Unfitted Bulk Finite Element Method for Solving Surface Partial - - PowerPoint PPT Presentation

Unfitted Bulk Finite Element Method for Solving Surface Partial Differential Equations

Spencer Patty

Texas A&M University Department of Mathematics Advisor: Andrea Bonito srobertp@math.tamu.edu

August 4, 2015

Spencer Patty (TAMU) Surface PDE August 4, 2015 1 / 4

White Blood Cell Motion

We model the motion of a free white blood cell in a liquid environment. The cell is represented implicitly as Ω = {x ∈ Λ | ϕ(x, t) ≥ 0}.

The Level Set Method + Incompressible Navier-Stokes

          

∂ϕ ∂t + u · ∇ϕ = 0

in [0, T] × Λ ρ ∂u

∂t + u · ∇u

• − ∇ · (2µ∇su) + ∇p = f

in [0, T] × Λ ∇ · u = 0 in [0, T] × Λ [2µ∇su − pI] · n = fΓ

• n [0, T] × Γ

(1) with stabilization terms.

Λ ∂Λ Γ = ∂Ω Ω ν n

Here fΓ represents the various physics that take place on the boundary of cell, for instance motion to minimize surface tension or minimize bending energy of membrane. In the more complicated cases fΓ must be calculated as a solution to a geometric pde that lives on the manifold Γ.

Spencer Patty (TAMU) Surface PDE August 4, 2015 2 / 4

Partial Differential Equation on Surface, Γ

Solve pde on bulk finite element with the mesh unfitted to the surface which is defined only implicitly such as by a level set method. In the case

• f Canham-Helfrich energy minimization in 2D (a simplified case), we need

to be able to solve for fΓ = k

• ∆ΓH + 1

3H3

• ,

x ∈ Γ (2) where H is the total curvature of the surface. Now, the vector curvature Hn can be written as a scale multiple of ∆ΓX where X is the identity

• perator on the surface, thus we study the surface Laplacian or

Laplace-Beltrami equation.

Classical Laplace-Beltrami Equation

Find u ∈ C2(Γ) such that −∆Γu + cu = f, x ∈ Γ

Spencer Patty (TAMU) Surface PDE August 4, 2015 3 / 4

Unfitted Bulk Finite Element Method

Figure : Example δǫ(x) for Γ = rotated capsule.

One direction of research is in using a smeared Dirac delta function δε(x) = 1

εψ

• φ(x)

ε

• ,

|ϕ(x)| < ε 0, else (3) with half-width ε = chβ and kernel ψ. Sadly when β = 1, there are O(1) errors using δε. But for example, with β = 3/4, we get O(h3/2) convergence using

Unfitted Bulk Finite Element Method with Smeared Dirac Function

Find uh ∈ Vh such that for all vh ∈ Vh = P1 (Th),

• Ω

δε(x) (∇uh · ∇vh + cuhvh) |∇ϕh|dx =

• Ω

δε(x)fevh|∇ϕh|dx

Spencer Patty (TAMU) Surface PDE August 4, 2015 4 / 4