194 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Comprehensive Three-Dimensional Dynamic
Modeling of Liquid Crystal Devices Using
Finite Element Method
Zhibing Ge, Student Member, IEEE, Thomas X. Wu, Senior Member, IEEE, Ruibo Lu, Xinyu Zhu, Qi Hong, and
Shin-Tson Wu, Fellow, IEEE
Abstract—In this paper, a comprehensive open-source three-di-
the dynamic behavior of liquid crystal (LC) directors in complex
structures. This dynamic model is based on interactively iterating
the vector representation of director profile and potential distribu-
tion. The director update formulations are derived in detail from
the Galerkin’s approach of FEM, including the weak form ap-
proach to simplify the highly nonlinear iteration equation. The po-
tential update formulations are derived from the Ritz’s approach
of FEM. A 2-D in-plane switching (IPS) structure is used as an ex-
ample to compare our approach with the FEM based commercial
software (2dimMOS).The results from both programs showan ex-
cellent agreement. Furthermore, our method also agrees well with
cell with zigzag electrodes.
Index Terms—Liquid crystal devices, liquid crystal displays,
modeling, finite element method.
able modeling of LC electro-optic behaviors is critical in both
developing novel devices and optimizing the current display
devices. Generally, the LC modeling includes two steps: eval-
uation of the LC directors’ deformation under external electric
fields and calculation of the optical properties thereafter. There-
fore, an accurate model of LC deformation is a prerequisite
for reliable device design and subsequent optical analysis. In
general, the dynamic modeling of the LC director calculations
can be implemented by finite-difference method (FDM) ,
, finite-element method (FEM) , , or a combination
of both. The success of these methods in accurate modeling
of LC devices has been demonstrated by the development of
LC modeling software, such as LC3D , DIMOS , LCD
Master , Techwitz LCD , LCD-DESIGN , LCQuest
, and others.
IQUID CRYSTAL (LC) materials have been used widely
in direct-view and projection displays. Accurate and reli-
Manuscript received June 6, 2005; revised July 24, 2005. This work is sup-
ported by Toppoly Optoelectronics Corporation (Taiwan), Taiwan, R.O.C.
Z. Ge, T. X. Wu, and Q. Hong are with the Department of Electrical and
Computer Engineering, University of Central Florida, Orlando, FL 32816 USA
(e-mail: email@example.com; firstname.lastname@example.org; email@example.com).
R. Lu, X. Zhu, and S.-T. Wu are with the College of Optics and Pho-
tonics, University of Central Florida, Orlando, FL 32816 USA (e-mail:
firstname.lastname@example.org; email@example.com; firstname.lastname@example.org).
Digital Object Identifier 10.1109/JDT.2005.858885
Further, as a direct solver (seeking solutions via directly solving
iterations) –, this method can generate accurate solutions
from solving large sparse matrix system, which is an inherent
property of FEM. Since efficient algorithms for large sparse
electromagnetics are well developed, they can be introduced to
development of computer aided symbolic derivation techniques
nowadays makes automatic FEM formula generation in LC
modeling possible, which was previously the main obstacle to
introduce FEM into accurate LC simulations. Several papers
have already exploited the FEM in the study of advanced
LC devices –. However, their focus is more device
physics oriented. To the best of our knowledge, no literature
has ever reported an open-source comprehensive mathematical
derivation and formulation of the FEM in LCD modeling. The
lack of this part in LCD simulators has hindered the research
as well as applications of LCD technologies. Therefore, the
main objective of this paper is to propose a comprehensive
a detailed derivation of the FEM formulations. In addition, a
special technique, the weak form method ,  to greatly
simplify formula derivations, is introduced as well.
In this paper, we first introduce the derivation of update
equations for LC director and potential profiles, as well as the
solve the above derived highly nonlinear equations based on the
,  for potential profile. Derivations of the nonlinear update
). Therefore, we introduce the weak
form method ,  to simplify the director update equations.
This technique makes the first-order 3-D FEM interpolation
function (tetrahedron shaped) accurate enough. Furthermore,
some effects which influence the director distribution such as
surface anchoring and flexoelectric polarization ,  are
discussed as well. To verify our derivations, a 2-D in-plane-
method and the FEM-based 2dimMOS . Finally, a 3-D super
IPS structure with zigzag electrodes  is simulated using
our FEM and results are compared to those derived from FDM
to validate our derivations as well.
1551-319X/$20.00 © 2005 IEEE
GE et al.: COMPREHENSIVE 3-D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM195
Fig. 1. Vector representation of a LC director.
II. MATHEMATICAL FORMULATIONS
A. Dynamic Model of LC Devices
Modeling of the dynamics of LC directors in response to ex-
ternal voltages generally starts off with minimizing the LC free
energy within the LC cell , . However, owing to the
high nonlinearity of the free energy expressions and the cou-
pling between elastic and electric energies, a direct solution of
the LC director deformation and potential distribution from the
free energy equation is almost impossible. Therefore, the dy-
namic modeling usually involves an iteration process : mini-
mizing the LC free energy to update the LC director profile and
solving the Gauss law (or minimizing the electric energy) to up-
date the potential profile interactively. In modeling the LC di-
rectors, both vector representation , as shown in Fig. 1, and
tensor – methods can be employed. Detailed com-
parison of these two methods is provided in . Owing to its
mathematical simplicity, fast computing speed, and reliability,
the vector method is widely employed in LCD simulations. In
this paper, we refer the vector method in our derivations.
1) Director Profile Update Theory: Numerical simulation
of LC transient state at each time step under constant voltage is
based on minimizing Gibbs free energy , , which is the
difference between Frank-Oseen strain free energy , ,
 and electric energy, by solving the Euler Lagrange equa-
tion , . The Gibbs free energy density in the LC cell can
be expressed as
in the given coordinates, as shown in Fig. 1;
wavenumber, which is a parameter specifying the equilibrium
of the LC materials; and
electric field, and potential function in the entire system region,
is the unit vector form of LC directors
is the chiral
are the electric displacement,
free energy (a volume integral of the energy density) by solving
Euler–Lagrange equation with inclusion of a Rayleigh dissipa-
tion function will result in , 
is the dielectric tensor of LC material which
and . It is known that to minimize the system
tiplier to constrain the unit length of the director
). Here the fluid flow effect is assumed to have
secondary importance and is thus ignored in (4). The Lagrange
is normalized as
each iteration step. Therefore, (4) specifying the iterative rela-
tion of LC directors under external applied voltages can be fur-
ther simplified to the following form:
represents, and, and is a Lagrange mul-
By taking a time-difference
can be further expressed as
Since the dielectric tensor
, thepotential and the directorfunctions in
coupled with each other as indicated by (1)–(3). Computer as-
sisted derivations show
has a highly nonlinear form con-
taining the highest spatial derivative of
up to the second order (e.g.,
use the first order 3Dtetrahedral basis function to evaluate these
second order derivatives, weak form method is introduced, as
will be discussed in later sections.
2) Potential Profile Update Theory: At given LC director
distribution and electrode voltage, the potential profile
be determined by solving the Gauss law
includes the LC director
with respect to
). In order to
as indicated in (2) and (3). Here solving the above
equation is equivalent to minimizing the electric energy
in the bulk from the variational
aspect. The bulk region defined here includes the LC cell and
other regions, such as the substrates. Both direct solver (e.g.,
FEM with Ritz’s method or Galerkin’s method) and iterative
approach (e.g., FDM with iterations) can be employed to solve
the desired potential distribution. Due to the simplicity in
implementation and the inherited accuracy of direct solver, we
will take FEM with Ritz’s method in the following derivations.
is expressed in terms of the poten-
196 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Fig. 2. Iteration flowchart of the dynamic modeling.
3) ModelingFlowchart: AsexplainedinSectionsII-A1and
II-A2, the coupling between director deformations and applied
electric fields makes it almostimpossible to obtainboth director
and potential profile solutions simultaneously. However, this
iterative scheme is illustrated in Fig. 2. The initial LC director
profile can usually be preset by its boundary conditions. The
stability of this iterative updating scheme is an important issue
B. FEM Implementation
The solution for the above-mentioned iteration scheme in-
cludes two iterative steps: 1) solving the potential distribution
at a given director profile and 2) solving the director profile at
a given potential distribution. FEM is employed to solve these
1) Director Update Formulations via Galerkin’s Method:
The high nonlinearity of (1) indicates that direct solution of the
LC director distribution is difficult to obtain. However,
Galerkin’s method can bypass this difficulty. The Galerkin’s
method belongs to the family of weighted residual methods,
which seek the solution by weighting the residual of the
differential equations. Suppose we divide the study region
into element regions with
order interpolation function). The local element has a specific
shape, such as a tetrahedral block in the 3-D FEM case. The
general solution of each director component
be approximated by interpolating from its values on the
global nodes using global interpolation functions as
nodes (considering the first
th node value of and interpolation function in the entire region,
respectively. The specific form of
thispaper. After inserting
into(5), the residual
interpolation, can be written as
is the approximate solution ofand are the
will be discussed later in
, error of the
The exact solution occurs only when
above approximation, the residual
value. However, with Galerkin’s method, the residual
minimized through weighting the residual
equals 0. Due to the
always leads to a nonzero
can be 1-D to 3-D integrals for different problems. Replacing
here by (9) yields
denotes the integral on the study domain , which
, we can obtainlinear equations with
. This equation group can be written in a matrix
denotes a matrix anddenotes a vector, with
as (7), we can obtain the iteration relation via
Galerkin’s method in a matrix representation as
Equation (16) is the final iteration equation of the dynamic
model, which specifies the director profile correlation between
two consecutive time steps.
However, the above-mentioned matrix
not easy to obtain because a good group of interpolation func-
defined on the entire domain is difficult to assign. Here
FEM can automatically generate an appropriate group of in-
terpolation functions. The detailed derivation with FEM in ob-
taining the matrix
The advantage of this method is evident that the global ma-
in (13) is uniquely determined by the mesh and the in-
terpolation functions, as indicated in (A-5) in the Appendix A.
As a result, in updating director profiles from (16), the matrix
and vector is still
is provided in Appendix A.
GE et al.: COMPREHENSIVE 3-D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM197
only needs to be calculated at the first iteration step and can
erty greatly reduces the computing time in the director updating
2) Potential Update Formulations via Ritz’s Method: From
the analysis of the potential update theory, the update of po-
tential is equivalent to solve the Gaussian law of (8), or mini-
mizing the system electric energy
a variational viewpoint. In our derivations, we use the Ritz’s
method to minimize the system electric energy, which inherits a
Galerkin’s method in solving the Gaussian law.
The free electric energy
in the study domain is equal to
and a scalar for isotropic media, such as the glass substrates.
Similarly, the approximate global potential solution
expressed by the global node value
is a tensor for anisotropic media, such as the LC layer,
and global interpolation
is a function of
might be different with the total node
number in the director update. From mathematical theory, its
value is minimized where the relative partial derivative of
with respect to equals zero. This leads toequations as
This equation group can be further expressed as a matrix repre-
element in matrix
denotes a matrix and
has a form of
denotes a vector. Each
The detailed derivation of
FEM is described in Appendix B. Here for the nematic LC ma-
terial, the tensor
is a symmetric matrix. This symmetric prop-
erty can further simplify
to the following form:
and further implementation by
3) Weak Form Technique: As explained in above sections,
for director and potential calculations have
definite forms and can be directly obtained from their cor-
responding equations using FEM. Still, the difficulty lies in
the complicated derivations of global vector
though approximation of the derivations can be applied ,
its accuracy in some complex device structures is not always
guaranteed. Therefore, the state-of-the-art computer software
such as MAPLE  is used to generate FEM-based formula-
tion derivations. We find the highest order of spatial derivative
with respect to, and
a result, the first order of interpolation/basis function [e.g.,
(A-2)] will lose accuracy due to the fact that its second order
spatial derivative equals to zero. For example, let us assume
there is one term
shown in (A-6). After substituting
with the first order interpolation function
diminishes to zero, which does not
A good solution of this issue is to introduce the weak form
method. Derivations of (6) based on commercial software show
can be summarized into a general expression
in (15). Al-
is up to the second order. As
, the second order
to the first order,
efficient. Neglecting the subscript index
in the following form as
stands for the terms with highest derivatives up
denotes these second order
we now rewrite (15)
from (23) into (24) yields
In (25), the first term with derivatives up to first order can be
accurately calculated by FEM with first-order interpolation
function . Although there are many terms with second-order
derivatives, they can always be divided into two categories: 1)
198JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Fig. 3. 2-D boundary planes from a 3-D bulk.
illustration of taking the weak form to accurately evaluate these
two categories: 1)
orequals to and 2) neithernor equals to . As an
. Using the integra-
tion by parts, we obtain
is fixed under strong anchoring assumption in the vertical
2-D integral term in (26) equals zero. This property further sim-
plifies (26) into
anddenote the boundary coordinates in the vertical
at these boundaries, the
lable by the first-order interpolation functions. On the other
in (27) is not always equal to zero on the boundaries in
direction. However, as illustrated in Fig. 3, the term
(a) 1-D interpolation function. (b) 2-D interpolation function. (c) 3-D
in (27) can be further
The 3-D interpolation function
terpolation function on the boundary planes (
and in this case), as shown in Fig. 4(b) and
(c). To evaluate this term, we can use 2-D FEM on the lateral
boundary planes to calculate (29). This is equivalent to weight
by 2-D FEM with 2-D interpolation functions
ontheboundary planes( - plane at
andin this case). The lateral derivative terms in
and directions (e. g.,
difference from the boundary point and its adjacent ones (e.g.,
decays to a 2-D in-
) can be calculated by a finite
). With the
assistance of FEM, these 2-D integral terms are calculable and
can be added to the 3-D integral terms evaluated by first order
interpolation functions. Other second order terms in (25) can be
calculated similarly in the same way as (26)–(29).
In other words, by taking the weak form method into deriva-
tions of (25), these second-order derivative terms will degrade
one order, which can be calculated by using the first-order
GE et al.: COMPREHENSIVE 3-D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM199
interpolation functions. Therefore, introducing the weak form
greatly facilitates the derivation of vector
approximations to sacrifice simulation accuracy.
4) Boundary Conditions and Other Topics: In solving the
LC director and potential profiles, their boundary conditions
in the study cell need to be well-defined. As shown in Fig. 5,
typical potential boundary conditions on electrodes are the
Dirichlet boundary conditions, i.e., the potential values on
electrodes 1 and 2 are fixed to assigned voltages. In the far
regions along the
direction, such as the outer boundaries
of the supporting substrates where no electrodes are present,
the Neumann conditions apply to the potential, i.e., the spatial
normal derivative of potential (e.g.,
the implementation of these boundary conditions for the poten-
tial, the fixed values on the electrodes can be forced by matrix
and vector manipulation in (20), and the Neumann condition
for potential in the far regions are naturally satisfied in FEM
with the assumption that the electric energy is totally confined
in the LC cells as well as the substrate regions.
For the LC director calculations, in the vertical
on the top and bottom alignment
layer—LC interfaces (at
ified by initial pretilt and azimuthal angle values. With strong
anchoring assumption, the director values on these interfaces
are fixed. A more general case is to take the surface anchoring
energy into consideration. The surface anchoring energy den-
sity from the Rapini-Papoular model ,  on the vertical
boundaries has the following forms:
without any further
) equals zero. In
and ) are usually spec-
coefficients (J/m ),
twist angle, respectively. Equation (30) can be expanded and
expressed in terms of the director components
and are the polar and azimuthal anchoring energy
and are the pretilt polar angle and
The surface anchoring energy from above density function can
be added to the total Gibbs free energy integrated from (1).
Here, one needs to beware the difference in unit between the
surface anchoring energy density (J/m ) and the bulk Gibbs
energy density (J/m ). In the lateral direction (
ditions, i.e., the
mann conditions, i.e.,
urations. Another type of lateral boundary condition frequently
employed in LCD simulation is the periodic boundary condi-
tion. The director
equal values at two later periodic boundaries, such as planes
andin Fig. 5. Because the LC cell patterns
in the pixel are usually periodically repeated in some devices
values are fixed, or by Neu-
and its derivatives have
such as in the IPS mode, the calculation results from one period
are enough to represent the entire pixel. The periodic boundary
size pixels with periodic configurations.
Those abovementioned boundary conditions are most widely
employed in the LC device simulations. However, boundary
conditions for the potential and director update are not confined
to these categories. In stead, they need to be designated in ac-
cordance to the specific problems studied.
In addition to the inclusion of surface anchoring energy,
flexoelectric polarization ,  is another important effect
that needs to be considered under certain conditions. Flexo-
electric effect describes the spontaneous polarization generated
by a deformation of the director in nematic phase composed of
molecules, which exhibit shape asymmetry and have permanent
dipole moments. The flexoelectricity induced polarization will
interact with the applied electric fields and deform the LC
directors. A phenomenological expression for this polarization
is given by
and are the splay and bend flexoelectric coefficients.
The coupling of this polarization with electric fields and LC
deformations will introduce an additional term to the Gibbs
free energy density in (1) as
by the FEM as described in above sections as well. Detailed
introduction and discussion of the flexoelectric effect in LC
modeling can be found in , , –, and references
Other effects that might affect the deformations LC directors
and potential distributions in a cell under certain conditions can
be referred to detailed discussion in the .
, thus it can be counted
III. RESULTS AND DISCUSSION
To illustrate and validate the FEM-based LC dynamic mod-
eling with the above derived formulations, we use a 2-D inter-
digital IPS structure with rectangular electrodes as an example
to compare our own developed method with the FEM-based
IPS with zigzag electrodes to compare our method with the
A. Two-Dimensional Conventional IPS Structure
Fig. 5 depicts a conventional 2-D IPS structure. The LC ma-
terial employed is MLC-6692 (from Merck) with its parameters
listed as follows:
Pa s. The LC cell gap is 4
LC pretilt angles on both substrates are 2 , and the rubbing an-
gles are 80 away from the
axis in the
the electrodes 1 and 2 are located in the same bottom planes at
, with an equal length of 4 m and a separation distance
In the simulation, the electrode 1 is applied with 6 V, and
the electrode 2 is grounded. The LC directors on the vertical
boundaries are assumed to be strongly anchored in our simu-
direction periodic boundary conditions in Fig. 5
are applied to both potential and LC director calculations. In the
2dimMOS, since no periodicboundary conditionsare available,
pN,pN, and rota-
plane. In Fig. 5,
200 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Fig. 5. A conventional 2-D in-plane switching structures.
we just expand two more periods in both left and right
tions to approximate the periodicity in the 2dimMOS (a total of
five periods). Here we take the center-truncated region (one pe-
riod) from the 2dimMOS to compare with our simulation based
on the derived method in Section II.
Fig. 6(a) and (b) plot the director distribution at 4 and 16 ms
with 6-V applied voltage, respectively. Apparently, the effect
of director tilt deformation near the interdigital electrode side
regions becomes more and more evident as the time changes
from 4 to 16 ms. The director twist deformation effect in
the inter-electrode regions also shows the same tendency.
The transmittance dependence on different
8, 12, and 16 ms are plotted in Fig. 7. At
the transmittance in the curve almost reaches the maximum
value which is 35% under real crossed polarizers. Due to
the structure symmetry, the directors in the regions above
each electrode’s surface have little twist in response to the
electric fields at different times, which results in dark regions
above the electrodes. From Fig. 7, our simulation results agree
well with those obtained from 2dimMOS, which indicates
our derivations are dependable.
positions at 4,
B. Three-Dimensional Super IPS Structure
The super IPSstructure with zigzag electrodesis then studied
by both derived FEM and FDM. The electrode shape and di-
mension in the bottom - plane are depicted in Fig. 8. And the
cross-section view of the cell in - plane is same as Fig. 5. In
our simulation, same LC material and cell gap as the 2-D sim-
ulation is taken in this configuration. The pretilt angles of LC
directors are 2 from the - plane, and the rubbing angles are
90 away from the
axis in the - plane. Six volts are applied
on the left electrode, while the right one is grounded to 0 V. The
LC directors on the vertical boundaries (in
sumed to be strongly anchored in our simulations. Fig. 8 only
shows a single period of the electrode structure within a pixel.
The periodic boundary conditions in the lateral directions (both
and directions) are employed in both potential and LC
Fig. 9(a)and (b) showsthecross-section view ofthepotential
and director distributions at
sampling the cross-section view of - plane, the
m, andm is taken for the sampling of
direction) are as-
and ms, respectively. In
is taken at
distribution at 16 ms with 6 V applied.
(a) LC director distribution at 4 ms with 6-V applied. (b) LC director
cross-section view. As can be seen from these figures, the LC
directors in the regions between the electrodes have been driven
to the parallel direction of electric fields in these regions at
ms, and in these regions not much difference is observed for
ms and ms. However, it takes a long time for
the directors in the regions above the electrode surfaces to be
fully rotated, as seen from
ms [see Fig. 10(a) and (b)]. This is because the electric
direction), while the electric fields in the inter-electrode regions
case, the less rotation in these regions generates dark regions as
illustrated in Fig. 11(a)–(c).
Fig. 11(a) to (c) shows the transmittance patterns at different
time steps of
(top) and FDM (bottom). It can be seen that these two methods
generate almost the same transmittance patterns. The main dis-
crepancy of these two method comes from the center positions
of electrodes (at
m), where the electrode angle is very
ms [see Fig. 9(a) and (b)] and
ms from the derived FEM
GE et al.: COMPREHENSIVE 3-D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM201
Fig. 7.Transmittance versus position at time equal to 4, 8, 12, and 16 ms from the derived FEM based method and the 2dimMOS.
Fig. 8. Super IPS structure with zigzag electrodes in simulation.
sharp. The FEM and FDM might generate different LC director
orientations, i.e., at these positions the LC director might rotate
either in the
direction or in the
other regions, the results of these two methods match well with
each other. Both methods show that the vertically distributed
electric fields above the electrodes make these regions have rel-
ative low transmittances. Here the asymmetric dark region pat-
terns in the
direction originate from the 2 pretilt angle in the
In addition to IPS structure described above, the derived
method is also applicable to other complex LCD structures,
such as multidomain vertical alignment (MVA)  and pat-
terned vertical alignment (PVA)  modes.
direction. However, in
We have derived a comprehensive dynamic LC director cal-
culation model based on FEM. The vector represented LC
director is calculated by iteration between the calculations of
potential and director distributions. Weak form is introduced
to simplify the derivation of director calculation formulations.
Excellent agreement is obtained between this method and the
commercial 2dimMOS software in calculating a 2-D IPS struc-
ture. Our method is also validated by the comparison between
FEM and FDM in simulating a 3-D super IPS structure, in
which good agreement is obtained. The derived method is also
applicable to other complex LC device designs and optimiza-
FEM IMPLEMENTATION FOR DIRECTOR UPDATE VIA
FEM is employed to get the solutions for the matrix
vector discussed in Sections II-B1. In FEM theory, a study do-
can be divided intoelement domains with
In each element domain, there are
tion functions (considering first-order interpolation functions)
as well. In any element domain (e.g., th element) the local di-
can be interpolated by its
local basis functions as
nodes with interpola-
local node values
tion in the th local element, respectively. The form of
and arethe thnodevalueandinterpolationfunc-
202 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
in ?-? plane at ? ? ? ?m at ? ? ?? ms with 6 V. (b) LC potential (top figure)
and director (bottom figure) distributions in ?-? plane at ? ? ? ?m at ? ? ??
ms with 6 V.
(a) LC potential (top figure) and director (bottom figure) distributions
tetrahedral element would be
coordinate values and the volume of the tetrahedron , .
Detailed expressions of these coefficients are provided in Ap-
pendix C. Similarly, the residual
can be weighted by the element basis function as
in each element domain
in ?-? plane at ? ? ? ?m at ? ? ?? ms with 6 V. (b) LC potential (top figure)
and director (bottom figure) distributions in ?-? plane at ? ? ? ?m at ? ? ??
ms with 6 V.
(a) LC potential (top figure) and director (bottom figure) distributions
time-difference scheme leads to
and expanding the above equation with a
By applying the assembling process of typical FEM procedure,
the global matrix
and global vector
from the element matrix
and element vector
in (13) can be obtained
GE et al.: COMPREHENSIVE 3-D DYNAMIC MODELING OF LIQUID CRYSTAL DEVICES USING FEM203
FDM (bottom figure) at ? ? ?? ms with 6 V. (b) The transmittance patterns
in the ?-? plane by FEM (top figure) and FDM (bottom figure) at ? ? ?? ms
with 6 V. (c) The transmittance pattern in the ?-? plane by FEM (top figure) and
FDM (bottom figure) at ? ? ?? ms with 6 V.
(a) Transmittance pattern in the ?-? plane by FEM (top figure) and
FEM IMPLEMENTATION FOR POTENTIAL UPDATE VIA RITZ’s
To derivethedefinite form of the matrix
we first express the electric field
in Sections II-B.2,
into a vector form as
Therefore, the free electric energy
in (17) can be further ex-
’s spatial derivative. Therefore once the
substituted in (19), the derivative of
be achieved by differentiation by parts
in (18) is
with respect to can
be obtained by differentiating the (B-3) with respect to
ofmatrix in(20) can
This will lead to the expression of (21), which can be uniquely
defined by FEM as below.
By applying the similar mesh and derivation as discussed in
the FEM implementation of director profile section, we can ob-
basis functions. The
can be written as
From the above equation, one can get a definite
and the coordinate values of the small local domain
204 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
specified, as illustrated in Appendix C. By applying the assem-
the element matrix
FEM INTERPOLATION FUNCTIONS
The local element shape for 1-D to 3-D cases is shown in
Fig. 4(a) to (c). For simplicity in the expression, we just ne-
glect the superscript element index
and the local interpolation functions
For the 1-D case in Fig. 4(a), the first-order interpolation (or
basis) function can be expressed as
for both the local coordi-
These two interpolation functions equal zero out of the local
region. As we can see that the
on the other one.
For the 2-D triangle shaped first order interpolation (or basis)
functions as shown in Fig. 4(b), they have the expression as
is equal to 1 on the th node,
is the area of the triangle, which is equal to
denotes the determinant of the inside matrix. And
, andare equal to
respectively. Here the interpolation function is equal to zero out
of the triangular region.
For the 3-D tetrahedron shaped first-order interpolation (or
basis) functions as shown in Fig. 4(c), they can be expressed as
is the volume of the tetrahedron, which is equal to
, and are equal to
respectively. The interpolation function is equal to zero out of
the tetrahedral region.
 A. Taflove, Computational Electrodynamics: The Finte-Difference
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Zhibing Ge received the B.S. and M.S. degrees
in electrical engineering from Zhejiang University,
Hangzhou, China, and University of Central Florida,
Orlando, in 2002 and 2004, respectively, and is
currently working toward the Ph.D. degree at De-
partment of Electrical and Computer Engineering,
University of Central Florida, Orlando. His Ph.D.
study concentration is in liquid crystal display
modeling, transflective liquid crystal displays, and
numerical analyses and optimization of liquid crystal
Thomas X. Wu received the B.S.E.E. and M.S.E.E.
degrees from the University of Science and Tech-
nology of China (USTC), Anhui, China, in 1988
and 1991, respectively, and the M.S. and Ph.D.
degrees in electrical engineering from the University
of Pennsylvania, Philadelphia, in 1997 and 1999,
From 1991 to 1995, he was with the faculty of the
Department of Electrical Engineering and Informa-
tion Science, USTC, as an Assistant and Lecturer. In
Fall 1999, he joined the Department of Electrical and
Computer Engineering, University of Central Florida (UCF), Orlando, as an
Assistant Professor. His current research interests and projects include complex
media, liquid crystal devices, electronic packaging of RF SAW devices, elec-
trical machinery, magnetics and EMC/EMI in power electronics, chaotic elec-
tromagnetics, millimeter-wave circuits, and CMOS/BiCMOS RFICs.
Dr. Wu was awarded the Distinguished Researcher Award from the College
of Engineering and Computer Science, University of Central Florida, in April
2004. Recently, he was listed in Who’s Who in Science and Engineering, Who’s
Who in America, and Who’s Who in the World.
Ruibo Lu received the M.S. degree in applied
physics from Department of Physics, East China
University of Science and Technology, Shanghai,
China, in 1995, and the Ph.D. degree in optics from
Department of Physics, Fudan University, Shanghai,
China, in 1998. His research work for the Ph.D.
degree focused on liquid crystal alignment and
ferroelectric liquid crystal devices for display and
advanced optical applications.
He was part of the faculty in Department of
Physics, and later in Department of Optical Science
and Engineering, Fudan University, Shanghai, China, from 1998 to 2001. He
was an optical engineer in Lightwaves 2020 Inc., San Jose, CA, from 2001 to
2002. Since then, he joined the School of Optics/CREOL (now as College of
Optics and Photonics), University of Central Florida, Orlando, as a research
scientist. His research interests include liquid crystal display technology, wide
viewing angle for liquid crystal TVs, liquid crystal components for optical
communications and optical imaging using liquid crystal medium.
Xinyu Zhu received the B.S. degree from Jilin
University, China, in 1996, and the Ph.D. degree
from Changchun Institute of Optics, Fine Mechanics
and Physics, Chinese Academy of Sciences, China in
2001. His Ph.D. research work involved mainly the
reflective liquid crystal display with single polarizer.
He is currently a research scientist at the College
of Optics and Photonics, University of Central
Florida, Orlando. His current research interests
include reflective and transflective liquid crystal
displays, liquid-crystal-on-silicon projection display,
wide view liquid crystal displays, and adaptive optics application with nematic
206 JOURNAL OF DISPLAY TECHNOLOGY, VOL. 1, NO. 2, DECEMBER 2005
Qi Hong received B.S. degree from the Nanjing
University of Aeronautics and Astronautics, Nan-
jing, China, in 1992, and the M.S.E.E. degree from
the University of Central Florida, Orlando, in 2002,
where he is currently working toward the Ph.D.
degree in the electrical engineering. His doctoral re-
search topics include liquid crystal device modeling,
wide viewing angle and fast response liquid crystal
He was design engineer at the Xiaxin Electronics
Company Ltd., Xiamen, China, from 1992 to 2000.
Shin-Tson Wu (M’98–SM’99–F’04) received the
B.S. degree in physics from National Taiwan Uni-
versity, and the Ph.D. degree from the University of
Southern California, Los Angeles.
He is currently a PREP professor at College of
Optics and Photonics, University of Central Florida
(UCF), Orlando. Prior to joining UCF in 2001, he
worked at Hughes Research Laboratories, Malibu,
CA, for 18 years. His studies at UCF concentrate
in foveated imaging, bio-photonics, optical commu-
nications, liquid crystal displays, and liquid crystal
materials. He has co-authored two books: Reflective Liquid Crystal Displays
(Wiley, 2001) and Optics and Nonlinear Optics of Liquid Crystals (World
Scientific, 1993), four book chapters, and over 220 journal papers.
Dr. Wu is a Fellow of the IEEE, Society for Infformation Display (SID), and
Optical Society of America (OSA).