International Conference on Computational Methods in Marine Engineering
MARINE 2009
T. Kvamsdal, B. Pettersen, P. Bergan, E. Oñate and J. García (Eds)
 CIMNE, Barcelona, 2009
ADVANCES IN THE DEVELOPMENT OF AN UNSTRUCTURED
FEM SOLVER FOR FLUID-STRUCTURE INTERACTION
PROBLEMS IN MARINE ENGINEERING
JULIO GARCÍA-ESPINOSA*†, ABEL COLL† AND EUGENIO OÑATE†
*
Compass Ingeniería y Sistemas (CompassIS)
C/ Tuset 8, 7º 2ª, 08006 Barcelona, España.
e-mail: info@compassis.com, web page: http://www.compassis.com
†
International Center for Numerical Methods in Engineering (CIMNE)
Universidad Politécnica de Cataluña
Campus Norte UPC, 08034 Barcelona, Spain
e-mail: congreso@cimne.upc.edu, web page: http://www.cimne.com
Key words: Marine Engineering, Finite Element Method (FEM), Finite Calculus (FIC),
Turbulence, Free Surface, Naval Hydrodynamics, Level Set, Domain Decomposition.
Summary. This work presents different advances in the development of a formulation
for fluid-structure interaction analysis in marine applications using the FIC stabilized
finite element method (FEM). The free surface evolution is solved using of an
overlapping domain decomposition concept in the statement of the problem. The work
also presents a new mesh generation algorithm for unstructured meshed including
boundary layer anisotropic refinement.
1 INTRODUCTION
The free surface flow problem can be considered as a particular case of the more
general problem of predicting the interface between two immiscible fluids: the flowing
liquid (typically water) and air. Computing the interface between two immiscible fluids
is difficult because neither the shape nor the position of the interface between the two
fluids is known a priori.
This paper shows a new stabilized Finite Element Method (FEM) for solving the
Navier-Stokes equations including free surface effects that overcomes most of the
difficulties of the existing methodologies. The main innovation is the introduction of an
overlapping domain decomposition concept in the statement of the problem for
increasing the accuracy in the capture of the free surface as well as in the resolution of
the governing equations in the interface between the two fluids. The general scheme has
been adapted in this work for the simulation of monophase flows. Furthermore, the
algorithm has been adapted to handle with unstructured meshes specially developed for
the accurately capturing of the boundary layer effects.
2 STATEMENT OF THE PROBLEM
The velocity and pressure fields of two incompressible and immiscible fluids moving in
the domain Ω  d (d=2,3) during the time interval  0,T  , can be described by the
incompressible Navier Stokes equations for multiphase flows, also known as the nonhomogeneous incompressible Navier Stokes equations [2]:
J. García-Espinosa, A. Coll and E. Oñate.
t     u   0
 t   u       u  u        f (1)
 u  0
where  is the fluid density field, u is the velocity field and  is the Cauchy stress
tensor defined by:
   pI  
(2)
    u  u T 
where μ is the dynamic viscosity and I is the identity matrix. Bold characters are used to
denote vector and tensor variables.
As we consider problems with a moving interface inside the domain, all sub domains
and functions defined hereafter are time dependent.
For t  0, T  , let 1  t   x   x is Fluid1 be the part of the domain  occupied by
fluid 1 and let 2  t   x   x is Fluid 2  be the part of the domain  occupied by
fluid 2. Therefore 1  t  and 2  t  are two disjoint sub domains of  . Then

  int 1  t  2  t 

t   0, T  (3)
where ‘int’ denotes the topological interior and the over bar indicates the topological
adherence of a given set. The system of equations (1) has to be completed with the
necessary initial and boundary conditions.
 ,  are defined in terms of the level set function  (signed distance to the free surface)
 1 , 1   0
(4)
  2 , 2   0
As shown in [2] the multiphase Navier Stokes problem (1) is therefore equivalent to
solve the following system of equations:
t  u      u  u       f
(5)
 u  0
coupled with the equation
t   u    0 (6)
Equation (6) defines the transport of the level set function due to the velocity field
obtained by solving the Eqs. (5).
,   
OVERLAPPING DOMAIN DECOMPOSITION METHOD
To overcome the drawbacks related to the discontinuity of fluid properties across the
interface and to impose the interfacial boundary condition we split the domain  into
two overlapping sub-domains. Based on this subdivision of the domain, we apply a
Dirichlet-Neumann overlapping domain decomposition technique with appropriate
boundary conditions in order to satisfying the interfacial condition.
Let  
#
e 1
N
e
, K e , e  be a finite element partition of domain  , where N e are
element nodes, K e is the element spatial domain, e are element shape functions and 
is the total number of elements in the partition.
2
J. García-Espinosa, A. Coll and E. Oñate.
Let us consider a domain decomposition of domain  into three disjoint sub domains
3  t  , 4  t  and 5  t  in such a way that 3  t   K3e , 5  K5e , where K e3
e
e
are the elements of the finite element partition  , such as x  K   x, t   0 and K e5
e
3
are the elements of the finite element partition such as x  K 5e   x, t   0 . The
geometrical domain decomposition is completed with
4  t    \  3  t  5  t   (7)
From this partition we define the two overlapping domains:



1  t  : int 3  t   4  t  , 2  t  : int 4  t   5  t 

(8)
In order to simplify the notation we will omit the time dependency of domains in the
rest of the section. Let i , i  1, 2 the boundary of  i , then:

 
 
i    i   D  i   N  i   M
iD
iN

(9)
iM
It has to be noted that  i contains an additional term  i that is not included into  .
This term comes from the presence of an interface inside  . Since we are using a
capturing technique, the position of the interface will not lay on mesh nodes, as usually
happens in a tracking technique. Therefore some elements can be intersected by the
interface. Then  i is defined as follows:
i  4  i  1 i1 (10)
 
In the following, we denote by xi a variable related to  i . We apply the Dirichlet
conditions on 1 and the Neumann conditions on  2 . For the boundary 1 , we make
use of the compatibility of velocities at the interface:
u1  u 2 on 1 (11)
For the boundary  2 we make use of the jump boundary condition. We are looking for
a traction vector t such that:
n 2   2  t on  2 (12)
and t must guarantee that the jump boundary condition in the free surface holds [3].
Let us write the variational problem considering the domain decomposition above
described.
 1 t u1 , v 
1
Domain 1
 1  u1   u1 , v
  t , v    t1g  t2s, v 
1M
1N
1
 1f , v
1
 r1d ,  h1d   q 1  0
2
on 1
 q,   u1 

1
u1  u 2
  1 , v  
3
1
1
1
 r1m ,  h1m   v 1 
2
(13)
J. García-Espinosa, A. Coll and E. Oñate.
  2  t u 2 , v 
Domain 2
2
  2  u 2   u 2 , v
  t , v    t1g  t2s, v 
2N
 q,   u 2 

2
2M
2
   2 , v   
2
  t , v    2f , v
2
1
 r2m ,  h 2m   v 2 
2
2
(14)
1
 r2d ,  h2 d   q 2  0
2
An iterative strategy between the two domains is used to reach a converged global
solution in the whole domain  . The global velocity solution obtained from Eq.(14) is
used as the convective velocity for the transport of the level set function.
The presented algorithm has been implemented in the software Tdyn (see
http://www.compassis.com for further information).
BOUNDARY LAYER- ANISOTROPIC MESH GENERATION
The accurately resolution of the boundary layer phenomena requires a specific
treatment. In this work, the scheme developed is based in the generation of an adapted
mesh in the boundary layer area, to increase the accuracy in the capturing of the velocity
gradients.
This development is based on an improvement of the existing frontal method mesh
generator of the GiD pre-processor. The standard frontal method mesh generator is
optimized for the generation of unstructured tetrahedral meshes and the algorithm
developed in this work is able to generate “a posteriori” boundary layer mesh of prisms
or tetrahedra. The philosophy of the method is based on the work presented in [4] but
adapted to the requirements of the existing tools.
ADAPTATION OF THE ODDLS METHOD FOR MONOPHASE FLOWS
It is usual in many practical applications to have only one fluid of interest. These
applications involve most of the naval / marine problems, where the density and
viscosity ratios between water and air are about 1000. It is important for these cases to
adapt the ODDLS technique to solve monophase problems, in order to reducing the
computational cost for capturing the free surface with the necessary accuracy, while
maintaining the advantages of the proposed method. In these cases, the computational
domain is reduced to the nodes in the fluid of interest plus those in the second fluid
being connected to the interface. The later nodes are used to imposing the pressure and
velocity boundary conditions on the interface.
REFERENCES
[1] E. Oñate, J. García-Espinosa and S. Idelsohn. Ship Hydrodynamics. Encyclopedia
of Computational Mechanics. Eds. E. Stein, R. De Borst, T.J.R. Hughes. John
Wiley & Sons, (2004)
[2] J. García-Espinosa, A. Valls and E. Oñate. ODDLS: A new unstructured mesh
finite element method for the analysis of free surface flow problems. Int. Jnl. Num.
Meth. Eng. 76 (9): 1297-1327, (2008)
[3] J. García Espinosa, E. Oñate. An Unstructured Finite Element Solver for Ship
Hydrodynamics. Jnl. Appl. Mech. Vol. 70: 18-26. (2003)
[4] Yasushi Ito y Kazuhiro Nakahashi, Proceedings of the 11th International Meshing
Roundtable (2002).
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