Computational Methods for Debonding in Composites

298 H. Miled et al.
Each member of Eq. 15.10 is multiplied by a test function a ∗ chosen in V and
integrated on the domain Ω which leads to the weak formulation. The hyperbolic
character of the orientation equation provides well known oscillating solutions when
the Standard Galerkin method is used. Stabilization techniques have been developed
to overcome this problem. Residual Free bubble (RFB) and Streamline Upwind
Petrov-Galerkin (SUPG) methods have been implemented and are used in this paper.
The Streamline Upwind Petrov-Galerkin method (SUPG) [8] consists in replacing
the test function a ∗ by another function ã ∗ defined as:
ã ∗ = a ∗ + ζv.∇a ∗
(15.12)
where ζ is a perturbation coefficient. For the consistency of the solution, ζ is taken
equal to hm/�2v� where hm is the mean mesh size. Since the main cause of oscillations
is a dominant convection term, the perturbation of the original a∗ function
decreases the effect of convection by adding a diffusive term. This scheme stabilizes
the solution and is also consistent.
The Residual Free Bubble method (RFB) is inspired by a class of multi-scale
methods using functional spaces enriched by bubble functions [7]. Its principle consists
in finding the solution of the weak problem on the functional space V RFB ,such
that:
V RFB = V ⊕VB
(15.13)
where VB is the bubble space. Each element a ∈ V
h RFB is then the sum of a linear
quantity a ∈ V and a bubble function a ∈ VB.
L B
a = a + a (15.14)
h L B
Brezzi et al. [7] showed that for convection equations, SUPG and RFB methods
are equivalents if the bubble function is condensed. These two methods were
compared to the Galerkin method for the flow of fiber reinforced Newtonian fluid
between parallel plates (Fig. 15.3).
η denotes the viscosity of the polymer which is supposed constant and equal
to 1,000 Pa.s. Profile of local velocity of the polymer is considered parabolic; we
suppose that orientation is initially isotropic and that the orientation tensor is maintained
isotropic at the entry. Results obtained by the Galerkin Standard method are
compared to the ones obtained using stabilization methods for the computation of
the first orientation tensor component a11 (Fig. 15.4).
Fig. 15.3 Two-dimensional Poiseuille flow