numerical solution:
All methods work well for all parameter values.
reason: The discretization method applied to the DAE yields to the same
method for the inherent dynamical part [Higueras, März, T. 2004].
C. Tischendorf
Modeling and Simulation with PDAEs
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What are PDAEs?
Applications
Modeling
Challenges
Numerical Simulation
Purely Imaginary Eigenvalues
Implementation Concept
element models
spatial discret.
network equations
time discretization
∂t f (y )+∂x g(y ) = h(y , t)
∂t f (y )+∆x g(y ) = h(y , t)
f (d 0 (y , t), y , t) = 0
f (∆t d(y , t), y , t) = 0
net control
parameters
topological
analysis
name=”reg1”, sp=30, ...
loops, connections, ...
nonlinear system
netlist
parser
initialization
pipe, nodeL=”name1”,
{n1, n2, ...},
homotopy providing y0
..., L=93, ...
{v 1, v 2, ...}
F (y ) = 0
Newton type method
yn = G(yn−1 )
linear system
postprocessing
Ay = b
pressures p, flows q, ...
C. Tischendorf
output
linear solver
preconditioning
y
LU, QR, CG, hybrid, ...
PAy = Pb
Modeling and Simulation with PDAEs
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Coupled Circuit-EM Simulation
The perturbation behavior (DAE index) depends on the choice of gauge
conditions for the vector potential A [Baumanns 2012].
• Lorenz gauge: ∇ · A = 0 ⇒ index 2
d
• Coulomb gauge: εµ dt
ϕ + ∇ · A = 0 ⇒ index 0/1/2
cross coupling effects in high frequency circuits
C. Tischendorf
Modeling and Simulation with PDAEs
Page 33/ 42
What are PDAEs?
Applications
Modeling
Challenges
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Purely Imaginary Eigenvalues
Blood Circuit Simulation
C. Tischendorf
heart chambers
blood flow
blood pressure
blood volume
Modeling and Simulation with PDAEs
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What are PDAEs?
Applications
Modeling
Challenges
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Blood Circuit Simulation
simulated aortic blood pressure: 117/82 mmHG (normal value)
increase/decrease
50% incr.
50% incr.
50% incr.
50% decr.
50% decr.
50% decr.
50% incr.+decr.
25% incr.+decr.
parameter
R1
R2
R3
C1
C2
C3
R1 , R2 , R3 + C1 , C2 , C3
R1 , R2 , R3 + C1 , C2 , C3
blood pressure
126/83
119/86
118/85
181/87
140/92
127/90
194/137
146/104
C1 , R1 : compliance and resistance in aortic vessels
C2 , R2 : compliance and resistance in arterioles
C3 , R3 : compliance and resistance in capillaries
C. Tischendorf
Modeling and Simulation with PDAEs
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What are PDAEs?
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Purely Imaginary Eigenvalues for Circuits
C. Tischendorf
CuC0
= iC
LiL0
= uL
0
= BC uC + BL uL + BR uR + BJ uJ + BV vs (t)
0
= QC iC + QL iL + QR iR + QJ js (t) + QV iV
0
= iR − GuR
Modeling and Simulation with PDAEs
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Purely Imaginary Eigenvalues for Circuits
CuC0
= iC
LiL0
= uL
0
= BC uC + BL uL + BR uR + BJ uJ + BV vs (t)
0
= QC iC + QL iL + QR iR + QJ js (t) + QV iV
0
= iR − GuR
Matrix Pencil
C. Tischendorf
λC
0
BC
0
0
0
−I
BL
0
0
−I
0
0
QC
0
0
λL
0
QL
0
0
0
BR
0
G
0
0
0
QR
−I
0
0
BJ
0
0
Modeling and Simulation with PDAEs
0
0
0
QV
0
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Matrix Pencils
Matrix Pencil (2)
Matrix Pencil (1)
λC 0 −I
0 −I 0
BC BL 0
0 0 Q
C
0 0 0
0 0 0 0
λL 0 0 0
0 BR 0 BJ
QL 0 QR 0
0 G −I 0
0
0
0
QV
0
λC 0 −I 0 0 0
0 −I 0 λL 0 0
Bˆ C Bˆ L 0 0 Bˆ R 0
ˆC Q
ˆL 0 Q
ˆR
0 0 Q
0 0 0 0 G −I
Theorem
Assume that a given circuit has neither V-loops nor I-cutsets. The
spectrum of the matrix pencil (1) coincides with that of the pencil (2),
provided that the latter is defined by the RLC circuit obtained after
open-circuiting current sources and short-circuiting voltage sources.
C. Tischendorf
Modeling and Simulation with PDAEs
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What are PDAEs?
Applications
Modeling
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Numerical Simulation
Purely Imaginary Eigenvalues
Purely Imaginary Eigenvalues (PIEs)
Proposition 1
Any eigenvector associated with a PIE verifies uR = iR = 0.
Proof Idea 1. If (u, i) = (uC , uL , uR , iC , iL , iR ) is an eigenvector to a PIE
λ then
u ∗ i = (Q T w )∗ i = w ∗ Qi = 0,
since Bu = 0 and Ker B = Im Q T .
2. Including the element related equations yields
¯ L∗ LiL + uR∗ GuR = 0
λuC∗ CuC + λi
3. Comparing real and imaginary parts concludes the assertion.
C. Tischendorf
Modeling and Simulation with PDAEs
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Purely Imaginary Eigenvalues (PIEs)
Proposition 2
All eigenvalues of an LC-circuit are PIEs.
C. Tischendorf
Modeling and Simulation with PDAEs
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Purely Imaginary Eigenvalues (PIEs)
Proposition 2
All eigenvalues of an LC-circuit are PIEs.
LC-block
Proposition 3
If the reduced circuit corresponding to pencil (2) has an LC-block then
the circuit has a PIE.
The converse of Proposition 3 is not true.
C. Tischendorf
Modeling and Simulation with PDAEs
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What are PDAEs?
Applications
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Purely Imaginary Eigenvalues
Purely Imaginary Eigenvalues (PIEs)
Theorem
A circuit has a PIE for all positive values of capacitances and inductances
if and only if the reduced circuit (after open-circuiting current sources
and short-circuiting voltage sources) has an LC-block.
C. Tischendorf
Modeling and Simulation with PDAEs
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What are PDAEs?
Applications
Modeling
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Summary
• modeling of flow networks leads to DAEs and PDAEs
• existence and uniqueness of PDAE solutions only known for certain
subclasses
• suitable spatial discretizations of PDAEs yield index-1 DAEs
• stability of the numerical solution for (P)DAEs depends on model
formulation
• network topology can and shall be used for well-posed network
(P)DAEs, consistent initialization and pre-conditioning
• network topological characterization of purely imaginary eigenvalues
C. Tischendorf
Modeling and Simulation with PDAEs
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What are PDAEs?
Applications
Modeling
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References
[1] L. Jansen and C. Tischendorf. A Unified (P)DAE Modeling Approach for Flow
Networks. To appear in Springer DAE-Forum Progress in Differential-Algebraic
Equations - Deskriptor 2013
[2] R. Lamour, R. März, and C. Tischendorf. Differential-algebraic equations: A
projector based analysis. Differential-Algebraic Equations Forum 1. Berlin:
Springer, 2013
[3] R. Riaza and C. Tischendorf. Structural characterization of classical and
memristive circuits with purely imaginary eigenvalues. Int. J. Circ. Theor. Appl.,
41:273-294, 2013
[4] G. Alí, N. Banagaaya, W.H.A. Schilders, and C. Tischendorf. Index-aware model
order reduction for linear index-2 DAEs with constant coefficients. SIAM J. Sci.
Comp., 35(3):A1487-A1510, 2013
[5] S. Grundel, L. Jansen, N. Hornung, P. Benner, T. Clees, and C. Tischendorf.
Model order reduction of differential algebraic equations arising from the
simulation of gas transport networks. To appear in Springer DAE-Forum
Progress in Differential-Algebraic Equations - Deskriptor 2013
[6] L. Jansen, M. Matthes, and C. Tischendorf. Global unique solvability for
memristive circuit DAEs of index 1. Int. J. Circ. Theor. Appl., DOI
10.1002/cta.1927, 2013
C. Tischendorf
Modeling and Simulation with PDAEs
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