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Challenges for finite volume models

Challenges for finite volume models. Haroldo F. de Campos Velho haroldo@lac.inpe.br http://www.lac.inpe.br/~haroldo. Outline. Few words on finite volume (FV) approach Patankar’s FV approach for CFD The driven cavity flow Investigations Mixed grids Fractal cavity

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Challenges for finite volume models

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  1. Challenges for finite volume models Haroldo F. de Campos Velho haroldo@lac.inpe.br http://www.lac.inpe.br/~haroldo

  2. Outline • Few words on finite volume (FV) approach • Patankar’s FV approach for CFD • The driven cavity flow • Investigations • Mixed grids • Fractal cavity • Step back: some FV discretizations

  3. Few words on finite volume • We can consider some strategies for solving PDE • Domain decomposition • Boundary decomposition • Spectral methods • Finite volume is a domain decomposition method • Partition the computational domain into control volumes (which are not necessarily the cells of the mesh) • Discretise the integral formulation of the conservation laws over each control volume (Gauss divergence theorem). • Solve the resulting set of algebraic equations or update the values of the dependent variables.

  4. FV: integral form • A key issue: integral form of conservation law: • applying the Gauss divergence theorem (there are some advantages for considering conservative form instead of non-conservative one)

  5. Source terms • It is difficult to maintain the balance of the form • applying standard finite volume approach: Gauss theorem can not be applied to the source term.

  6. An application: incompressible fluid • We starting from the fluid dynamics formulation • Initial and boundary conditions

  7. Some remarks on incompressible fluid • Incompressible fluid is consider a simpler version of the N-S equation. • Is it the above statement true? • Yes and no: Yes: there are less equations to be solved. No: Who is the integration factor?

  8. Integration factor • Simple ODE: • the integration factor is: , therefore

  9. Integration factor • Matrix ODE: • If P-1 exists:

  10. Integration factor • Incompressible fluid: who is the pressure? • Is there an integration factor? • Clearly: M-1 does not exist. Mathematical tools: • Drazin generalized inverse (ODE) - after discretization • Application of a type singular semi-group (PDE).

  11. Integrating incompressible fluid • Deriving a Poisson equation for pressure • with Neumann boundary conditions

  12. FV: discrete version 1. The flux terms are discretised by where is the numerical flux.

  13. FV: discrete version 1. For 2D flow:

  14. FV: discrete version 1. For 2D flow:

  15. FV: discrete version For 2D flow (pressure: white; velocities: blue): Patankar staggered grid control volume (red) Grid cell node

  16. FV: discrete version – a question: Looking at the figure: Why should not we use a simple triangular control volume (black), instead of red one? And then, the bad dream started …

  17. What would we want? • We were trying to study a fluid flow inside of a fractal domain. • The first idea is to use unstructured grids • We were also interested to investigate a mixed grid (combining structured grid + unstructured grid). • Why is someone interested in mixed grid? • Obvious reason: improve computational performance.

  18. Driven cavity flow • This is easier fluid dynamical problem that it must be solved by numerical process.

  19. Studing fractal cavity flow • Koch curve generator: • Two pre-fractals:

  20. Studing fractal cavity flow • Fractal cavity + finite volume decomposition

  21. Mixed grids (Chimera/Dragon grid)

  22. FV: unstructured grids baricenter

  23. FV: unstructured grids • Grid-(a): work! • Grid-(b): doesn’t work • Grid-(c): doesn’t work • Grid-(d): doesn’t work Why?

  24. Answering the question • S. Abdallah (J. Comput. Phys., 70, 1987) has shown that structured grids obey the compatibility equation. • How about the unstrutured grids?

  25. Answering the question • The compatibility equation: it is an identity in fluid dynamics. • The solution for the Poisson eqution for pressure exists if the compatibility condition is verified:

  26. Discrete compatibility equation • Using the discrete pressure Poisson equation • Assuming (Sa edge size):

  27. Discrete compatibility condition

  28. Discrete compatibility equation

  29. Fractal cavities properties • Attractors for fractal cavities: (a) standard cavity (b) fractal cavity

  30. Co-localized approaches • Someone should be surprised with the results, where cell-center and cell-vertex did not work. • Beause, some authors have used cell-center and cell-vertex, and such procedures presented good results. • What’s wrong?

  31. Co-localized approaches • Actually, nothing is wrong. • However, the evil is in details: • Co-localized variables and cell-center: - Frink’s approach (AIAA, 1994): he use a different interpolation scheme. - Marthur-Marthy (Num. Heat Transf., 1997): they uses a different scheme to compute the gradient. • Velocities located at vertex, and pressure at barycenter: - Thomadakis-Leschziner (Num. Methods Fluids, 1996): The control volume is computed with union of barycenter, we used median-dual scheme.

  32. Spectral schemes x Finite volume • Finite volume approaches can be used in a more complex domain (geometry). • Recent results are indicating that for high resolution, spectral schemes have a bigger computational effort than finite volume. • Which is the future for the spectral schemes? • Maybe, it depends on the type of computing used.

  33. Spectral schemes x Finite volume • Transforms (FFT/Legendre) could be implemented in a hybrid computing: hardware and software for processing. • Hybrid computing: FPGA, GPU.

  34. Spectral schemes x Finite volume • Enhancing the computing: appering the first optical processors. • Optical processors (FPGA): (a) Lenslet (Mar/2003) (b) Intel (Feb/2006)

  35. Final remarks • Structured grid the compatibility equation is always verified. • Compatibility condition can be used to select a type of unstrutured grid. • Nothing is neutral in numerical approximation. • Remember: the evil is in details.

  36. Preliminar Analysis • Smallest cube: L = 120 h-1 Mpc • 1.7107  2.2106 particules runing time ~ 60 h • redshift computed • z = 10.0, 1.0, 0.1 e 0.0 • Edge effect • L z <n> • 239.5 Mpc todos 1.22 • 120 Mpc 10.0 1.23 • 120 Mpc 1.0 1.27 • 120 Mpc 0.1 1.29 • 120 Mpc 0.0 1.30

  37. Percolation (FoF) • Percolation radius • Rperc = b <R>  f = n / <n>  2/ b3 • Mass scales:M(Np) M(M) class • 1  71010 small and galaxies • LMC 2 1010 M • 2-50 21011—31012 “regular” galaxies • Via-Láctea 7 1011 M • M87 3 1012 M • 50-15k 31012 —11015 groups or clusters • Grupo Local 4 1012 M • Coma 1 1015 M • > 15k > 11015 superaclusters • SA Local 2 1015 M • Rperc b f Np • Gal. 0.1 0.11 1500 2-50 VL  10 part. (R=100-150 kpc) • Cluster 0.184 0.2 250 50-15000 Gr. Local  55 part. • SA 1.25 1.15 2 > 15000 SA Local  30000 part.

  38. Obrigado pela atenção.

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