Finite Volume Methods G. Manzini 1 1 IMATI - CNR, Pavia Padova - PowerPoint PPT Presentation

Finite Volume Methods G. Manzini 1 1 IMATI - CNR, Pavia Padova 16/4/2007

Outline Contents: • part 1: an informal introduction to Finite Volume Methods (FVMs); • part 2: application to steady transport problems in convection-dominated regimes and layer calculation; Other collaborations: • application to strongly anisotropic diffusion models: the locking effect (joint work with M. Putti); • application to partially saturated multi-layered soils: mass conservation (joint work with. S. Ferraris);

R + , such that R + , The basic model problem: R d and the convection-diffusion equation • Find a function u ( x , t ) defined on Ω × ∂ u ∂ t + div ( β u − K ∇ u ) = G , Ω × in with Ω ⊂ • u, conserved quantity; • β , velocity field, K , diffusion tensor; • f , production/consumption source term;

The basic model problem: the convection-diffusion equation R + , R + , • u 0 ( x ) initial condition, R d . u = u 0 in Ω , t=0 , • g D , g N , boundary condition data defined on ∂ Ω = Γ D ∪ Γ N Γ D × u = g D , on Γ N × n · K ∇ u = g N , on with Ω ⊂

R + , R + , R + , Sub-problems ∂ u ∂ t + div ( β u − K ∇ u ) = G , Ω × in u = u 0 Ω , t=0 , in Γ D × u = g D , on Γ N × n · K ∇ u = g N , on • ∂ u /∂ t = 0, steady-state case; • | β | > > K , advection-dominated case, layers ; • K = diag ( 1 , δ ) , with δ< < 1, or δ> > 1 (and possibly | β | = 0), strongly anisotropic diffusion tensors. • Γ N (Γ D ) = ∅ , pure Dirichlet (Neumann) problem.

Four steps towards finite volume discretizations: � � • introduce a mesh partitioning of the domain T h = T , T, control volumes; • re-formulate the equation by integrating over each control volume T as a balance of physical fluxes; • mimic the conservation property of the underlying divergence operator by a discrete balance of properly defined numerical fluxes; • solve for an approximation of cell averages of the conserved quantity and eventually reconstruct piece-wise linear approximate solutions.

Deeper inside. . . • we reformulate the model problem as an integral equation on each control volume; • we represent the integral equation for each control volume as an ordinary differential equation (ODE); • collecting ODEs from all control volumes we get a system of (possibly non-linear) algebraic equations. . . • . . . that are (hopefully) amenable to a solution using computational methods. Thus, we need to approximate volume integrals and surface integrals to form algebraic expressions.

Domain, Zone, Grid, and Cell Prior to discussing the Finite Volume approximation, let us examine the control volumes on which volume and surface integrals will be approximated. . . The control volumes exists at several levels: • flow domain , extent of CFD analysis • zone , divide domain for convenience, if needed • grid , divides each zone into cells • cell , smallest control volume, but finite

Domain, Zone, Grid, and Cell domain zone 1 grid 2, 32 × 16 cells zone 2 grid 1, 16 × 8 cells

General Classification of Grids Grids can be classified as • structured , • unstructured , • hybrid . • Structured grids use a structured grid topology : • cells are arranged in an array structure • location of neighbor items (cells, faces, vertices) is implicit in the array indices (i,j,k) Use of structured grids allows efficient storage and book-keeping of grid items information.

Example of a structured grid i,j+1 i+1,j+1 � � i,j+1 � � � � � � � � i+1,j i,j i−1,j � � � � � � � � i+1,j i,j i,j−1

General Classification of Grids • unstructured grids do not have an underlying “regular” grid topology: • no inherent ordering of grid items, • the arrangement of grid items must be specified explicitly (by using lists). Use of unstructured grids allows greater flexibility in generating and adapting grids at the expense of greater storage of cell information. • hybrid grids contain both structured and unstructured grids, usually in separate zones

Example of an unstructured grid 27 41 � � 6 � � � � � � � � 38 14 2 � � � � � � 24 � � 37 51 Control volume T 14 is formed by vertices labeled by { 37 , 24 , 41 , 6 } and has neighbours labeled by { 2 , 51 , 38 , 27 } .

Remarks Grids of quadrilaterals may be considered as structured or unstructured, but when control volumes of more general shape are considered, we must use unstructured grids:

Anatomy of a finite volume cell As integral equation are approximated on the finite volume cell to form an algebraic relation, we point out that: • any cell can take on a generalized shape, but they are usually • non-overlapping; • convex shaped (not always required); • any cell contains a finite (positive) volume; • the size of cells, usually denoted by the symbol h , indicates the level of computational resolution of error analysis;

For 2-D finite volume cells: • a volume cell is a polygon defined by a control poly-line; • the control poly-line is subdivided into a finite number of edges (2-D); • edges are usually straight lines; • an edge is bounded by two non-coincident vertices.

For 3-D finite volume cells: • a volume cell is a polyhedron defined by a control surface; • the control surface is faceted into a finite number of faces (3-D); • faces can take on a variety of shapes; • a face is bounded by edges; • edges are usually straight lines.

I , and integrate the equation over the Integral conservation formulation • We take K = ν control volume T, � � � ∂ u ∂ t dV + div ( β u − ν ∇ u ) dV = G dV , T T T • assume that control-volume shapes be time-independent, � � � d u dV + div ( β u − ν ∇ u ) dV = G dV , dt T T T • and apply the Gauss divergence theorem, � � � d n T · ( β u − ν ∇ u ) dS = G dV . u dV + dt T ∂ T T

Integral conservation formulation • Some remarks on the equation structure: � � � d n T · ( β u − ν ∇ u ) dS u dV + = G dV . dt T ∂ T T � �� � � �� � � �� � time variation of flux balance of u through source of u u in volume T surface ∂ T in volume T • We divide by | T | and split face contributions to fluxes � � � � d 1 u dV + 1 1 n e · β u dS | e | | T | | T | | e | dt T e e ∈ ∂ T � � � − 1 = 1 n e · ν ∇ u dS G ( t ) dV | e | | T | e T • to obtain � � � dt u T ( t ) + 1 d | e | F e ( u ) − G e ( u ) = G T , | T | e ∈ ∂ T

Summarizing: • finite volume methods search for an approximation of � � � dt u T ( t ) + 1 d | e | F e ( u ) − G e ( u ) = G T ( t ) , for all T ∈ T h | T | e ∈ ∂ T where � u T = 1 u dV , cell average of u over T | T | T � F e = 1 n e , T · β u dS , advective flux | e | e � G e = 1 n e , T · ν ∇ u dS , diffusive flux | e | e � G T = 1 G dV , cell average of source term over T | T | T

Some general remarks � � � dt u T ( t ) + 1 d | e | F e ( u ) − G e ( u ) = G T , for all T ∈ T h | T | e ∈ ∂ T • u T is the conserved quantity representing the flow; • the integral conservation equation applies for each control volume; • control surface bounds the control volume; • flow is driven through the control surface by fluxes F e and G e ; • flow can be time-varying (unsteady), i.e. F e and G e depend on u T ( t ) ; • further, we may consider time-dependent control volumes and control surfaces as well.

Finite Volume Approximation In the integral equation � � � dt u T ( t ) + 1 d | e | F e ( u ) − G e ( u ) = G T , for all T ∈ T h | T | e ∈ ∂ T we approximate the continuous flux balance summation term by the discrete flux balance � � � � � � | e | F e ( u ) − G e ( u ) ≈ | e | F e ( u h ) − G e ( u h ) , e ∈ ∂ T e ∈ ∂ T where • u h ≈ { u T } collects the approximation of cell averages of u; � n e , T · β u dS , numerical advective flux ; 1 • F e ( u h ) ≈ | e | e � n e , T · ν ∇ u dS , numerical diffusive flux . 1 • G e ( u h ) ≈ | e | e

Underlying assumptions on solution approximation R N T , N T is the number of cells Note that within the finite volume framework we are approximating � 1 u h ( t ) | T i = u i ( t ) ≈ u T i ( t ) = u dV , | T i | T i which is a constant (in space) quantity over T i . Thus, • u h = { u i } is a piecewise constant function over Ω ; • u h = { u i } is (also) a vector in forming T h ; • the position of the solution point in the cell is not yet defined (compare with FEM or FDM methods). Two possible interpretations: • we are approximating cell averages; • we are approximating the value of u at some point inside T (barycenters work up to second-order of accuracy).

Location of the solution in the finite volume cell The location of the flow solution and geometry of the finite volume cell with respect to the grid can be of two types: • cell-centered cell : the flow solution is located at the centroid of the cell volume defined by the grid lines (primary grid); • cell-vertex cell (also node-centered cell): • the flow solution is located at the vertices of the grid; • the finite-volume cell is formed about the vertex (dual grid). Each approach has its advantages and disadvantages, but if things are done right, both approaches do well.