Recent Advances in PDEs: Analysis, Numerics and Control Sevilla, January 25-27, 2017 Numerical solution of fluid mechanics problems with pure Lagrange-finite element methods Marta Ben´ ıtez (1) , Alfredo Berm´ udez (2) and Pedro Font´ an (3) (1) Department of Mathematics. Universidade da Coru˜ na and ITMATI. Spain (2) Department of Applied Mathematics. Universidade de Santiago de Compostela and ITMATI. Spain (3) Instituto Tecnol´ogico de Matem´ atica Industrial (ITMATI). Spain
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Recent Advances in PDEs: Analysis, Numerics and Control
Sevilla, January 25-27, 2017
Numerical solution of fluid
mechanics problems with pure
Lagrange-finite element methods
Marta Benıtez(1), Alfredo Bermudez(2) and Pedro Fontan(3)
(1) Department of Mathematics. Universidade da Coruna and ITMATI. Spain
(2) Department of Applied Mathematics. Universidade de Santiago deCompostela and ITMATI. Spain
(3) Instituto Tecnologico de Matematica Industrial (ITMATI). Spain
Contents
1 Introduction
2 The framework of continuum mechanics3 Navier-Stokes equations
Test 1. Academic problem: order of convergenceTest 2. Free boundary flow. Sloshing modes.Test 3. Free boundary flow. Dam-break problemTest 4. Flow past a cylinder
Introduction
Convection-diffusion equations
Continuum thermo-mechanics: energy and, in the case of a mixture,mass of species conservation equations (scalar)
Continuum thermo-mechanics: motion or momentum equation(vector)
Particle finite element method (PFEM)S.R. Idhelson E. Onate, F. del Pin, 2004.
Pure Lagrangian methods (PL)
The equations are rewritten in the reference configuration by using themotion to change from Eulerian to Lagrangian variables.Then they are solved by using discretization methods.
M. Benıtez, A. Bermudez. Numerical Analysis of a second-orderpure Lagrange-Galerkin method for convection-diffusion problems. PartI: time discretization. SIAM J. Numer. Anal., 50:858-882, 2012.
M. Benıtez, A. Bermudez. Numerical Analysis of a second-orderpure Lagrange-Galerkin method for convection-diffusion problems. PartII: fully discretized scheme and numerical results. SIAM J. Numer.Anal., 50:2824-2844, 2012.
M. Benıtez, A. Bermudez. Pure Lagrangian and semi-Lagrangianfinite element methods for the numerical solution ofconvection-diffusion problems. Int. Numer. Anal. Mod., 11:271-287,2014.
Pure Lagrangian methods (PL) (cont)
M. Benıtez, A. Bermudez. Pure Lagrangian and semi-Lagrangianfinite element methods for the numerical solution of Navier-Stokesequations. Appl. Numer. Math., 95:62-81, 2015.
M. Benıtez, A. Bermudez. Second order pure Lagrange-Galerkinmethods for fluid-structure interaction problems. SIAM J. Sci. Comput.37:744–777, 2015.
The formalism of continuum mechanics is used to introduce both PLand SL methods for fluid and fluid-structure problems
Navier-Stokes equations in Lagrangian coordinates (p ∈ Ω)
Find two functions u : Ω× [0, T ] −→ V and πm : Ω× [0, T ] −→ R such that
ρ0u− Div((
− πmI + ηm(∇uF−1 + F−t(∇u)t))detFF−t
)
= bm detF in Ω× (0, T ),
detFF−t : ∇u = gm detF in Ω× (0, T ),
u(p, 0) = v0(p) in Ω,
u(p, t) = vd,m(p, t) on ΓDτ × [0, T ],
(−πmI + ηm(∇uF−1 + F−t
(∇u)t
))F−t
m = |F−t(p, t)m(p)|hm
on ΓN × [0, T ],
where ρ0(p) = ρ(X(p, t), t) det F (p, t) is the (given) reference density.
Recall that F (p, t) := ∇X(p, t) = I +∇u(p, t).
Weak formulation
Find two functions u(·, t) ∈ H1(Ω) and πm(·, t) ∈ L2(Ω) such that
∫
Ωρ0u · z+
∫
Ω
(−πmI + ηm
(∇uF−1 + F−t (∇u)t
))F−t
: ∇zdetF =
∫
Ωbm · zdetF
+
∫
ΓN
|F−tm|detFhm · z dAp ∀z ∈ H
1ΓD(Ω),
∫
ΩdetFF−t : ∇u q =
∫
Ωgm detF q ∀q ∈ L2(Ω).
Pure Lagrange-Galerkin schemes
Approximation of the second time derivative (three-point central formula)
u(p, t) ≈u(p, t+∆t)− 2u(p, t) + u(p, t−∆t)
∆t2+O
(∆t2
)
Approximation of the first time derivative (two-point central formula)
∇u(p, t) ≈∇u(p, t+∆t)−∇u(p, t−∆t)
2∆t+O
(∆t2
)
1. Central Pure Lagrangian-Galerkin: (CPLG) −→ t = tn+1/2 (Linear scheme)
∫
Ω
ρ0
(un+3/2∆t − 2u
n+1/2∆t + u
n−1/2∆t
∆t2
)· z−
∫
Ω
πn+1/2m,∆t detF
n+1/2∆t (F
n+1/2∆t )−t
: ∇z+ η
∫
Ω
∇u
n+3/2∆t −∇u
n−1/2∆t
2∆t(F
n+1/2∆t )−1
+(Fn+1/2∆t )−t (∇u
n+3/2∆t )t − (∇u
n−1/2∆t )t
2∆t
detF
n+1/2∆t (F
n+1/2∆t )−t : ∇z
=
∫
Ω
bn+1/2 X
n+1/2∆t · zdetF
n+1/2∆t
+
∫
ΓN
|(Fn+1/2∆t )−t
m| detFn+1/2∆t h
n+1/2 Xn+1/2∆t · z ∀z ∈ H
1ΓD (Ω).
where Xn+1/2∆t (p) := p+ u
n+1/2∆t (p), F
n+1/2∆t (p) := I +∇u
n+1/2∆t (p) and
0 ≤ n ≤ N − 1.
Time discretization (divergence condition)
∫
ΩdetF
n+1/2∆t (F
n+1/2∆t )−t :
(∇u
n+3/2∆t −∇u
n−1/2∆t
2∆t
)q = 0
∀q ∈ L2(Ω).
Boundary condition for the displacement
un+3/2∆t (p) = u
n−1/2(p) + 2∆tvn+1/2d
(X
n+1/2∆t (p)
)on ΓD,
and 0 ≤ n ≤ N − 1.
Initial conditions for the displacement
We take Ω−∆t/2 as reference configuration instead of Ω0. Then,
u−1/2∆t (p) := 0,
u1/2∆t (p) := ∆tv0
(p+ v
0(p)∆t
2
).
In order to keep the global second order of the scheme it is importantto build an approximation of u1/2 of order 3 as the previous one.Further details can be seen in
→ M. Benıtez, A. Bermudez. Pure Lagrangian and semi-Lagrangian
finite element methods for the numerical solution of Navier-Stokes
equations. Appl. Numer. Math., 95:62-81, 2015.
⋆ Post-processing. Second-order central formulas
Approximation of the material description of the velocity
vn+1m (p) = u
n+1(p) ≃ vn+1m,∆t(p) :=
un+3/2∆t (p)− u
n+1/2∆t (p)
∆t.
Motion approximation at times tn+1N−1n=0
Xn+1(p) ≃ Xn+1∆t (p) :=
un+3/2∆t (p) + u
n+1/2∆t (p)
2+ p.
⋆ Post-processing. Second-order central formulas
Approximation of the material description of the velocity
vn+1m (p) = u
n+1(p) ≃ vn+1m,∆t(p) :=
un+3/2∆t (p)− u
n+1/2∆t (p)
∆t.
Motion approximation at times tn+1N−1n=0
Xn+1(p) ≃ Xn+1∆t (p) :=
un+3/2∆t (p) + u
n+1/2∆t (p)
2+ p.
Approximation of the spatial description of the velocity
vn+1(Xn+1(p)) = v
n+1m (p) ≃ v
n+1m,∆t(p).
⋆ Post-processing. Second-order central formulas
Approximation of the material description of the velocity
vn+1m (p) = u
n+1(p) ≃ vn+1m,∆t(p) :=
un+3/2∆t (p)− u
n+1/2∆t (p)
∆t.
Motion approximation at times tn+1N−1n=0
Xn+1(p) ≃ Xn+1∆t (p) :=
un+3/2∆t (p) + u
n+1/2∆t (p)
2+ p.
Approximation of the spatial description of the velocity
vn+1(Xn+1(p)) = v
n+1m (p) ≃ v
n+1m,∆t(p).
Approximate pressure in spatial coordinates
πn+1/2(Xn+1/2(p)) = πn+1/2m (p) ≃ π
n+1/2m,∆t (p).
Space discretization
Numerical experiments in 2D (3D in progress).
P1+bubble/P1 finite elements for displacement/pressure in the fluid
Restarting the CPLG scheme
Let us assume that we decide to restart the scheme at time tr−1/2 being
1 ≤ r ≤ N − 1. Firstly we also compute ur+1/2∆t,h and then v
r∆t,h.
The new reference domain: Ωtr−1/2(a priori unknown) −→ is
determined by using the approximate motion Xr−1/2∆t,h .
Initial conditions have to be carefully chosen in order to not to loseaccuracy:
u(y, tr−1/2) = 0,
u(y, tr+1/2) = ∆tvr∆t,h
(y + v
r∆t,h(y)
∆t
2
).
Further details in → M. Benıtez, A. Bermudez. Pure Lagrangian and
semi-Lagrangian finite element methods for the numerical solution of
2. Linearized Newmark Pure Lagrange-Galerkin schemes(LNPLG) −→ t = tn+1 (linear schemes)∫
Ω
ρ0
(un+1∆t − u
n∆t −∆t(1− γ)un
∆t
∆tγ
)· z−
∫
Ω
πn+1m,∆t det F
n+1∆t (Fn+1
∆t )−t : ∇z
+
∫
Ω
ηn+1 Xn+1∆t ∇u
n+1∆t (Fn+1
∆t )−1 det Fn+1∆t (Fn+1
∆t )−t : ∇z
+
∫
Ω
ηn+1 Xn+1∆t (Fn+1
∆t )−t(∇un+1∆t )t det Fn+1
∆t (Fn+1∆t )−t : ∇z
=
∫
Ω
bn+1 Xn+1
∆t · zdet Fn+1∆t
+
∫
ΓN
|(Fn+1∆t )−t
m| det Fn+1∆t h
n+1/2 Xn+1∆t · z ∀z ∈ H
1ΓD(Ω).
whereXn+1
∆t (p) := p+ un+1∆t (p), Fn+1
∆t (p) := I +∇un+1∆t (p),
un+1∆t (p) = u
n∆t(p) + ∆tun
∆t(p) +∆t2
2un∆t(p), (Newmark with β = 0)
Linearized Newmark Pure Lagrangian schemes(LNPLG) −→ t = tn+1
Divergence condition.
∫
Ω
det Fn+1∆t (Fn+1
∆t )−t : ∇un+1∆t q =
∫
Ω
gn+1 Xn+1∆t det Fn+1
∆t q ∀q ∈ L2(Ω).
Boundary condition for the material velocity
un+1∆t (p) = v
n+1d
(Xn+1
∆t (p))
on ΓD.
Update acceleration and displacement at time tn+1 by using un+1∆t
un+1∆t =
un+1∆t − u
n∆t −∆t(1− γ)un
∆t
∆tγ,
un+1∆t = u
n∆t +∆tun
∆t +∆t2(βun+1
∆t +
(1
2− β
)un∆t
).
Initial conditions
Displacement and velocity:
u0∆t(p) = 0, u
0∆t(p) = v
0(p).
Initial acceleration is computed from equations at initial time:
∫
Ωρ0u0
∆t · z+
∫
Ω
(−π0
∆tI + η0(gradv0 +
(grad v0
)t))
: grad z =
∫
Ωb0 · z ∀z ∈ H
10(Ω),
∫
Ω
(Div u0
∆t − grad v0 · ( grad v0)t)q
=
∫
Ω
((∂g
∂t
)0
+ grad g0 · v0
)q ∀q ∈ L2(Ω).
Notice that F 0 = I and ((F−t)·)0 = −(gradv0
)t.
For spatial discretization we also use piecewise P1-bubble/P1 finite elements
Motion approximation
Xn+1(p) = p+ un+1(p) ≃ Xn+1
∆t,h(p) := p+ un+1∆t,h(p).
Motion approximation
Xn+1(p) = p+ un+1(p) ≃ Xn+1
∆t,h(p) := p+ un+1∆t,h(p).
Approximation of the spatial description of the velocity
vn+1(Xn+1(p)) = u
n+1(p) ≃ vn+1(Xn+1
∆t,h(p)) = un+1∆t,h(p).
Motion approximation
Xn+1(p) = p+ un+1(p) ≃ Xn+1
∆t,h(p) := p+ un+1∆t,h(p).
Approximation of the spatial description of the velocity
vn+1(Xn+1(p)) = u
n+1(p) ≃ vn+1(Xn+1
∆t,h(p)) = un+1∆t,h(p).
Approximation of the spatial description of the acceleration
an+1(Xn+1(p)) = u
n+1(p) ≃ an+1(Xn+1
∆t,h(p)) = un+1∆t,h(p).
Motion approximation
Xn+1(p) = p+ un+1(p) ≃ Xn+1
∆t,h(p) := p+ un+1∆t,h(p).
Approximation of the spatial description of the velocity
vn+1(Xn+1(p)) = u
n+1(p) ≃ vn+1(Xn+1
∆t,h(p)) = un+1∆t,h(p).
Approximation of the spatial description of the acceleration
an+1(Xn+1(p)) = u
n+1(p) ≃ an+1(Xn+1
∆t,h(p)) = un+1∆t,h(p).
Approximate pressure in spatial coordinates
πn+1(Xn+1(p)) = πn+1m (p) ≃ πn+1(Xn+1
∆t,h(p)) = πn+1m,∆t,h(p).
Let us assume that we decide to restart the scheme at time tr being1 ≤ r ≤ N − 1. Then we also compute u
r∆t,h.
The new reference domain: Ωtr (a priori unknown) is approximated byusing Xr
∆t,h.
Xl∆t,hXr
∆t,h
Xltr,∆t,h
p
Ω
y
Ωtr
x
Ωtl
New initial conditions:
ur∆t,h(y) = 0,
ur∆t,h(y) = v
r∆t,h(y) in Ωtr ,
ur∆t,h(y) = a
r∆t,h(y) in Ωtr .
Remark
We get a Linearized Newmark Semi-Lagrange-Galerkin scheme (LNSLG)by restarting the LNPLG scheme at each time step
Remark
For β = 0 and γ = 1/2 the linearized Newmark pureLagrange-Galerkin (LNPLG) scheme (which is conditionally stable)can be rewritten only in terms of displacements.
It coincides with the CPLG scheme. However, the reference domainand the initial conditions are slightly different.
Computer implementation
FEniCS project (https://fenicsproject.org)
FEniCS project is a set of computational libraries focusing on the solutionof partial differential equations by finite element methods. Main features:
Open source.
Domain specific language UFL (Unified Form Language), built on topof the Python language, to represent variational formulations in anotation close to mathematical notation.
FFC (FEniCS Form Compiler) compiler for finite element variationalforms. Generates efficient low level C++ code from high level UFLdescription.
Automatic functional differentiation.
M. S. Alnaes, J. Blechta, J. Hake, A. Johansson, B. Kehlet,
A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N.
Wells, The FEniCS Project Version 1.5, Arch. of Num. Soft., vol. 3,2015.
Test 1: an academic example
Ω = (0, 1)× (0, 1), T = 2
ρ = 1, η = 10−3. Functions b, g, v0 and vd are taken such that theexact solution is
For this test we analyze the rates of convergence of four linear methods:three pure-Lagrangian and one semi-Lagrangian
Central Pure Lagrange-Galerkin scheme (CPLG)
Second order for displacement and velocity
First order for pressure
Linearized Newmark Pure Lagrange-Galerkin scheme for β = 1/6 andγ = 1/2 (LNPLG2) and LNSLG
Second order for displacement, velocity and acceleration
Second order for pressure
Linearized Newmark Pure Lagrange-Galerkin scheme for β = 1/6 andγ = 1 (LNPLG1)
First order for displacement, velocity and acceleration
First order for pressure
Velocity. Time discretization error in l∞(L2h) norm (Nx = Ny = 125)
101
102
103
104
105
10−10
10−8
10−6
10−4
10−2
100
N: number of time step
Err
or
l∞(Lh2(Ω
tn
)) error curve (velocity)
CPLGLNPLG
2
LNSLG2
y=C/N2
LNPLG1
y=C/N
Pressure. Time discretization error in l∞(L2h) norm (Nx = Ny = 125)
101
102
103
104
105
10−10
10−8
10−6
10−4
10−2
100
N: number of time step
Err
or
l∞(Lh2(Ω
tn
)) error curve (pressure)
CPLGLNPLG
2
LNSLG2
y=C/N2
LNPLG1
y=C/N
Velocity. Spatial discretization error in l∞(L2h) norm (Nx = Ny = 125)
101
102
10−5
10−4
10−3
10−2
10−1
100
Nx=N
y: number of degrees of freedom in each spatial direction (h=1/N
x)
Err
or
l∞(L2h(Ω
tn
)) error curve (velocity)
CPLGLNPLG
2
LNSLG2
y=C/Nx2
y=C/Nx
Pressure. Spatial discretization error in l∞(L2h) norm (Nx = Ny = 125)
101
102
10−5
10−4
10−3
10−2
10−1
100
Nx=N
y: number of degrees of freedom in each spatial direction (h=1/N
x)
Err
or
l∞(L2h(Ω
tn
)) error curve (pressure)
CPLGLNPLG
2
LNSLG2
y=C/Nx2
y=C/Nx
Total computing time per 1 s simulation
Test 2. Free boundary flow: sloshing modes
Water in a rectangular container
Body force:
b1(x1, x2, t) = ρAg sin(2πNt)
b2(x1, x2, t) = −ρg
where the frequency N has been taken close to resonant frequencies:N = 0,89 Hz, 1,67 Hz. and A = 0,01 m
A. Huerta, K. Liu, Viscous flow with large free surface motion,Comp.Meth. Appl. Mech. and Eng. 69 (1988) 277-324.
H. Zhou, J. F. Li and T. S. Wang , Simulation of liquid sloshing incurved-wall containers with arbitrary LagrangianEulerian method, Int. J.Numer. Meth. Fluids 57 (2008) 437452.
Numerical results
Pure Lagrangian methods without remeshing
Numerical results. N = 0,89 Hz, ρ = 1000, η = 0.001. CPLG.
Figura: Modulus of the velocity and streamlines
Numerical results. N = 0,89 Hz, ρ = 1000, η = 0.001. CPLG.
0 1 2 3 4 5 6 7 8 9−0.25
0
0.25
0.5
t: time
Dim
ensi
onle
ss w
ave
heig
ht
Time history of the wave height at the wall for µ=1.d−3
left wallright wall
Figura: Time histories of the heights at the walls
Numerical results. N = 1,67 Hz, ρ = 1000, η = 0.001. CPLG.
Figura: Modulus of the velocity and streamlines
Numerical results. N = 1,67 Hz, ρ = 1000, η = 0.001. CPLG.
0 2 4 6 8 10−0.06
−0.04
−0.02
0
0.02
0.04
0.06
t: time
Dim
ensi
onle
ss w
ave
heig
ht
Time history of the wave height at the wall
left wallright wall
Figura: Time histories of the heights at the walls
Numerical results. N = 0,89 Hz, ρ = 1000, η = 1. CPLG.
0 5 10 15 20 25 30 35 40
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
t: time
Dim
ensi
onle
ss w
ave
heig
ht
Time history of the wave height at the wall
left wallright wall
Figura: Time histories of the heights at the walls. Longer time interval
Numerical results with Newmark Pure Lagrange-Galerkin (NPLG).
Figura: Time histories of the heights at the walls
Test 3. Free boundary flow: dam break problem
Data:
Cavity: [0, 3.5] × [0, 7].L = 3,5 cm
ρ = 1 g/cm3,η = 0.01 g/(cm · s),g = 980 cm/s2
Initial velocity: 0 cm/s
at t = 0
v2 = 0
v1 = 0
Problem definition
References⋆ J.C. Martin, W.J. Moyce. An experimental study of the collapse of
liquid columns on a rigid horizontal plane. Philos Trans R Soc London, SerA. 244, 312-324 (1952)
⋆ C.W. Hirt, B.D. Nichols. Volume of fluid (VOF) method for thedynamics of free boundaries. J. Comput. Phys. 39,201-225 (1981)
⋆ B. Ramaswamy, M. Kawahara. Lagrangian finite element analysisapplied to viscous free surface fluid flow. Int J Numer Methods Fluids.7,953-984 (1987)
⋆ P. Hansbo. The characteristic streamline diffusion method for thetime-dependent in compressible NavierStokes equations. Comput Meth ApplMech Eng 99,171-186 (1992)
⋆ E. Walhorn, A. Kolke, B. Hubner, D. Dinkler. Fluid-structurecoupling within a monolithic model involving free surface flows. Computersand Structures. 83, 2100-2111 (2005)
⋆ Wolfgang A. Wall, Steffen Genkinger, Ekkehard Ramm. A
strong coupling partitioned approach for fluidstructure interaction with free
surfaces. Computers and Fluids. 36, 169-183 (2007)
Numerical results
Pure Lagrangian methods without remeshing
Dam break problem (2860 vertices, ∆t = 0.001)
0 0.5 1 1.5 2 2.5 3 3.51
1.5
2
2.5
3
3.5
4
4.5
time t (2g/L)1/2
x(t)
/L
History of the displacement
CPLGWolfgang A. Wall et al.E. Walhorn et al.P. HansboRamaswamy & KawaharaHirt & Nichols (experimental)Martin & Moyce (experimental)
Pressure
Modulus of the velocity
Dam break problem (2860 vertices, ∆t = 0.001)
Figure: Deformed mesh after 0.2 s.
Test 4. External flows: flow past a cylinder
In order to work with the same particles all the time we redefine theproblem with periodic boundary condition at the input/exitboundaries of a large enough computational domain.
The input velocity is imposed on the upper and lower boundaries
Flow past a cylinder
Density= 1. Dynamic viscosity: 0.01, 0.001
Flow past a cylinder of diameter 0.1
Computational domain: [0, 20] × [0, 0.5]
Input velocity: 1
Flow past a cylinder (η = 0.01, n. of vertices: 38910, ∆t = 0.001)
Figure: Modulus of the velocity, streamlines
Flow past a cylinder (η = 0.01, n. of vertices: 38910, ∆t = 0.001)
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
v1(0.2,⋅): horizontal velocity for x=0.2
y
(LG) methodFluent
0.86 0.88 0.9 0.92 0.94 0.96 0.98 1−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
v1(1,⋅): horizontal velocity for x=1
y
(LG) methodFluent
Figure: Comparison with ANSYS-FLUENT
Flow past a cylinder (η = 0.01, n. of vertices: 38910, ∆t = 0.001)
0.85 0.9 0.95 1−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
v1(−0.2,⋅): horizontal velocity for x=−0.2
y
(LG) methodFluent
0.9 0.92 0.94 0.96 0.98 1−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
v1(−1,⋅): horizontal velocity for x=−1
y
(LG) methodFluent
Figure: Comparison with ANSYS-FLUENT
Flow past a cylinder (η = 0.001, n. of vertices: 38910, ∆t = 0.0001)
Figure: Modulus of the velocity, streamlines at t = tn, t = tn + 2,respectively
Flow past a cylinder (η = 0.001, n. of vertices: 38910, ∆t = 0.0001)
Figure: Modulus of the velocity, streamlines at t = tn + 4, t = tn + 6,respectively
Recent Advances in PDEs: Analysis, Numerics and Control
Sevilla, January 25-27, 2017
Numerical solution of fluid
mechanics problems with pure
Lagrange-finite element methods
Marta Benıtez(1), Alfredo Bermudez(2) and Pedro Fontan(3)
(1) Department of Mathematics. Universidade da Coruna and ITMATI. Spain
(2) Department of Applied Mathematics. Universidade de Santiago deCompostela and ITMATI. Spain
(3) Instituto Tecnologico de Matematica Industrial (ITMATI). Spain