Taylor-Hood element

Figure 2.11 Nine node Taylor-Hood element...

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

To demonstrate the capability of the presented model, a free swelling experiment on hydrogel is simulated by the FE tool PANDAS. Therefore, a 3-dimensional block is discretized with hexagonal extended Taylor-Hood elements with a quadratic approximation of us and linear approximations of p and Cm- To initiate swelling, the concentration of the external solution is decreased from 0.15 mol/l to 0.125 mol/1 within 10 sec. 

The governing equations (9), (10) and (11) are spatially diseretized using a Galerkin finite-element approach in conjunction with P2-P1 tetrahedral Taylor-Hood elements, with ten nodes for velocity and temperature, and four nodes for pressure. A finite element solver for three-dimensional non-Newtonian fluid flow and advection-diffiision heat transfer has been developed. The unknown veloeity, pressure and temperature flelds can be expressed in terms of the shape fimctions and, and nodal... 

To use non-standard elements belonging to the Taylor-Hood or Crouzeix-Raviart groups that satisfy the BB condition. Examples of useful elements in this category are given in Table 3.1, for further explanations about the properties of these elements see Pittman (1989). 

Taylor, C. and Hood, P., 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73-100. 

C. Taylor and P. Hood, A Numerical Solution of the Navier-Stokes Equations Using the Finite Element Method, Computers and Fluids 1 73-100, 1973. 

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