Problem Statements Problem 1: Minimize the OseenFrank Energy E . u - - PowerPoint PPT Presentation

Problem Statements

Problem 1: Minimize the Oseen–Frank Energy E. u : Ω ⊂ R3 → S2, E(u) =

• Ω

W(u, ∇u)dx, 2W(u, ∇u) = k1(div u)2 + k2(u · curl u)2 + k3|u × curl u|2 + (k2 + k4)(tr(∇u)2 − (div u)2) Problem 2: Minimize the Helfrich Energy H. u : Σ ⊂ R3 → S2, n : Σ ⊂ R3 → S2 H(u) =

• Σ

KS 2 (divS u)2 + KT 2 |u × n|2 + KE 2 (stretching term)dS + other terms (van der Waal, surface tension, etc)

Fordham University

• J. Hineman

Present Work

One-constant approximation: WHM(u, ∇u) = K|∇u|2

2

; WHM is the harmonic map energy density. Alouges Algorithm:

• F. Alouges 1994, F. Alouges, J.M. Ghidaglia 1997, S. Bartel 2005

◮ Initialize: choose u(0) in H1(Ω, S2) ◮ Minimize: Compute w(j) ∈ H1(Ω, R3) to satisfy

EHM(u(j) − w(j)) ≤ EHM(u(j) − v), ∀v ∈ H1(Ω, R3)

◮ Project: set u(j+1) = u(j) − w(j)

|u(j) − w(j)| Direct Minimization:

• R. Ryham et al 2013, 2015

◮ min µx − x(0) + 2∆tH(x) ◮ Gradient descent µ(x − x(0)) = −∆t∇xH(x) ◮ Fully-implicit backwards Euler (solved efficiently via Newton

iteration) (∆t∇2

xH(x) + µ)δx = µ(x(0) − x) − ∆t∇xH(x), x ←

− x + δx (1)

Fordham University

• J. Hineman

Open Problems

• 1. Does finite element procedure of Bartels adapt to

quads/hexes for WHM? As written, a regular acute triangular mesh is required.

• 2. Minimization of E. There are few numerical results treat the

full energy.

• 3. Complete description of (nematic) liquid crystals requires a

coupled Navier–Stokes/Transported heat flow of harmonic maps system. There are few results using full Oseen–Frank energy and full constitutive relations.

• 4. Implement iterations like (1). This would be especially useful

for the most general problem which allow the surface Σ to evolve (say using phase-field). So far, our simulations are axisymmetric.

4.1 Full disclosure: really, non-equilibrium problem, and thus requires a minimum energy path formulation = ⇒ string method = ⇒ parallelism. 4.2 Automatic mesh refinement probably a necessity for 3d.

Fordham University

• J. Hineman