A second order finite volume scheme on space-adaptive staggered - - PowerPoint PPT Presentation

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A second order finite volume scheme

• n space-adaptive staggered grids in 3D

Wolfram Rosenbaum

• Prof. Dr. Sebastian Noelle

RWTH Aachen, Germany

Staggered grid construction in 1D

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How to construct dual cells primal grid dual grid primal cell dual cell

• split volume of a primal cell
• assign volume parts to the nodes of the primal cell
• assemble dual cells around each primal node

⇒ split primal cell into halves = only one splitting rule in 1D

Staggered grid construction in 2D

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Splitting a primal cell 1. primal cell → 16 squares 2. inner squares → 2 triangles 3. assign atoms → nodes on primal cell

Staggered grid construction in 2D

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Assembling local patterns six local patterns dual grid

• connecting patterns yields to planar dual faces

Staggered grid construction in 3D

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Primal grid properties

• structured adaptive octtree
• 1-level transition over common faces and edges

Staggered grid construction in 3D

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Splitting a primal cell 1. primal cell → 64 cubes 2. face-inner cubes → 2 prisms 3. interior cubes → 6 tetrahedra 4. assign atoms → nodes on primal cell

Staggered grid construction in 3D

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227 combinatorially different local patterns

. . .

and many others

Staggered grid construction in 3D

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Assembling local patterns

• connecting patterns yields

to planar dual faces

• dual

faces composed

• f

simple geometric shapes (triangles, rectangles)

Numerical example: First Order Rotating Cone

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initial data at t = 0 exact solution at t = π/2

Numerical example: First Order Rotating Cone

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t = 0 t = π/6

Numerical example: First Order Rotating Cone

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t = π/3 t = π/2

Towards a second order scheme

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Algorithmical demands

• pw. linear reconstruction of cell centered conservative data

→ solve local data-dependent least-square problem with built-in slope limitation [Ollivier-Gooch, 1996]

• execute second order integration in space (over volumes and

faces) and time → midpoint rule for time integration and space integration over volumes → trapezoidal rule for space integration over faces

Towards a second order scheme

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Flux integration

p

p1 p2 p3 v1 v2 v3 v4 v5

• composed n-sided dual faces
• # dual face parts ≫ 3 # dual nodes
• cog often on primal boundary
• evaluate fluxes in dual nodes
• weights for 2nd order quadrature

Towards a second order scheme

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Barycentric weight splitting

composed dual face w0 w1 w2 w3 w4 w5 w6 w7 T1 R1 R2 R3 w0 = R1

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w1 = R1+R2

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w7 = R1+R2+R3

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+ T1

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w0 w0 w0 w1 w1 w2 w2 w2 w3 w3 w4 w4 w4 w5 w5 w5 w6 w6 w6 w7 T1 R1 R2 R3

Numerical example: Second Order Rotating Cone

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t = 0 t = π/6

Numerical example: Second Order Rotating Cone

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t = π/3 t = π/2

Comparison of First and Second Order Rotating Cone

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first order second order

Constructive Solid Geometry

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AND = OR =

Constructing a Nozzle

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→ modify FV scheme in cut cells without CFL-number limitation [Pember, Bell, Colella, Crutchfield, Welcome, 1995]

Conclusion

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• 3D Riemann solver free second order Finite Volume scheme
• higher local effort, lower global cost
• CSG-styled adaptive cartesian grids
• pattern-based construction of dual mesh
• patterns generated on demand and stored in look-up tables

Outlook

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in progress:

• adapt numerical scheme for cut cell handling
• incorporate various boundary conditions (inflow, outflow, . . . )

in order to:

• run complex applications
• compare the scheme to competing methods