Geometrical design of thermoelectric generators based on ...€¦ · novel thermoelectric generators, there is a need to study the underlying optimal geometry of the devices. Moreover,

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

Copyright c© 2000 John Wiley & Sons, Ltd. Revised

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

Copyright c© 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1–6

Prepared using nmeauth.cls

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




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.




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)




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.




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.




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)




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.




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




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




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




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


+ (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.




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.




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




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




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)




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




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.




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




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.




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.




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).



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




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


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


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


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


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.




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.




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




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.




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.




(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%




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




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.




(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




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




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.




(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




-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




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.




(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




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




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




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




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




(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




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




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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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




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)




⟨∂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 ·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


∫Ω

A′(T ) ·U · ∇pwdx+

∫Ω

A ·K · ∇pdx+

∫Ω

A ·L · ∇pdx

=0

(68)




∫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 ·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


∫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




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 ·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)




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 ·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


+ (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)




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 ·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)




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 ·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


+ (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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