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 X i ( α, β, t ) : R 2 → S from the plane onto the free surface. Associated partition of unity { Ψ i } , � Ψ i = 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: Ψ t = X t · ∇ s Ψ Preserve physical quantities on material lines: µ t = − U · ∇ s µ Upwind considerations require: (Ψ µ ) t = µ Ψ t + Ψ µ t = µ X t · ∇ s Ψ − Ψ U · ∇ s µ With reconstruction as: � � µ = ( Ψ i ) µ = (Ψ i µ ) C. Carlo Fazioli (Drexel University) Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 3 / 8

b b b b b Interpolation At a point X ∗ on one patch, need the value of Ψ µ from other patches. X ∗ has unknown preimage α = α ij + ∆ α . Once α is known, can easily interpolate Ψ µ 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: t + U ± · ∇ U ± + ∇ p = 0 U ± in D ± ∇ · U ± = 0 in D ± ∇ × U ± = 0 on S U + · n = U − · n on S Vortex sheet with strength µ induces vector potential A ( x ) = 1 � � 1 � µ ( x ′ ) n ( x ′ ) × ∇ x ′ dS ( x ′ ) | x − x ′ | 4 π S Physical velocity: U · n = ( n · ∇ × A ) n = [( A · T 1 ) 2 − ( A · T 2 ) 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 G δ G δ = − 1 erf ( r /δ ) = G ( x ) erf ( r /δ ) 4 π r C. Carlo Fazioli (Drexel University) Overlapping Patches for Dynamic Surface Problems 11 Jan 2014 6 / 8

Correction Terms � �� � � �� � � � − δ = − + − δ δ δ Regularization correction: � n ( x ′ ) × ∇ x ′ G δ ( x − x ′ ) − ∇ x ′ G ( x − x ′ ) µ ( x ′ ) dS ( x ′ ) � � ǫ = S δ 2 √ π { T 2 µ 1 − T 1 µ 2 } + O ( δ 3 ) = 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