Finite-Volumes II: Non Cartesian Sauro Succi. Finite Volumes Real-life geometries: coordinate-free Courtesy of Prof. M. Porfiri, NYU Poly.

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Presentation transcript:

Finite-Volumes II: Non Cartesian Sauro Succi

Finite Volumes Real-life geometries: coordinate-free Courtesy of Prof. M. Porfiri, NYU Poly

Structured Non-Cartesian Geometrical data

Co/Contravariant/Cartesian

CEV = Centers/Edges/Vertices Non-cartesian: structured S W N Ε NΕ SΕ NW SW ne se C ew n s Non-orthogonal Still structured

Non-structured: diffusive flux Non-orthogonality: S W N Ε NΕ SΕ NW SW ne se C ew n s

CEV = Centers/Edges/Vertices Staggered S W N Ε NΕ SΕ NW SW ne se C ew n s Non-orthogonal

Navier-Stokes (Compressible) Staggered FV

NW NΕ SΕ SW n e w s P E N W S Vertex-centered staggered

Discretized Gauss: Continuity Discretized Convective Fluxes Same for north,west, south … Non-orthogonality issues (!) S W N Ε NΕ SΕ NW SW ne se C ew n s

Discretized Gauss: Continuity Discretized midpoint (2 nd order 8 neigh) Discretized Simpson (4 th order, 8 neigh)

Discretized Convective Fluxes

Discretized Gauss: Momentum_x Convective and Dissipative Fluxes

Non-Linear (outer) iteration Nonlinear (outer) iteration, k=0,1…

Real-life geometries Courtesy of Prof. M. Porfiri, NYU Poly

Example: Global: Cylindrical, Spherical, Local: Oblique

Unstructured FV~FEM

Reconstruction: Cell Centered

Mean square residual Minimize error functional:

Mean square residual

Exercise: Construct gradient on Regular cells Trapezoidal cells

Vertex control elements

Gradient computation: Gauss-Green

P E

Finite Volumes: summary Intuitive and physically sound Round-off Conservative (fluxin=-fluxout) Geo-topological ahead, laborious Interpolation to be decided (unlike FEM) Structured: Finite-Difference with non-smooth coordinates Unstructured: Close to FEM No-singularity (1/r for sherical coordinates) Commercially dominant (STAR-CD, FLUENT…)