1. Topology optimization of
Lithium-ion batteries
How-to maximize the discharge capacity
by changing the geometry
Tianchi Xu
FacultyMechanical,MaritimeandMaterialsEngineering
2.
3. PREFACE
It would not have been possible to finish the project and the thesis without the support of friends and
family. Hereby, I would like to express my gratitude to the people who contributed directly and indirectly to
my thesis:
I would like to thank my supervisor Matthijs Langelaar for showing me which direction I should go for
study and helping me to solve my confusion. I am privileged that I have such an excellent supervisor.
I would like to thank Floris van Kempen as my daily supervisor. Floris is responsible for checking my
weekly tasks , giving advice on my research and correcting my final thesis. He is careful, patient and precise.
I am very grateful for his useful suggestions to my work.
At last, I would like to thank my parents and my girlfriend for their support. Without them I could not
have gone through those tough times.
Tianchi Xu
Delft, November 2015
iii
4.
5. ABSTRACT
Discharge capacity is an important factor that determines the performance of lithium ion battery. The
internal resistance of the electrodes influence the discharge capacity. As the electrode geometry influences
its resistance, topology optimization can be applied to determine the electrode shape such that it has a mini-
mal internal resistance and thus obtain the maximum discharge performance. The optimizations are carried
out for situations where time-dependent effects can be ignored, but also for cases where the transient phe-
nomena are considered. The results are compared with a battery model with reference geometry design(non-
optimized shape with the same electrode volume). The optimization studies ignoring time dependent effects
are verified using a time-dependent simulation , but also a time dependent optimization study was devel-
oped that can be used although at a high computational cost. The mathematical model and the deduction of
the optimization model are illustrated. The influences of different design parameters (mesh density, volume
range, design domain shape, etc.) have been analysed by means of numerical case studies. Several penalty
techniques have been used to make the final topology more realistic and easy to manufacture. The discharge
capacity and capacity fade under different discharge current,different design structures and different opti-
mization constraints are imposed and analysed. A three dimensional electrode model based on the topology
optimization is built and simulated. The comparison between different optimization methods has been stud-
ied. The suitable usage conditions of different optimization methods have been discussed. After the study, it
shows that topology optimization can be used in design of batteries.Optimizing current density will help to
get a larger capacity in low discharge rate and optimizing electrode potential will help to get a larger capacity
in high discharge rate.
v
9. 1
INTRODUCTION
1.1. INTRODUCTION OF LITHIUM ION BATTERY
A battery is a device that consists of one or more electrochemical cells that convert chemical energy into
electrical energy. Each cell contains one positive terminal (cathode), one negative terminal (anode) and elec-
trolyte which allows ions to move between the positive and negative electrode. There are various types of
batteries based on different metals such as Lead battery, Nickel battery or the Lithium-based battery. Among
those types of batteries, the lithium-based battery is relatively new and got commercially available in the past
few decades. Much research focuses on this battery because of the wide range of use. The general interest
is based on the intrinsic property of the element, lithium is the lightest metal and has the highest electro-
chemical potential and the largest specific energy per unit weight. This property makes that Li/Li ion battery
systems can provide the highest energy density[1]. Although lithium possesses these positive properties, it
becomes unstable during the charging process. In order to maintain the stability, lithium ion batteries are
built with small sacrifice of specific energy. This keeps manufactures and packers safe because it keeps the
voltage and currents below prescribed,secure levels. As battery techniques developed, it became possible
to produce lithium ion batteries with low maintenance and high capacity. Nowadays, lithium ion battery is
widely used in portable devices and increasingly used in electric trains and vehicles.
Lithium ion battery composes of three parts which are positive electrode, negative electrode and elec-
trolyte like other batteries [2]. Positive electrode is made up from various types of metal oxide [3]. Negative
electrode is made of carbon , most commonly in the form of graphite. The electrolyte is lithium salt dissolved
in organic solvent, A schematic working process of a typical lithium ion battery is shown in Figure 1.1.
Figure 1.1 Schematic working process of typical lithium ion battery
In the lithium ion battery, the processes of insertion and deinsertion of lithium ions in the electrodes (an-
ode and cathode) are simultaneous. This process is the source of the current flowing in the circuit. When
the lithium ions are inserted into the electrode, the battery is being charged. In contrast, when the lithium
ions are deinserted out of the electrode, the battery will be charged. The complete discharge-charge cycle
is known as “rocking chair” because the cycling movement of lithium ions between the two opposite elec-
trodes[4]. During the discharging process, the anode undergoes oxidation (loss of electrons) and the cathode
undergoes reduction (gain of electrons). During the charge process, the anode undergoes reduction (gain of
electrons) and the cathode undergoes oxidation (loss of electrons).[2] The lithium ion battery with LixC6 and
1
10. 2 1. INTRODUCTION
Li2MnO4 as the electrode material,for example,charges and discharges acording to the following reactions:
Discharging process
LixC6 =⇒ xLi+
+xe−
+C6 Anode(oxidation)
Li1−xMn2O4+xLi+
+xe−
=⇒ Li2MnO4 Cathode(reduction)
LixC6+Li1−xMn2O4 =⇒ Li2MnO4+C6 Overall(cell)reaction
Charging process
xLi+
+xe−
+C6 =⇒LixC6 Anode(reduction)
Li2MnO4 =⇒ Li1−xMn2O4+xLi+
+xe−
Cathode(oxidation)
Li2MnO4+C6 =⇒ LixC6+Li1−xMn2O4 Overall(cell)reaction
1.2. IMPROVING THE PERFORMANCE OF LITHIUM ION BATTERIES
The essential property existing in a battery is the trade-off relation between energy capacity and the power
capability[5]. Based on this property, a battery can not achieve a high discharge capacity under a high dis-
charge current. A battery can be designed for different situations for different that require different perfor-
mance criteria. In this thesis, the capacity is chosen to be a main goal and optimized.
Some factors which influence the battery capacity are listed.
1. Electrode material intrinsic property.
Electrode material energy capacity is an important factor determining the discharge capacity of a battery.
The performance of a battery can be optimized by inducing new electrode material. For example, silicon can
be induced in the electrode material of modern lithium ion batteries[6-9] because electrode with silicon can
be composited to a structure with bigger energy density. LiFePO4 is also a suitable material because it has
large theoretical capacity and exhibits excellent thermal stability in the fully charged. Although LiFePO4 has
a cope with low electronic conductivity, some techniques such as modification of the LiFePO4 particles with
carbon or reducing the size of the particles can compensate for this defect [10]. Now new materials that are
possible to make the battery combining high charge/discharge rate with high energy density are tested, while
it is still a long way to go to build such a battery.[11].
2. Manufacturing method of the electrode.
Manufacturing technology can also optimize the capacity performance of a battery. For example, Mn3O4
can be made to selectively to grow on reduced graphene oxide sheets. After the growth process, it is wired
up to a current collector through a graphene network which underlies the sheet. Such a design will obtain a
larger capacity because the micro structure is more beneficial to capture particles. [12].
3.Electrode internal resistance
The internal resistance of a battery is also an important factor influencing the discharge capacity of a
battery. A battery will stop working while its voltage drops to a specific cut-off voltage. Energy can only be
delivered before the time when the battery reached its cut off voltage. A battery with low impedance can
provide bigger current flow and deliver more energy. A battery with high impedance cab only deliver limited
energy because of the restricted path, and the circuit may cut off prematurely. The influence of the internal
resistance is illustrated in Figure 1.2. In the figure the dashed line is the ideal battery discharge curve without
internal resistance and the solid line is the one with internal discharge curve. The curve with bigger internal
resistance reaches the end voltage earlier, so it stops working earlier and delivers less energy.
Figure 1.2 Influence of the internal resistance
11. 1.3. TOPOLOGY OPTIMIZATION DESIGN ON LITHIUM ION BATTERY 3
1.3. TOPOLOGY OPTIMIZATION DESIGN ON LITHIUM ION BATTERY
Topology optimization is a mathematical approach which optimizes material layout within a given de-
sign domain, under certain sets of loads and boundary conditions such that the resulting layout meets a
prescribed set of performance targets.It is implemented based on finite element methods for the analysis,
and optimization techniques based on the method of moving asymptotes. Each mesh will be defined as dif-
ferent type of material in different iteration steps. A strictly convex approximating sub-problem is generated
and solved in each step of the iterative process until the material distribution achieves the best performance.
Engineers can find the best designs which meet the requirement by using topology optimization.
1.3.1. AVAILABILITY OF TOPOLOGY OPTIMIZATION DESIGN ON BATTERY
Up to now, there is no topology optimization on lithium ion battery or other kinds of batteries.With the
development of micro fabrication, the micro-nano scale structure on the battery thin film is possible to be
realized. With this technique electrode and electrolyte with different surface features are possible to make ef-
fects on the performance of a battery. For electrochemical cells, the geometry is able to change the resistance
of cathode thus significantly influencing the performance of the electrochemical cell. This is a direction for
optimizing lithium ion battery because the internal resistance is an important parameter which determines
the available capacity and the end voltage during the process of discharge of a battery [13]. Based on this
theory, topology optimization has the possibility on improving the performance of battery by optimizing the
shape of the electrode.
The schematic figure of a battery film is shown in Figure 1.3. The components of the battery are polymers
with small thickness. The optimization is carried on with a two dimension case in the y-z plane. The influence
of the potential distribution in the x-y plane will not be taken into consideration because it is too small to be
counted[14].
Figure 1.3 Schematic figure of a battery thin film
Such an optimization strategy turns the topology optimization into a two dimensional domain. In order to
continue the optimization, the mathematical model should describe how lithium ion battery works in the y-z
domain. Based on such a study, the objective function, control variables and constraints will be determined
and the optimization process will be carried out.
1.3.2. APPROACH OF BATTERY DESIGN BASED ON TOPOLOGY OPTIMIZATION
The approach in this thesis is to optimize the topology and shape of the electrodes using simplified and
mostly steady-state models. And then, to verify the optimized designs using a complete transient model. The
internal resistance of the battery will be by certain methods so that the battery can reach the cut off voltage
after a longer time and thus have a bigger discharge capacity. In the first step, the electrode shape will be op-
timized to make the battery has a better performance on conducting current. This step will be optimized for
steady-state conditions because static topology optimization is a general method and it is not that much time
consuming compared with time dependent topology optimization. In the second step, the optimized elec-
trode shape will be analysed using a time-dependent simulation. The discharge capacity will be compared
12. 4 1. INTRODUCTION
with those from batteries with the conventional film design. The comparison will show whether topology
optimization is suitable for lithium ion battery and if it is useful, how much improvement will topology opti-
mization bring to lithium ion batteries.
13. 2
MATHEMATICAL MODEL
The mathematical model of the lithium ion battery is an important factor for the topology optimization
because it contains the relations between electrode impedance, current density and potential.The objective
function, design variables and constraints can be determined by analysing the mathematical model and the
shape that influencing the battery capacity performance will be determined. By simplifying the model, the
combination between the model and the topology optimization problem will be formed in a proper method.
Description of the mathematical model is split into two parts. First, we develop a simplified steady-state
model that is used to optimize the electrode structure. And second, a full transient model that better approx-
imates the real performance of the battery and is used to verify and analyse the optimized design in detail.
2.1. SIMPLIFIED MODEL FOR OPTIMIZATION STUDIES
The schematic figure for describing the physical working process of a battery is illustrated in Figure 2.1.
During discharge process, Lithium ions will first diffuse out of the electrode material particle and then
ions get through electrode and electrolyte forming current. Because of the porosity property of the porosity,
porous electrode is flowed through by both electrode current and electrolyte current. While in electrolyte
phase, there is only electrolyte current. The figure shows the porous property of the electrode material and
the diffusion of lithium ions into the electrode materials.
Figure 2.1 Schematic figure of lithium ion battery working process
The mathematical model used in topology optimization problem focuses on the relation between the
current density, battery material resistance and potential distribution because current density and terminal
potential are indicators of the value of the internal resistance.Also, the lithium concentration in the electrode
material is included in the model. All the definitions of the parameters in section 2.1 are present in table 2.1.
Table2.1 Parameters in topology optimization mathematical model
5
14. 6 2. MATHEMATICAL MODEL
Parameter Description Unit
c concentration of Li ions in the electrolyte mol/m3
Cs concentration of Li in the electrode mol/m3
Ds diffusion coefficient of Li in the electrode m2
/s
is current density in the electrode A/m2
il current density in the electrolyte A/m2
f± activity of the salt in the electrolyte mol/m3
R gas constant(8.314J/(mol K))
r radius of spherical particles µm
T temperature of the system K
t0
+ transport number of the positive ion
t time of operation s
κ ionic conductivity of electrolyte S/m
σ electronic conductivity of solid matrix S/m
φs potential in electrode V
φl potential in electrolyte V
2.1.1. CURRENT DENSITY AND LITHIUM ION CONCENTRATION
Batteries always work with an external circuit to connect the electrodes electrically. Typically, in that
circuit the current is generated to charge the battery or used, in order to discharge the battery. Current density
distribution in electrode differs from that in the electrolyte. In electrode, the current density can be described
with simple form of Ohm’s law:
is = −σ∇φs (2.1)
The current density in the electrolyte is more complicated because the concentration gradient in the elec-
trolyte heavily influences the transport of lithium ions and thus influences the current density. In order to
take this factor into consideration, the current density in the electrolyte is expressed using Ohm’s law, but
modified to take the ion concentration into account [15]:
il = −κ∇φl −
2κRT
F
(1+
∂ln f±
∂ln c
)(1− t0
+)∇lnc (2.2)
Lithium ion concentration in the electrode material particle can be expressed as the equation describing
the molecular diffusion in the spherical coordinates system. This process is modelled under several assump-
tions:
1. The anode and cathode electrode material are both porous.
2. The particles of the material are spherical as illustrated in the scheme.
3. The solid phase diffusion coefficient is independent of concentration.
Besides such assumptions,the particles are assumed to migrate in spherical coordinates. Lithium ion
concentration in spherical material particles can be described as:
∂Cs
∂t
= Ds[
∂2
Cs
∂t2
+
2
r
∂Cs
∂r
] (2.3)
The equation above is only used to demonstrate the diffusion process of lithium ions into/out of the
electrode particles. The confusion caused by the existence of the time-dependent equation in a static envi-
ronment will be explained later.
2.1.2. MODEL SIMPLIFICATION
Topology optimization in this case is applied to achieve an electrode shape more beneficial for conduct-
ing.The model can be simplified to make the relations between resistance, current density and potential
clearer without decreasing the accuracy of the model. The simplification is based on the constant and null
terms[4] in Table 2.2 in the LixC6-Li2MnO4 electrochemical system.
Table2.2 Constant and null terms
15. 2.1. SIMPLIFIED MODEL FOR OPTIMIZATION STUDIES 7
Term Value Explanation
t0
+ constant
Transport number is a function of the lithium-ions concentration.
Because of the lack of reliable data, here it is defined as a constant.
∂ln f±
∂ln c
null
This term is defined as null because there is no specific experiment
data to evaluate the term value
Based on Table 2.2, the equation(2.2) that describes current density in the electrolyte:
il = −κ∇φl −
2κRT
F
(1+
∂ln f±
∂ln c
)(1− t0
+)∇lnc
can be simplified to
il = −κ∇φl −
2κRT
F
(1− t0
+)∇lnc (2.4)
As mentioned previously, the topology optimization model is based on conditions, while the concentra-
tion term c is a time dependent variable. So the concentration value should be a specific value at specific
moment. We also assume that the Li-ion concentration is homogenous which means that Li-ions are dis-
tributed evenly in the electrolyte domain. The term ∇lnc then becomes equal to zero because the gradient
of a constant is zero. Based on this reasoning, equation(2.4)can be simplified to
il = −κ∇φl , (2.5)
which is simply form of Ohm’s law again.
The process of diffusion in the molecular particle cam be neglected because this is a time dependent
process and the topology optimization focuses on static condition. So the equation(2.3) describing particle
diffusion in electrode material:
∂Cs
∂t
= Ds[
∂2
Cs
∂t2
+
2
r
∂Cs
∂r
]
is neglected.
These assumptions lead to the following mathematical models governing the current density in electrode
phase and electrolyte phase stated in equation(2.1) and equation(2.5)respectively:
Current density in electrode phase
is = −σ∇φs
,
il = −κ∇φl
,
Current density in electrolyte phase
il = −κ∇φl
.
The schematic graph of the current location is illustrated in Figure 2.2. The two equations transfer the
optimization problem to a simple primary current distribution condition. The internal resistance can be
calculated by the total current and the potential gradient. So optimizing the resistance can be implemented
by optimizing the current density or the potential distribution.
Figure 2.2 Schematic figure of lithium ion current description
16. 8 2. MATHEMATICAL MODEL
2.2. MODEL FOR VERIFICATION STUDIES
The mathematical model here is built based on the electrochemical cell composed with LixC6 and LiMnO4.
As discussed before, the topology optimized outcomes based on the model in the previous section will be ver-
ified in a time dependent simulation. A complex model simulating the battery working process is necessary
and it will be discussed in this section.
The battery working process contains electrical discharge-charge and oxidation-reduction reaction is
a Faradic process.Formulas governing this process is composed with normal/modified Ohm’s law, inser-
tion/deinsertion of Lithium ion into electrode material via molecular diffusion and lithium ion transportation
in the porous material.The relation between the process of insertion/deinsertion of the lithium ions and the
charge/discharge current can be expressed by Faraday’s Law. All the definitions of the parameters in section
2.2 are present in table 2.3.
Table2.3 Parameters in mathematical model for time dependent simulation
Variable Description Unit/Value
a specific interfacial area m2
/m3
c concentration of Li ions in the electrolyte mol/m3
Cs concentration of Li in the electrode mol/m3
D diffusion coefficient of Li in the electrolyte m2
/s
Ds diffusion coefficient of Li in the electrode m2
/s
D diffusion coefficient of the salt in the electrolyte m2
/s
F Faraday constant(96487C/mol)
f± activity of the salt in the electrolyte mol/m3
is current density in the electrode A/m2
il current density in the electrolyte A/m2
jLi pore wall flux of Li ions mol/(cm2
s)
N mass transport flux mol/m2
R reaction term of the mass balance equation mol/(m3
s)
R gas constant( 8.314J/(mol K))
r radius of spherical particles µm
Rf film resistance Ωm2
Rs radius of electrode spherical particle m
T temperature of the system K
t0
+ transport number of the positive ion
t time of operation s
u0
open circuit voltage V
δ x-axis direction length µm
δy y-axis direction length µm
porosity of porous electrode
η over-potential V
κ ionic conductivity of electrolyte S/m
σ electronic conductivity of solid matrix S/m
φs potential in electrode V
φl potential in electrolyte V
Index Description
a anode
s separator
c cathode
T maximum value for lithium ion in electrode material
0 initial condition
2.2.1. GOVERNING EQUATIONS
Some transformation is necessary in order to transfer the initial forms of governing equations into more
general and convenient forms. Those forms describe the process with the concentration change of the lithium
ions in time domain. The transformation procedure will be shown in the following paragraph.
17. 2.2. MODEL FOR VERIFICATION STUDIES 9
TRANSFORMATION OF GOVERNING EQUATIONS
The current density in the electrolyte can be described with Faraday’s Law:
∇·il = Fa jLi (2.6)
According to the law of charge conservation, the gradient of the summation of electrolyte and electrode
current density should be zero:
∇·(is +il ) = 0 (2.7)
Therefor the current density in the solid phase electrode can be described as the following equation:
∇·is = −Fa jLi (2.8)
Current densities in electrode and electrolyte are following the simple Ohm’s law and modified Ohm’s law
respectively as they are shown in the previous section :
is = −σ∇φs (2.9)
il = −κ∇φl −
2κRT
F
(1+
∂ln f±
∂ln c
)(1− t0
+)∇lnc (2.10)
In lithium ion battery, the Li ion flux is related to the discharge and charge current. So it is more conve-
nient to relate the current density to the migration of Li ions than to relate the current density to potential
gradient. Substituting equation(2.6) into equation(2.10) and substituting equation(2.8) into equation(2.9),
the relation between current density gradient and Li ion flux based on Faraday’s law is represented as the
following equations:
−σ∇2
φs = −Fa jLi (2.11)
−κ∇2
φl = Fa jLi +
2κRT
F
(1+
∂ln f±
∂lnc
)(1− t0
+)∇2
lnc (2.12)
Lithium ion concentration in electrode material is also the same as it is in the previous section:
∂Cs
∂t
= Ds[
∂2
Cs
∂t2
+
2
r
∂Cs
∂r
] (2.13)
Both the positive electrode and the negative electrode are porous. This fact means that electrolyte is in
all the three battery regions and Li-ions migrate between different electrode via electrolyte. This process is
governed by the mass balance condition deduced from the concentration solution theory.The mass balance
condition is represented as:
∂c
∂t
= −∇·N+R (2.14)
The total ion flux is represented as:
N = −D(1−
dlnc0
dlnc
)·∇c +
il t0
+
F
(2.15)
The reaction term combining the oxidation-reduction reaction with the flux of Li ions is represented as:
R =
α
ν+
(1− t0
+)jLi (2.16)
Substituting equation(2.15) and equation(2.16) into equation(2.14), the general form describing the change
of concentration Lithium ions over time is represented as:
∂c
∂t
= D(1−
dlnc0
dlnc
)∇2
c −
il t0
+
F
+
α
ν+
(1− t0
+)jLi (2.17)
Equation(2.17) is an ideal equation which is only available in continuous battery region such as elec-
trolyte. For porous electrode region in which both electrode and electrolyte material exist, this equation is
not valid. In order to make this mass balance equation applicable to all batteries regions, a porosity factor
18. 10 2. MATHEMATICAL MODEL
is induced to make the equation available in more region. After the correction, the general form of equa-
tion(2.17) is represented as:
∂c
∂t
= D(1−
dlnc0
dlnc
)∇2
c −
il t0
+
F
+
α
ν+
(1− t0
+)jLi (2.18)
SUMMARY OF GOVERNING EQUATIONS
After the transformation,all the governing equations are represented as the following equations:
Ohm’s law in electrode
−σ∇2
φs = −Fa jLi
Ohm’s law(modified) in electrolyte
−κ∇2
φl = Fa jLi +
2κRT
F
(1+
∂ln f±
∂lnc
)(1− t0
+)∇2
lnc
Lithium ion diffusion in active electrode material sphere
∂Cs
∂t
= Ds[
∂2
Cs
∂t2
+
2
r
∂Cs
∂r
]
Lithium ion diffusion in electrolyte through porous material
∂c
∂t
= D(1−
dlnc0
dlnc
)∇2
c −
il t0
+
F
+
α
ν+
(1− t0
+)jLi
2.2.2. SIMPLIFICATION
The general governing equations presented above contain some constant or null terms.Considering this
fact, they can be simplified by inducing those terms to the equations. Constant and null terms are illustrated
in Table 2.4
Table2.4 Constant and null terms
Term Value Explanation
t0
+ constant
Transport number is a function of the lithium-ions concentration.
Because of the lack of reliable data, here it is defined as a constant.
ν+ constant Dissociation of electrolyte here is 1:1.
dln c0
dln c
null
This term is defined as null because there is no specific experiment
data to evaluate the term value
∂ln f±
∂ln c
null
This term is defined as null because there is no specific experiment
data to evaluate the term value
According to Table(2.4), the governing equations can be rewritten as new simplified forms. The final sim-
plified equations are represented as:
Ohm’s law in electrode
−σ∇2
φs = −Fa jLi (2.19)
Ohm’s law(modified) in electrolyte
−κ∇2
φl = Fa jLi +
2κRT
F
(1− t0
+)∇2
lnc (2.20)
Lithium ion diffusion in active electrode material sphere
∂Cs
∂t
= Ds[
∂2
Cs
∂t2
+
2
r
∂Cs
∂r
] (2.21)
Lithium ion diffusion in electrolyte through porous material
∂c
∂t
= D∇2
c +α(1− t0
+)jLi (2.22)
19. 2.2. MODEL FOR VERIFICATION STUDIES 11
2.2.3. BOUNDARY CONDITION
Assuming that the battery materials are distributed homogeneously in the corresponding region and cur-
rent flows from one terminal plane to another. For one single region with specific potential gradient, the the
current density flow can be represented as Figure 2.2.
Figure 2.2 Schematic figure of boundaries. Contour: electric potential. Arrows: current density vector.
In order to make the boundary conditions more intuitionistic and trenchant,the model can be assumed
as a simple one dimensional system because the current density flows along only one direction between both
ending and it does not change as the vertical location changes. Under such a simplification, the boundary
condition is only considered in specific x-axis locations.The schematic plot illustrating the location of each
boundary is in Figure 2.3. In the figure, the positions of electrode-electrolyte interfaces, anode and cathode
current collector are marked.
Figure 2.3 Schematic figure of boundaries
For both normal and Ohm’s law, the boundary condition is set to define the potential at current collectors
and the electrode-electrolyte interfaces. For the Li ions diffusion in active material along the spherical coor-
dinates, the concentration of Li ions along the sphere radius is defined. For Li ions diffusion in electrolyte, the
concentration at both current collectors and different interfaces is defined. The summary of the boundary is
in Table 2.5 and Table 2.6.
Table2.5 Boundary conditions along x-axis
20. 12 2. MATHEMATICAL MODEL
Equation x = 0 x=δa x = δa +δs x = L
Eqution(2.19) φs = 0
dφs
dx
= 0 φs = φs,0
dφs
dx
= −
I
δ
Eqution(2.20) φl =φl,0 Continuity Continuity
∂φl
∂x
= 0
Eqution(2.22)
∂c
∂x
= 0 Continuity Continuity
∂c
∂x
= 0
Table2.6 Boundary conditions in spherical coordinate
Equation r = 0 r = Rs
Equation(2.20)
∂Cs
∂r
= 0
∂Cs
∂r
= −
jLi
Ds
2.2.4. SUMMARY OF GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
In the previous paragraphs, the general form of governing equations and boundary conditions have been
deduced. In order to make it clear, they are collected in Table 2.7.
Table2.7 Summary of mathematical model for time dependent simulation
Govening equation Boundary condition
Anode x = 0 x = δa
−
σ
x2
∂2
φs
δ2
= −Fa jLi φs = 0
dφs
dx
= 0
−
κ
δ2
∂2
φl
∂x2
= Fa jLi +
2κRT
δ2F
(1− t0
+)∇
∂2
lnc
∂x2
φl = φl,0 continuity
∂c
∂t
=
D
δ2
∂2
c
∂x2
+α(1− t0
+)jLi
∂c
∂x
= 0 continuity
Separator x = δa x = δa +δs
−
κ
δ2
∂2
φl
∂x2
= Fa jLi +
2κRT
δ2F
(1− t0
+)∇
∂2
lnc
∂x2
continuity continuity
∂c
∂t
=
D
δ2
∂2
c
∂x2
continuity continuity
Cathode x = δa +δs x = L
−
σ
x2
∂2
φs
δ2
= −Fa jLi continuity
dφs
dx
= −
I
δ
−
κ
δ2
∂2
φl
∂x2
= Fa jLi +
2κRT
δ2F
(1− t0
+)∇
∂2
lnc
∂x2
φl = φl,0
∂φl
∂x
= 0
∂c
∂t
=
D
δ2
∂2
c
∂x2
+α(1− t0
+)jLi continuity
∂c
∂x
= 0
Active electrode material r = 0 r = Rs
∂Cs
∂t
= Ds[
∂2
Cs
∂t2
+
2
r
∂Cs
∂r
]
∂Cs
∂r
= 0
∂Cs
∂r
= −
jLi
Ds
21. 3
TOPOLOGY OPTIMIZATION PROBLEM
In the previous section, the mathematical model has been introduced. The model is the basis of the
topology optimization problem. Following the governing equation introduced, the objective function will be
set and the corresponding design variable and constraints will also be introduced. The parameters to be used
in this chapter are summarized in Table 3.1.
Table3.1 Parameters in topology optimization problem
Parameter Description Unit/Value
A area of single mesh element m2
C conductivity for single finite element mesh S/m
is current density in the electrode A/m2
il current density in the electrolyte A/m2
I integration of current along base boundary A/m
P penalty factor
W total amount of electrode material area m2
WU upper bound of total amount of electrode material area m2
κ ionic conductivity of electrolyte S/m
σ electronic conductivity of solid matrix S/m
φs potential in electrode V
φl potential in electrolyte V
ρ design variable
3.1. MODEL OF TOPOLOGY OPTIMIZATION PROBLEM
According to the previous section, the simplified mathematical model for topology optimization is repre-
sented by:
Current density in the electrode phase
is = −σ∇φs, il = −κ∇φl (3.1)
Current density in the electrolyte phase
il = −κ∇φl (3.2)
A schematic figure of one electrode under the equation is shown in Figure 3.1.
13
22. 14 3. TOPOLOGY OPTIMIZATION PROBLEM
Figure 3.1 Schematic figure for topologyu model
The white marked A1 domain is filled with electrolyte material. The black marked A0 domain is filled
with electrode material. The green-marked bar-shaped domain is the current collector. In the case of the
example in the figure the ions are inserted into the electrode, the battery will be charged and the charging
current will flow to the current collector. B0 and B1 are the boundaries at which the potential is set to provide
the potential gradient. Vector n is the normal vector of the boundary B0 and it defines the direction of the
current. The charge current into the current collector can be described as:
I =
B0
σ∇φs ×dl (3.3)
The optimization modelling needs a design parameter to be completed. Here a design variable ρ is in-
duced to control the material property in the whole design domain. A solid isotropic material with penal-
ization(SIMP) method [16] approach is used in this case to interpolate the material properties from a give
design variable ρ. The design variable ρ of electrode material is [0,1] and for electrolyte material it is [1,0]
piece-wisely. When rho is 1, it means the mesh is filled with the electrode material, when rho is 0, it means
the mesh is not filled with the electrode material but with electrolyte material. At a single point, the conduc-
tivity of the mesh is defined as :
C = σρP
+κ(1−ρ)p
(3.4)
3.2. OBJECTIVE FUNCTION
In the previous discussion, internal resistance of the battery is a influential factor of the battery. The goal
of the topology optimization is to find the proper shape that minimizes the resistance of the electrode. This
is the same as finding the shape that maximizes the conductivity of the cathode. According to the equation
above, the resistance can be calculated using a current source or a potential gradient. So the problem of opti-
mizing the conductivity is equivalent to optimizing the current or the potential. Under such a transformation,
the objective function will have two forms:
1. Maximizing the total current outward B0 boundary under certain potential gradient.
Maxmize.I = σ∇φs ×dl (3.5)
2. Minimizing the terminal potential at B1 boundary under certain current density distribution.
Mimimize.φs =
B1
B0
is
σ
×dl (3.6)
23. 3.3. CONSTRAINT 15
3.3. CONSTRAINT
Regardless of the shape, the conductivity of the electrode is to a large extend determined by the amount of
material. More conducting material will make the electrode more conductive and thus decreasing the inter-
nal resistance of the electrode. The total amount of porous electrode material can be defined by combining
the element area and the element density[17]:
W = ΣρA (3.7)
For the design domain, electrode and electrolyte both exist. In order to satisfied such a condition, the
amount of electrode should be constrained by setting a upper bound as represented in the following equation:
W = ΣρA ≤ WU (3.8)
Using upper bound to control the shape works well for cases using SIMP methods. Besides, a filter is also
commonly introduced, but this will lead to a smoother shape, especially for the edge of the electrode. This
may have a negative influence for the next step of transient analysis.
3.4. SUMMARY OF THE TOPOLOGY OPTIMIZATION PROBLEM
In the previous discussion, the design variable, objective function and constraint have been developed.
The objective function can have two different forms which are both feasible approaches. The following opti-
mization designs and the simulation processes will also refer to the corresponding objective goal respectively.
TOTAL CURRENT MAXIMIZATION
For the first case which maximizes the total current outward B0 boundary under certain potential gradi-
ent, the problem can be described as:
Find ρ that:
Maxmize.I(φs,ρ) = σ∇φs ×dl
Subject to:
W ≤ WU
0 ≤ ρ ≤ 1
TERMINAL POTENTIAL MINIMIZATION
For the second case which Minimizes the terminal potential at B1 boundary under certain current density
distribution, the problem can be described as :
Find ρ that:
Mimimize.φs(is,ρ) =
B1
B0
is
σ
×dl
Subject to:
W ≤ WU
0 ≤ ρ ≤ 1
24.
25. 4
TOPOLOGY OPTIMIZATION FOR MAXIMIZING
TOTAL CURRENT
Maximizing the current density along the current collector is the way easy to be figured out because lower
internal resistance leads to a higher current density.This optimization is applied on the shape of one elec-
trode(anode or cathode). The influence of the other electrode is not taken into consideration.
4.1. SETTINGS IN TOPOLOGY OPTIMIZATION
The topology optimization procedure is implemented in the software Comsol Multiphysics which is a
finite element analysis, solver and Simulation software for various physics and engineering applications.
Boundary conditions, material definitions and optimization equations mentioned in the previous section
should be set in the proper form so that the topology optimization can function properly.
4.1.1. BOUNDARY CONDITION
In the following topology optimization, the boundary conditions are shown in Figure 4.1.
Figure 4.1 Boundary conditions
The blue marked domain is the design domain. An electrode current source term:
QS = ∇is
is added in the design domain to simulate the initial condition that with the chemical reaction and the
migration of lithium ions, the domain is filled with current. The electric ground is added on the black marked
boundary. Here electric ground is not a fixed value; any prescribed electric potential value will be proper
because the factor which determines the current density is the potential gradient instead of the potential at a
certain boundary.
17
26. 18 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
4.1.2. MATERIAL DEFINITION
The electrode material is porous. So the combination of the porosity (also can be called volume fraction)
and the conductivity determined the effective conductivity of the electrode. With the design parameterρ, the
expression of the electrode and electrolyte in the design domain will be defined as:
Electrode conductivity: σρP
Electrode volume fraction:FsρP
Electrolyte conductivity: κ
Electrolyte volume fraction: 1−Fs +Fl (1−ρ)
With such definition:
ρ = 1:Mesh element is filled with porous material with electrode conductivityσ, electrode volume fraction
Fs, electrolyte conductivity κ and electrolyte volume fraction 1−Fs.
ρ = 0:Mesh element is filled with electrolyte conductivity κ and electrolyte volume fraction 1−Fs +Fl .
4.1.3. OBJECTIVE FUNCTION AND CONSTRAINTS
The location where objective function is applied on is shown in Figure 4.2.
Figure 4.2 Objective function location
The black marked boundary is the current collector and the integral objective function:
Maxmize.I(φs,ρ) = σ∇φs ×dl
is added on the boundary
Here the objective function boundary is a little shorter than the side of the design domain. This is because
when the boundary is the same length as the side of the domain, the optimization process will not converge.
So the selection of the boundary should avoid picking the ending point of the side, thus leading to the result
that the objective function boundary is shorter than the side of the design domain.
Besides the design variable ρ is set to be available in the whole design domain with upper bound 1 and
lower bound 0. Another constraint is that the total amount of electrode should be less than a certain value.
This can be solved by adding an integral inequality constraint in the design domain. The integration of ρ
represents for the amount of electrode and the upper bound of the material volume can be set according to
different requirement.
The values and the explanations of the parameters in the previous formulas are illustrated in Table 4.1
Table4.1 Parameters in topology optimization setting
Parameter Description Value Unit
Fl volume fraction of electrolyte 0.5
Fs volume fraction of electrode 0.5
P penalty factor 2
Qs electrode current source 1.75×108
A/m3
κ conductivity of electrolyte 0.2 S/m
σ conductivity of electrode 3.8 S/m
ρ Electrode material density [0,1]
27. 4.2. OPTIMIZATION OUTCOMES 19
Because in the following transient analysis, the conductivity of the electrode and the electrolyte will be set
again, so the conductivity values used in topology are set in priority for being beneficial for optimization.
4.2. OPTIMIZATION OUTCOMES
Topology optimization will be operated under different parameter settings and the out comes will be
compared with a reference design to figure out the improvement.
4.2.1. REFERENCE DESIGN
In order to evaluate if the topology optimization will lead to a better performance, and deserve for the
further transient analysis, a reference design is induced to be compared with the optimized shape.For exam-
ple, if the optimized design domain is a 100µm × 100µm square with the volume constraint 40% and the base
width 60% of the side length, the corresponding reference design shape is shown in Figure 4.3.
Figure 4.3 Reference design
The blue domain is defined as porous electrode with area 100µm × 100µm ×40%, base width 60µm.The
grey domain is defined as a porous electrode with 0 volume fraction of electrode and 1 volume fraction of
electrolyte which can be treated as electrolyte approximately. The area of the grey domain is100µm × 100µm
×60%. The materials of electrode and electrolyte have the same conductivity and volume fraction as they do
in the optimized optimization. Boundary conditions and other settings are also same as those of the opti-
mizations. The current will be integrated along the current collector boundary as a reference value compared
with the results from optimization designs.
4.2.2. TOPOLOGY OPTIMIZATION DESIGN OUTCOMES
The sample topology optimization design is with the mesh resolution 40× 40 and electrode material
amount 40% of the total design domain. Result shape is shown in Figure 4.4.
Figure 4.4 Sample result of topology shape. Blue: electrolyte material. Red:porous electrode material
The shape is similar to the outcome of a heat conductor. Because a current source is added in the whole
domain, the electrode will stretch itself as much as possible to get more access to the current and drain them
to the base. The shape also changes the distribution of the electric field. This comparison is shown in Figure
4.5.
28. 20 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
The red arrows in the left figure represent for the current flow. Compared with the graph without any de-
sign on the right, the optimized design shows a big difference. The equal potential lines are along the surface
of the electrode and the gradient of potential is very big at the surface of the electrode. This is beneficial for
the conduction of the current. At the domain adjacent to the current collector, the potential gradient is also
very large thus making a positive influence of the current conduction.
Figure 4.5 Electric field plot.
Left: optimized design. Right: Reference design.
Black contours : electrical potential. Colour scale : potential gradient
The current flow shown in Figure 4.6 shows that most of the current are induced from the branches of the
electrode and gathered to the current collector.
Figure 4.6 Current flow in optimized shape.
Black contours : electrical potential. Colour scale : potential gradient
For the performance of the optimized design, the current density outward the current collector boundary
is calculated as 4.615 A/m2
. Compared with the 1.217A/m2
, the improvement is almost three times. But this
does not mean that such a design is three times better than the reference design because this comparison is
under a static condition, the concentration of the lithium ions, the migration of lithium ions and other facts
in transient process are not taken into consideration. Up to now, the improvement only shows that topology
optimization can be used for optimizing the shape of electrode.
4.2.3. RESULTS ANALYSIS
For the optimized design, besides those constant describing the material property and the upper/lower
bound of the design variable, a lot of parameters such as the number of the mesh(determining the resolution
of the graph), the selected length of the current collector, the upper bound of the electrode volume constraint
are not fixed. The influence of those unfixed parameters will be studied in the following section.
29. 4.2. OPTIMIZATION OUTCOMES 21
INFLUENCE OF DIFFERENT RESOLUTION
The resolution of the meshed domain determines the sharpness of the optimized graph and the perfor-
mance of the optimized shape. The larger the resolution is, the more details the graph will illustrate. But
a high resolution graph will have a larger time for each step of iteration. So the influence of the resolution
should be researched to have a good balance between performance and efficiency. The comparison is car-
ried out under five resolutions: 20×20, 40×40, 60×60, 80×80 and 100×100. Besides the resolution, other
settings are fixed in order to eliminate the unpredicted affects. The graph of different and the corresponding
results are shown in Table 4.2 below. The improvement is relative to the corresponding reference design.
Table4.2 Shapes with different mesh resolution
Resolution Figure Result.opt(A/m2
) Result.ref(A/m2
) Improvement(%)
20×20 4.520 1.217 271.4
40×40 4.615 1.217 279.2
60×60 4.643 1.217 281.5
80×80 4.645 1.217 281.7
100×100 4.655 1.217 282.5
Besides the 20 by 20 resolution which is too small to be adopted, the results under other resolutions are
approximately the same. But the improvement of the resolution gives a huge improvement on the quality of
the graph. In the 20 by 20 graph, the optimization only clear to show a trend that the electrode should have a
tree-shaped scheme.
INFLUENCE OF VOLUME CONSTRAINT
Volume constraint is another factor to influence the performance. In the comparison group, the volume
constraint is set as 20%, 25%, 30%, 35%, 40% respectively, the current collector width is 60µm and the resolu-
tion is 80×80. The result is shown in Table 4.3.
Table4.3 Shapes with different volume constraint
30. 22 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
Vol.Constraint(%) Figure Result.opt(A/m2
) Result.ref(A/m2
) Improvement(%)
20 4.579 1.205 280.0
25 4.605 1.211 280.2
30 4.624 1.214 280.8
35 4.634 1.216 281.1
40 4.645 1.217 281.6
The higher volume fraction will induce a higher outward current in the base both in the optimized design and
the reference design. Under different volume constraint, the improvement is nearly the same.
INFLUENCE OF BASE WIDTH
Base width determines the optimized shape and the performance of the electrode. In order to research
the influence of the base width, other parameters have to be set as the same. Because the optimized domain
is square, when the base width is so small, the length of the electrode may be beyond the domain and cause
troubles, the volume constraint should be selected as a low value. In this case, the volume fraction is selected
as 25% and the resolution is also 80×80. The width is 30µm, 40µm, 50µm, 60µm, 70µm, 80µm. The results
are illustrated in Table 4.4.
Table4.4 Shapes with different base width
31. 4.2. OPTIMIZATION OUTCOMES 23
Base width(µm) Figure Result.opt(A/m2
) Result.ref(A/m2
) Improvement(%)
30 3.952 0.722 447.4
40 4.230 0.903 368.4
50 4.436 1.064 316.9
60 4.605 1.211 280.3
70 4.753 1.346 253.1
80 4.872 1.471 231.2
As the Base widths increase, results from both optimization design and reference design increase while the
improvements decrease. This means that optimization is more effective with low base width situations.
INFLUENCE OF THE DESIGN DOMAIN
The design domain is another influence factor of the topology optimization. Here the influence is studied.
The design domain x-axis length/y-axis length ratio is set to be 1/0.5, 1/1, 1/1.5, 1/2, 1/3. The resolution is
80×80, the base width is 40µm and the volume constraint is 25% the design domain area. Results are shown
in Table 4.5.
Table4.5 Shapes with different design domain
32. 24 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
x/y ratio Figure Result.opt(A/m2
) Result.ref(A/m2
) Improvement(%)
1/0.5 4.310 0.923 367.0
1/1 4.230 0.903 368.4
1/1.5 4.103 0.878 367.3
1/2 3.969 0.855 364.2
1/3 3.696 0.813 354.6
The table above shows that as the ratio between x-axis length and the y-axis length decreases, both the
performance of the optimized design and the reference design decrease. Besides the situation with ratio 1/3,
which is not sufficiently optimized due to the graph, the other groups have an approximately same improve-
ment on the performance.
4.2.4. RESULTS ANALYSIS CONCLUSION
RESOLUTION
The difference between the performances of the optimized shape shows up only when the resolution
difference is extremely large. When the resolution reaches a certain decent range, it will have tiny influence
on the result.
VOLUME UPPER BOUND
Volume upper bound will influence both the optimized result and the reference result. But the improve-
ment of the optimized result is only under tiny influence of the volume.
BASE WIDTH
Bigger base width will leads to a larger current density in optimized design and reference design. The
smaller the base width is, the larger the improvement will be. Base width has a significant influence on the
improvement.
SHAPE OF DESIGN DOMAIN
The shape of the design domain has a small influence on the improvement. Only when the ratio between
width and length is too big, the improvement will change obviously.
4.3. SPECIFIC TECHNIQUE FOR CERTAIN PROBLEMS
Because of the simplification of the mathematical model and the simple objective function and boundary,
the previous design has some defects in physical and manufacturing aspects. In order to eliminate such de-
33. 4.3. SPECIFIC TECHNIQUE FOR CERTAIN PROBLEMS 25
fects and make the design more practically feasible, some methods will be induced to penalise the formation
of the unpredicted shape.
4.3.1. ELIMINATING GAP IN THE ELECTRODE
In the previous topology, the basic trend of the shape of the electrode has been clear. The branch like
shape can help the electrode to collect more current. But in the region ambient to the current a gap always
exists as the previous Figure 4.4.Such a gap is not allowed because the battery electrode should be totally
connected and a gap with such a small scale is difficult to be manufactured. Before using some specific
method to eliminate the gap, the influence of the gap has to be studied. The study is under a simplified
model illustrated in Figure 4.7.
Figure 4.7 Simplified model for gap study
GAP INFLUENCE IN SIMPLIFIED MODEL
The current along the current collector boundary will change as the gap length changes. By studying the
change, the influence of the gap will be figured to some extent. The relation between total current and the
gap length is in the interpolation graph form as shown in Figure 4.8.The x-axis of the graph represents for the
length of the gap, the y-axis represents for the current density in the current collector.
Figure 4.8 Relation between total current and gap length
The plot shows that when the length of the gap changes from 0µm to 10µm, the current becomes smaller
and smaller. The previous study has shown that the current density is related to the potential gradient. This
relation in the simplified model is shown in Figure 4.8.
34. 26 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
Figure 4.8 Relation between total current and potential gradient under different gap length
The plot above shows that the current changing trend is same as the potential gradient changing trend.
The function of the length of the gap is changing the potential gradient in the gap as well as the potential gra-
dient in the electrode. Because the potential gradient is related to the current, the gap changes the potential
distribution thus influencing the current.
GAP INFLUENCE IN OPTIMIZATION DESIGN
The influence is compared between two optimized shape with and without gap as shown in Figure.4.9.
Figure 4.9 Comparison samples.
Blue: electrolyte material. Red:porous electrode material
In the right graph, the integration of the current density along the boundary is 2.28A/m. In the right
graph, the outcome drops to 0.96A/m. Such a phenomenon is different from the conclusion in simplified
model in which the show up of gap will decrease the total current along the current collector boundary. As
mentioned previously, current density is relevant to electrode conductivity and potential gradient. In this
case, the current conductivity is constant. So the potential gradient under different gap could be the effective
factor.
EXPLANATION OF THE INCONSISTENCY
The electric field is the key to explain the inconsistency. The plots of the potential gradient in simplified
model are in Figure 4.10.
35. 4.3. SPECIFIC TECHNIQUE FOR CERTAIN PROBLEMS 27
Figure 4.10 Potential gradient in simplified model.
Contour:electrode potential. Streamlines:current density
The plots above show that the appearance of the gap changes the potential distribution. But as the gap
appears, the potential between the electrode and electrolyte in the up and down domain diverge the current
so that the current inward the electrode is less than the situation without gap. In other words, under such
a situation with insufficiently optimized situation, the difference of conductivity is dominant. Current will
chose the path with larger conductivity. The no gap electrode can gain a large current because its conductivity
is more continuous and the more continuous conductivity is more beneficial for the current conducting.
While for the case in the optimized design shown in Figure 4.11, conductivity is no longer dominant. Both
situations are beneficial for current conducting, so the potential gradient in the domain near the current
collector dominates the outward current.
Figure 4.11 potential gradient in optimization design.
Contour:electrode potential.Red arrows:current density.Black:electrolyte.White:porous electrode.
The plots in the optimized model above show that optimized shape with gap has an equipotential line
near the base so that the potential gradient in the base is larger and the current is larger. One fact can prove the
difference above is that in the simplified model, the potential gradient difference in the base is approximately
14%. But in the topology optimized shape, the difference of potential gradient with and without the gap in the
base changes a lot (from 0.999A/m to 0.449A/m). This extent is similar to the corresponding current change
from 2.28A/m to 0.96A/m.
METHODS TO PENALIZE THE GAP
The first method is a simple approach by filling the gap in painting software and importing the image
into Comsol again as a function. After the import, one more iteration step in the optimization with the initial
value same as the figure function is implemented. Then the optimized shape will show up without gap.This
method has a high requirement of the painting quality and some tiny shapes will burr at the surface of the
electrode.
The second method is changing the objective function from the integration of current along the base
boundary to the current density in a domain closed to the boundary. The schematic plot of this method is in
36. 28 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
Figure 4.12.
Figure 4.12 Changing objective function domain
For the optimized shape in this study, results of the both methods are 0.96A/m and 0.95A/m respectively.
Both methods can achieve the same outcome. While between the two methods, the second method is pre-
ferred because filling the gap in graph processing software is very complicated and the induction of the graph
function will induce some image error and the error will cause calculation mistakes in the afterwards iteration
steps.
4.3.2. ELIMINATING THE FORMATION OF ENCLOSED AREAS
The optimization with low resolution will show less detail and will resist the formation of some features to
be formatted in high mesh resolution, one of the uncertain feature is the enclosed area shown in Figure 4.13.
Figure 4.13 Formation of Enclosed area.
Blue: electrolyte material. Red:porous electrode material
In the lithium ion battery, an enclosed area inside the electrode region is not admissible. One reason
is that such hole will not lead ions to transmit to the other electrode because ions will be trap in it. The
other reason is that the enclosed area is in irregular shape that is hard to be fabricated. Although there is
technique available for the fabrication of enclosed nano-channel, the cross section is regular rectangular[18]
using proton beam writing and thermal bonding ). However, the area will be defined as a desirable feature
according to the topology optimization model because the current source boundary condition will result in
current production in the hole and this will be a condition beneficial for the maximization of the objective
function even though the feature is no admissible.
Two methods can be used to solve this problem.One method is adding a negative source in certain part of
the design domain, the other method is changing the design domain and the region where objective function
is applied.
METHOD 1: ADDING NEGATIVE SOURCE
The key factor in solving this problem is adding an artificial negative current density source[17] in certain
defined domain:
37. 4.3. SPECIFIC TECHNIQUE FOR CERTAIN PROBLEMS 29
QN = −Qc (1−ρH
)
in which:
QN : negative current density source term Qc : prescribed constant defining the amplitude of the current
density ρ: effective density H:penalty factor
By adding this penalty function, when there is a region defined as no electrode material, the negative
current source will be added in the region thus weakening the performance of the domain shape. So this
term prevent the generation of the trapped electrode because the negative current will be all induced into the
electrode material and the optimized process will eliminate such a phenomenon to maximize the outward
current density along the electrode base.
In the design of the battery, because the enclosed areas are all ambient the current collector area, so the
negative source can be added only in those affected areas. Two negative sources with different strength will
be added in two different domain illustrated in Figure 4.14.
Figure 4.14 Application domain of negative current source
In the left plot, the amplitude of the current source is 1.75×108
A/m3
and the penalty factor is 2. This strong
negative source makes sure that the domain is filled with electrode material. In the domain where enclosed
area occurs, the amplitude of the current source is 0.88×108
A/m3
, which is about half of the amplitude added
in the previous region. This amplitude makes sure that the holes will be penalized and the shape will be
optimized adequately. The shape without enclosed area is shown in Figure 4.15.
Figure 4.15 Electrode without enclosed area.
Blue: electrolyte material. Red:porous electrode material
Because the negative current source will change the initial condition, the newly optimized shape will be
a little different from the old one. The most obvious difference is in the region negative current added, more
material will be filled in order to counteract the effect of the negative source term. But the scheme is similar
in the gross view. As the conclusion drew in the previous section, resolution and volume factor are not the
key issue in this topology problem, the graphs with different base width and deign domain will be studied.
38. 30 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
INFLUENCE OF BASE WIDTH
In this case, the volume fraction is selected as 40% and the resolution is 80×80. The widths are 40µm,
60µm, 80µm. The results are illustrated in Table 4.6. The improvement is a relative improvement which is
corresponding to how much the optimized design performance increases compared to the reference value.
All the improvements in the following chapters are the same meaning.
Table4.6 Shapes with different base width
Base width(%) Figure Result.opt(A/m2
) Result.ref(A/m2
) Improvement(%)
40 1.507 0.899 67.6
60 1.608 1.208 33.1
80 1.681 1.481 13.5
As the Base widths increase, results from both optimization design and reference design increase while
the improvements decrease. But the improvements are smaller than those brought by optimized shapes with
gaps.
INFLUENCE OF THE DESIGN DOMAIN
In this case, the resolution is 80×80, the base width is 40% of the y-axis length and the volume constraint
is 40% the design domain area. Results are shown in Table 4.7.
Table4.5 Shapes with different design domain
x/y ratio Figure Result.opt(A/m2
) Result.ref(A/m2
) Improvement(%)
1/1 1.507 0.899 67.6
1/2 1.402 0.858 63.4
1/3 1.308 0.880 63.3
As the ratio between width (x-axis) and the length (y-axis) decreases, both the performance of the opti-
mized design and the reference design decrease. Because the gap is filled, the potential gradient along the
base area goes down and the corresponding current density integration along the current collector boundary
goes down. This result is identical to the conclusion mentioned in the section discussing the gap.
39. 4.3. SPECIFIC TECHNIQUE FOR CERTAIN PROBLEMS 31
METHOD 2 CHANGING OBJECTIVE FUNCTION FEASIBLE DOMAIN
The second method to eliminate the gap and the hole is by changing objective function feasible domain.
The feasible domain is illustrated in Figure 4.16.
Figure 4.16 Feasible domain of objective function
The orange marked domain is defined as the current collector. The objective function is also defined as
maximize the current density in the current collector. Negative current source is added to eliminate enclosed
in the current collector domain. This method can be only used in the situation in which the base width is not
so large. If the width is very big, holes will also show up. While in certain design condition, this method is
very handy and the time cost for each step of iteration is lower than the previous method.
INFLUENCE OF BASE WIDTH
In this case, the volume fraction is selected as 40% and the resolution is 80×80. The widths are 40µm,
60µm, 80µm. The results are illustrated in Table 4.8.
Table4.8 Shapes with different base width
Base width(%) Figure Result.opt(A/m2
) Result.ref(A/m2
) Improvement(%)
40 1.497 0.803 86.4
60 1.568 0.978 60.3
80 1.624 1.142 42.2
As the Base widths increase, results from both optimization design and reference design increase while
the improvements decrease. But the improvements are smaller than those brought by optimized shapes with
gaps.
INFLUENCE OF THE DESIGN DOMAIN
In this case, the resolution is 80×80, the base width is 40% of the y-axis length and the volume constraint
is 40% the design domain area. Results are shown in Table 4.7.
Table4.9 Shapes with different design domain
40. 32 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
x/y ratio Figure Result.opt(A/m2
) Result.ref(A/m2
) Improvement(%)
1/1 1.497 0.803 86.1
1/1.5 1.469 0.825 78.1
1/2 1.433 0.828 73.1
1/3 1.362 0.813 67.5
As the ratio between width (x-axis) and the length (y-axis) decreases, both the performance of the optimized
design and the reference design decrease. Because the gap is filled, the potential gradient along the base
area goes down and the corresponding current density integration along the current collector boundary goes
down.
Compared with the two methods, the method with negative source is preferred. Because for the method
simply switching the objective function, the application condition is so limited. Besides, the convex objective
function domain is a square with tiny length and it is hard to be realized on an electrode film.
4.4. TIME DEPENDENT SIMULATION USING TOPOLOGY OPTIMIZED SHAPE
In the previous section the optimization focuses on the single electrode. So the result should be applied
on one electrode in the transient analysis. Since the parameters are based on the battery whose cell capacity
is determined by the initial state of charge of the negative electrode [15], the negative electrode will be the
domain on which the optimized design is applied. The analysis method is illustrated in Figure 4.17.
Figure 4.17 Time dependent simulation method
The optimized shape will work as an image function. The red marked shape is defined as electrode and
the blue marked region is defined as electrolyte. The optimized deign will be compared with the reference
design. The internal resistance will influence the time at which the battery reach the cut-off voltage thus
affecting the running time and the total capacity of the battery, so running time and discharge capacity will
be studied emphatically.
41. 4.4. TIME DEPENDENT SIMULATION USING TOPOLOGY OPTIMIZED SHAPE 33
4.4.1. IMAGE FUNCTION ACCURACY ANALYSIS
Before the analysis goes on, the accuracy of the image function should be studied. As mentioned pre-
viously, the image function will have some edge problems and material definition uncertainties which may
have negative effects on the final result. In order to make sure that the simulation result is realistic, the accu-
racy should be checked.Two models are shown in Figure 4.18.
Figure 4.18 Models with different building method.
Left:direct build model.Gray:electrolyte.Blue:porous electrode
Right:Image function.Blue:electrolyte.Red:porous electrode
The process of the comparison is simple. The model build directly by Comsol is illustrated in the left,
and the image function is on the left. The charge time, discharge time and unworked time are the same
for the both case. The positive voltage is recorded at each iteration step. By comparing the voltage and the
corresponding gain/loss of the capacity, the accuracy of the image function will be figured out.
For the comparison illustrated above, the x-length and y-length of the electrode is 67µm and 60µm corre-
spondingly. The charge and discharge current densities are both 17.5A/m2
. The discharge terminate voltage
is set at 3.5V and the charge terminate voltage is set at 4.4V. The comparison result is shown in Figure 4.19.
Figure 4.19 Comparison result on the accuracy of image function
X-axis:running time(s).Y-axis:absolute value of power density(W/m2
)
The figure contains the absolute value of battery power .The blue line represents for the image function
and the green line represents for the built model. As shown in the plot, the two lines almost coincide. Such
phenomenon means that the error between both results is very tiny.Model with different electrode shape are
compared in order to make the conclusion more convinced. The average image function error for different
models is recorded in Table 4.10.
Table4.10 Image function error
42. 34 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
Model base width(y-axis) Electrode area Discharge process average error Charge process average error
80µm 400µm2
-0.12% 0.46%
70µm 400µm2
-0.40% 0.60%
60µm 400µm2
-0.17% 0.52%
50µm 400µm2
-0.44% 0.61%
As the table shows, the error between the two models is small. This means the image function has a
relatively acceptable accuracy and if the improvement of the performance is big enough, the model error will
not be contained as a factor.
4.4.2. TIME DEPENDENT SIMULATION AND RESULTS COMPARISON
With the elimination of the algorithm disturbance, the performance competence can be implemented
under a relative ideal condition. The comparison focuses on the influence of the internal resistance on the
state of charge. By comparing the time at which the battery reaches its charge and discharge cut-off poten-
tial, the improvement of topology optimized battery will show up in an intuitionistic way. The charge and
discharge current density is 17.5A/m2
. Charge and discharge cut-off density are 4.4V and 3.4V respectively.
The parameters in detrail are shown in Table 4.11.
Table4.11 Parameters in simulation
Parameters Description Unit Value
I Discharge current density (A/m2
) 17.5
T Simulation temperature K 298
Ce,0 Initial concentration of electrolyte mol/m3
2000
Csa,0 Initial concentration of lithium ions in the anode mol/m3
14870
Csc,0 Initial concentration of lithium ions in the cathode mol/m3
3900
δa Width of anode µm 100
δc Width of cathode µm 100
δs Width of separator µm 20
δy Length of the base current collector µm 100
εa Porosity of anode 0.357
εc Porosity of cathode 0.444
Ca
T Maximum lithium ions concentration in anode mol/m3
26390
Cc
T Maximum lithium ions concentration in cathode mol/m3
22860
Da
s Diffusivity coefficient of particles in anode m2
/s 3.90× 10−14
Da
s Diffusivity coefficient of particles in cathode m2
/s 1.00× 10−13
D Diffusivity coefficient of electrolyte m2
/s 7.50× 10−11
ka
r Reaction rate constant in anode m5.5
/mol0.5
s 2.00× 10−11
kc
r Reaction rate constant in cathode m5.5
/mol0.5
s 2.00× 10−11
Ra
p Radius of anode active material sphere µm 12.5
Rc
p Radius of negative active material sphere µm 8
t+
0 Transport number of positive ions 0.363
σa Conductivity of anode material S/m2 100
σc Conductivity of cathode material S/m2 3.8
Under such a condition, the full discharge-charge cycle of the battery is illustrated in Figure 4.20.
43. 4.4. TIME DEPENDENT SIMULATION USING TOPOLOGY OPTIMIZED SHAPE 35
Figure 4.20 Full working cycle comparison between optimized and reference design(terminal voltage)
X-axis:running time(s).Y-axis:positive terminal voltage(V)
X-axis is running time and y-axis is the cathode potential as the mark of the state of charge. The blue
line is the charge-discharge curve of optimized deign. Green line is the curve of reference design. The plot
shows that both discharge and charge process of the battery with optimized anode shape are prolonged. The
discharging time is increased from 765s to 845s. As the result of the increasing capacity, the charging time
is also increased from 700s to 795s. The lower internal resistance can explain such an improvement because
lower internal resistance will make the battery reach a higher discharge/charge level. The absolute value of
the discharge and charge power in time domain is illustrated in Figure 4.21.
Figure 4.21 Full working cycle comparison between optimized and reference design(battery power)
X-axis:running time(s).Y-axis:absolute value of power density((W/m2
)).
X-axis is absolute value of power density and y-axis is the cathode potential as the mark of the state of
charge. The blue line is the charge-discharge curve of optimized deign. Green line is the curve of reference
design. The figure shows that the charge power is larger than the discharge. This phenomenon coincides with
the result that the discharging process takes longer time than the charging process. The discharge capacity
which determines the life time of the battery can be calculated by integrating the power density on the time
domain. In this case, the capacity of the battery increases from 47383J/m2
to 54206J/m2
. The increase is
approximately 14.4%.
44. 36 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
CAPACITIES WITH DIFFERENT DISCHARGE RATES
For lithium, high discharge capacity cannot be obtained in high discharge rate condition. As current den-
sity increased, voltage drop rapidly, discharge time decreases and finally the capacitance of the capacitor is
decreased [19].The reason of this capacity loss is that high discharge rate is able to lead the salt concentration
deplete fast, thus resulting in the loss of capacity. If the discharge rate is extremely fast, the salt concentration
will drop to null at almost the beginning stage of the discharge[15] . The electro-chemistry theory inside is
not an emphasis in this thesis. How does the optimized battery perform under different discharge rate will
be studied next.
The comparison is carried out under five different discharge current densities which are 10A/m2, 15
A/m2, 20 A/m2, 25 A/m2, 30 A/m2 respectively. Some idealized conditions are also set in the study. First,
the influence of temperature on discharge capacity is neglected because as the temperature increases the
capacity of the battery will change with time. Second, the capacity fade is neglected. Capacity fade is a phe-
nomenon that after multi times of working cycle, the capacity of lithium ion battery will drop to a lower degree
because of several reasons such as lithium deposition, electrolyte decomposition, active material dissolution,
phase transition inside the insertion electrode materials, and passive film formation on the electrode and cur-
rent collectors [20]. The neglecting of this condition is under the condition of easy calculation, otherwise the
capacity under any working cycle will be taken into record and calculation.
Capacities under different discharge rates are in the Table 4.12. The improvement is calculated between
the optimized capacity and reference capacity under the same discharge rate.
Table4.12 Capacities under different discharge rates
Discharge rate(A/m2
) Capacity opt(J/m2
) Capacity ref(J/m2
) Improvement (%)
10 69724 59390 17.4
15 59500 50850 17.0
20 50320 44587 12.9
25 42598 39349 8.2
30 35757 33364 7.1
The table shows that as the discharge rate increases, the capacity of the battery decreases. This coincides
with the theory and the experiment result of lithium ion battery. The improvement under high discharge rate
is smaller than that under low discharge rate.This phenomenon can be explained that when current is low, the
optimization on current flow conduction has large effect because at this time the current plays a big role in the
performance of the battery. While when current becomes large the improvement of total current occupies a
smaller part than the previous situation in which the current is small. As the capacity loss happens regardless
of the situation whether the battery is optimized, the effect of optimized shape on resisting the capacity loss
at high discharge deserves to be researched.
Table 4.13 contains such information. The discharge capacity at 10A/m2
is set as the reference capacity.
Table4.13 Capacities loss different discharge rates
Discharge rate(A/m2
) Capacity opt(J/m2
) Capacity loss opt(%) Capacity ref(J/m2
) Capacity loss ref(%)
10 69724 0 59390 0
15 59500 14.7 50850 14.3
20 50320 27.8 44587 24.9
25 42598 38.9 39349 33.7
30 35757 48.7 33364 43.8
In the table below, the listed data shows that as the discharge rate increases, the capacity loss of optimized
battery is bigger than the capacity loss of reference designed battery. On mathematical aspect. This is reason-
able because the improvement at high discharge rate is smaller than that at low discharge rate. So the portion
of remaining energy at high discharge rate will be less than that at low discharge compared with the reference
design. On the other hand, although the discharge current in the out circuit are the same, inside the battery,
the electrode gaining more current will get more heat and when battery is heated, the capacity will loss faster.
The heat dissipation figure shown in Figure 4.22 can prove this.
45. 4.4. TIME DEPENDENT SIMULATION USING TOPOLOGY OPTIMIZED SHAPE 37
Figure 4.22 Heat dissipation of optimized design and reference design(Current:30A/m2
)
In the figure, at high discharge rate, before the battery reaches the cut-off voltage, the heat dissipation
of optimized design is larger than that from the reference design. More generated heat will cause a bigger
increase of the internal resistance and the increasing resistance will in return make the low capacity at high
discharge rate even lower[20]. So the capacity loss of optimized design at high discharge rate is larger.
CAPACITIES WITH DIFFERENT BASE WIDTH
Optimization with different shape has been studied. The results are shown in Table 4.14a.
Table4.14a Simulation of Shapes with different base widths
Figure
Base width(µm) 0.8 1.2 1.6
Discharge rate(A/m2
) Capacity(J/m2
) Capacity(J/m2
) Capacity(J/m2
)
10A 72694 69724 67488
15A 60321 59500 56729
20A 50829 50320 47181
25A 43282 42598 41330
30A 35832 35757 34596
It is drawn that with a larger base width, the discharge capacity decreases. This is not the same as the
previous static optimization. The reason is that in the static optimization, the improvement is based on the
improvement of the total current density. So a larger base width means a larger total current density along the
boundary. While in the time dependent optimization, the improvement is based on the value of total power
density. So the difference of the selected parameter leads to the different rule.
The improvement of shapes with different base width are recorded in Table 4.14b.
Table4.14b Capacities under different base width
Discharge rate(A/m2
) Improvement (%)(0.8µm) Improvement (%)(1.2µm) Improvement (%)(1.6µm)
10 22.4 17.4 13.6
15 18.6 17.0 11.6
20 14.0 12.9 5.8
25 9.9 8.2 5.0
30 7.4 7.1 3.7
It is drawn that with a larger base width, the improvement decreases. This is similar to the conclusion
drawn from the previous static optimization that a higher base width have a lower improvement.
46. 38 4. TOPOLOGY OPTIMIZATION FOR MAXIMIZING TOTAL CURRENT
CAPACITIES WITH DIFFERENT AMOUNT OF MATERIAL
The improvement of shapes with different amount of are recorded in Table 4.15.
Table4.15a Simulation of Shapes with different volume constraint
Figure
volume constraint(%) 30 35 40
Discharge rate(A/m2
) Capacity(J/m2
) Capacity(J/m2
) Capacity(J/m2
)
10A 64730 67230 69724
20A 46589 48500 50320
30A 33101 34462 35757
When the volume constraint increases, because there are more electrode material, the total discharge
capacity increases. This is the same with the previous static optimization.
The improvement of shapes with different base width are recorded in Table 4.15b.
Table4.14b Capacities under different volume constraint
Discharge rate(A/m2
) Improvement (%)(30%) Improvement (%)(35%) Improvement (%)(40%)
10 17.3 17.3 17.4
20 12.4 12.7 12.9
30 6.7 7.0 7.1
When the volume constraint increases, because there are more electrode material, the total discharge
capacity increases. While the improvement at different volume constraint is nearly the same. This is the
same with the previous static optimization.
47. 5
TOPOLOGY OPTIMIZATION FOR MINIMIZING
TERMINAL POTENTIAL
Minimizing the potential for a specific current density is another method to minimize the internal resis-
tance of the battery. In the model for this optimization problem, the boundary conditions are similar to that
of the first model, while the material definition and objective function is in a different form.
5.1. SETTINGS IN TOPOLOGY OPTIMIZATION
Before optimization, boundary conditions, material definitions and optimization equations mentioned in
the previous section should be set to the proper form so that the topology optimization can function properly.
5.1.1. BOUNDARY CONDITION
In the following topology optimization, the boundary conditions are shown in Figure 5.1.
Figure 5.1 Boundary conditions
The blue marked domain is the design domain. It contains positive electrode, negative electrode and
electrolyte. An electrode current source term:
QS = ∇·is
is added in the design domain to simulate the initial condition that with the chemical reaction and the
migration of lithium ions, the domain is filled with current. The electric ground is added on the black marked
boundary. This is the same as that in the previous topology optimization chapter.
5.1.2. MATERIAL DEFINITION
The optimized potential is the positive electrode voltage while the electric ground boundary condition is
added to the negative electrode. In this case the shapes of both electrodes will be optimized simultaneously.
39
48. 40 5. TOPOLOGY OPTIMIZATION FOR MINIMIZING TERMINAL POTENTIAL
A method used for multiple phase material in one design domain[21] is involved here. The definition of the
material are represented as:
Electrode conductivity: σspρ
P1
p +σsnρ
P1
n (1−ρp)P2
Electrode volume fraction:Fs
Electrolyte conductivity: κ
Electrolyte volume fraction: 1−Fs +Fl (1−ρp)(1−ρn)
With the following definitions for the design variables:
ρp = 1,ρn = 0or1:Mesh element is filled with porous positive material with electrode conductivityσsp,
electrode volume fraction Fs, electrolyte conductivity κ and electrolyte volume fraction 1−Fs.
ρp = 0,ρn = 1:Mesh element is filled with porous positive material with electrode conductivityσsn, elec-
trode volume fraction Fs, electrolyte conductivity κ and electrolyte volume fraction 1−Fs.
ρp = 0,ρn = 0:Mesh element is filled with electrolyte conductivity κ and electrolyte volume fraction 1 −
Fs +Fl .
5.1.3. OBJECTIVE FUNCTION AND CONSTRAINTS
The optimization problem is described in Figure 5.2.
Figure 5.2 Objective function location
The black marked boundary is the positive electrode boundary which acts as the current collector for the
positive electrode. This is the same as the negative electrode demonstrated before. The integral objective
function:
Mimimize.φs(is,ρ) =
is
σ
×dl
is added on the boundary.
Both design variables can be set in the whole design domain with an upper bound 1 and lower bound of
0. Another constraint is that the total amount of each electrode should be less than certain value.
The values and the explanations of the parameters in the previous formulas are illustrated in Table 5.1
Table5.1 Parameters in topology optimization setting
Parameter Description Value Unit
Fl volume fraction of electrolyte 0.5
Fs volume fraction of electrode 0.5
P1 penalty factor for positive electrode 6
P1 penalty factor for negative electrode 2
Qs electrode current source 1.75×108
A/m3
κ conductivity of electrolyte 0.2 S/m
σsp conductivity of positive electrode 3.8 S/m
σsn conductivity of negative electrode 100 S/m
ρp Positive electrode material density [0,1]
ρn Negative electrode material density [0,1]
49. 5.2. OPTIMIZATION RESULTS 41
5.2. OPTIMIZATION RESULTS
The optimization is carried for the design domain with a rectangular shape with a length of 200µm and a
height of 100µm. The mesh is uniform and consists of 80 by 40 rectangular elements. The upper bound for
the total amount of positive electrode material is 25% of the design domain area. The upper bound for the
negative electrode material is 20% of the design domain area. The result is shown in Figure 5.3.
Figure 5.3 Result of topology optimization
Blue:electrolyte. Red:positive porous electrode. Green:negative porous electrode
The blue region is the negative electrode, the red region is the positive electrode and the green region is
the electrolyte. The electrode shape optimized for minimum terminal potential shows a dramatic difference
from the shape optimized for current. The reason is that when the potential is fixed on one edge, the method
to minimize the potential at the opposite edge is to fill the path between the two edges with conductive
material because the path with conductive material is the path with minimum potential drop. The potential
distribution is shown in Figure 5.4.
Figure 5.4 potential distribution in optimization design
Red lines:outline of positive electrode. Black lines: outline of negative electrode.
Red arrows:current density.Contour:electrode potential
The red arrows represent the charge flow. The colour scale indicates the strength of the potential. The
black outline shows the edge of the negative electrode and the red line outlines the positive electrode. As one
can see from the picture, the potential change in the electrode region is very small. The electrode shape con-
centrates the high potential area to the central part of the electrolyte. This minimizes the resistance between
the two electrode boundaries.
5.3. TIME DEPENDENT SIMULATION OF THE OPTIMIZED DESIGN
The transient analysis for this case is similar to the transient analysis applied to the current optimiza-
tion electrode of the previous chapter. The difference is that the optimized shape contains both electrodes
and they cannot be defined with one image function. The positive aspect in the case is that the optimized
shapes for both electrodes are simply-connected and have a convex shape. So the transient analysis and the
performance comparison can be carried by drawing the geometry manually. Here, a two-dimensional and a
three-dimensional model is created and studied respectively.
5.3.1. TWO-DIMENSION MODEL AND SIMULATION
The model built basing on topology optimization result is shown in Figure 5.5.
50. 42 5. TOPOLOGY OPTIMIZATION FOR MINIMIZING TERMINAL POTENTIAL
Figure 5.5 Model based on topology optimization result and reference model
Blue:electrolyte. Orange:positive electrode. Gray:negative electrode.
The left figure shows the optimized design. The right figure is the reference design. The grey domain is
the negative electrode, the blue domain is electrolyte and the orange domain is the positive electrode. The
discharge capacity is calculated and compared with a reference with same amount of material and discharge
rate. The parameters used for the battery are shown in Table 5.1.
Table5.1 Parameters in simulation
Parameters Description Unit Value
Ce,0 Initial concentration of electrolyte mol/m3
2000
Csa,0 Initial concentration of lithium ions in the anode mol/m3
14870
Csc,0 Initial concentration of lithium ions in the cathode mol/m3
3900
εa Porosity of anode 0.357
εc Porosity of cathode 0.444
Ca
T Maximum lithium ions concentration in anode mol/m3
26390
Cc
T Maximum lithium ions concentration in cathode mol/m3
22860
Da
s Diffusivity coefficient of particles in anode m2
/s 3.90× 10−14
Da
s Diffusivity coefficient of particles in cathode m2
/s 1.00× 10−13
D Diffusivity coefficient of electrolyte m2
/s 7.50× 10−11
ka
r Reaction rate constant in anode m5.5
/mol0.5
s 2.00× 10−11
kc
r Reaction rate constant in cathode m5.5
/mol0.5
s 2.00× 10−11
Ra
p Radius of anode active material sphere µm 12.5
Rc
p Radius of negative active material sphere µm 8
t+
0 Transport number of positive ions 0.363
σa Conductivity of anode material S/m2
100
σc Conductivity of cathode material S/m2
3.8
Capacity under different discharge rates are in Table 5.2. The improvement is calculated between the
optimized capacity and reference capacity under the same discharge rate.
Table5.2 Capacities under different discharge rates
Discharge rate(A/m2
) Capacity opt(J/m2
) Capacity ref(J/m2
) Improvement (%)
10 64164 62768 2.2
15 56837 55029 3.3
20 51228 49063 4.4
25 46732 44157 5.8
30 42687 39339 8.5
The table shows that as the discharge rate increases, the capacity of the battery decreases. The improve-
ment under high discharge rate is larger than that under low discharge rate. The difference from the first
optimization result is that the improvement at high discharge rate is much bigger than the improvement at
low discharge rate.
The capacity loss is also shown in the Table 5.3. The discharge capacity at 10A/m2
is set as the reference
capacity.
Table5.3 Capacities loss different discharge rates
51. 5.3. TIME DEPENDENT SIMULATION OF THE OPTIMIZED DESIGN 43
Discharge rate(A/m2
) Capacity opt(J/m2
) Capacity loss opt(%) Capacity ref(J/m2
) Capacity loss ref(%)
10 64164 0 62768 0
15 56837 11.4 55029 12.3
20 51228 20.2 49063 21.8
25 46732 27.2 44157 29.7
30 42687 33.5 39339 37.3
As the discharge rate increases, the capacity loss of optimized battery is less than the capacity loss of refer-
ence designed battery. This is because at high discharge rate, the heat dissipation power density of reference
design is bigger than the heat dissipation power density of optimization.The comparison is shown in Figure
5.6.
Figure 5.6 Heat dissipation of optimized design and reference design(Current:30A/m2
)
In all optimized cases, the negative electrode shows the trend that it is moved as close to the base area as
possible. So the closest distance along current direction should be researched to find out if it has a large effect
on the performance of the battery.The schematic graph shows this study is shown in Figure 5.7.
52. 44 5. TOPOLOGY OPTIMIZATION FOR MINIMIZING TERMINAL POTENTIAL
5.7(A)Reference closest distance
5.7(B)Short closest distance
5.7(C)Shorter closest distance
Figure 5.7 Schematic graph of the closest distance
The distance and corresponding discharge capacity is in the Table 5.4. (discharge rate is 30A/m2
)
Table5.4 Electrode distance and discharge capacity
Distance(µm) Capacity opt(J/m2
) Improvement(%)
REF 39339 0
160 39589 0.6
140 39835 1.3
120 40302 2.4
100 40772 3.6
80 41137 4.6
60 41489 5.5
40 41725 6.1
20 42132 7.1
The table above shows that as the distance between the end of the negative electrode and the positive
electrode decreases, the capacity of the battery will increase. It is clearly a trend seen when optimizing the
battery, that the closest distance should be as small as possible.
5.3.2. THREE-DIMENSION MODEL AND SIMULATION
With the help of for instance the Nano-imprint technique[22], it is possible to create simple but 3D struc-
tures that can be used in actual thin film batteries. This micro structure has been shown to be available for
Lithium ion batteries using silicon as the anode material. The process of nanoimprint lithography(NIL) is
shown in Figure 5.8.
53. 5.3. TIME DEPENDENT SIMULATION OF THE OPTIMIZED DESIGN 45
Figure 5.8 Outline of the NIL process for Si nanowall anode
This shape is similar to the result drawn in the previous optimization, but now in a way that it can be
readily implemented in a real battery design. With the help of topology optimization as inspiration, but
simplifying the geometry such that it can be manufactured a three-dimension model can be built to show
the impact on actual lithium ion batteries.
The positive electrode film and the detail graph are in Figure 5.9.
Figure 5.9 Positive electrode 3D model.Unit:µm
The left Figure is the zoom out view of the lithium ion battery positive electrode film and the right Figure
is a single cell of the film. The structure of the electrode surface is similar to the previously optimized design.
The negative electrode film and a detail are given in Figure 5.10.
54. 46 5. TOPOLOGY OPTIMIZATION FOR MINIMIZING TERMINAL POTENTIAL
Figure 5.10 Negative electrode 3D model.Unit:µm
The left Figure is the zoom out view of the lithium ion battery negative electrode film and the right Figure
is the single cell of the film.The positive and negative films are made such that they fit on top of each other,
with the teeth in between the layers. Compared to conventional batteries with two flat polymers, the smallest
distance between anode and cathode is smaller in the new design. A better performance is expected with
such a nano-structure.
The capacity for different discharge rates are stated in Table 5.5. The improvement is calculated between
the optimized capacity and reference capacity for the same discharge rates, respectively.
Table5.5 Capacities under different discharge rates
Discharge rate(A/m2
) Capacity opt(J/m2
) Capacity ref(J/m2
) Improvement (%)
10 75181 73701 2
15 67879 65496 3.6
20 60737 57214 6.2
25 54068 49568 9.1
30 49039 44332 10.6
The table shows that as the discharge rate increases, the capacity of the battery decreases. The improve-
ment under high discharge rate is larger than that under low discharge rate. The difference from the first
optimization result is that the improvement at high discharge rate is much bigger than the improvement at
low discharge rate.
The capacity loss is also shown in the Table 5.6. The discharge capacity at 10A/m2
is set as the reference
capacity.
Table5.6 Capacities loss different discharge rates
Discharge rate(A/m2
) Capacity opt(J/m2
) Capacity loss opt(%) Capacity ref(J/m2
) Capacity loss ref(%)
10 75181 0 73701 0
15 67879 9.7 65496 11.1
20 60737 19.2 57214 22.4
25 54068 28.1 49568 32.7
30 49039 34.8 44332 39.8
The improvement agrees with the conclusion above. Such a micro structure is available on Lithium with
silicon wire as anode material. And using the optimized shape design on lithium ion batteries with conven-
tional material is theoretically able to improve the performance of lithium ion battery substantially.
55. 6
TIME DEPENDENT OPTIMIZATION
As discussed above, by the time a battery reaches its cut-off voltage, the battery will stop to discharge. The
discharge curve shows that the terminal potential decreases monotonically during the discharge process.
This means that, if the potential is higher at any point in time before the cut-off, the battery will take more
time to reach its cut-off voltage (shown in Figure 6.1). So maximizing the potential at a single instance is a
nice first start to simplify the optimization problem and reduce the computational cost of the time dependent
analysis.
Figure 6.1 Theory of the time dependent optimization
In the previous chapters, the optimization is carried out based on the results of non-transient analyses.
Although time dependent optimization takes much time, it potentially allows a much better design to be
optimized since the physical phenomena do not behave in a steady state manner.
6.1. SETTINGS IN TOPOLOGY OPTIMIZATION
Before optimization, boundary conditions, material definitions and optimization equations mentioned in
the previous section should be set in the proper form so that the topology optimization can function properly.
The procedure of the time optimization design is similar to the previous design. The different aspect is that in
this case, the optimization is carried out under a time dependent and multiple design domain environment.
The values and the explanations of the parameters in the following formulas are illustrated in Table 6.1
Table6.1 Parameters in topology optimization setting
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