### Tech Briefs

CFD with the FCBI Elements

The flow-condition-based interpolation — FCBI — elements for CFD solutions in ADINA are now very widely used (for some applications see ADINA News of Oct. 30, 2006 and May 26, 2005) and it is appropriate to summarize once the important properties of these elements.

The approach used in the element formulations was first published in Ref. [1], see also Refs. [2-4]. The approach is using features of formulation that are employed in finite element and control volume methods, in order to obtain discretization schemes that are practical and more effective. The name FCBI elements is based on using judicious element interpolation functions that depend on the flow within the element.

The major attributes of the FCBI elements are

• The elements are stable in low Reynolds number incompressible flows; excellent flow and pressure predictions are obtained.

• The elements are stable in high Reynolds and Peclet number flows.

• The elements pass the patch test when using distorted element meshes (the patch test is described in Ref. [5], p. 263).

• The elements satisfy local momentum and mass conservation (traditional finite element methods do not satisfy this condition).

• The element formulations do not contain any factors to be selected for stability, or otherwise.

• The elements are available in ADINA for fully incompressible flows, slightly compressible flows and low-speed compressible flows, including Darcy flows, and of course for all fluid structure interaction — FSI — problems involving such flows (see for example the ADINA News of Nov 15, 2008 and Apr 15, 2003).

In the figures below we show some simple test applications of the 4-node 2D and 8-node 3D elements.

First, the performance of the elements in a patch test of Darcy flow is shown in Figures 1, 2 and 3. The elements are quite distorted and, clearly, the patch test is passed.

Figure 1  Patch Test of Darcy Flow, 2D

Figure 2  Patch Test of Darcy Flow, 3D

Figure 3  Patch Test of Darcy Flow, 3D, 2 materials
(a very thin practically invisible element layer is used to connect element groups 1 and 2 )

Then the performance of the elements in the well-known driven cavity flow problem is shown in Figure 4 below. This is merely a numerical experiment to show that even though a rather coarse mesh is used, stable solutions are obtained even at high Reynolds numbers. In practice, of course, for higher Reynolds number flows, finer meshes and a turbulence model would be employed.

Figure 4  2D Driven Cavity Flow Problem, 12x12 mesh

It is clear that the FCBI elements in ADINA are very valuable for many analyses. However, the FCBI approach is quite general and allows of course for various additional element developments, see for example Ref. [6].

Keywords:
FCBI, flow-condition-based interpolation, CFD, Darcy flow, driven cavity flow, patch test

References

1. K.J. Bathe and H. Zhang, "A Flow-Condition-Based Interpolation Finite Element Procedure for Incompressible Fluid Flows", Computers & Structures, 80, 1267-1277, 2002.

2. K.J. Bathe and J. P. Pontaza, "A Flow-Condition-Based Interpolation Mixed Finite Element Procedure for High Reynolds Number Fluid Flows", Mathematical Models and Methods in Applied Sciences, 12, no. 4, 525-539, 2002.

3. H. Kohno and K.J. Bathe, "Insight into the Flow-Condition-Based Interpolation Finite Element Approach: Solution of Steady-State Advection-Diffusion Problems", Int. J. for Numerical Methods in Eng., 63, 197-217, 2005.

4. H. Kohno and K.J. Bathe, A Flow-Condition-Based Interpolation Finite Element Procedure for Triangular Grids, Int. J. Num. Meth. in Fluids, 49, 849-875, 2005.

5. B. Banijamali and K.J. Bathe, "The CIP Method Embedded in Finite Element Discretizations of Incompressible Flows", Int. J. for Numerical Methods in Eng., 71, 66-80, 2007.