Preconditioning Navier Stokes Problem Dis cretized by Discrete Duality Finite Volume schemes

Sarah Delcourte

Description

Preconditioning Navier-Stokes Problem Dis- cretized by Discrete Duality Finite Volume schemes 1 Sarah Delcourte * — Delphine Jennequin ** * NACHOS Project, INRIA Sophia-Antipolis, France, ** DASSAULT SYSTEM, Suresnes, France, ABSTRACT. We focus on the Discrete Duality Finite Volume (DDFV) method whose particular- ity is to allow the use of unstructured or nonconforming meshes. We discretize the non-linear Navier-Stokes problem, using the rotational formulation of the convection term, associated with the Bernoulli pressure. With an iterative algorithm, we are led to solve a saddle-point problem at each iteration. We give a particular interest to this linear problem by testing some precondi- tioners issued from finite elements, which we adapt to the DDFV method. KEYWORDS: finite volumes, preconditioners, saddle-point, Navier-Stokes equations 1. Introduction Let be ? an open bounded connected domain of R2 with a Lipschitz boundary denoted by ?. We consider the numerical resolution of the bidimensional stationnary Navier-Stokes equations: given f , find (u, p) such that 8 > < > : ??∆u+ u ·?u+?p = f in ?, ? · u = 0 in ?, u = 0 on ?, R ? p(x) dx = 0.

• uj ·


• ?ti2 ? ?ti1


• dual cells


• discrete duality


• navier stokes equations


• finite volume


• delaunay-voronoi meshes

Sujets

• Navier?Stokes equations
• Finite volume method

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 26
• Langue: English

Extrait

Preconditioning Navier Stokes Problem Dis
cretized by Discrete Duality Finite Volume
1schemes
* **Sarah Delcourte — Delphine Jennequin
* NACHOS Project, INRIA Sophia Antipolis, France, [e-mail]
** DASSAULT SYSTEM, Suresnes, France, [e-mail]
ABSTRACT. We focus on the Discrete Duality Finite Volume (DDFV) method whose particular-
ity is to allow the use of unstructured or nonconforming meshes. We discretize the non linear
Navier Stokes problem, using the rotational formulation of the convection term, associated with
the Bernoulli pressure. With an iterative algorithm, we are led to solve a saddle point problem
at each iteration. We give a particular interest to this linear problem by testing some precondi
tioners issued from finite elements, which we adapt to the DDFV method.
KEYWORDS: finite volumes, preconditioners, saddle point, Navier Stokes equations
1. Introduction
2Let be › an open bounded connected domain ofR with a Lipschitz boundary
denoted by¡. We consider the numerical resolution of the bidimensional stationnary
Navier Stokes equations: givenf, find(u;p) such that
¡”¢u+u¢ru+rp = f in›;
r¢u = 0 in›;
[1]
u = 0 on¡;
p(x)dx = 0:
›
The discretization of the Navier Stokes equations by finite volume schemes has at
tracted interest these last years but classical finite volume methods work on meshes
with orthogonal constraints like rectangular grids (see Harlow & Welch [HAR 65])
or so called "admissible meshes" (see [EYM 00, Def. 9.1]), which can be seen as a
generalization of Delaunay Voronoi meshes.
1. This work was performed when the authors were at the CEA Saclay,
DANS/DM2S/SFME/LMPE, Gif sur Yvette, France.
8>>>>:R<In what follows, we focus on the DDFV approach described in [DOM 05] for the
Laplace equation. The main interest of this staggered finite volume method is that it
applies on almost all meshes (unstructured and non conforming meshes) without any
orthogonality constraint. However, the extension of this method to three dimensional
meshes need some adaptation. Here, we present the DDFV scheme for fluid dynamics
equations and this one can be seen as a generalization of Nicolaides’ scheme [NIC 95]
developed on Delaunay Voronoi meshes.
This paper is organized as follows: in section 2, we present the construction of the
primal, dual and diamond meshes. Then, we define discrete gradient, divergence and
curl operators on these meshes. In section 3, we focus on the Navier Stokes equations
and present its discretization. Section 4 is devoted to the description of several kinds
of preconditioners adapted to the DDFV method and some numerical comparisons of
these solvers are given.
2. Definitions and notations
We consider a first partition of› (named primal mesh) composed of elementsT ,i
withi2[1;I], supposed to be convex polygons.
Further, we denote byS , withk2[1;K], the nodes of the polygons of the primalk
mesh. With each of these points, we associate a polygon denoted byP , obtained byk
joining the centers of gravity G associated to the elements of the primal mesh (andi
possibly to midpoints of the boundary sides) of whichS is a vertex to the midpointsk
of the edges of whichS is an extremity. TheP s constitute a second partition of›,k k
referenced as dual mesh. Figure 1(a) displays an example of a primal mesh and its
associated dual mesh.
With each edge of the primal mesh, denoted by A = [S S ], with j 2j k (j) k (j)1 2
[1;J], we associate a quadrilateral named “diamond cell” and denoted byD . Whenj
A is not on the boundary, this cell is obtained by joining the pointsS andS ,j k (j) k (j)1 2
which are the two nodes of A , with the gravity centers G and G of thej i (j) i (j)1 2
elements of the primal mesh sharing this side. WhenA is on the boundary¡, the cellj
D is obtained by joining the two nodes ofA with the pointG associated with thej j i (j)1
only element of the primal mesh of whichA is a side. The cellsD constitute a thirdj j
partition of›, named “diamond mesh”. Such cells are displayed in Figures 1(b) and
01(c). The unit normal vector toA andA = [G G ] are respectively denotedj i (j) i (j)j 1 2
0 0byn andn . More precisely,n points outwardT whilen points outwardP . Atj ji i kj jk
last, the area of the cellsT ,P andD is denoted byjTj,jP j andjD j.i k j i k j
¡T P I+J K DDefinition 1 Given any` = (` ;` )2R £R , the discrete gradientr isi k h
defined by its values over the diamond cellsD :j
1D P P 0 0 T T(r `) := ` ¡` jA jn + ` ¡` jA jn : [2]j j jh k k j j i i2 1 2 12jD jj
hihiTi
Gi
S Gik 2P
k
Gi1DSk j
1
S
k2
D
S jk2
SG G k1i i(a) (b) 1 (c) 2
Figure 1. (a) An example of a primal mesh and its associated dual mesh. (b) An inner diamond
cell. (c) A boundary diamond cell.
‡ ·T
@† @†In the very same way, we may approach the vector curl operatorr£†= ;¡
@y @x
Dby a discrete vector curl operatorr £ on the diamond cells:h
1D P P 0 0 T T
(r £`) :=¡ ` ¡` jA j¿ + ` ¡` jA j¿ ; [3]h j k k j j i i j j2 1 2 12jD jj
0 0 0where the unit vectors ¿ and ¿ are such that (n ;¿ ) and (n ;¿ ) are orthogonalj j jj j j
2positively oriented bases ofR .
T;P2J T PDefinition 2 Given anyu=(u )2R , the discrete divergencer ¢:=(r ¢;r ¢)j h h h
is defined by its values over the primal cellsT and the dual cellsP :i k
1T(r ¢u) := jA ju ¢n ;i j j jih jTji
j2V(i) [4]
1 1P 0 0(r ¢u) := jA ju ¢n + jA ju ¢n :k j j j jh j jjP j 2k
¡j2E(k) j2E(k)\[J¡J +1;J]
‡ ·
@†y @†xIn the very same way, we may approach the scalar curl operatorr£†= ¡@x @y
T;P T Pby a discrete scalar curl operatorr £:=(r £;r £) replacing the normal unith h h
vectorn by the tangential unit vector¿ in [4].
¡ ¢ ¡ ¢2 2I K 2JDefinition 3 If (`;ˆ) 2 R £R and (u;v) 2 R , then we define the fol
lowing scalar products:
(u;v) := jD ju ¢v ; [5]D j j j
j2[1;J]
1 T T P P
(`;ˆ) := jTj` ˆ + jP j` ˆ : [6]T;P i i i k k k
2
i2[1;I] k2[1;K]
0X@hXXiXhi11XAA0X@¡J I+J KWe also define the trace ofu2R and`2R £R on the boundary¡ by
1 P T P(u;`) := jA ju £ ` +2` +` : [7]¡;h j j k (j) i (j) k (j)1 2 24
j2¡
Proposition 1 The following discrete analogues of the Green formulae hold:
T;P D(r ¢u;`) =¡(u;r `) +(u¢n;`) ; [8]T;P D ¡;hhh
T;P D(r £u;`) =(u;r £`) +(u¢¿;`) ; [9]T;P D ¡;hh h
¡ ¢2 ¡J T P I+J Kfor allu2 R and all`=(` ;` )2R £R .
3. Discretization of the Navier Stokes equations
We are interested in the approximation of non linear problem [1]. For continuous
operators,¡¢u can be rewritten as¡¢u =r£r£u¡rr¢u: On the other
hand, to avoid a problem of definition of the convective term on staggered meshes, we
use the rotational formulation ofu¢ru which reads:
2u
u¢ru=(r£u)u£ez +r ; [10]
2
Twhereu£e =(¡u ;u ) withu andu the two components ofu, and we intro z y x x y
2u
duce the Bernoulli pressure: … = p+ : At last, in order to ensure the uniqueness
2R
of…, we set …(x)dx=0:
›
With an iterative process to solve the non linearity (the fixed point method for
example), we are led to solve the following linear system, called Oseen equations:
givenf andu , find(u;…) such thatG
¡” [r£r£u¡rr¢u]+(r£u )u£e +r… = f in›;G z
r¢u = 0 in›;
[11]
u = 0 on¡;
…(x)dx = 0:
›
Hypothesis 1 We assume that each boundary primal cell has only one edge which
belongs to the boundary¡.
We look for the approximation(u ) of the velocityu on the diamond cells andj j2[1;J]
T Pthe approximation(… ) ;(… ) of the Bernoulli pressure… on the primali2[1;I] k2[1;K]i k
and dual cells respectively.
We discretize the first equation of [11] on the interior diamond cells and the sec
ond equation of [11] both on the primal and dual cells. Then, the boundary condition
u = 0 is discretized on the boundary diamond cells while the condition of vanishing
mean pressure is discretized on the primal and dual cells.
:>>R<>X>8We shall suppose that the locations of the values ofu are the same as those ofu,G
that is on the diamond cells. Therefore, we may easily calculater£u on the primalG
T;P
and dual cells according to the discrete operator r £. However, since the firsth
equation in [11] is discretized on diamond cells, we shall use the following quadrature
formula to calculater£u over anyD :G j
T T P P(r £u ) +(r £u ) +(r £u ) +(r £u )G i G i G k G kh 1 h 2 h 1 h 2(r£u ) … : [12]G jDj 4
Then, for all diamond cells, we set:
h i
D D T;P D T;P
¡ ¢ u =(r £r £u) ¡(r r ¢u) :j jh h hh h
j
Now, we can discretize the continuous problem [11] by the following system:
D D D
¡” ¢ u +(r£u ) u £e +(r …) = f ; 8D 2= ¡; [13a]h G j j z h j j jDjj
T;P(r ¢u) = 0; 8T ;8P ; [13b]ii;k kh
Du = f ; 8D 2¡; [13c]j jj
T P
jTj… = jP j… = 0; [13d]i i k k
i2[1;I] k2[1;K]
R
D 1 Dwhere we have setf = f(x)dx; 8j2= ¡ and f =0; 8j2¡:j jjD j Dj j
Property 1 F