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   � ∂ u  − ∇ · (2 µ ∇ s u ) + ∇ p = f  � ρ ∂t + u · ∇ u in [0 , T ] × Λ  (1) ∇ · u = 0 in [0 , T ] × Λ    [2 µ ∇ s u − pI ] · n = f Γ  on [0 , T ] × Γ  with stabilization terms. Here f Γ represents the various physics that take Λ place on the boundary of cell, for instance motion Γ = ∂ Ω to minimize surface tension or minimize bending Ω energy of membrane. In the more complicated n cases f Γ must be calculated as a solution ∂ Λ to a geometric pde that lives on the manifold Γ . ν 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 of Canham-Helfrich energy minimization in 2D (a simplified case), we need to be able to solve for � � ∆ Γ H + 1 3 H 3 f Γ = k , x ∈ Γ (2) where H is the total curvature of the surface. Now, the vector curvature H n can be written as a scale multiple of ∆ Γ X where X is the identity operator on the surface, thus we study the surface Laplacian or Laplace-Beltrami equation. Classical Laplace-Beltrami Equation Find u ∈ C 2 (Γ) such that − ∆ Γ u + cu = f, x ∈ Γ Spencer Patty (TAMU) Surface PDE August 4, 2015 3 / 4

Unfitted Bulk Finite Element Method One direction of research is in using a smeared Dirac delta function � 1 � � φ ( x ) ε ψ , | ϕ ( x ) | < ε ε δ ε ( x ) = (3) 0 , else with half-width Figure : Example δ ǫ ( x ) ε = ch β and kernel ψ . Sadly when β = 1 , for Γ = rotated capsule. there are O (1) errors using δ ε . But for example, with β = 3 / 4 , we get O ( h 3 / 2 ) convergence using Unfitted Bulk Finite Element Method with Smeared Dirac Function Find u h ∈ V h such that for all v h ∈ V h = P 1 ( T h ) , � � δ ε ( x ) f e v h |∇ ϕ h | d x δ ε ( x ) ( ∇ u h · ∇ v h + cu h v h ) |∇ ϕ h | d x = Ω Ω Spencer Patty (TAMU) Surface PDE August 4, 2015 4 / 4