A STRUCTURE PRESERVING FEM FOR UNIAXIAL NEMATIC LIQUID CRYSTALS - - PowerPoint PPT Presentation

A STRUCTURE PRESERVING FEM FOR UNIAXIAL NEMATIC LIQUID CRYSTALS

Ricardo H. Nochetto Department of Mathematics and Institute for Physical Science and Technology University of Maryland Joint work with Juan Pablo Borthagaray, University of Salto, Uruguay Shawn Walker, Louisiana State University Wujun Zhang, Rutgers University Numerical Methods and New Perspectives for Extended Liquid Crystalline Systems ICERM, December 9-14, 2019

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Outline The Landau - de Gennes Model Structure Preserving FEM Γ-Convergence Discrete Gradient Flow Simulations Electric and Colloidal Effects Conclusions and Open Problems

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Outline The Landau - de Gennes Model Structure Preserving FEM Γ-Convergence Discrete Gradient Flow Simulations Electric and Colloidal Effects Conclusions and Open Problems

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Liquid Crystals with Variable Degree of Orientation

• Nematic liquid crystal (LC) molecules are often idealized as elongated rods.

Modeling is further simplified by an averaging procedure to replace local arrangement of rods by a few order parameters.

Left: Thermotropic LC; Right: Schlieren texture of liquid crystal nematic phase with surface point defects (boojums). Picture taken under a polarization microscope.

• Defects are inherent to LC modeling, analysis and computation. They can

be orientable (±1 degree) or non-orientable (±1/2 degree).

• Computation of LCs should allow for defects and yield convergent

approximations of relevant physical quantities.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Half-integer order defects

Figure: Singularities (defects) of degree (b) 1/2 and (c) −1/2. Taken from S´ anchez et al., Nature, 2012.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Orientability

• Head-to-tail symmetry: director fields (unit vectors n) introduce an
• rientational bias into the model that is not physical (Ericksen’s model).
• Defects of degree ±1/2 are not orientable (cannot be described by director

fields but rather by line fields); Landau-DeGennes model.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Ensemble Averaging

• Probability distribution ρ: for x ∈ Ω and p ∈ S2 the unit sphere, ρ satisfies

ρ(x, p) ≥ 0,

• S2 ρ(x, p)ds(p) = 1,

ρ(x, p) = ρ(x, −p)

• First moment:
• S2 p ρ(x, p)ds(p) = −
• S2 p ρ(x, −p)ds(p) = 0
• Second moment:

M(x) =

• S2 p ⊗ p ρ(x, p)ds(p) ∈ R3×3

⇒ tr(M) = 1.

• Isotropic uniform distribution:

ρ(x, p) = 1 4π ⇒ M = 1 3I.

• The Q-tensor: measures deviation from isotropic uniform distribution

Q := M − 1 3I, ⇒ Q = QT , tr(Q) = 0.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

The Q-tensor

• Biaxial form of Q: for n1, n2 ∈ S2 director fields and s1, s2 ∈ R scalar order

parameters, Q reads Q = s1

• n1 ⊗ n2 − 1

3I

• + s2
• n2 ⊗ n2 − 1

3I

• The signs of n1 and n2 have no effect on Q (head-to-tail symmetry).
• Eigenvalues λi(Q) of Q: probability distribution implies − 1

3 ≤ λi(Q) ≤ 2 3

λ1(Q) = 2s1 − s2 3 , λ2(Q) = 2s2 − s1 3 λ3(Q) = −s1 + s2 3

• Uniaxial form of Q: either s1 = 0, s2 = 0 or s1 = s2 and

Q = s

• n ⊗ n − 1

3I

• ,

−1 2 ≤ s ≤ 1.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

The One-Constant Landau - de Gennes Model

• One-constant LdG energy: sum of elastic and potential energies

E[Q] := E1[Q] + 1 ǫ E2[Q] where ǫ > 0 is small (the nematic correlation length).

• Elastic (Frank) energy:

E1[Q] := 1 2

• Ω

|∇Q|2dx.

• Double-well potential energy: E2[Q] :=
• Ω ψ(Q)dx where

ψ(Q) = A tr(Q2) + B tr(Q3) + C tr(Q2)2 for A, B, C ∈ R. The minimizer of ψ(Q) is a uniaxial state.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Uniaxial vs Biaxial States

• Biaxial: for thermotropic LCs, the nematic biaxial phase remained elusive

for a long period until

◮ Acharya, et al, PRL, 2004; ◮ Madsen, et al, PRL, 2004; ◮ Prasad, et al, J. Amer. Chem. Soc., 2005.

• Uniaxial: 2012 book by Sonnet and Virga (sec. 4.1) says

The vast majority of nematic liquid crystals do not, at least in homogeneous equilibrium states, show any sign of biaxiality.

• LdG-model: does not enforce uniaxiality which, however, is still prevalent in

many situations.

• FEM: we present a finite element method for uniaxial LCs.
• Comparisons: we compare uniaxial and biaxial behavior near a Saturn-ring

defect at the end of this talk.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Q-Model with Uniaxial Constraint

• One-constant energy: E[Q] = E1[Q] + E2[Q] where Q = s(n ⊗ n

=Θ

− 1

dI)

E1[Q] =

• Ω

|∇Q|2, E2[Q] =

• Ω
• A tr(Q2) + B tr(Q3) + C
• tr(Q2)

2 .

• Property 1:

∇Q = ∇s ⊗

• Θ − 1

dI

• + s∇Θ,

∇Θ : Θ = ∇Θ : I = 0 |∇Q|2 = |∇s|2

• Θ − 1

dI

• 2
• = d−1

d

+s2|∇Θ|2 + 2s∇s ·

• ∇Θ :
• Θ − 1

dI

• =0

.

• Property 2: A direct calculation gives

s2 = C1tr(Q2), s3 = C2tr(Q3), s4 = C3

• tr(Q2)

2.

• Equivalent one-constant energy: κ = d−1

d , ψ suitable double-well potential

E1[Q] =

• Ω

κ|∇s|2 + s2|∇Θ|2, E2[Q] =

• Ω

ψ(s).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Q-Model with Uniaxial Constraint

• One-constant energy: E[Q] = E1[Q] + E2[Q] where Q = s(n ⊗ n

=Θ

− 1

dI)

E1[Q] =

• Ω

|∇Q|2, E2[Q] =

• Ω
• A tr(Q2) + B tr(Q3) + C
• tr(Q2)

2 .

• Property 1:

∇Q = ∇s ⊗

• Θ − 1

dI

• + s∇Θ,

∇Θ : Θ = ∇Θ : I = 0 |∇Q|2 = |∇s|2

• Θ − 1

dI

• 2
• = d−1

d

+s2|∇Θ|2 + 2s∇s ·

• ∇Θ :
• Θ − 1

dI

• =0

.

• Property 2: A direct calculation gives

s2 = C1tr(Q2), s3 = C2tr(Q3), s4 = C3

• tr(Q2)

2.

• Equivalent one-constant energy: κ = d−1

d , ψ suitable double-well potential

E1[Q] =

• Ω

κ|∇s|2 + s2|∇Θ|2, E2[Q] =

• Ω

ψ(s).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

One-Constant Ericksen’s Model

• Model: The equilibrium state minimizes (one-constant Ericksen’s model):

E[s, n] :=

• Ω

κ|∇s|2 + s2|∇n|2dx

• :=E1[s,n]

+

• Ω

ψ(s)dx

• :=E2[s]

where κ > 0 and ψ is a double well potential with domain (− 1

2, 1).

• Director field: |n| = 1 (unit vector).

n n n θ well-aligned local defect perpendicular s ≈ 1 s ≈ 0 s ≈ −1/2

• Scalar order parameter: s is the degree of orientation (−1/2 < s < 1).

◮ s = 1: perfect alignment with n. ◮ s = 0: no preferred direction (isotropic). This defines the set of defects:

S = {x ∈ Ω, s(x) = 0}.

◮ s = −1/2: perpendicular to n. A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Ericksen Model vs. Oseen-Frank Model

• When s = s0 > 0, the energy reduces to the Oseen-Frank energy:

E :=

• Ω

|∇n|2dx.

• One disadvantage of Oseen-Frank model is that the director field with finite

energy is inconsistent with both line and plane defects.

r z

C

◮ Line defects:

n = r |r| , |∇n| = 2 |r| .

◮ Compute

• C |∇n|2dx :

1 4 r2 rdr = ∞.

◮ Compute

• C κ|∇s|2 + s2|∇n|2dx :

1 s2 4 r2 rdr < ∞.

• Ericksen’s model allows for defects to have finite energy (regularization).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Ericksen Model vs. Oseen-Frank Model

• When s = s0 > 0, the energy reduces to the Oseen-Frank energy:

E :=

• Ω

|∇n|2dx.

• One disadvantage of Oseen-Frank model is that the director field with finite

energy is inconsistent with both line and plane defects.

r z

C

◮ Line defects:

n = r |r| , |∇n| = 2 |r| .

◮ Compute

• C |∇n|2dx :

1 4 r2 rdr = ∞.

◮ Compute

• C κ|∇s|2 + s2|∇n|2dx :

1 s2 4 r2 rdr < ∞.

• Ericksen’s model allows for defects to have finite energy (regularization).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Ericksen’s Model: Point Defect of Degree 1 in 2d

• Dirichlet boundary conditions: s = s∗,

n = x

|x|

• Director field: κ = 2

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Ericksen’s Model: Point Defect of Degree 1 in 2d

• Scalar order parameter: smin = 2.02 · 10−2
• Movie: Director field nh and order parameter sh

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Q-Tensor Model: Point Defect of Degree 1 in 2d

• Dirichlet boundary conditions: s = s∗,

n = x

|x|

• Line field Θ = n ⊗ n: κ = 1

2

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Q-Tensor Model: Point Defect of Degree 1 in 2d

• Scalar order parameter sh: smin = 0.02, Eh = 9.18
• Movie: Line field Θh and order parameter sh

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Q-Tensor Model: Point Defects of Degree ±1/2 in 2d

• Dirichlet boundary conditions: s = s∗ corresponds to absolute minimum of

double well potential ψ and n = n∗ corresponds to defect of degree 3/2 at (0., 0.).

• Scalar field s and line field Θ = n ⊗ n:
• 5 defects: 4 defects of degree 1

2 on the diagonals of Ω = (0, 1)2 and 1

defect of degree − 1

2 at the origin.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Primary Goals

• Structure preserving FEM: Design discrete energies Eh[Qh] = Eh[Θh, sh] to

approximate the Landau-deGennes energy E[Θ, s] =

• Ω

κ|∇s|2 + s2|∇Θ|2

• =E2[Q]

+

• Ω

ψ(s)dx

• =E2[s]

, Q = s

• n ⊗ n

=Θ

−1 dI

• Γ-convergence: if Qh are absolute minimizers of Eh

◮ Qh converge to absolute minimizers Q of E as h → 0 ◮ Eh[Qh] → E[Q] as h → 0.

• Gradient flow: Design a gradient flow to find a minimizer (equilibrium state
• r stationary point) Qh of the discrete energy Eh.
• Monotonicity: Energy decrease property of the gradient flow.
• Simulation of defects: Line and plane defects, saturn ring, degree ±1 and

±1/2 defects, etc (caused by electric fields and/or colloidal inclusions).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

(Incomplete) Literature Review

Modeling and PDE Hardt Kinderlehrer Lin (1986), Kinderlehrer (1991), Kinderlehrer, Ou (1992) Brezis (1987, 1989), Ericksen (1991) Virga (1994), DeGennes-Prost (1995) Ambrosio (1990) Lin (1989, 1991) Ball Zarnescu (2011); Ball Majumdar (2010). Numerical methods for director fields: Cohen Lin Luskin (1989) Alouges (1997) Liu Walkington (2000) Calderer Golovaty Lin Liu (2002) Bartels (2010); Bartels, Dolzmann, Nochetto (2012) Yang Forest Li Liu Shen Wang (2013) Numerical method for Ericksen’s model and Q-tensor model Barrett Feng Prohl (2006) (2D-FEM via regularization) James Willman Fern´ andez (2006) (Q tensor method) Shin Cho Lee Yoon and Won (2008) (Q tensor method) Lin Lin Wang (2010) Bartels Raisch (2014) Walkington (2011, 2017).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Outline The Landau - de Gennes Model Structure Preserving FEM Γ-Convergence Discrete Gradient Flow Simulations Electric and Colloidal Effects Conclusions and Open Problems

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

PDE Structure of Ericksen’s Model

• One constant model:

E[s, n] :=

• Ω

κ|∇s|2 + s2|∇n|2dx

• :=E1[s,n]

+

• Ω

ψ(s)dx

• :=E2[s]

Euler-Lagrange equation: (degenerate elliptic) div (s2∇n) − s2|∇n|2n = 0. −κ∆s − s|∇n|2 + 1 2ψ′(s) = 0.

• Energy identity: since |n| = 1, we have ∇|n|2 = 2nT (∇n) = 0.
• Ω

|∇ (sn)

• :=u

|2dx =

• Ω

|n ⊗ ∇s + s∇n|2dx =

• Ω

|∇s|2 + s2|∇n|2dx.

• Equivalent energy: [Ambrosio 90, Lin 91] E1[s, n] =

E1[s, u] with

• E1[s, u] :=
• Ω

(κ − 1)|∇s|2 + |∇u|2dx, (u = sn) i.e. a simple quadratic functional, but with a negative term (κ < 1).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

PDE Structure of Ericksen’s Model

• One constant model:

E[s, n] :=

• Ω

κ|∇s|2 + s2|∇n|2dx

• :=E1[s,n]

+

• Ω

ψ(s)dx

• :=E2[s]

Euler-Lagrange equation: (degenerate elliptic) div (s2∇n) − s2|∇n|2n = 0. −κ∆s − s|∇n|2 + 1 2ψ′(s) = 0.

• Energy identity: since |n| = 1, we have ∇|n|2 = 2nT (∇n) = 0.
• Ω

|∇ (sn)

• :=u

|2dx =

• Ω

|n ⊗ ∇s + s∇n|2dx =

• Ω

|∇s|2 + s2|∇n|2dx.

• Equivalent energy: [Ambrosio 90, Lin 91] E1[s, n] =

E1[s, u] with

• E1[s, u] :=
• Ω

(κ − 1)|∇s|2 + |∇u|2dx, (u = sn) i.e. a simple quadratic functional, but with a negative term (κ < 1).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

PDE Structure of the Q-Model: Relation to Ericksen’s Model

• One-constant energy: If κ = d−1

d

and ψ is a suitable double-well potential, then E1[Q] =

• Ω

κ|∇s|2 + s2|∇Θ|2, E2[Q] =

• Ω

ψ(s).

• Auxiliary variable: If U = sΘ = s
• n ⊗ n
• , then

∇U = ∇s ⊗ Θ + s∇Θ ⇒ |∇U|2 = |∇s|2 + s2|∇Θ|2 because ∇Θ : Θ = 0. Therefore E1[Q] = E1[s, U] where

• E1[s, U] =
• Ω

(κ − 1)|∇s|2 + |∇U|2.

• Structural conditions:

−1 2 < s < 1, U = s n ⊗ n

=Θ

, n ∈ Sd−1.

• Admissible class:

A :=

• (s, U) ∈ H1(Ω)×[H1(Ω)]d×d : (s, U) satisfy structural conditions a.e.
• .

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

PDE Structure of the Q-Model: Relation to Ericksen’s Model

• One-constant energy: If κ = d−1

d

and ψ is a suitable double-well potential, then E1[Q] =

• Ω

κ|∇s|2 + s2|∇Θ|2, E2[Q] =

• Ω

ψ(s).

• Auxiliary variable: If U = sΘ = s
• n ⊗ n
• , then

∇U = ∇s ⊗ Θ + s∇Θ ⇒ |∇U|2 = |∇s|2 + s2|∇Θ|2 because ∇Θ : Θ = 0. Therefore E1[Q] = E1[s, U] where

• E1[s, U] =
• Ω

(κ − 1)|∇s|2 + |∇U|2.

• Structural conditions:

−1 2 < s < 1, U = s n ⊗ n

=Θ

, n ∈ Sd−1.

• Admissible class:

A :=

• (s, U) ∈ H1(Ω)×[H1(Ω)]d×d : (s, U) satisfy structural conditions a.e.
• .

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Numerical Difficulties

• PDEs: Degenerate nonlinear elliptic PDEs.
• Line field: Construct a vector field Θh = nn ⊗ nh such that nh has unit

length at nodes (locking otherwise).

• Scalar field: Construct sh such that − 1

2 < sh < 1 (at the nodes).

• Auxiliary tensor field: Construct a rank-one tensor field Uh that satisfies

the structural condition at the nodes Nh Uh(xi) = sh(xi)

• nh(xi) ⊗ nh(xi)
• ∀ xi ∈ Nh.
• Γ-convergence: theory without regularization that allows for defects.
• Computation of minimizers: discrete gradient flow.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Finite Element Spaces

• Meshes: Let Th = {T} be a conforming, shape-regular triangulation of Ω,

with set of nodes (vertices) denoted by Nh.

• Finite element spaces:

Uh := {uh ∈ [H1(Ω)]d : uh|T is linear} Sh := {sh ∈ H1(Ω) : sh|T is linear} Nh := {nh ∈ Uh : nh|T is linear |nh(xi)| = 1 at all nodes xi ∈ Nh} Th := Nh ⊗ Nh.

• Weakly acute mesh: the entries of the stiffness matrix {kij} satisfy

kij = −

• Ω

∇φi · ∇φjdx ≥ 0 for i = j, where {φi} denotes the continuous piecewise linear “hat” basis functions which satisfy φi(xj) = δij for all nodes xj ∈ Nh.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Structure Preserving FEM

• Dirichlet forms: For a piecewise linear function sh, we have
• Ω

|∇sh|2dx = 1 2

• i,j

kij(si − sj)2,

• Ω

|∇Θh|2dx = 1 2

• i,j

kij(Θi − Θj)2, where si = sh(xi) and Θi = Θh(xi) for all nodes xi ∈ Nh.

• Discrete energy: approximate E1[s, n] =
• Ω κ|∇s|2 + s2|∇Θ|2dx

by Eh

1 [sh, Θh] := κ

2

N

• i,j=1

kij (si − sj)2

• =κ
• Ω |∇sh|2dx

+ 1 2

N

• i,j=1

kij s2

i + s2 j

2

• |Θi − Θj|2
• ≈
• Ω s2

h|∇Θh|2dx

.

• Nodal structural conditions: let

U i = siΘi = si

• ni ⊗ ni
• ∀xi ∈ Nh.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Structure Preserving FEM

• Dirichlet forms: For a piecewise linear function sh, we have
• Ω

|∇sh|2dx = 1 2

• i,j

kij(si − sj)2,

• Ω

|∇Θh|2dx = 1 2

• i,j

kij(Θi − Θj)2, where si = sh(xi) and Θi = Θh(xi) for all nodes xi ∈ Nh.

• Discrete energy: approximate E1[s, n] =
• Ω κ|∇s|2 + s2|∇Θ|2dx

by Eh

1 [sh, Θh] := κ

2

N

• i,j=1

kij (si − sj)2

• =κ
• Ω |∇sh|2dx

+ 1 2

N

• i,j=1

kij s2

i + s2 j

2

• |Θi − Θj|2
• ≈
• Ω s2

h|∇Θh|2dx

.

• Nodal structural conditions: let

U i = siΘi = si

• ni ⊗ ni
• ∀xi ∈ Nh.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Stability of the Discrete Energy

• Dirichlet form: Let uh be piecewise linear with nodal values U i = siΘi

2

• Ω

|∇U h|2dx =

• i,j

kij(siΘi − sjΘj)2 =

• i,j

kij si + sj 2 (Θi − Θj) + (si − sj)Θi + Θj 2 2

• Orthogonality: Exploiting the relation (Θi − Θj) : (Θi + Θj) = 0 yields

2

• Ω

|∇U h|2dx =

• i,j

kij si + sj 2 2 (Θi − Θj)2 + (si − sj)2 Θi + Θj 2 2 =

• i,j

kij s2

i + s2 j

2

• Θi − Θj

2 +

• i,j

kij(si − sj)2 −

• i,j

kij(si − sj)2

• Θi − Θj

2

• 2
• Energy inequalities: U i = siΘi,

si = |si|, U i = siΘi (structural conditions) Eh

1 [sh, Θh] ≥ (κ − 1)

• Ω

|∇sh|2dx +

• Ω

|∇U h|2dx =: Eh

1 [sh, U h]

Eh

1 [sh, Θh] ≥ (κ − 1)

• Ω

|∇ sh|2dx +

• Ω

|∇ U h|2dx =: Eh

1 [

sh, U h].

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Stability of the Discrete Energy

• Dirichlet form: Let uh be piecewise linear with nodal values U i = siΘi

2

• Ω

|∇U h|2dx =

• i,j

kij(siΘi − sjΘj)2 =

• i,j

kij si + sj 2 (Θi − Θj) + (si − sj)Θi + Θj 2 2

• Orthogonality: Exploiting the relation (Θi − Θj) : (Θi + Θj) = 0 yields

2

• Ω

|∇U h|2dx =

• i,j

kij si + sj 2 2 (Θi − Θj)2 + (si − sj)2 Θi + Θj 2 2 =

• i,j

kij s2

i + s2 j

2

• Θi − Θj

2 +

• i,j

kij(si − sj)2 −

• i,j

kij(si − sj)2

• Θi − Θj

2

• 2
• Energy inequalities: U i = siΘi,

si = |si|, U i = siΘi (structural conditions) Eh

1 [sh, Θh] ≥ (κ − 1)

• Ω

|∇sh|2dx +

• Ω

|∇U h|2dx =: Eh

1 [sh, U h]

Eh

1 [sh, Θh] ≥ (κ − 1)

• Ω

|∇ sh|2dx +

• Ω

|∇ U h|2dx =: Eh

1 [

sh, U h].

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Discrete Admissible Class

• Discrete structural conditions:

−1 2 < si < 1, U i = si

• ni ⊗ ni
• ,

|ni| = 1 ∀xi ∈ Nh.

• Discrete admissible class:

Ah :=

• (sh, U h) ∈ Sh×Uh : (sh, Uh) satisfies discrete structural conditions
• .
• Discrete restricted admissible class:

Ah(Ihg, IhR) :=

• (sh, U h) ∈ Ah :

sh = Ihg, U h = IhR

• n ∂Ω
• ,

where Ih is the Lagrange interpolation operator.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Outline The Landau - de Gennes Model Structure Preserving FEM Γ-Convergence Discrete Gradient Flow Simulations Electric and Colloidal Effects Conclusions and Open Problems

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Assumptions for Γ-Convergence

• Meshes: Let {Th}h≥0 be a sequence of weakly acute meshes.
• Lim-sup inequality (consistency or recovery sequence): Given (s, U) ∈ A,

namely s, U = sΘ ∈ H1(Ω), and satisfy Dirichlet boundary conditions s = g, U = R, there exists (sh, U h) ∈ Ah(Ihg, IhR) such that (sh, U h) → (s, U) in H1(Ω) and lim sup

h→0

Eh

1 [sh, Θh] ≤ E1[s, Θ].

• Lim-inf inequality (stability or lower semi-continuity): Given a sequence

(sh, U h) ∈ Ah(Ihg, IhR), let ˜ sh = Ih|sh|, U h = Ih[˜ shΘh] be so that (˜ sh, U h) converges weakly in H1(Ω) to (˜ s, U). Then we have

• E1[˜

s, U] ≤ lim inf

h→0

• Eh

1 [˜

sh, U h].

• Equi-coercivity: There exists a constant C > 0 such that

shH1(Ω), U hH1(Ω) ≤ CEh

1 [sh, Θh].

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Theorem: Γ-Convergence for the Q-Model Let {Th}h≥0 be a sequence of weakly acute meshes. Then the discrete energy Eh

1 [sh, Θh] satisfies the lim-sup, lim-inf, and equi-coercivity properties.

Moreover,

• If Eh[sh, Θh] ≤ Λ uniformly, then there exist subsequences (not relabeled)

(sh, U h), (˜ sh, ˜ U h), and Θh such that

◮ (sh, Uh) converges to (s, U) weakly in H1 and strongly in L2; ◮ (˜

sh, ˜ Uh) converges to (˜ s, ˜ U) weakly in H1 and strongly in L2;

◮ The limits satisfy ˜

s = |s| = |U| = | ˜ U|;

◮ There exists a line field Θ defined in the complement of the singular set

S = {s = 0} such that Θh converges to Θ in L2(Ω \ S) and U = sΘ and ˜ U = ˜ sΘ;

◮ Θ admits a Lebesgue gradient ∇Θ and |∇

U|2 = |∇ s|2 + s2|∇Θ|2 a.e. in Ω \ S.

• If (sh, Θh) is a sequence of global minimizers of Eh, then every cluster point

(s, Q) is a global minimizer of E.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Consistency or Recovery Sequence: Regularization of Functions in A(g, R)

• Regularization: Given ǫ > 0 and (s, u) ∈ A(g, R), i.e. satisfying s = g,

U = R on ∂Ω, there exists (sǫ, U ǫ) ∈ A(g, R) ∩ [W 1

∞(Ω)]d+1 such that

(s, U) − (sǫ, U ǫ)H1(Ω) ≤ ǫ.

• Difficulties: pair (sǫ, U ǫ) must satisfy

◮ Structural conditions

− 1 2 < sǫ < 1, rank(Uǫ) ≤ 1, tr(Uǫ) = sǫ a.e. in Ω;

◮ Dirichlet boundary conditions

(sǫ, Uǫ) = (g, R)

• n ∂Ω;

◮ Approximate the pair (s, U) in H1(Ω).

• Lim-sup equality: For any ǫ > 0, we obtain (Ihsǫ, IhU ǫ) ∈ Ah(Ihg, IhR) by

nodewise interpolation of (sǫ, Uǫ) and E1[sǫ, Θǫ] = lim

h→0 Eh 1 [Ihsǫ, IhΘǫ] = lim h→0

• Eh

1 [Ihsǫ, IhU ǫ] =

E1[sǫ, U ǫ].

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Consistency or Recovery Sequence: Regularization of Functions in A(g, R)

• Regularization: Given ǫ > 0 and (s, u) ∈ A(g, R), i.e. satisfying s = g,

U = R on ∂Ω, there exists (sǫ, U ǫ) ∈ A(g, R) ∩ [W 1

∞(Ω)]d+1 such that

(s, U) − (sǫ, U ǫ)H1(Ω) ≤ ǫ.

• Difficulties: pair (sǫ, U ǫ) must satisfy

◮ Structural conditions

− 1 2 < sǫ < 1, rank(Uǫ) ≤ 1, tr(Uǫ) = sǫ a.e. in Ω;

◮ Dirichlet boundary conditions

(sǫ, Uǫ) = (g, R)

• n ∂Ω;

◮ Approximate the pair (s, U) in H1(Ω).

• Lim-sup equality: For any ǫ > 0, we obtain (Ihsǫ, IhU ǫ) ∈ Ah(Ihg, IhR) by

nodewise interpolation of (sǫ, Uǫ) and E1[sǫ, Θǫ] = lim

h→0 Eh 1 [Ihsǫ, IhΘǫ] = lim h→0

• Eh

1 [Ihsǫ, IhU ǫ] =

E1[sǫ, U ǫ].

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Weak Lower Semi-continuity: Lim-Inf Inequality

• Energy inequality: Recall that

Eh

1 [sh, Θh] ≥

Eh

1 [

sh, U h] = −1 d

• Ω

|∇ sh|2dx +

• Ω

|∇ U h|2dx, where sh = Ih|sh| and uh = Ih[ shΘh].

• Difficulties: is the right-hand side convex in ∇

U h? Note negative first term!

◮ True if |

sh| = | Uh| in Ω

◮ But this relation |

sh(xi)| = | Uh(xi)| is only valid at the nodes xi ∈ Nh.

• Weak lower semi-continuity: Let W h in Uh converge weakly to W in the

H1-norm. Then lim inf

h→0

• Ω

−1 d

• ∇Ih|W h|
• 2 +
• ∇W h
• 2 ≥
• Ω

−1 d

• ∇|W |
• 2 +
• ∇W
• 2.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Weak Lower Semi-continuity: Lim-Inf Inequality

• Energy inequality: Recall that

Eh

1 [sh, Θh] ≥

Eh

1 [

sh, U h] = −1 d

• Ω

|∇ sh|2dx +

• Ω

|∇ U h|2dx, where sh = Ih|sh| and uh = Ih[ shΘh].

• Difficulties: is the right-hand side convex in ∇

U h? Note negative first term!

◮ True if |

sh| = | Uh| in Ω

◮ But this relation |

sh(xi)| = | Uh(xi)| is only valid at the nodes xi ∈ Nh.

• Weak lower semi-continuity: Let W h in Uh converge weakly to W in the

H1-norm. Then lim inf

h→0

• Ω

−1 d

• ∇Ih|W h|
• 2 +
• ∇W h
• 2 ≥
• Ω

−1 d

• ∇|W |
• 2 +
• ∇W
• 2.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Equi-Coercivity For any (sh, U h) ∈ Ah(Ihg, IhR) we have U h = Ih[shΘh], U h = Ih[ shΘh] and

• Estimate for (sh, U h):

Eh

1 [sh, Θh] ≥ d − 1

d max

• Ω

|∇U h|2,

• Ω

|∇sh|2

• Estimate for (

sh, U h): Eh

1 [sh, Θh] ≥ d − 1

d max

• Ω

|∇ U h|2,

• Ω

|∇ sh|2

• .

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Outline The Landau - de Gennes Model Structure Preserving FEM Γ-Convergence Discrete Gradient Flow Simulations Electric and Colloidal Effects Conclusions and Open Problems

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Discrete Gradient Flow for the Q-Model

• Three-step algorithm: Given s = sh and Θ = Θh = Ih[nh ⊗ nh], iterate

◮ Weighted gradient flow for Θ: find T k = nk ⊗ tk + tk ⊗ nk such that tk is

tangent to nk and for all v tangent to nk (Bartels, Raisch (2014)) 1 τ

• Ω

s2

k∇tk : ∇v + tk · v+δΘEh 1 [sk, Θk+T k; V ] = 0

∀V = nk⊗v+v⊗nk.

◮ Projection: update Θ by

Θk+1 := nk + tk |nk + tk| ⊗ nk + tk |nk + tk| .

◮ Gradient flow for s: find sk+1 such that

1 2τ

• Ω

|sk+1 − sk|2 + δsEh

1 [sk+1, Θk+1; z] + δsEh 2 [sk+1; z]

• convex splitting

= 0 ∀z, where δSE2[s; z] =

• Ω ψ′(s)z and ψ is a double-well potential.
• Relaxation: the effect of weight s2

k is to allow for larger variations of tk.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Discrete Gradient Flow for the Q-Model

• Three-step algorithm: Given s = sh and Θ = Θh = Ih[nh ⊗ nh], iterate

◮ Weighted gradient flow for Θ: find T k = nk ⊗ tk + tk ⊗ nk such that tk is

tangent to nk and for all v tangent to nk (Bartels, Raisch (2014)) 1 τ

• Ω

s2

k∇tk : ∇v + tk · v+δΘEh 1 [sk, Θk+T k; V ] = 0

∀V = nk⊗v+v⊗nk.

◮ Projection: update Θ by

Θk+1 := nk + tk |nk + tk| ⊗ nk + tk |nk + tk| .

◮ Gradient flow for s: find sk+1 such that

1 2τ

• Ω

|sk+1 − sk|2 + δsEh

1 [sk+1, Θk+1; z] + δsEh 2 [sk+1; z]

• convex splitting

= 0 ∀z, where δSE2[s; z] =

• Ω ψ′(s)z and ψ is a double-well potential.
• Relaxation: the effect of weight s2

k is to allow for larger variations of tk.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Energy Decrease Property for the Q-Model

• Variations in line field: avoid nonlinear term t ⊗ t (Bartels & Raisch 2014).
• Projection step: is energy decreasing (Alouges 1997, Bartels 2010).
• Convex splitting: to treat the double-well potential term in the gradient

flow for sk.

• Theorem (energy decrease property) If the meshes Th are weakly acute

and the time step τ ≤ C0hd/2, then there holds Eh

1 [sN, ΘN] + 2

τ

N−1

• k=0
• sk∇tk2

L2(Ω) + tk2 L2(Ω) + sk+1 − sk2

≤ Eh

1 [s0, Θ0]

∀N ≥ 1. Thus, the algorithm stops in a finite number of steps for any tolerance ε.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Why do we have a CFL condition?

• Tangential variations: In Step 1 we update Θk = Ih(nk ⊗ nk) according to

Θk + Tk, T k = nk ⊗ tk + tk ⊗ nk with tk(xi) · nk(xi) = 0 (nodal tangential update).

• Rank-one update: In Step 2 we create the rank-1 update
• nk + tk
• ⊗
• tk + nk
• = Θk + Tk + tk ⊗ tk

and next normalize it (projection into discrete line fields update).

• Correction term: We must thus accound for tk ⊗ tk
• i,j

kij

• sk(xi)2 + sk(xj)2

δij(tk ⊗ tk)

• 2 ≈ h−d

Ω

s2

k|∇tk|2 + |tk|22

. An induction argument together with τ ≤ C0hd/2, with C0 proportional to Eh[s0, Θ0], allows for control of this term.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Outline The Landau - de Gennes Model Structure Preserving FEM Γ-Convergence Discrete Gradient Flow Simulations Electric and Colloidal Effects Conclusions and Open Problems

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Ericksen’s Model: “Propeller” Defect κ = 0.1

• Director field at 4 slices: z = 0.2, 0.4, 0.6, 0.8

0.5 1 0.5 1 Z = 0.2 Slice X Y 0.5 1 0.5 1 Z = 0.4 Slice X Y 0.5 1 0.5 1 Z = 0.6 Slice X Y 0.5 1 0.5 1 Z = 0.8 Slice X Y

• Movie: Director field at z = 0

Movie: 3d defect is marked in red.

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Q-Model: 3-D Line Defect of Degree 1

• Line field at 4 slices: z = 0.1, 0.35, 0.65, 0.9; smin = 1.0 × 10−2.
• Movie: Line field at z = 0.5. Gradient flow starts with line defect of degree

1 located at (0.25, 025).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ-Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems

Q-Model: 3-D Line Defect of Degree 1/2

• Line field at 4 slices: z = 0.1, 0.35, 0.65, 0.9; smin = 6.8 × 10−2.
• Movie: Line field at z = 0.5. Gradient flow starts with line defect of degree

1 at (0.25, 025).

A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto