Geometrical design of thermoelectric generators based on ...€¦ · novel thermoelectric...
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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 2000; 00:1–6 Prepared using nmeauth.cls [Version: 2002/09/18 v2.02]
Geometrical design of thermoelectric generators based on
topology optimization
A. Takezawa1,∗,† and M. Kitamura1
1 Division of Mechanical System and Applied Mechanics, Faculty of Engineering, Hiroshima University,
1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima, Japan
SUMMARY
This paper discusses an application of the topology optimization method for the design of
thermoelectric generators. The proposed methodology provides the optimized geometry in accordance
with various arbitrary conditions such as the types of materials, the volume of materials, and the
temperature and shape of the installation position. By considering the coupled equations of state for
the thermoelectric problem, an analytical model subject to these equations is introduced that mimics
the closed circuit composed of thermoelectric materials, electrodes and a resistor. The total electric
power applied to the resistor and the conversion efficiency are formulated as objective functions to
be optimized. The proposed optimization method for thermoelectric generators is implemented as a
geometrical optimization method using the solid isotropic material with penalization (SIMP) method
used in topology optimizations. Simple relationships are formulated between the density function of
the SIMP method and the physical properties of the thermoelectric material. A sensitivity analysis for
the objective functions is formulated with respect to the density function and the adjoint equations
∗Correspondence to: Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, Hiroshima, Japan†E-mail: [email protected]
Received
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2 A. TAKEZAWA AND M. KITAMURA
required for calculating it. Depending on the sensitivity, the density function is updated using the
method of moving asymptotes (MMA). Finally, numerical examples are provided to demonstrate the
validity of the proposed method. Copyright c© 2000 John Wiley & Sons, Ltd.
key words: thermoelectricity; multiphysics; topology optimization; finite element method;
sensitivity analysis
1. INTRODUCTION
Thermoelectric generators have tremendous potential in engineering fields. They are composed
of thermoelectric materials exhibiting the Seebeck effect, a phenomenon that produces an
electric potential difference from the temperature difference between two material junctions.
Inherently, such devices are capable of being implemented as energy-harvesting devices
installable in power generators that use waste heat from vehicles or plants, and in maintenance-
free electrical sources for small wireless devices, implanted medical devices and some aerospace
devices. For the fundamental mechanical aspects of thermoelectric generators and other
application examples, the reader is referred to various text books and comprehensive reviews
(e.g. [1, 2, 3, 4]).
The performance of a thermoelectric generator can be improved in several ways: the
performance of thermoelectric materials, the material selection, the sophistication of the
device geometry and the situation within the entire system, including the coolant system
and installation site, to name a few. In this research, a method is considered for analyzing
thermoelectric device geometries to improve power-generating characteristics. Recently, there
has been an increase in the development of new thermoelectric devices with various
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 3
geometries (e.g. [5, 6, 7, 8, 9, 10, 11]). Moreover, some elementary parameterization studies
and optimization studies based on experimental and simplified numerical models have
been performed [5, 12, 13, 14]. However, these methodologies have only provided sizing
optimizations, and thus strongly depend on the quality of the initial defined geometry.
Therefore, novel high-performance geometrical design continues to be elusive. To develop
novel thermoelectric generators, there is a need to study the underlying optimal geometry
of the devices. Moreover, as these devices come into general use in the future, the geometry of
these devices may need to have complex designs depending on the situation. Typical shapes
may not be adequate, opening up the need for an effective optimization method to determine
appropriate geometrical shapes.
The search for optimal device geometry can be assisted by accurate numerical performance
analysis using finite element analysis. Although the equations of state describing the
thermoelectric effect in continuum mechanics are coupled in a highly nonlinear fashion,
several methods for solving this problem using finite element analysis have been proposed
[14, 15, 16]. Geometrical optimization can be performed by integrating such finite element
methods and some optimization methods. However, these finite element methods have not
yet been applied to the detailed optimization of device geometries. Thus, this study presents
a new design procedure for the thermoelectric generator based on continuum mechanics, the
finite element method, and numerical optimization methods. Since material properties related
to the thermoelectric effect usually exhibit a strong dependency on the temperature, and the
mechanics of the effect are very complicated, this geometrical optimization of thermoelectric
devices is both challenging and interesting. The topology optimization method [17, 18] is
often used in engineering fields and is applied to various physical problems, including difficult
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4 A. TAKEZAWA AND M. KITAMURA
coupled nonlinear problems (e.g. [19, 20]). This method has an advantage over conventional
shape optimization methods (e.g. [21, 22, 23]) in that it can achieve fundamental geometrical
optimization, including a topology change in the target geometry. Recently, this methodology
was extended to the electrical power generation problem from a mechanical input using
piezoelectric material, which is regarded as one of the most important energy harvesting
technologies for thermoelectric generators [24, 25, 26, 27, 28].
In this paper, we apply the topology optimization method to the design problem of
thermoelectric generators. The coupled equations of state for the thermoelectric problem are
first considered. An analytical model subjected to the equations of state is realized by a closed
circuit, consisting of p and n-type thermoelectric materials, electrodes and a load resistor.
The total electric power applied to the resistor (electric power output) and the conversion
efficiency (= electric power output / total heat flow from the source) is formulated as an
objective function. The proposed optimization of the thermoelectric generator is implemented
as a geometrical optimization using the solid isotropic material with the penalization (SIMP)
method of topology optimization [29, 30, 31]. The relationships between the density function
of the SIMP method and the physical properties of the thermoelectric material are formulated
by simple equations. The sensitivities for each objective function with respect to the density
function, and the adjoint equations required to calculate it are formulated. Based on the
sensitivity, the density function is updated using the method of moving asymptotes (MMA)
[32]. The MMA has numerous benefits in various optimization problems by virtue of its
combination with topology optimization. Finally, numerical examples are provided as a
validation of the proposed methodology.
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 5
2. Formulation
2.1. Equations of state
Initially we considered the equations of state pertaining to the thermoelectric effect. Ignoring
time-dependent effects, only the equilibrium state was considered, and all materials were
assumed to be isotropic with respect to the conduction of electric charges and heat, and the
Seebeck and Peltier effects. The basic equations of state for electrical and thermal conduction
are written as follows:
∇ · j = 0 (1)
∇ · q = f (2)
where j and q are the electric current density and the heat flux density vectors, and f is the
volume heat source. In the thermoelectric phenomena, j and q are coupled by the following
equations [33]:
j = σ(E − α∇T ) (3)
q = βj − λ∇T (4)
where E is the electric field vector, T is temperature, σ and λ are the electrical and thermal
conductivities, and α and β(= Tα) are the Seebeck and Peltier coefficients. The heat source is
assumed to be given by the electric power, that is, f = j ·E. Introducing the electric potential
V (with −∇V = E) and substituting Eqs.(3) and (4) into Eqs.(1) and (2), the equations of
state are expressed with respect to the state variables V and T :
∇ · (−σ∇V − ασ∇T ) = 0 (5)
∇ · −(α2σT + λ)∇T − ασ∇V T = σ∇V · ∇V + α∇T · ∇V . (6)
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6 A. TAKEZAWA AND M. KITAMURA
By adding the following boundary conditions,
V = V0 on ΓDV, n · j = j0 on ΓNj (7)
T = T0 on ΓDT, n · q = q0 on ΓNq (8)
Eqs.(5) and (6) can be solved with respect to V and T . ΓDV and ΓNj are the boundaries on
which the Dirichlet and Neumann conditions are imposed with respect to the state variable
V , and similarly for ΓDT and ΓNq with respect to the state variable T .
2.2. Modeling of the devices
The analysis domain in which the state is represented as the solution to the equations of
state Eqs.(5) and (6) is set up. The simplest actual thermoelectric device is composed of
thermoelectric materials, a resistor and electrodes as depicted in Fig.1. Th and Tc (Th > Tc)
represent the temperatures of the hot and cold junctions of the device. The heat flow results
from the temperature difference between these junctions. The current flow then takes place
according to the equations of states. Since p and n-type thermoelectric materials have positive
and negative Seebeck coefficients respectively and the current flow occurs in the opposite
direction to the heat flow, series coupling of these materials leads to the full use of the heat
flow for power generation. The consequent current flow is supplied to the outside electrical
load corresponding to an electrical device.
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 7
Heat flow Cold junction: Tcp-type thermoelectric material n-type thermoelectric material
ElectrodeElectrical load
CurrentHot junction: Th
Figure 1. Schematic view of a simple thermoelectric generator
A simplified 2D domain modeling of this device is shown in Fig. 2. This representation is
used in the 2D finite element analysis. The whole domain Ω has been categorized into three
sub-domains ΩTE, ΩE and ΩL, which correspond to the thermoelectric materials, the electrodes
and the electrical load (resistor). As thermoelectric materials are classified according to two
carrier types, ΩTE is further divided into ΩTE-p for a p-type material and ΩTE-n for an n-type
material. In the same way, ΩE is further divided into four types, ΩE-pc and ΩE-ph for the
cold and hot junctions of p-type material, ΩE-nc and ΩE-nh for the cold and hot junctions of
n-type material. ΓDTc and ΓDThindicate the Dirichlet boundary conditions with respect to
the temperature corresponding to the cold and the hot junctions. To evaluate the performance
of thermoelectric generator in the closed circuit as shown in Fig.1, the current flow must be
continuous on the side of the resistor domain and the opposite side of the electrode domain.
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8 A. TAKEZAWA AND M. KITAMURA
The direct approach is to introduce a periodic boundary condition for the electric potential
on these sides. However, in this research, closed-circuit representation is simply obtained by
introducing ground boundary conditions (a zero electric potential boundary condition).
ΩTE-p ΩTE-nΩE-pc ΩE-ncΩLΩ cDTΓ hDTΓ0 cDV ,DTΓ0DVΓ ΩE-ph ΩE-nh
Figure 2. A schematic diagram of a 2D analysis domain for a simple thermoelectric generator
2.3. Objective function and optimization problem
In this study, the performance of thermoelectric generators is optimized by maximizing the
following two indexes, the electric power output applied to the resistor (hereinafter the electric
power output) and the conversion efficiency, which is calculated as the ratio between the electric
power output and the total heat flow from the source. Only the shape of ΩTE is considered as
the design variable for the optimization, while keeping the shape of ΩE and ΩL fixed. These
indexes are formulated as the function of the shape of the thermoelectric domain ΩTE and the
state functions V and T which are varied during the optimization process as follows:
F1(ΩTE, V, T ) =
∫Ω
H1j ·Edx (9)
F2(ΩTE, V, T ) =
∣∣∣∣∫Ω
H2 · qdx∣∣∣∣ (10)
F3(ΩTE, V, T ) =F1(ΩTE, V, T )
F2(ΩTE, V, T )(11)
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 9
where F1 is the electric power output, F2 is the total heat flow from the source, F3 is the
conversion efficiency determined by the ratio of F1 to F2, H1(x) is a generalized Heaviside step
function that represents local integration over the resistor domain with a value of 1 if x ∈ ΩL,
and H2(x) is a two-dimensional vector that represents local integration over the electrode
domain to calculate the input heat flow from the heat flux vector. Since the direction of heat
flow is almost completely decided by the boundary conditions, it does not vary in the process
of optimization. Thus, F2 cannot be non-smooth during optimization although its expression
includes an absolute value. Note that the above objective functions are formulated as the
integration over the whole domain since the sensitivity analysis of the proposed methodology
is based on the assumption that the objective functions have this form.
The optimization problem of the thermoelectric domain is formulated by setting F1 or F3
multiplied by −1 as the objective function with the addition of a volume constraint that limits
the total material cost:
minimizeΩTE
J(ΩTE, V, T ) = −F1(ΩTE, V, T ) (12)
or
minimizeΩTE
J(ΩTE, V, T ) = −F3(ΩTE, V, T ) (13)
where
∫ΩTE
dx ≤ UV (14)
where UV is the upper limit of the volume.
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10 A. TAKEZAWA AND M. KITAMURA
2.4. Topology optimization
The topology optimization method is used to optimize the geometry of the thermoelectric
domain ΩTE, because this method can perform more fundamental optimizations over arbitrary
domains including shape and topology, viz. the number of holes. The fundamental idea is to
introduce a fixed, extended design domain D that includes a priori, the optimal shape Ωopt
and the utilization of the following characteristic function:
χ(x) =
1 if x ∈ Ωopt
0 if x ∈ D \ Ωopt
(15)
Using this function, the original design problem of Ω is replaced by a material distribution
problem incorporating a physical property, χA, in the extended design domain D, where A is a
physical property of the original material of Ω. Unfortunately, the optimization problem does
not have any optimal solutions [34]. A homogenization method is used to perform the relaxation
of the solution space [17, 34]. In this way, the original material distribution optimization
problem with respect to the characteristic function is replaced by an optimization problem
of the “composite” consisting of the original material and a material with very low physical
properties, e.g. Young’s modulus or thermal conductivity, mimicking voids with respect to the
density function. This density function represents the volume fraction of the original material
and can be regarded as a weak limit of the characteristic function. In the optimization problem,
the relationship between the material properties of the composite and the density function
must be defined. The most popular approach, which sets a penalized proportional material
property [29, 31], is the “solid isotropic material with penalization” (SIMP) method. In the
case of the elasticity problem, the physical and mechanical backgrounds of the methodology
were clarified employing the homogenization method [30]. In this paper, employing the concept
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 11
of the SIMP method, the relationships between the three material properties of the composite
used in thermoelectric analysis (i.e., the Seebeck coefficient α, electrical conductivity σ and
thermal conductivity λ) and the density function are set according to the following simple
equation with the penalized material density:
α∗ = ρpααo (16)
σ∗ = ρpσσo (17)
λ∗ = ρpλλo (18)
with
0 ≤ ρ(x) ≤ 1, x ∈ ΩTE (19)
where the upper suffix ∗ signifies that the material property relates to the composite, the lower
suffix o to the original material, and pα, pσ, and pλ are positive penalization parameters. The
above density function is only set over thermoelectric domain ΩTE which can be regarded as
domain D in Eq.(15). Note that, the above relationships between the density function and
physical properties are simply artificial numerical interpolations.
2.5. Sensitivity analysis
To perform optimizations, the method of moving asymptotes (MMA) [32] is used, which
requires first-order sensitivity analysis of the objective function with respect to the design
variable ρ. Since the derivation is quite lengthy, only the results are shown here and the
detailed derivation is outlined in the Appendix.
The two adjoint variables p and q are introduced to calculate the sensitivity of the objective
function, which depends on the two state variables V and T . The sensitivity of the objective
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12 A. TAKEZAWA AND M. KITAMURA
function with respect to function ρ is represented as an independent type of objective function:
J ′(ρ) =− σ′(ρ)∇V + (α′(ρ)σ + ασ′(ρ))∇T · (∇p−∇V q)
− (α′(ρ)σ + ασ′(ρ))∇V T + (2αα′(ρ)σ + α2σ′(ρ))T∇T + λ′(ρ)∇T · ∇q
(20)
To calculate the sensitivity of the electric power output F1, the adjoint variables p and q must
be obtained by solving the following coupled adjoint equations:
∇ · −σ∇p− ασT∇q + σ(2∇V + α∇T )q − 2H1σ∇V −H1ασ∇T = 0
p = 0 on ΓDV
(21)
∇ · −(α2σT + λ)∇q + ασ∇V q − ασ∇p+H1ασ∇V
+ ασ∇V + (α2σ + λ′(T ))∇T + (α′(T )σ + ασ′(T ))∇V T
+ (2αα′(T )σ + α2σ′(T ))T∇T · ∇q
− α′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇V q
=− σ′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇p
q = 0 on ΓDT
(22)
Since the physical properties of thermoelectric material, α, σ, and λ, are strongly dependent
on temperature, their derivatives with respect to T , α′(T ), σ′(T ), and λ′(T ) are included in
Eq.(22).
To calculate the sensitivity of the conversion efficiency F3, the sensitivity of the total heat
flow from the source F2 is first calculated. In this case, the adjoint variables p and q are
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 13
solutions of the following coupled adjoint equations:
∇ · (−σ∇p− ασT∇q + σ(2∇V + α∇T )q ∓ C1ασTH2) = 0
p = 0 on ΓDV
(23)
∇ · −(α2σT + λ)∇q + ασ∇V q − ασ∇p∓ C1(α2σT + λ)H2
+ ασ∇V + (α2σ + λ′(T ))∇T + (α′(T )σ + ασ′(T ))∇V T
+ (2αα′(T )σ + α2σ′(T ))T∇T · ∇q
− α′(T )∇V + (α′(T )σ + ασ′(T ))∇T ) · ∇V q
+ (σ′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇p
=∓ C1H2 · (ασ∇V + α2σ∇T )
q = 0 on ΓDT
(24)
where
C1 =
1 if
∫H2 · qdx ≥ 0
−1 if
∫H2 · qdx < 0
(25)
Using the above sensitivities, the sensitivity of the conversion efficiency F2 is calculated as
follows:
F ′2(ρ) =
(F1
F3
)′
=F ′1(ρ)F3 − F1F
′3(ρ)
F32 (26)
All the above sensitivities apply when the objective functions F1, F2 and F3 are without a
negative sign. Thus, they are used in the optimization procedure after multiplying by −1.
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14 A. TAKEZAWA AND M. KITAMURA
3. Numerical implementation
3.1. Algorithm
The optimization procedure is performed using a basic algorithm composed of a sensitivity
calculation and updating design variable using MMA. However, in the optimization of the
thermoelectric generator, the electric power output and the conversion efficiency are dependent
on the electrical conductivity of the material of the resistor domain in addition to the design
variable. Their maximum value can be obtained when the resistor values of the resistor
domain and the thermoelectric material domain are nearly equal (see e.g [1, 2]). However, the
resistor value of the thermoelectric material domain varies during the optimization process.
For this reason, the electrical conductivity of the resistor domain needs to be adjusted to the
optimal value in the iteration of each optimization. Thus, the inner optimization of electrical
conductivity of the resistor material is inserted into the optimization algorithm. Since the
inner optimization problem is an unconstrained problem with a single design variable, it can
easily be solved using the golden section search method. Based on the above, the optimization
algorithm is as shown in Fig.3.
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 15
Set an initial value of density function ρ
Calculate the state variables V and T by solving Eqs.(5) and (6)using the finite element method.
Calculate the objective function and the constraints.
Update the electrical conductivity of the resistor materialusing the golden section search method.
Converged?
Calculate the adjoint variables p and q by solving Eqs.(20)-(25)using the finite element method.
Calculate the sensitivities of the objective function and the constraint.
Converged?
End
Yes
No
Yes
No
Figure 3. The flow chart of the optimization algorithm
3.2. Finite element analysis for the equations of state
Finite element analysis is used in this study to solve the equations of state. To perform finite
element analysis, the equations of state in Eqs.(3) and (4) are first formulated in following
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16 A. TAKEZAWA AND M. KITAMURA
weak form: ∫Ω
−j(V, T ) · ∇vdx = 0 (27)∫Ω
−q(V, T ) · ∇wdx+
∫Ω
j(V, T ) · ∇V wds = 0
for V ∈ X1, ∀v ∈ X1, T ∈ X2, ∀w ∈ X2 (28)
where X1 and X2 are Sobolev spaces for functions satisfying the Dirichlet boundary conditions
and v and w are the test functions. The internal approximations of these equations are then
formulated as follows: ∫Ω
−jh · ∇vhdx = 0 (29)∫Ω
−qh · ∇whdx+
∫Ω
jh · ∇Vhwhds = 0 (30)
The lower suffix h indicates the discretized values. For simplicity of explanation, the discretized
value of the state variables are approximated using the shape functions Nj of the first order
Lagrange finite elements corresponding to j-th nodes as follows:
Vh(x) =Nd∑j=1
Nj(x)Vh(x) (31)
Th(x) =Nd∑j=1
Nj(x)Th(x) (32)
where xj is the position of j-th node. The discretized current and heat flux are also formulated
as follows:
jh(x) = σh(x)(−∇Vh(x)− αh(x)∇Th(x)) (33)
qh(x) = −αh(x)jh(x)− λh(x)∇Th(x) (34)
Finally, by combining the state equations into one equation and introducing the discretized
test functions v(x) = w(x) = Ni(x), they are formulated in their vector and matrix form as
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 17
follows:
KU = b (35)
where
K =
KV V KV T
KTV KTT
(36)
KV V =
(∫Ω
σh∇Nj · ∇Nidx
)1≤i,j≤Nd
(37)
KV T =
(∫Ω
αhσh∇Nj · ∇Nidx
)1≤i,j≤Nd
(38)
KTV =
(∫Ω
αhσh∇Nj · ∇Nidx
)1≤i,j≤Nd
(39)
KTT =
(∫Ω
(αh2σh + λh)∇Nj · ∇Nidx
)1≤i,j≤Nd
(40)
U =((Vh(xj))1≤j≤Nd
, (Th(xj))1≤j≤Nd
)(41)
b =
(0,
(∫Ω
σh(∇NiVh · ∇NiVh + αh∇NiVh · ∇NiTh)Nidx
)1≤i≤Nd
)(42)
Since the discretized equation of state in Eq.(35) are nonlinear because of the dependency
of the coefficient on the state variables, an iterative approach must be used to solve them.
The residual minimization approach based on the Newton method is used in this study. The
residual of the discretized equation of state in Eq.(35) is formulated as follows:
R = (RV ,RT ) = KU − b (43)
Using the Newton method, the update of the discretized state vector at the n-th iteration Un
is performed by the following equation.
Un+1 = Un +∆Un = Un −(∂R(Un)
∂U
)−1
R(Un) (44)
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18 A. TAKEZAWA AND M. KITAMURA
The increment ∆Un is calculated by solving the following linear system:
∂R(Un)
∂U∆Un = −R(Un) (45)
The matrix ∂R(Un)∂U is calculated as follows:
∂R(Un)
∂U=
∂RV
∂V∂RV
∂T
∂RT
∂V∂RT
∂T
(46)
∂RV
∂V=
(∫Ω
σh∇Nj · ∇Nidx
)1≤i,j≤Nd
(47)
∂RV
∂T=
(∫Ω
αhσh∇Nj · ∇Ni + (α′h(T
nh )σh + αhσ
′h(T
nh ))T
nh ∇Nj · ∇Nidx
)1≤i,j≤Nd
(48)
∂RT
∂V=
(∫Ω
αhσh∇Nj · ∇Ni − (2σh∇Nj · ∇NjVh + αh∇Nj · ∇Nj)Nidx
)1≤i,j≤Nd
(49)
∂RT
∂T=
∫Ω
(αh2σh + λh + (2α′
h(Tnh )αhσh + αh
2σ′h(T
nh ) + λ′
h(Tnh ))T
nh )∇Nj · ∇Ni
− (σ′h(T
nh )∇NjVh · ∇NjVh + (σ′
h(Tnh )αh + σhα
′h(T
nh ))∇NjVh · ∇NjTh
+ σhαh∇NjVh · ∇Nj)Nidx
1≤i,j≤Nd
(50)
The adjoint equations in Eqs.(21)-(25) are solved by the finite element method in the same
way. Note that the material properties α, σ, and λ are independent of adjoint variables p and
q, and the equations are linear. Thus, no iterations are required to obtain a solution to the
adjoint equations.
3.3. Setting the density function in the finite element method
The density function used in the SIMP method is usually set over the domain as realized by
the finite element mesh in the finite element analysis. A typical choice of density function
is a piecewise discontinuous function over each finite element, and the whole finite element
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 19
mesh is constructed with consideration of the required resolution of the density function in
mind. Moreover, a square mesh is a useful structure in discretizing the density function, as
we can easily resolve the optimal configuration as a bitmap graphic from an optimal density
distribution. On the other hand, in the analyses of coupled equations of state with strong
non-linearities like Eqs.(5) and (6), obtaining a stable convergence of solutions in a reasonable
computational time is in general quite sensitive to how fine the finite element mesh is. Thus,
they are preferably set independently to achieve both a required resolution and a stable short-
time finite element analysis. The need for the such an approach is discussed in [35].
To achieve this, in this study, ρ is assumed as a C0 continuous function and sets the equally-
spaced control point of the density function independently of the finite element mesh, as shown
in Fig.4. The density function is constructed as the linear interpolation function of the values
of the function at these control points. Finite element analysis is then performed using the
density function in an arbitrarily set finite element mesh. The update of the density function
is performed by updating the function value of the control points based on the sensitivity
evaluated at these control points.
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20 A. TAKEZAWA AND M. KITAMURA
Control point of density functionEvaluation point of sensitivityControl point of density functionEvaluation point of sensitivityFinite element mesh
Grid fordensity function
Figure 4. Relationship between control points of the density function and the evaluation points of the
sensitivity
3.4. Filtering method
A topology optimization has a fundamental numerical instability, the so-called checker board
problem [36, 37], which is caused by errors in the sensitivity analysis. Since the evaluation
points involved in calculating the sensitivity are decided, in our method, independently of the
finite element mesh, these errors can be more serious than in normal topology optimizations.
Thus, the projection method is used as a filter to reduce the error effect [38]. In addition
to the density function, this method sets the design variables and “projects” these onto the
density function using a projection function. By adjusting the effective range and shape of the
function, the checkerboard problem can be avoided. With this method, a new projected density
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 21
function µ is established, and its value, calculated at the i-th control point, is as follows:
µ(xi) =
∑j∈Si
ρ(xj)wi(xj − xi)∑j∈Si
wi(xj − xi)(51)
where:
Si = j | xj ∈ Ωpi (52)
Ωpi(xi) = x | ||x− xi|| ≤ rmin, x ∈ D (53)
where xi is the location of the i-th control point, Ωpi is the effective circular area of the
projection function wi set on the i-th control point, Si is the set of indices for the control
points in Ωpi, ρ(xj) is the value of the original density function at the j-th control point, and
rmin is the radius of the effective area of the projection function. The function wi is a linear
weighting function defined as:
w(x− xi) =
rmin − ||x− xi||
rminif x ∈ Ωpi
0 if x ∈ D \ Ωpi
(54)
The above functions are calculated at each control point of the domain of the density function
and µ is used as a new filtered density function. In the original paper [38], the discretized
design variable function was set on the nodes of the finite element mesh. In contrast, in this
study the function is independent of the finite element mesh set, and the filtered new density
function is evaluated at the same control points as the original density function. Moreover,
since the approximated Heaviside step function is not introduced, the filtering effect is the
same as that for the density filter proposed in [39]. However, the filtering method used here is
described as a projection method, since it is a more comprehensive methodology.
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22 A. TAKEZAWA AND M. KITAMURA
4. Numerical example
The following numerical examples are provided to establish the utility of the proposed method.
In all examples, finite element analysis is performed using the commercial software COMSOL
Multiphysics 3.5a for quick implementation of the proposed methodology, and to first solve
the equations of state and adjoint with a multi-core processor. All finite element analysis is
performed using quadratic elements. The penalization parameter of the SIMP method used in
Eqs.(16)-(18), pα, pσ and pλ, are all set to unity. The electric power output and the conversion
efficiency are designated by EC and CE for short in some figures showing results.
4.1. Material properties
The material properties used in the numerical examples are first determined. (Bi0.2Sb0.8)2Te3
and Bi2(Te0.97Se0.03) are used as p-type and n-type materials. For all temperature-dependent
quantities of the Seebeck coefficient α, electrical conductivity σ and thermal conductivity λ,
the material properties are taken into account from experimental data of [16], which have
been plotted in Fig.5. The electrical power factor as calculated using α2σ, an important
index in terms of power generation, and the figure of merit as calculated using α2σ/λ, an
important index in terms of conversion efficiency, are presented. The temperature-dependent
characteristics are considered only for thermoelectric materials. The material chosen for the
electrode domain is copper, giving the associated properties, α = 6.5 × 10−6(V/oC), σ =
5.9×104(S/mm), λ = 3.5×10−1(W/mm/oC). The material for the resistor domain is a virtual
material with no Seebeck effect but the same thermal conductivity as copper and the optimized
electrical conductivity as discussed in Section 3.1.
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 23
5x10-5
1x10-4
1.5x10-4
2x10-4
2.5x10-4
0 50 100 150 200 250 300 350
p-typen-type
Temperature (oC)
See
beck
coe
ffic
ient
(V
/o C)
(a) Seebeck coefficient α
3x101
4x101
5x101
6x101
7x101
8x101
9x101
1x102
1.1x102
0 50 100 150 200 250 300 350
p-typen-type
Temperature (oC)
Ele
ctri
cal c
ondu
ctiv
ity
(S/m
m)
(b) Electrical conductivity σ
8x10-4
1x10-3
1.2x10-3
1.4x10-3
1.6x10-3
1.8x10-3
2x10-3
0 50 100 150 200 250 300 350
p-typen-type
Temperature (oC)
The
rmal
con
duct
ivit
y (W
/mm
/o C)
(c) Thermal conductivity λ
0
5x10-7
1x10-6
1.5x10-6
2x10-6
2.5x10-6
0 50 100 150 200 250 300 350
p-typen-type
Temperature (oC)
Ele
ctri
c po
wer
fac
tor
(W/o C
2 /mm
)
(d) Electrical power factor α2σ
0
5x10-4
1x10-3
1.5x10-3
2x10-3
2.5x10-3
3x10-3
0 50 100 150 200 250 300 350
p-typen-type
Temperature (oC)
Figu
re o
f m
erit
(1/
o C)
(e) Figure of merit α2σ/λ
Figure 5. Physical properties of thermoelectric materials
4.2. Validation of sensitivity analysis
The validity of the analytical sensitivity derived in Eqs.(20)-(26) was established by comparing
the results with sensitivities obtained numerically. The simple 1D model used in the analysis
is shown in Fig.6 where the domain is composed of thermoelectric domains with p-type and
n-type material, the electrode domains and a resistor domain, each with a 1mm2 cross-
sectional area. The temperatures for both ΓDThand ΓDTc were set to one of two cases:
(Th, Tc) = (150, 50), (300, 200) and (300, 50).
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24 A. TAKEZAWA AND M. KITAMURAΩL ΩTE-p5mm 5mm 20mm 5mmxo0DVΓ ΩE-pccDTΓ ΩE-ph ΩE-nh ΩTE-n5mm 20mm ΩE-nc5mm 0 cDV ,DTΓhDTΓFigure 6. 1D domain for sensitivity analysis
The domain was discretized into 0.5mm 1D finite elements. The value of the density function
was set to 0.5 in the thermoelectric domain. The analytical sensitivities of the electric power
output and the conversion efficiency in Eqs.(20)-(26) were first calculated. Then, the numerical
sensitivities were calculated with a tolerance of 1×10−5 at 80 points in the center of the finite
elements in the thermoelectric material domain. Both sets of data were normalized and are
shown in Fig.7. The curve of analytical sensitivity coincided with the discrete plot of numerical
sensitivities in each temperature condition, which established the validity of the proposed
sensitivity analysis.
Some physical considerations can be obtained from these sensitivity plots. The first
characteristic is that each sensitivity state of the electric power output and the conversion
efficiency has slightly different shapes under different temperature conditions. This might come
from the temperature dependency of the thermoelectric materials and lead to slightly different
optimal configurations under different temperature conditions. According to the plot of the
electrical power factor and figure of merit shown in Fig.5, the material properties which strongly
affect the electric power output and the conversion efficiency, the lower the temperature, the
higher the properties become. This can cause higher sensitivity in low temperature areas.
The second characteristic is that in each temperature condition, both sensitivity curves have
similar outlines, which are higher in their absolute values in the low temperature domain. In
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 25
an extreme case, the n-type material has an almost identical shape. Consequently, the optimal
configurations obtained by the proposed methodology under the same temperature conditions
will result in similar shapes, even when different objective functions are used.
-1
-0.9
-0.8
-0.7
-0.6
-0.5
0 5 10 15 20
Analytical sensitivity of EP
Numerical sensitivity of EP
Analytical sensitivity of CE
Numerical sensitivity of CE
x
Sens
itiv
ity
(a) p-type,
Tc = 50oC, Th = 150oC
-1
-0.998
-0.996
-0.994
-0.992
-0.99
-0.988
-0.986
30 35 40 45 50
Analytical sensitivity of EPNumerical sensitivity of EPAnalytical sensitivity of CENumerical sensitivity of CE
x
Sens
itiv
ity
(b) n-type, Tc =
50oC, Th = 150oC
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
0 5 10 15 20
Analytical sensitivity of EPNumerical sensitivity of EPAnalytical sensitivity of CENumerical sensitivity of CE
x
Sens
itiv
ity
(c) p-type, Tc =
200oC, Th = 300oC
-1
-0.9995
-0.999
-0.9985
-0.998
-0.9975
30 35 40 45 50
Analytical sensitivity of EPNumerical sensitivity of EPAnalytical sensitivity of CENumerical sensitivity of CE
x
Sens
itiv
ity
(d) n-type, Tc =
200oC, Th = 300oC
-1
-0.8
-0.6
-0.4
-0.2
0
0 5 10 15 20
Analytical sensitivity of EPNumerical sensitivity of EPAnalytical sensitivity of CENumerical sensitivity of CE
x
Sens
itiv
ity
(e) p-type,
Tc = 50oC, Th = 300oC
-1
-0.998
-0.996
-0.994
-0.992
-0.99
30 35 40 45 50
Analytical sensitivity of EPNumerical sensitivity of EPAnalytical sensitivity of CENumerical sensitivity of CE
x
Sens
itiv
ity
(f) n-type, Tc =
50oC, Th = 300oC
Figure 7. Comparison between analytical sensitivities and numerical sensitivities
4.3. Optimization of the electric power output
As a first numerical study of the optimization of thermoelectric generators, an optimization
aimed at the maximization of the electric power output is performed. The design domain
composed of two reverse L-shaped thermoelectric materials and electrodes and a resistor is
illustrated in Fig.8. The thickness of the domain is set to 1mm, to emulate a plate-like device.
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26 A. TAKEZAWA AND M. KITAMURA
Although the layouts of the p-type and n-type materials have been decided for ease of 2D
modeling and are impractical in actuality, the results can be applied to general devices of
bi-thermoelectric materials set parallel, as depicted in Fig.1. The reverse L shape might differ
considerably from the conventional thermoelectric generator with the straight bar shaped
thermoelectric material. Since the proposed methodology is intended for use in designing the
shape of thermoelectric devices under complicated thermal conditions, such strange shapes are
used in the numerical example. In the design of conventional straight bar-like thermoelectric
materials with simple heat flow, the conventional sizing optimization is adequately effective.
The density function in ΩTE is optimized by minimizing the objective function in Eq.(12), i.e.
total electric power applied to ΩL, which multiplied by −1. Altogether 15000 control points
of the density function are set in ΩTE at regular intervals of 0.5 mm. The analysis domain is
discretized by 2240 triangular elements.
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 27
ΩE-pcΩTE-p
ΩL 45mm5mm 5mm 10mm 10mm
50mm0DVΓ
10mm10mmΩTE-n10mm10
mmhDTΓ
ΩE-ncΩE-phΩE-nh 0 cDV ,DTΓ
cDTΓ5mm5mm
Figure 8. A 2D design domain
4.3.1. Optimization under different volume constraints The optimization is performed under
different volume constraints, which are set at 25%, 50% and 75% of the full volume of ΩTE. The
common volume constraint is set at the total volume of the p-type and n-type thermoelectric
materials. Thus, volume rates of these materials are valid during optimization. The initial value
of the density function is uniformly set to 0.25, 0.5 or 0.75 in ΩTE according to the volume
constraint. The temperatures of ΓDThand ΓDTh
are set to 50oC and 150oC.
To confirm that the validity of the results of the finite element analysis was maintained
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28 A. TAKEZAWA AND M. KITAMURA
during the optimization process, we needed to ensure enough converged results were obtained
in the non-linear finite element analysis of the relationship between electric power output and
the rate of heat flow in the electrode domain. Theoretically, the electric power output in Eq.(9)
is equal to the lost heat that is calculated as the difference between the heat flow from the hot
junction and the heat flow rejected at the cold junction. The former is calculated as the sum
of the average value of y directional heat flow on the domains ΩE-ph and ΩE-nh by setting the
vector H2 as follows:
H2(x) =
[0, 0.2] if x ∈ ΩE-ph ∪ ΩE-nh
0 if x ∈ Ω \ (ΩE-ph ∩ ΩE-nh)
(55)
In the same way, the latter is calculated as the sum of the average value of x directional heat
flow in the domains ΩE-pc and ΩE-nc, by setting the vector H2 as follows:
H2(x) =
[0.2, 0] if x ∈ ΩE-pc ∪ ΩE-nc
0 if x ∈ Ω \ (ΩE-pc ∩ ΩE-nc)
(56)
The ratio between the electric power output and the lost heat (designated as LH) in
the optimization under 50% volume constraint is shown in Fig.9. Almost the same values
were maintained during the optimization. The same results were obtained in other volume
constraints. Thus, the validity of the results obtained by the finite element analysis during the
optimization was confirmed.
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 29
0.99985
0.99986
0.99987
0.99988
0.99989
0.9999
0.99991
0.99992
0.99993
0 10 20 30 40 50
Iteration
EP/
LH
Figure 9. The ratio between the electric power output and the lost heat
Figure 10 shows the optimal configuration obtained after 50 iterations in each case. The
p-type and n-type materials are shown in separate figures after rotating n-type material
around 180 degrees. In all cases, the volume constraints finally became active. Figure 11 shows
the convergence history of the objective functions. The objective functions for each volume
constraint are compared in Table I. Table II gives the ratio of volumes of both thermoelectric
materials to the total volume in the optimal configurations.
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30 A. TAKEZAWA AND M. KITAMURA
(a) p-type, 25% volume con-
straint
(b) n-type, 25% volume con-
straint
(c) p-type, 50% volume con-
straint
(d) n-type, 50% volume con-
straint
(e) p-type, 75% volume con-
straint
(f) n-type, 75% volume con-
straint
Figure 10. Optimal configurations under different volume constraints
Table I. Comparison of objective functions
V.C. 25% 50% 75% Full volume
Objective function −1.31× 10−3 −2.30× 10−3 −3.03× 10−3 −3.38× 10−3
Ratio of objective functionto the full volume one 38.9% 68.1% 89.7% 100%
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 31
Table II. The ratio of the volume of p-type and n-type materials to total volume
V.C. 25% 50% 75%
p-type 58.9% 60.0% 56.1%
n-type 41.1% 40.0% 43.9%
-3.5x10-3
-3x10-3
-2.5x10-3
-2x10-3
-1.5x10-3
-1x10-3
-5x10-4
0
0 10 20 30 40 50
V.C. 25%V.C. 50%V.C. 75%
Iteration
Obj
ecti
ve f
unct
ion
(W)
Figure 11. Convergence history
It seems that in each case clear optimal configurations were obtained after stable
convergences. The tendency is for smooth heat flow and smooth current flow. As with fluid
flow, the heat flow will stagnate at sharp corners and not contribute to electrical generation.
According to Table I, lower objective functions were obtained for higher volume constraints.
Since the Dirichlet boundary conditions were introduced in this optimization, the gradient of
temperature is decided independently of the material distribution. In that case, to increase the
heat flow that contributes to power generation, there should be more material. However, under
the full-volume condition, the heat flow stagnated on the corner as mentioned previously. Thus,
the optimal configurations with the volume constraint achieved more efficient power generation
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32 A. TAKEZAWA AND M. KITAMURA
than the full-volume configuration. In all cases, the ratio of the volume of p-type material to the
total volume is higher than the same ratio for n-type material. Two heat flows for the p-type
and n-type materials resulted from the hot junction between these two materials, while the
electrical flow was through both materials from the ground to the other side. Thus, the power
generation and electrical conductivity can be adjusted in both materials within the limited
volume. Since the n-type material has higher electric power factor and electrical conductivity
than p-type material according to Fig.5, more p-type material seems to be required to create
a balance.
Fortunately, a gray unclear domain with intermediate densities was not obtained even though
the proposed methodology is constructed employing the SIMP method in Fig.10. The reason
seems to be that the thermoelectric optimization problem is basically dominated by the heat
conduction problem as we mentioned above. Previous research in this field has usually obtained
relatively clear shapes (e.g., [40, 41])
4.3.2. Optimization under different temperature conditions This section focuses on the
difference between the optimal configurations obtained given different temperature settings.
The temperatures for both ΓDThand ΓDTc are set to one of two cases: (Th, Tc) = (300, 200)
and (300, 50). The volume constraint is set to 50% of the full volume of ΩTE. The initial value
of the density function is uniformly set to 0.5 in ΩTE according to the volume constraint.
Figure 12 shows the optimal configuration obtained after 50 iterations in each case. In all
cases, the volume constraints finally became active, as in the previous example. Table III
gives the ratio of the volumes of both thermoelectric materials to the total volume and the
value of the objective function in the optimal configurations. The results under the condition
(Th, Tc) = (300, 50) calculated in the previous example are also shown in this table.
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 33
(a) p-type, (Th, Tc) =
(300, 200)
(b) n-type, (Th, Tc) =
(300, 200)
(c) p-type, (Th, Tc) =
(300, 50)
(d) n-type, (Th, Tc) =
(300, 50)
Figure 12. Optimal configurations under different temperature conditions
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34 A. TAKEZAWA AND M. KITAMURA
Table III. The ratios of the volumes of p-type and n-type materials to total volume and the optimum
value of the objective function.
(Th, Tc) (150, 50) (300, 200) (300, 50)
p-type 58.9% 60.0% 56.1%
n-type 41.1% 40.0% 43.9%
Objective function −2.3× 10−3 −1.02× 10−3 −9.88× 10−3
Although the shapes obtained were almost the same, Figs.10 and 12 and Table III show
the difference in the volume of both materials under different temperature conditions. The
temperature dependency of the physical properties of the thermoelectric material can induce
differences in the volume allocation of these materials for different temperature ranges. In both
types of materials, the electric power factor is lower in the low temperature domain than in
the high temperature domain. This is why the higher electric power output was obtained in
the high temperature case. The highest output was of course obtained in the case where the
temperature gap was the highest.
4.4. Optimization of the conversion efficiency
The optimization of another important performance index, the conversion efficiency, is dealt
with next. It is maximized during optimization using the objective function in Eq.(13). The
design domain illustrated in Fig.8 is used as in the first example.
4.4.1. Optimization under the different volume constraints The optimization is performed
under different volume constraints, which are set 25%, 50% and 75% of the full volume of
ΩTE. The initial value of the density function is uniformly set to 0.25, 0.5 or 0.75 in ΩTE
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 35
according to the volume constraint. The temperatures of ΓDTcand ΓDTh
are set to 50oC and
150oC.
Figure 13 shows the optimal configuration obtained after 50 iterations in each case. In all
cases, the volume constraints finally became active, similar to the optimization of the electric
power output. Figure 14 shows the convergence history of the objective functions. The objective
functions for each volume constraint are compared in Table IV. Table V gives the ratio of the
volumes of both the thermoelectric materials to the total volume in the optimal configurations.
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36 A. TAKEZAWA AND M. KITAMURA
(a) p-type, 25% volume con-
straint
(b) n-type, 25% volume con-
straint
(c) p-type, 50% volume con-
straint
(d) n-type, 50% volume con-
straint
(e) p-type, 75% volume con-
straint
(f) n-type, 75% volume con-
straint
Figure 13. Optimal configurations under different volume conditions
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 37
-5x10-2
-4x10-2
-3x10-2
-2x10-2
-1x10-2
0
0 10 20 30 40 50
V.C. 25%V.C. 50%V.C. 75%
Iteration
Obj
ecti
ve f
unct
ion
Figure 14. Convergence history
Table IV. Comparison of objective functions
V.C. 25% 50% 75% Full volume
Objective function −4.06× 10−2 −4.17× 10−2 −4.17× 10−2 −4.04× 10−2
Ratio of objective functionto the full volume one 100.3% 103.0% 103.1% 100%
Table V. The ratio of the volume of p-type and n-type materials to the total volume
V.C. 25% 50% 75%
p-type 58.9% 61.7% 58.3%
n-type 41.1% 38.3% 41.7%
In each case, clear optimal configurations were obtained after stable convergence, as with
the first objective function. These shapes were quite similar to the optimal configurations of
the first example in Fig.10. The optimal configurations of this example have the same physical
aspects as those in the first example, which are for smooth heat flow and current flow. Only
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38 A. TAKEZAWA AND M. KITAMURA
small differences can be found in the optimal configurations. For example, in Fig.13(c) more p-
type material is located on the cold junction side than on the hot junction side. This is because
the figure of merit for p-type material depends more strongly on the temperature than that for
n-type material, as shown in Fig.5. Thus, the volumes of both sides are adjusted to generate
the un-wasted current flow in the p-type material. Although the volume constraints become
active, no clear improvement in the objective function was observed with an increase in the
volume, unlike in the first example as shown in Table IV. Thus, more than 50% of the volume
can be regarded as redundant when only considering the conversion efficiency in this problem.
4.4.2. Optimization under different temperature constraints The difference between the
optimal configurations obtained under different temperature settings is focused on next, as
in the case of the electric power output maximization problem. The temperatures for both
ΓDThand ΓDTc are set to one of two cases: (Th, Tc) = (300, 200) and (300, 50). The volume
constraint is set to 50% of the full volume of ΩTE. The initial value of the density function is
uniformly set to 0.5 in ΩTE according to the volume constraint.
Figure 15 and Table VI show the optimal results. They correspond to Fig.12 and Table III
of the electric power output maximization problem.
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 39
(a) p-type, (Th, Tc) =
(300, 200)
(b) n-type, (Th, Tc) =
(300, 200)
(c) p-type, (Th, Tc) =
(300, 50)
(d) n-type, (Th, Tc) =
(300, 50)
Figure 15. Optimal configurations under different temperature conditions
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40 A. TAKEZAWA AND M. KITAMURA
Table VI. The ratio of the volume of p-type and n-type materials to the total volume and the optimum
value of the objective function.
(Th, Tc) (150, 50) (300, 200) (300, 50)
Volume ratio of p-type 61.7% 56.1% 59.8%
Volume ratio of n-type 38.3% 43.9% 40.2%
Objective function −4.27× 10−2 −1.28× 10−2 −5.83× 10−2
The differences in the shapes of the edges of p-type material can also be found in this
example. The difference in volume between the p-type material on the hot junction side and
on the cold junction side is smaller in the case of (Th, Tc) = (300, 200) than in the case of
(Th, Tc) = (150, 50), because the gap in the figure of merit of p-type material in the specified
temperature range is smaller as shown in Fig.5. The largest difference can be observed in
the case of (Th, Tc) = (300, 50), because the gap in the figure of merit is the highest. A
slight difference can also be found in the shape of n-type material, which has less of a gap
in the physical property than the p-type material. In Table VI, the volume ratio of p-type
material is larger in the case of (Th, Tc) = (150, 50) than in the case of (Th, Tc) = (300, 200),
while in Table III the opposite occurs. This can come from the temperature dependency of
thermal conductivity. The value of heat flow under Dirichlet boundary conditions with respect
to temperature, which should be small for high conversion efficiency, becomes larger as the
thermal conductivity of the material becomes large. In the low temperature range, p-type
material clearly has lower thermal conductivity than n-type material, while the difference
becomes smaller in the high temperature range. Thus, the p-type material was regarded as
effective for conversion efficiency in the lower temperature range and was accordingly allocated
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 41
a larger volume.
4.4.3. Comparison of the results obtained using the maximization of the electric power
output The performance of the optimal configuration obtained using the conversion efficiency
maximization is compared with that obtained using the electric power output optimization.
Table VII shows the comparison of the performances obtained under the condition of 50%
volume constraint and (Th, Tc) = (150, 50), (300, 200), (300, 50). In all temperature conditions,
the higher results were obtained in each performance index corresponding to the objective
function.
Table VII. The comparison of the performance of optimal configurations obtained using the different
objective functions
(Th, Tc) (150, 50) (300, 200) (300, 50)
Electric power maximization Electric power (W) 2.303 × 10−3 1.025 × 10−3 9.876 × 10−3
Coversion effeciency 4.146 × 10−2 1.283 × 10−2 5.806 × 10−2
Coversion effeciency maximization Electric power (W) 2.239 × 10−3 9.975 × 10−4 9.655 × 10−3
Coversion effeciency 4.168 × 10−2 1.284 × 10−2 5.832 × 10−2
The conversion efficiency is important when the heat flow is not produced by the temperature
gap, but rather directly by the heat flux. To confirm this, we re-analyzed each optimal
configuration obtained using different objective functions under the Neuman boundary
conditions for the temperature. The re-analysis was performed by replacing the boundary
ΓDThby the Neuman boundary condition in the design domain shown in Fig.8. The optimal
configurations used were those obtained under the conditions of (Th, Tc) = (150, 50) and 50%
volume constraint. The value of heat flux was set to five patterns, which were from 1.0× 10−3
to 5.0× 10−3 at 1.0× 10−3 intervals. Figure 16 shows the resulting electric power output. An
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42 A. TAKEZAWA AND M. KITAMURA
advantage of about 2.5%-3.5% was clearly obtained using the optimal configuration obtained
from the conversion efficiency optimization.
0
5x10-4
1x10-3
1.5x10-3
2x10-3
1.022
1.024
1.026
1.028
1.03
1.032
1.034
1.036
0 0.01 0.02 0.03 0.04 0.05 0.06
The result of EP maximization
The result of CE maximization
The ratio between two results
The
out
put e
lect
ric
pow
er (
W)
Ratio betw
een both electric powers
Input heat flux (W)
Figure 16. Comparison of the electric power output under the Neumann boundary condition
4.5. Optimization under the Neumann boundary condition
As a final example, the optimization under the Neumann boundary condition for temperature
was carried out. This was the direct optimization for the condition where the heat flow was
produced by a heat flux on the boundary. As in the last re-analysis numerical example, the
optimization was performed by replacing the boundary ΓDThby the Neumann boundary
condition in the design domain shown in Fig.8. Under this condition, the two types of objective
function in Eqs.(12) and (13) were almost equivalent, because the heat flow from the source
was fixed under the Newman boundary condition. The optimization was performed only for
maximizing the electric power output with the heat flux of 1.0× 10−3 and the various volume
constraints of 5%, 10%, 20%, 30%, 40% and 50% of the full volume of ΩTE. The initial value of
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 43
the density function was uniformly set to 0.05-0.5 in ΩTE according to the volume constraints.
Figure 17 shows the relationship between the volume of the optimal configurations and the
electric power output. Figure 18 shows the optimal configurations with the lowest and highest
volumes. The volume constraint did not become active and the results converged with very
low volumes. Moreover, because of the initial dependency of the problem, different optima
were obtained in each case. According to Fig.17 the higher the electric power the smaller the
volume becomes and the theoretical global optima seems to be an infinitely thin bar. A higher
temperature gap can be obtained in the structure with the smaller volume, which is reasonable
considering the physics of heat flow. However, these optimal configurations are impractical.
While the resistance value of the electrical load is adjusted to near the resistance value of the
thermoelectric material in the proposed method, the electrical load of the device is specified at
some level in the practical design of the thermoelectric generator. Thus, too thin a structure
will not be effective.
-3.95x10-4
-3.9x10-4
-3.85x10-4
-3.8x10-4
-3.75x10-4
-3.7x10-4
-3.65x10-4
-3.6x10-4
-3.55x10-4
25 30 35 40 45 50
Volume (mm2)
Obj
ecti
ve f
unct
ion
(W)
Figure 17. Comparison of the electric power output under the Neumann boundary condition
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44 A. TAKEZAWA AND M. KITAMURA
(a) p-type with the lowest
volume
(b) n-type with the lowest
volume
(c) p-type with the highest
volume
(d) n-type with the highest
volume
Figure 18. Optimal configurations under under the Neumann boundary condition
5. Conclusion
The topology optimization method was applied to the design problem of thermoelectric
generators. Coupled equations of state for the thermoelectric problem were first considered. An
analytical model subject to the equations of state was introduced that mimicked a closed circuit
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 45
composed of thermoelectric materials, electrodes and an electrical load. The total electric
power drawn by the resistor and the conversion efficiency were formulated as an objective
function. The proposed thermoelectric generator optimization method was implemented as a
geometrical topology optimization method, using the solid isotropic material with penalization
(SIMP) method. The relationship between the density function of the SIMP method and the
physical properties of the thermoelectric material were formulated by simple equations. The
sensitivity analysis of the objective function with respect to the density function and adjoint
equations required to calculate it, were formulated. Depending on the sensitivity, the density
function was updated using the method of moving asymptotes (MMA). Numerical examples
were provided to demonstrate the validity of our method.
In addition to optimal configurations, some mechanical aspects of the geometry
of thermoelectric generators were obtained from the numerical examples. First, and
fundamentally, smooth heat flows and current flows were the most important mechanical
aspect of optimal configurations in both types of optimization. However, the detailed shape
and volume of p-type and n-type materials should be adjusted, depending on the types
of objective function and thermoelectric material, to generate an optimal geometry. The
proposed methodology effectively helped this process. Second, optimal configurations were
slightly different according to temperature conditions. That is, the temperature dependence
of physical properties of materials used in this research had a slight effect on the optimal
configuration. However, depending on the materials, this effect could become large. Moreover,
the development of high performance thermoelectric materials is advancing and they can have
strong temperature dependency. In this event, the proposed method that derives detailed
optimization of the device can be effective in designing a thermoelectric generator using the
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46 A. TAKEZAWA AND M. KITAMURA
material. Third, there was no topology change and there were no voids in the examples studied
in this paper, even when the topology was optimized. We have thus confirmed mechanically
that voids are not required in optimal thermoelectric devices with a simple bent heating route.
However, in the design of the device under a more complicated heat condition, such as in
the case of multiple heat sources and multiple heat sinks, the topology would greatly affect
performance.
Following on from the work described in this paper, there are some opportunities for further
research, although challenging. First, the so-called Peltier electrical cooling devices are another
important application of thermoelectric material. Although the purposes of the devices are
different from those of the thermoelectric generators, the mechanical background is identical.
Thus, the proposed methodology can easily be applied to the design of these systems. Second,
more practical boundary conditions about the heat conduction problem, such as the convective
heat transfer boundary condition and the radiation heat transfer boundary condition, can be
introduced. Since such boundary conditions depend on the shape of the boundary, additional
techniques such as in [42] need to be considered in topology optimization. Third, the concept of
functionally graded material (e.g. [43]) can be applied to the design of thermoelectric material
(see Chapter 6 of [1]). Since topology optimization methodology is compatible with the design
of FGM [44], the proposed methodology can be extended to it.
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50 A. TAKEZAWA AND M. KITAMURA
APPENDIX
In this appendix, the detailed derivation of the sensitivity in Eq.(20) and the adjoint equations
in Eqs.(21)-(25) is outlined. The derivatives of the objective functions with respect to the
density function are based on the procedure shown in Chapter 5 of [45]. The word “derivative”
is used in the sense of the directional derivative. First, the electric vector j and the heat
flux vector q are represented using vectors A and B and matrixes V and M for simplicity as
follows:
j = −σ∇V − ασ∇T
= A ·U(57)
q = αTj − λ∇T
= −ασ∇V T − α2σT∇T − λ∇T
= B · V
(58)
where
A = [−σ,−ασ], B = [−ασ,−α2σ,−λ], U =
∇V
∇T
, V =
∇V T
T∇T
∇T
(59)
The general objective function of thermoelectric problem is defined as J(ρ) =∫Ωl(V, T )dx.
The derivative of this function in the direction θ is then
〈J ′(ρ), θ〉 =∫Ω
j′(V )〈V ′(ρ), θ〉dx+
∫Ω
j′(T )〈T ′(ρ), θ〉dx =
∫Ω
j′(V )vdx+
∫Ω
j′(T )wdx (60)
where v = 〈V ′(ρ), θ〉, w = 〈T ′(ρ), θ〉. Here, the equations of state in Eqs.(1) and (2) are
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 51
considered in the weak form as follows:
∫Ω
j · ∇pdx =
∫Ω
A ·U · ∇pdx = 0 (61)∫Ω
q · ∇qdx−∫Ω
j · ∇V qds =
∫Ω
B · V · ∇qdx−∫Ω
A ·U · ∇V qdx = 0 (62)
where p and q are test functions. Using p and q as adjoint states, the Lagrangian is formulated
as follows:
L(ρ, V, T, p, q) =
∫Ω
j(V, T )dx+
∫Ω
J · ∇pdx+
∫Ω
q · ∇qdx−∫Ω
J · ∇V qdx
=
∫Ω
j(V, T )dx+
∫Ω
A ·U · ∇pdx+
∫Ω
B · V · ∇qdx−∫Ω
A ·U · ∇V qdx
(63)
Using this, the derivative of the objective function can be expressed as
〈j′(ρ), θ〉 =⟨∂L
∂ρ(ρ, V, T, p, q), θ
⟩+
⟨∂L
∂V(ρ, V, T, p, q), 〈V ′(ρ), θ〉
⟩+
⟨∂L
∂T(ρ, V, T, p, q), 〈T ′(ρ), θ〉
⟩=
⟨∂L
∂ρ(ρ, V, T, p, q), θ
⟩+
⟨∂L
∂V(ρ, V, T, p, q), v
⟩+
⟨∂L
∂T(ρ, V, T, p, q), w
⟩ (64)
Consider the case where the second and third terms are zero. These terms are calculated as
follows:
⟨∂L
∂V, v
⟩=
∫Ω
j′(V )vdx+
∫Ω
A · 〈U ′(V ), v〉 · ∇pdx+
∫Ω
B · 〈V ′(V ), v〉 · ∇qdx
−∫Ω
A · 〈U ′(V ), v〉 · ∇V qdx−∫Ω
A ·U · ∇vqdx
=
∫Ω
j′(V )vdx+
∫Ω
A ·K · ∇pdx+
∫Ω
B ·M · ∇qdx
−∫Ω
A ·K · ∇V qdx−∫Ω
A ·U · ∇vqdx
(65)
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52 A. TAKEZAWA AND M. KITAMURA
⟨∂L
∂T,w
⟩=
∫Ω
j′(T )wdx+
∫Ω
A′(T ) ·U · ∇pwdx+
∫Ω
A · 〈U ′(T ), w〉 · ∇pdx
+
∫Ω
B′(T ) · V · ∇qwdx+
∫Ω
B · 〈V ′(T ), w〉 · ∇qdx
−∫Ω
A′(T ) ·U · ∇V qwdx−∫Ω
A · 〈U ′(T ), w〉 · ∇V qdx
=
∫Ω
j′(T )wdx+
∫Ω
A′(T ) ·U · ∇pwdx+
∫Ω
A ·L · ∇pdx
+
∫Ω
B′(T ) · V · ∇qwdx+
∫Ω
B ·N · ∇qdx
−∫Ω
A′(T ) ·U · ∇V qwdx−∫Ω
A ·L · ∇V qdx
=0
(66)
where
K = 〈U ′(V ), v〉, L = 〈U ′(T ), w〉, M = 〈V ′(V ), v〉, N = 〈V ′(T ), w〉 (67)
When the adjoint states p and q satisfy the above adjoint equations, the second and third
terms of Eq.(64) can be ignored. On the other hand, the derivatives of Eqs.(61) and (62) with
respect to ρ in the direction θ are
∫Ω
A′(ρ) ·U · ∇pθdx+
∫Ω
A′(T ) ·U · ∇p〈T ′(ρ), θ〉dx
+
∫Ω
A · 〈U ′(V ), v〉 · ∇pdx+
∫Ω
A · 〈U ′(T ), w〉 · ∇pdx
=
∫Ω
A′(ρ) ·U · ∇pθdx+
∫Ω
A′(T ) ·U · ∇pwdx+
∫Ω
A ·K · ∇pdx+
∫Ω
A ·L · ∇pdx
=0
(68)
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 53
∫Ω
B′(ρ) · V · ∇qθdx+
∫Ω
B′(T ) · V · ∇q〈T ′(ρ), θ〉dx
+
∫Ω
B · 〈V ′(V ), v〉 · ∇qdx+
∫Ω
B · 〈V ′(T ), w〉 · ∇qdx
−∫Ω
A′(ρ) ·U · ∇V qθdx−∫Ω
A′(T ) ·U · ∇V q〈T ′(ρ), θ〉dx
−∫Ω
A · 〈U ′(V ), v〉 · ∇V qdx−∫Ω
A · 〈U ′(T ), w〉 · ∇V qdx−∫Ω
A ·U · ∇vqdx
=
∫Ω
B′(ρ) · V · ∇qθdx+
∫Ω
B′(T ) · V · ∇qwdx+
∫Ω
B ·M · ∇qdx+
∫Ω
B ·N · ∇qdx
−∫Ω
A′(ρ) ·U · ∇V qθdx−∫Ω
A′(T ) ·U · ∇V qwdx
−∫Ω
A ·K · ∇V qdx−∫Ω
A ·L · ∇V qdx−∫Ω
A ·U · ∇vqdx
=0
(69)
Substituting Eqs.(65) and (66) into Eqs.(68) and (69) and combining them into one equation,
the following equation is obtained:∫Ω
j′(V )vdx+
∫Ω
j′(T )wdx
=
∫Ω
A′(ρ) ·U · ∇pθdx+
∫Ω
B′(ρ) · V · ∇qθdx−∫Ω
A′(ρ) ·U · ∇V qθdx
(70)
Substituting Eq.(70) into Eq.(60), the derivative of the objective function is
J ′(ρ) = A′(ρ) ·U · ∇p+B′(ρ) · V · ∇q −A′(ρ) ·U · ∇V q
= A′(ρ) ·U · (∇p−∇V q) +B′(ρ) · V · ∇q
(71)
Substituting Eq.(59) into Eq.(71) yields following equation:
J ′(ρ) =− σ′(ρ)∇V + (α′(ρ)σ + ασ′(ρ))∇T · (∇p−∇V q)
− (α′(ρ)σ + ασ′(ρ))∇V T + (2αα′(ρ)σ + α2σ′(ρ))T∇T + λ′(ρ)∇T · ∇q
(72)
Next, the adjoint equations are calculated from Eqs.(65) and (66). When the electric power
output in Eq.(9) is considered as the objective function, it is formulated using Eq.(59) as
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54 A. TAKEZAWA AND M. KITAMURA
follows:
J(ρ) =
∫Ω
j(V, T )dx =
∫Ω
H1j ·Edx = −∫Ω
H1A ·U · ∇V dx (73)
Thus,
∫Ω
j′(V )vdx = −∫Ω
H1A · 〈U ′(V ), v〉 · ∇V dx−∫Ω
H1A ·U · ∇vdx
= −∫Ω
H1A ·K · ∇V dx−∫Ω
H1A ·U · ∇vdx
(74)
∫Ω
j′(T )wdx = −∫Ω
H1A · 〈U ′(T ), w〉 · ∇V dx
= −∫Ω
H1A ·L · ∇V dx
(75)
Note that, the temperature dependency of the vector A can be ignored because the Heaviside
function H1 becomes zero in ΩTE . Substituting Eqs.(59), (67) and (74) into Eq.(65) gives
following equation
−∫Ω
H1A ·K · ∇V dx−∫Ω
H1A ·U · ∇vdx
+
∫Ω
A ·K · ∇pdx+
∫Ω
B ·M · ∇qdx−∫Ω
A ·K · ∇V qdx−∫Ω
A ·U · ∇vqdx
=
∫Ω
−σ∇p− ασT∇q + σ(2∇V + α∇T )q + 2H1σ∇V +H1ασ∇T · ∇vdx
=0
(76)
Converting the above equation from the weak to the strong form, the first adjoint equation is
obtained as follows:
∇ · −σ∇p− ασT∇q + σ(2∇V + α∇T )q − 2H1σ∇V +H1ασ∇T = 0 (77)
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 55
Substituting Eqs.(59), (67) and (75) into Eq.(66) the following equation is obtained:
−∫Ω
H1A ·L · ∇V dx+
∫Ω
A′(T ) ·U · ∇pwdx+
∫Ω
A ·L · ∇pdx
+
∫Ω
B′(T ) · V · ∇qwdx+
∫Ω
B ·N · ∇qdx
−∫Ω
A′(T ) ·U · ∇V qwdx−∫Ω
A ·L · ∇V qdx
=
∫Ω
−(α2σT + λ)∇q + ασ∇V q − ασ∇p+H1ασ∇V · ∇wdx
−∫Ω
ασ∇V + (α2σ + λ′(T ))∇T + (α′(T )σ + ασ′(T ))∇V T
+ (2αα′(T )σ + α2σ′(T ))T∇T · ∇qwdx
+
∫Ω
α′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇V qwdx
−∫Ω
σ′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇pwdx
=0
(78)
Converting the above equation from the weak to the strong form, the second adjoint equation
is obtained as follows:
∇ · −(α2σT + λ)∇q + ασ∇V q − ασ∇p+H1ασ∇V
+ ασ∇V + (α2σ + λ′(T ))∇T + (α′(T )σ + ασ′(T ))∇V T
+ (2αα′(T )σ + α2σ′(T ))T∇T · ∇q
− α′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇V q
=− σ′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇p
(79)
When the total heat flow from the source in Eq.(10) is considered as the objective function,
it is formulated as the objective function using Eq.(59) as follows:
J(ρ) =
∫Ω
j(V, T )dx =
∣∣∣∣∫ H2 · qdx∣∣∣∣ = ∫
Ω
C1H2 · qdx =
∫Ω
C1H2 · (B · V )dx (80)
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56 A. TAKEZAWA AND M. KITAMURA
where
C1 =
1 if
∫H2 · qdx ≥ 0
−1 if
∫H2 · qdx < 0
(81)
Thus,
∫Ω
j′(V )vdx =
∫Ω
C1H2 · (B · 〈V ′(V ), v〉)dx
=
∫Ω
C1H2 · (B ·M)dx
(82)
∫Ω
j′(T )wdx =
∫Ω
C1H2 · (B · 〈V ′(T ), w〉)dx
=
∫Ω
C1H2 · (B ·N)dx
(83)
The temperature dependency of the vector B can be ignored because the vector H2 becomes a
zero vector in ΩTE . Substituting Eqs.(59), (67) and (82) into Eq.(65) gives following equation
∫Ω
C1H2 · (B ·M)dx
+
∫Ω
A ·K · ∇pdx+
∫Ω
B ·M · ∇qdx−∫Ω
A ·K · ∇V qdx−∫Ω
A ·U · ∇vqdx
=
∫Ω
−σ∇p− ασT∇q + σ(2∇V + α∇T )q − C1ασTH2 · ∇vdx
=0
(84)
Converting the above equation from the weak to the strong form, the first adjoint equation is
obtained as follows:
∇ · −σ∇p− ασT∇q + σ(2∇V + α∇T )q − C1ασTH2 = 0 (85)
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GEOMETRICAL OPTIMIZATION OF THERMOELECTRIC GENERATORS 57
Substitutin Eqs.(59), (67) and (83) into Eq.(66) the following equation is obtained:∫Ω
C1H2 · (B ·N)dx+
∫Ω
A′(T ) ·U · ∇pwdx+
∫Ω
A ·L · ∇pdx
+
∫Ω
B′(T ) · V · ∇qwdx+
∫Ω
B ·N · ∇qdx
−∫Ω
A′(T ) ·U · ∇V qwdx−∫Ω
A ·L · ∇V qdx
=
∫Ω
−(α2σT + λ)∇q + ασ∇V q − ασ∇p− C1(α2σT + λ)H2 · ∇wdx
−∫Ω
ασ∇V + (α2σ + λ′(T ))∇T + (α′(T )σ + ασ′(T ))∇V T
+ (2αα′(T )σ + α2σ′(T ))T∇T · ∇qwdx
+
∫Ω
α′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇V qwdx
−∫Ω
σ′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇pwdx
−∫
C1H2 · (ασ∇V + α2σ∇T )wdx
=0
(86)
Converting the above equation from the weak to the strong form, the second adjoint equation
is obtained as follows:
∇ · −(α2σT + λ)∇q + ασ∇V q − ασ∇p− C1(α2σT + λ)H2
+ ασ∇V + (α2σ + λ′(T ))∇T + (α′(T )σ + ασ′(T ))∇V T
+ (2αα′(T )σ + α2σ′(T ))T∇T · ∇q
− α′(T )∇V + (α′(T )σ + ασ′(T ))∇T · ∇V q
=− C1H2 · (ασ∇V + α2σ∇T )
(87)
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