Challenges for finite volume models Haroldo F. de Campos Velho

Outline 1. Few words on finite volume (FV) approach 2. Patankar’s FV approach for CFD 3. The driven cavity flow 4. Investigations a) Mixed grids b) Fractal cavity 5. Step back: some FV discretizations

Few words on finite volume 1. We can consider some strategies for solving PDE a) Domain decomposition b) Boundary decomposition c) Spectral methods 2. Finite volume is a domain decomposition method a) Partition the computational domain into control volumes (which are not necessarily the cells of the mesh) b) Discretise the integral formulation of the over each control volume (Gauss divergence theorem). b) Discretise the integral formulation of the conservation laws over each control volume (Gauss divergence theorem). c) Solve the resulting set of algebraic equations or update the values of the dependent variables.

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

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

An application: incompressible fluid 1. We starting from the fluid dynamics formulation 2. Initial and boundary conditions

Some remarks on incompressible fluid 1. Incompressible fluid is consider a simpler version of the N-S equation. 2. Is it the above statement true? 3. Yes and no: Yes: there are less equations to be solved. Yes: there are less equations to be solved. No: Who is the integration factor? No: Who is the integration factor?

Integration factor 1. Simple ODE: 2. the integration factor is:, therefore

Integration factor 1. Matrix ODE: 2. If P -1 exists:

Integration factor 1. Incompressible fluid: who is the pressure? 2. Is there an integration factor? 3. Clearly: M -1 does not exist. Mathematical tools: a) Drazin generalized inverse (ODE) - after discretization b) Application of a type singular semi-group (PDE).

Integrating incompressible fluid 1. Deriving a Poisson equation for pressure 2. with Neumann boundary conditions

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

FV: discrete version 1. For 2D flow:

FV: discrete version 1. For 2D flow:

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

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 …

What would we want? 1. We were trying to study a fluid flow inside of a fractal domain. 2. The first idea is to use unstructured grids 3. We were also interested to investigate a mixed grid (combining structured grid + unstructured grid). 4. Why is someone interested in mixed grid? 5. Obvious reason: improve computational performance.

Driven cavity flow 1. This is easier fluid dynamical problem that it must be solved by numerical process.

Studing fractal cavity flow 1. Koch curve generator: 2. Two pre-fractals:

Studing fractal cavity flow 1. Fractal cavity + finite volume decomposition

Mixed grids (Chimera/Dragon grid)

FV: unstructured grids baricenter

FV: unstructured grids 1. Grid-(a): work! 2. Grid-(b): doesn’t work 3. Grid-(c): doesn’t work 4. Grid-(d): doesn’t work Why?

Answering the question 1. S. Abdallah (J. Comput. Phys., 70, 1987) has shown that structured grids obey the compatibility equation. 2. How about the unstrutured grids?

Answering the question 1. The compatibility equation: it is an identity in fluid dynamics. 2. The solution for the Poisson eqution for pressure exists if the compatibility condition is verified:

Discrete compatibility equation 1. Using the discrete pressure Poisson equation 2. Assuming (S a edge size):

Discrete compatibility condition

Discrete compatibility equation

Fractal cavities properties 1. Attractors for fractal cavities: (a) standard cavity (b) fractal cavity (a) standard cavity (b) fractal cavity

Co-localized approaches 1. Someone should be surprised with the results, where cell-center and cell-vertex did not work. 2. Beause, some authors have used cell-center and cell- vertex, and such procedures presented good results. 3. What’s wrong?

Co-localized approaches 1. Actually, nothing is wrong. 2. However, the evil is in details: a) Co-localized variables and cell-center: - Frink’s approach (AIAA, 1994): he use a different interpolation scheme. - 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. - Marthur-Marthy (Num. Heat Transf., 1997): they uses a different scheme to compute the gradient. b) 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. - Thomadakis-Leschziner (Num. Methods Fluids, 1996): The control volume is computed with union of barycenter, we used median-dual scheme.

Spectral schemes x Finite volume 1. Finite volume approaches can be used in a more complex domain (geometry). 2. Recent results are indicating that for high resolution, spectral schemes have a bigger computational effort than finite volume. 3. Which is the future for the spectral schemes? 4. Maybe, it depends on the type of computing used.

Spectral schemes x Finite volume 1. Transforms (FFT/Legendre) could be implemented in a hybrid computing: hardware and software for processing. 2. Hybrid computing: FPGA, GPU. FPGA, GPU.

Spectral schemes x Finite volume 1. Enhancing the computing: appering the first optical processors. 2. Optical processors (FPGA): (a) Lenslet (Mar/2003) (b) Intel (Feb/2006)

Final remarks 1. Structured grid the compatibility equation is always verified. 2. Compatibility condition can be used to select a type of unstrutured grid. 3. Nothing is neutral in numerical approximation. 4. Remember: the evil is in details.

Preliminar Analysis Smallest cube: L = 120 h -1 Mpc 1.7  10 7  2.2  10 6 particulesruning time ~ 60 h redshift computed z = 10.0, 1.0, 0.1 e 0.0 Edge effect L z Mpctodos Mpc Mpc Mpc Mpc

Percolation (FoF) Percolation radius R perc = b f = n /  2/ b 3 Mass scales: M(N p ) M(M  ) class 1  7  small and galaxies LMC 2  M   —3  “regular” galaxies Via-Láctea 7  M  M87 3  M  50-15k 3  —1  groups or clusters Grupo Local 4  M  Coma 1  M  > 15k > 1  superaclusters SA Local 2  M  R perc b f N p Gal VL  10 part. (R= kpc) Cluster Gr. Local  55 part. SA > SA Local  part.

Obrigado pela atenção.