Overlapping Patches for Dynamic Surface Problems C. Carlo Fazioli - - PowerPoint PPT Presentation

Overlapping Patches for Dynamic Surface Problems

• C. Carlo Fazioli

Drexel University

11 Jan 2014

• C. Carlo Fazioli (Drexel University)

Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 1 / 8

Coordinate Charts, or Patches

A collection of mappings Xi(α, β, t) : R2 → S from the plane onto the free surface. Associated partition of unity {Ψi}, Ψi = 1 on S. Advantages: adaptivity complex surfaces isothermal coordinates

• C. Carlo Fazioli (Drexel University)

Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 2 / 8

Time Evolution of Patches

Preserve partition of unity by preserving on normal lines: Ψt = Xt · ∇sΨ Preserve physical quantities on material lines: µt = −U · ∇sµ Upwind considerations require: (Ψµ)t = µΨt + Ψµt = µXt · ∇sΨ − ΨU · ∇sµ With reconstruction as: µ = (

• Ψi)µ =
• (Ψiµ)
• C. Carlo Fazioli (Drexel University)

Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 3 / 8

Interpolation

At a point X∗ on one patch, need the value of Ψµ from other patches.

b b b b b

X∗ has unknown preimage α = αij + ∆α. Once α is known, can easily interpolate Ψµ there (say, bicubic).

• C. Carlo Fazioli (Drexel University)

Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 4 / 8

Physical Problem

Vortex sheet (S) motion in ideal flow: U±

t + U± · ∇U± + ∇p = 0

in D± ∇ · U± = 0 in D± ∇ × U± = 0

• n S

U+ · n = U− · n

• n S

Vortex sheet with strength µ induces vector potential A(x) = 1 4π

• S

µ(x′)n(x′) × ∇x′

• 1

|x − x′|

• dS(x′)

Physical velocity: U · n = (n · ∇ × A)n = [(A · T1)2 − (A · T2)1] n

• C. Carlo Fazioli (Drexel University)

Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 5 / 8

Kernel Smoothing

Regularize the fundamental solution to Gδ Gδ = − 1 4π erf(r/δ) r = G(x)erf(r/δ)

• C. Carlo Fazioli (Drexel University)

Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 6 / 8

Correction Terms

• −
• δ =
• −
• δ
• +
• δ

−

• δ
• Regularization correction:

ǫ =

• S

n(x′) ×

• ∇x′Gδ(x − x′) − ∇x′G(x − x′)
• µ(x′)dS(x′)

= δ 2√π {T2µ1 − T1µ2} + O(δ3) Discretization correction based on estimates of the Fourier coefficients of the regularized kernel, but won’t fit onto a slide.

• C. Carlo Fazioli (Drexel University)

Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 7 / 8

Acknowledgements and References

Joint work with M. Siegel and M. Booty (NJIT), and D. Ambrose (Drexel). References: Beale: “A Grid-Based Boundary Integral Method for Elliptic Problems in Three Dimensions” Bruno and Kunyansky: “A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: ...” Caflisch and Li: “Lagrangian Theory for 3D Vortex Sheets” Baker and Nie: “Application of Adaptive Quadrature to Vortex Sheet Motion”

• C. Carlo Fazioli (Drexel University)

Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 8 / 8