Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
The Effect of Dissipation on the
Transformation-Based Approximate
Electromagnetic Cloaking Scheme
Andy Thaler
Department of Mathematics
University of Utah
Math 5440
December 8, 2010
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Outline
1
Helmholtz Equation
In 2D, models the propagation of TE and TM waves
In 3D, models the propagation of acoustic waves
2
Cloaking for the 2D Helmholtz Equation
Boundary Measurements Map
Definition of Cloaking
Main Idea Behind Transformation-Based Cloaking
Dissipation and Results
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Wave Equation
Maxwell’s Equations imply that the electric field satisfies the wave
equation, namely
∂2E
∇2 E = µ0 0 2
∂t
Assuming the waves are traveling in the y −direction, the solution
is of the following form:
h
i
e 0 e i(ky −ωt)
E (y , t) = Re E
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Helmholtz Equation
∇ · [A(x)∇u(x)] + ω 2 q(x)u(x) = 0
1
When derived from Maxwell’s Equations, A(x) and q(x) are
related to µ(x) and (x).
2
u represents the longitudinal component of the electric field
(TM waves) or the magnetic field (TE waves).
3
In 2D, it models the propagation of monochromatic
electromagnetic transverse plane waves.
4
In 3D, it models the propagation of acoustic waves.
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
What is Cloaking?
1
2
Assume radial symmetry
Cloak against plane waves
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Boundary Measurements Map
∇ · [A(x)∇u(x)] + ω 2 q(x)u(x) = 0
1
2
3
We consider B2 .
Maps field to flux for the Helmholtz equation.
We used the inverse of this map (more stable numerically).
Definition (Boundary Measurements Map)
The boundary measurements map associated with the Helmholtz
equation is defined by:
ΛA,q (u) = (A∇u) · ν ,
(1)
x
where ν is the outward normal vector of B2 , and ν(x) = for
2
x ∈ ∂B2 .
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Invariance Principle
∇x · [A(x)∇x u(x)] + ω 2 q(x)u(x) = 0
F : B2 → B2, smooth enough, F(x) = x on ∂B2 .
Changing variables with y = F(x) gives
∇y · [F∗ A(y)∇y v (y)] + ω 2 q∗ (y)v (y) = 0 ,
where v (y) = u[F−1 (y)].
Also,
ΛA, q = ΛF∗ A, q∗
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Invariance Principle
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Transformation-Based Cloaking Proposed by Pendry
ΛF∗ I ,1∗ = ΛI ,1
A(x), q(x) =
F∗ I , 1∗ in B2 \ B1/2
arbitrary real in B1/2
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Transformation-Based Approximate Cloaking Scheme
ΛA,q = ΛAρ ,qρ



2ρ y
F−1
(y)
=
ρ
4 (2 − ρ) 2 (2 − ρ)
y

 2−
+
|y|
3
3
|y|
Andy Thaler
1
for |y| <
2
1
for ≤ |y| ≤ 2 .
2
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Transformation-Based Approximate Cloaking Scheme
ΛA,q = ΛAρ ,qρ 6= ΛI ,1



2ρ y
F−1
(y)
=
ρ
4 (2 − ρ) 2 (2 − ρ)
y

 2−
+
|y|
3
3
|y|
Andy Thaler
1
for |y| <
2
1
for ≤ |y| ≤ 2 .
2
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Kohn, Onofrei, Vogelius, and Weinstein
1
2
1
.
| log(ρ)|
In 3D, error ∼ ρ.
In 2D, error ∼
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Dissipation
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Effect of Dissipation
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Effect of Dissipation
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Step 1–Mapping
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Step 2–Setup
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Step 2–Setup

qρ

(a) ∇2 u1 (x) + ω 2 u1 (x) = 0


Aρ






(b) ∇2 u2 (x) + ω 2 (1 + iβ)u2 (x) = 0







(c) ∇2 u3 (x) + ω 2 (1 + iδ)u3 (x) = 0



∂u1
∂u2


(d) u1 (ρ, θ) = u2 (ρ, θ); Aρ
(ρ, θ) =
(ρ, θ)


∂r
∂r




∂u2
∂u3



(2ρ, θ) =
(2ρ, θ)
(e) u2 (2ρ, θ) = u3 (2ρ, θ);


∂r
∂r





 (f) ∂u3 (2, θ) = ψ
∂r
Andy Thaler
in Bρ
in B2ρ \ Bρ
in B2 \ B2ρ
on ∂Bρ
on ∂B2ρ
on ∂B2
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Step 2–Setup
where
(1) 0
(1) 0
(2ωze ) − J00 (2ωze ) H0
(ωze ρ)
ze J0 (ωρ) J00 (ωze ρ) H0
qρ =
(1) 0
(1)
0
0
J0 (ωρ) J0 (ωze ρ) H0
(2ωze ) − J0 (2ωze )H0 (ωze ρ)
Aρ = qρ I
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Step 2–Setup
where
(1) 0
(1) 0
(2ωze ) − J00 (2ωze ) H0
(ωze ρ)
ze J0 (ωρ) J00 (ωze ρ) H0
qρ =
(1) 0
(1)
0
0
J0 (ωρ) J0 (ωze ρ) H0
(2ωze ) − J0 (2ωze )H0 (ωze ρ)
Aρ = qρ I
ze = 1 + iδ
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Step 2–Setup
where
(1) 0
(1) 0
(2ωze ) − J00 (2ωze ) H0
(ωze ρ)
ze J0 (ωρ) J00 (ωze ρ) H0
qρ =
(1) 0
(1)
0
0
J0 (ωρ) J0 (ωze ρ) H0
(2ωze ) − J0 (2ωze )H0 (ωze ρ)
Aρ = qρ I
ze = 1 + iδ
δ > 0 denotes the dissipation level in the annulus B2 \ B2ρ
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Step 2–Setup
where
(1) 0
(1) 0
(2ωze ) − J00 (2ωze ) H0
(ωze ρ)
ze J0 (ωρ) J00 (ωze ρ) H0
qρ =
(1) 0
(1)
0
0
J0 (ωρ) J0 (ωze ρ) H0
(2ωze ) − J0 (2ωze )H0 (ωze ρ)
Aρ = qρ I
ze = 1 + iδ
δ > 0 denotes the dissipation level in the annulus B2 \ B2ρ
ψ is the normal derivative of the incoming plane wave on ∂B2
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Separation of Variables for u1
Let u1 (r , θ) = R(r )Θ(θ). Then we have
∇2 u1 + ω 2
qρ
u1 = 0
Aρ
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Separation of Variables for u1
Let u1 (r , θ) = R(r )Θ(θ). Then we have
qρ
u1 = 0
Aρ
qρ
∂ 2 u1 1 ∂u1
1 ∂ 2 u1
⇔
+
+
+ ω 2 u1 = 0
2
2
2
∂r
r ∂r
r ∂θ
Aρ
∇2 u1 + ω 2
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Separation of Variables for u1
Let u1 (r , θ) = R(r )Θ(θ). Then we have
qρ
u1 = 0
Aρ
qρ
∂ 2 u1 1 ∂u1
1 ∂ 2 u1
⇔
+
+
+ ω 2 u1 = 0
2
2
2
∂r
r ∂r
r ∂θ
Aρ
q
1
1
ρ
⇔ R 00 Θ + R 0 Θ + 2 RΘ00 + ω 2 RΘ = 0
r
r
Aρ
∇2 u1 + ω 2
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Separation of Variables for u1
Let u1 (r , θ) = R(r )Θ(θ). Then we have
qρ
u1 = 0
Aρ
qρ
∂ 2 u1 1 ∂u1
1 ∂ 2 u1
⇔
+
+
+ ω 2 u1 = 0
2
2
2
∂r
r ∂r
r ∂θ
Aρ
q
1
1
ρ
⇔ R 00 Θ + R 0 Θ + 2 RΘ00 + ω 2 RΘ = 0
r
r
Aρ
qρ
R 00
R 0 Θ00
⇔ r2
+r
+
+ ω2 r 2 = 0
R
R
Θ
Aρ
∇2 u1 + ω 2
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Separation of Variables for u1
Let u1 (r , θ) = R(r )Θ(θ). Then we have
qρ
u1 = 0
Aρ
qρ
∂ 2 u1 1 ∂u1
1 ∂ 2 u1
⇔
+
+
+ ω 2 u1 = 0
2
2
2
∂r
r ∂r
r ∂θ
Aρ
q
1
1
ρ
⇔ R 00 Θ + R 0 Θ + 2 RΘ00 + ω 2 RΘ = 0
r
r
Aρ
qρ
R 00
R 0 Θ00
⇔ r2
+r
+
+ ω2 r 2 = 0
R
R
Θ
Aρ
qρ
R 00
R0
Θ00
⇔ r2
+r
+ ω2 r 2 = −
=K.
R
R
Aρ
Θ
∇2 u1 + ω 2
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
K = 0.
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
K = 0.
Θ00 = 0.
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
K = 0.
Θ00 = 0.
Θ(θ) = C + Dθ.
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
K = 0.
Θ00 = 0.
Θ(θ) = C + Dθ.
2π−periodic iff D = 0
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
K = 0.
Θ00 = 0.
Θ(θ) = C + Dθ.
2π−periodic iff D = 0
K > 0 e.g. K = µ2 .
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
K = 0.
Θ00 = 0.
Θ(θ) = C + Dθ.
2π−periodic iff D = 0
K > 0 e.g. K = µ2 .
Θ00 + µ2 Θ = 0.
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
K = 0.
Θ00 = 0.
Θ(θ) = C + Dθ.
2π−periodic iff D = 0
K > 0 e.g. K = µ2 .
Θ00 + µ2 Θ = 0.
Θ(θ) = Ee iµθ + Fe −iµθ .
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
3 Cases
K < 0 e.g. K = −µ2 .
Θ00 − µ2 Θ = 0.
Θ(θ) = Ae µθ + Be −µθ .
NOT 2π−periodic
K = 0.
Θ00 = 0.
Θ(θ) = C + Dθ.
2π−periodic iff D = 0
K > 0 e.g. K = µ2 .
Θ00 + µ2 Θ = 0.
Θ(θ) = Ee iµθ + Fe −iµθ .
2π−periodic iff µ = k, where k is a non-negative integer
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Bessel’s Equation
The equation for R becomes
2
2 00
0
2 qρ 2
r − k R = 0.
r Rk + rRk + ω
Aρ
(The boundary conditions will be dealt with momentarily). The
solution to Bessel’s Equation of order k is
r
r
qρ
qρ
(1)
Rk (r ) = Pk Jk ω
r + Qk H k
ω
r
Aρ
Aρ
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
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Superposition
Thus
u1,k (r , θ) = Rk (r )Θk (θ)
r
r
qρ
qρ
(1)
= Pk Jk ω
Ee ikθ + Fe −ikθ .
r + Qk H k
ω
r
Aρ
Aρ
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
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Cloaking for Helmholtz
Introduction to Dissipation
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Results
Superposition
Thus
u1,k (r , θ) = Rk (r )Θk (θ)
r
r
qρ
qρ
(1)
= Pk Jk ω
Ee ikθ + Fe −ikθ .
r + Qk H k
ω
r
Aρ
Aρ
Superposing these solutions gives
r
r
∞ X
qρ
qρ
(1)
r + bk Hk
ω
r
e ikθ + e −ikθ
u1 (r , θ) =
ak Jk ω
Aρ
Aρ
k=0
r
r
∞
X
qρ
qρ
(1)
=
ak Jk ω
r + bk Hk
r
e ikθ .
ω
Aρ
Aρ
k=−∞
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
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Introduction to Dissipation
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Results
Solution

r
∞
X
qρ



ωr e ikθ
ak Jk
(a) u1 (r , θ) =


A
ρ


k=−∞




∞ h
i
X
(1)
(b)
u
(r
,
θ)
=
c
J
(ωz
r
)
+
d
H
(ωz
r
)
e ikθ
2
k k
b
k k
b



k=−∞


∞ h

i
X

(1)


(c)
u
(r
,
θ)
=
e
J
(ωz
r
)
+
f
H
(ωz
r
)
e ikθ

e
e
3
k k
k k

in Bρ
in B2ρ \ Bρ
in B2 \ B2ρ
k=−∞
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
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Cloaking for Helmholtz
Introduction to Dissipation
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Results
Error
The error we measured was the following normalized error:
R
∂B2
|u3 − u0
R
∂B2
|ψ|2
|2
1
2
1
2
or, equivalently,
hR
2π
θ=0 |u3 (2, θ)
hR
− u0
(2, θ) |2 dθ
2π
2
θ=0 |ψ (2, θ) | dθ
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i1
2
i1
2
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
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Results
Dependence on ρ and δ
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Dependence on ρ and δ
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The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Electromagnetic Wave
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The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
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Results
Contour Plot of Electromagnetic Wave
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The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
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Results
Shadow
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The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Cloaking
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Zoom-in
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
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Cloaking for Helmholtz
Introduction to Dissipation
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Results
Bessel’s Equation of Order 0
1
R 00 (r ) + R 0 (r ) + R(r ) = 0
r
We assume a series solution of the form
R(r ) =
∞
X
cn r n
n=0
Assuming we can differentiate term-by-term, we have
R 0 (r ) =
∞
X
ncn r n−1
n=1
R 00 (r ) =
∞
X
n(n − 1)cn r n−2
n=2
Andy Thaler
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Bessel’s Equation of Order 0
Inserting the series into Bessel’s equation gives
∞
X
n(n − 1)cn r
n=2
∞
X
X
1X
cn r n = 0
+
ncn r n−1 +
r
n(n − 1)cn r n−2 +
n=2
∞
X
∞
∞
n−2
n=1
n=0
∞
X
∞
X
ncn r n−2 +
n=1
n(n − 1)cn r n−2 + c1 r −1 +
n=2
∞
X
c1
+
r
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n=0
ncn r n−2 +
n=2
cn r n = 0
∞
X
cn−2 r n−2 = 0
n=2
∞
X
n2 cn + cn−2 r n−2 = 0
n=2
The Effect of Dissipation on the Transformation-Based Approxim
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Introduction to Dissipation
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Results
Bessel’s Equation of Order 0
This implies that
c1 = 0
cn = −
cn−2
n2
Using induction, we find that the solution to the above recurrence
relation is

0 for n odd


cn =
n
(−1)k c0


for n even, where k =
2
2k
2
2 (k!)
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Bessel’s Equation of Order 0
Thus our solution is
R(r ) =
∞
X
(−1)k c0
k=0
22k
= c0 +
(k!)
∞
X
2
r 2k
(−1)k c0
22k (k!)2
k=0
r 2k
The ratio test shows that the above series converges for all r .
We thus define
∞
X
(−1)k 2k
J0 (r ) =
r
2k (k!)2
2
k=0
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Bessel’s Equation of Orders 0, 1, 2, and 10
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Clown
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim
Introduction
Cloaking for Helmholtz
Introduction to Dissipation
Solution
Results
Thank you!!!
Also, thank you to Professor Daniel Onofrei.
Andy Thaler
The Effect of Dissipation on the Transformation-Based Approxim