Raviart

Crouzeix, M. and Raviart, P. A., 1973. Conforming and non-conforming finite elements for solving the stationary Navier-Stokes equations. RAIRO, Seric Rouge 3, 33 -76. 

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). 

Triangular Crouzeix-Raviart Quadratic Constant Vertices and Centre... 

Rectangular Crouzeix-Raviart Bi-quadratic Linear Corners, mid-sides Centre... 

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. 

The proposed model consists of a biphasic mechanical description of the tissue engineered construct. The resulting fluid velocity and displacement fields are used for evaluating solute transport. Solute concentrations determine biosynthetic behavior. A finite deformation biphasic displacement-velocity-pressure (u-v-p) formulation is implemented [12, 7], Compared to the more standard u-p element the mixed treatment of the Darcy problem enables an increased accuracy for the fluid velocity field which is of primary interest here. The system to be solved increases however considerably and for multidimensional flow the use of either stabilized methods or Raviart-Thomas type elements is required [15, 10]. To model solute transport the input features of a standard convection-diffusion element for compressible flows are employed [20], For flexibility (non-linear) solute uptake is included using Strang operator splitting, decoupling the transport equations [9],... 

A discontinuous Galerkin Q method based on the Lesaint-Raviart techniques was used in [16]. An interesting point in this formulation is the use of an element by element basis if a... 

Lesaint-Raviart (discontinuous Galeitin) quasi-Newton method... 

Upwinding methods as well as discontinuous Galerkin (Lesaint-Raviart) methods have l n introduced to account for transport terms in the diiferential constitutive equation [62]. 

In this paragraph, a Lesaint-Raviart method is presented. A Newton algorithm allowing fixed values of the viscoelastic extra-stress components outside the finite elements is used. A fixed-point algorithm on those exter extra-stress components is also involved. Tffis quasi-Newton method needs a storage requirement of the same size as that related to a classical decoupled method, but allows improved convergence [39]. 

Triangular Crouzeix-Raviart elements [63] were selected, which verify the so-called Babuska-Brezzi compatibility conditions (quadratic approximation for the velocity discontinuous linear approximation for the isotropic part of the stress tensor discontinuous quadratic approximation for the stresses) (Fig. 18). 

The Lesaint-Raviart (discontinuous Galerkin) method [62] is based on discontinuous approximations of the extra-stress components and is characterized by an element per element treatment of the constitutive equation. This leads to the introduction of a stress step in the Galerkin formulation, from the preceding... 

Finite volume or finite element methods will in the authors opinion survive since they are able to deal with arbitrary geometries. For more details on these methods see books by (Zinkiewicz 1997) or (Girault and Raviart 1986). Also common multi-purpose-codes like ANSYS or ANSYS/CFX are based on finite element and finite volume methods. 

Girault V, Raviart PA (1986) Finite Element Methods for Navier-Stokes-Equations. Springer-Verlag Berlin... 

Rabinovitch M, Daux JC, Raviart JL, Mevrel R, Carbon fiber reinforced magnesium and aluminium composite fabricated by liquid hot pressing, 4th European Conf on Composite Materials, Stuttgart, 405-410, Sep 25-28, 1990. 

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