Bi-linear

In this element the velocity and pressure fields are approximated using biquadratic and bi-linear shape functions, respectively, this corresponds to a total of 22 degrees of freedom consisting of 18 nodal velocity components (corner, mid-side and centre nodes) and four nodal pressures (corner nodes). 

Rectangular Taylor-Hood Bi-quadratic Bi-linear Comers, mid-sides and centre Corners... 

Depending on the type of elements used appropriate interpolation functions are used to obtain the elemental discretizations of the unknown variables. In the present derivation a mixed formulation consisting of nine-node bi-quadratic shape functions for velocity and the corresponding bi-linear interpolation for the pressure is adopted. To approximate stres.ses a 3 x 3 subdivision of the velocity-pressure element is considered and within these sub-elements the stresses are interpolated using bi-linear shape functions. This arrangement is shown in Edgure 3.1. 

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. 

The outlined scheme is shown to yield stable solutions for non-zero Weissenberg number flows in a number of benchmark problems (Swarbric and Nassehi, 1992b). However, the extension of this scheme to more complex problems may involve modifications such as increasing of elemental subdivisions for stress calculations from 3 x 3 to 9 x 9 and/or the discretization of the stress field by biquadratic rather than bi-linear sub-elements. It should also be noted that satisfaction of the BB condition in viscoelastic flow simulations that use mixed formulations is not as clear as the case of purely viscous regimes. 

SHAPE. Gives the shape functions in terms of local coordinates for bi-linear or bi-quadratic quadrilateral elements. 

Figure 7.5. Examples of one-dimensional bi-linear basis functions ga (x). 

Alternatively, the mean composition fields can be estimated only at grid-cell centers or grid nodes, and then these knot values can be interpolated to the particle locations (Wouters 1998 Subramaniam and Haworth 2000 Jenny et al. 2001). For example, using bi-linear basis functions ga x) for each grid node (denoted by a), the estimated mean composition at grid node xa is given by (Jenny et al. 2001)... 

The bi-linear basis functions have the following properties (see Fig. 7.5) ... 

In constant-density flows,48 the particle fields must be used to find estimates for the Reynolds stresses uu X and the composition means 0 X4. The latter can be found using any of the methods described earlier (e.g., LCME or LLME). Estimates for the Reynolds-stress tensor are needed at the grid nodes in the FV code. As in (7.46), bi-linear basis functions can be employed to estimate each component at grid node xa ... 

Has sigma term probably corrects a Has small bi-linear component (7.) calculated log P. includes DCMU also. ... 

The extension of the formulas to degenerate accepting modes which occur when a Jahn-Teller effect in the excited state is present is relatively easy. In this case the products of distribution functions can be rewritten by convolution into fundamental distributions which does not change the overall expression of Eq. (29) [38, 66]. Also, it is possible to consider an intermixing of modes in the excited state by virtue of the bi-linear term in Eq. (1) (Duschinsky effect [67]). Since it is difficult to decide from most of the spectra if this effect is really observed in the case of the present complex compounds, we will not consider it here and refer to the literature [68,69]. This is justified as long as we can explain the experimental spectra satisfactorily applying the parallel mode approximation leading to the line shape function of Eq. (29) as it has been described in the method above. 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

An attempt to solve the difficulties and inconsistencies arising from an approximated derivation of quantum-classical equations of motion was made some time ago [15] to restore the properties that are expected to hold within a consistent formulation of dynamics and statistical mechanics, and are instead missed by the existing approximate methods. We refer not only to the properties that the Lie brackets, which generate the dynamics, satisfy in a full quantum and full classical formulation, e.g., the bi-linearity and anti-symmetry properties, the Jacobi identity and the Leibniz rule12, but also to statistical mechanical properties, like the time translational invariance of equilibrium correlation functions [see eq.(8)]. 

The collision operators C are bi-linear in the distribution functions, and the summation indices run over all charged species / and neutral species m in the plasma, respectively. 

The bi-linear collision term resulting from neutral neutral interaction Cnm is simplified in present edge models to non-linear BGK-like model collision expressions [15]. 

A generalization to reactive and chemical processes is straight forward and indicated by the brackets in the previous two statements. The collision integrals remain bi-linear in the two distribution functions of the two particles entering the collision. We must introduce appropriate Kronecker-Delta s and allow for more than two post-collision particles. We obtain, e.g., for three post-collision particles (in processes such as e + H2 —> e I II I II, dissociation, or e + H —> e + IT1 + e, ionization) for the gain term in the equation for species j ... 

All systems examined show the same fundamental result found in previous works the critical stress intensity factor, Kc, bears a bi-linear relationship with the factor characterising fibre orientation, with different slopes over different ranges of the orientation factor, suggesting a transition between different fracture mechanisms at a critical angle. 

Note that each equation, (10.41) and (10.42), is bi-linear because of the product of two unknowns. Ad and E. However, if we specify one of the unknowns, the equations become linear. The iterative Born inversion is based on subsequently determining Ad from equation (10.42) for specified E, and then updating E from equation (10.41) for predetermined Ad, etc. Within the framework of this method the Green s tensors Ge and Gh, and the background field stay unchanged. 

These filters are usually employed to suppress strong and undesirable resonances, e.g. parent signals of isotopically diluted spin systems. In a typical experiment the undesirable magnetization is inverted and allowed to relax until the equilibrium state is reached. At this point the inverted magnetization is not observable and the basic experiment may be started. In the simplest case a jump and return inversion pulse or binomial pulses can provide the necessary selective inversion. A simple example of application of a T relaxation filter is the J/-BIRD (BI linear Rotation Decoupling) HMQC... 

Consider first the model described by Eqs (13.6), (13.7), (13.10), and (13.11) where the hannonic oscillator under study is coupled bi-linearly to the harmonic bath. The relevant correlation functions ( F(Z),.F(0) ) and (F(Z)F(0))c should be calculated with F = and its classical countei part, where Uj are coordin-... 

---