Penalty schemes

The penalty method is based on the expression of pressure in terms of the incompressibility condition (i.e. the continuity equation) as 

9 Elimination of the pressure term from the equation of motion does not automatically yield a robust scheme for incompressible flow and it is still necessary to satisfy the BB stability condition by a suitable technique in both forms of the penalty method. 

76 FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES The continuous penalty technique 

In the continuous penalty technique prior to the discretization of the governing equations, the pressure in the equation of motion is substituted from Fquation (3.6) to obtain 

Equation (3.8) is the basic working equation of the continuous penalty method. 

---

The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

The described application of Green s theorem which results in the derivation of the weak statements is an essential step in the formulation of robu.st U-V-P and penalty schemes for non-Newtonian flow problems. 

Working equations of the continuous penalty scheme in Cartesian coordinate systems... 

After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and algebraic manipulations the working equations of the continuous penalty scheme are obtained as... 

As it can be seen the working equations of the penalty scheme are more compact than their counterparts obtained for the U-V-P method. 

In some applications it may be necessary to prescribe a pressure datum at a node at the domain boundary. Although pressure has been eliminated from the working equations in the penalty scheme it can be reintroduced through the penalty terms appearing in the boundary line integrals. 

In conjunction with the discrete penalty schemes elements belonging to the Crouzeix-Raviart group arc usually used. As explained in Chapter 2, these elements generate discontinuous pressure variation across the inter-element boundaries in a mesh and, hence, the required matrix inversion in the working equations of this seheme can be carried out at the elemental level with minimum computational cost. 

MODELLING OF STEADY-STATE VISCOMETRIC FLOW -WORKING EQUATIONS OF THE CONTINUOUS PENALTY SCHEME IN CARTESIAN COORDINATE SYSTEMS... 

Temperature variations are found by the solution of the energy equation. I he finite element scheme used in this example is based on the implicit 0 time-stepping/continuous penalty scheme described in detail in Chapter 4, Section 5. 

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

In the continuous penalty scheme used here the penalty parameter is defined as... 

CONTINUOUS PENALTY SCHEME IN CONJUNCTION VJITH SELECTIVELY REDUCED... 

---