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Shape function representation

Following the discretization of the solution domain Q (i.e. line AB) into two-node Lagrange elements, and representation of T as T = Ni(x)Ti) in terms of shape functions A, (.v), i = 1,2 within the space of a finite element Q, the elemental Galerkin-weighted residual statement of the differential equation is written as... [Pg.55]

Equation (2) is identified as a second-order, nonlinear differential equation once, the curvature is expressed in terms of a shape function of the melt/crystal interface. The mean curvature for the Monge representation y = h(x,t) is... [Pg.303]

But there was no significant further development of the ideas set forth by Parr and Bartolotti until 1994, when one of the present authors (A.C.) realized that the shape function provided the key to resolve a mathematical difficulty that is inherent in the different Legendre-transform representations of chemical DFT [20]. [Pg.272]

Equation 19.17 is not the original way the N, p(r) ambiguity was resolved [20]. As mentioned previously, the original paper on the shape function in the isomorphic representation performed the Legendre transform on the energy per particle. This gives an intensive, per electron, state function [20] ... [Pg.273]

In this chapter, we have derived the two-dimensional finite element penalty formulation for creeping flows where the pressure was eliminated by assuming a compressible flow. Here, we will use a mixed formulation, where the pressure is included among the unknown variables. In the mixed formulation, we use different order of approximation for the pressure as we will for the velocity. For instance, if tetrahedral elements are used, we can use a quadratic representation for the velocity (10 nodes) and a linear representation for the pressure (4 nodes). Hence, we must use different shape functions for the velocity and pressure. For such a formulation we can write... [Pg.491]

A functional representation of the cloud chemistry simulated chamber is shown in Figure 1. The chamber is constructed of a cylindrical aluminum inner shell 1.8 m in diameter, 2.5 m in height with dome-shaped ends that provide a total volume of 6.6 m. The... [Pg.184]

It has been shown how shape-related molecular description can be extracted starting from the GRID positive MIF. Similarly, the nodes of attractive interaction between the probe and the molecule (negative MIF) can be handled. The red isocontour surfaces in Fig. 5.4 represent the interaction of methotrexate with the water probe. In this case the application of the procedure described above will highlight the relative position on the molecular surface of the chemical moiety interacting with the water probe. The representation in Fig. 5.4(b) is not a strict shape function but more a new pharmacophoric representation of the molecule. [Pg.107]

We imagine discretizing the region between 0 and a with A - - 1 nodes as illustrated in fig. 3.4. The representation of the wave function is now restricted entirely to the values of the wave function at the nodes, i/f,, with the value between nodes determined by interpolation via V (x) = rfciNiix). The particular choices of shape functions used in the present analysis are shown in fig. 3.4 as well. Because of fhe fad thaf the shape functions are highly localized, the matrices Kij and Mij introduced above will be nonzero only for the cases in which j = i or... [Pg.96]

D desCTiptors for spectrum-like representation of a molecular structure, defined by 3-D coordinates of its atoms, are obtained by a projection of all atomic centers of a molecule onto a sphere in the center of which is the molecule [97,98], The projection ray from the central point of the sphere defines a pattern of points on the sphere, and each point represents a particular atom. Then, the center of each atom is represented by a beU-shaped function, the intensity of which is determined by the distance between the coordinate origin and a particular atom. The Mulliken charge of the particular atom is an additional parameter incorporated into this function. As the beU-shaped function of atoms has been taken Lorentzian curve with the form ... [Pg.46]

It is known that the dipolar moment d(t) of a molecule osdllating with time is a source d electro-magnetic radiation. If this oscillation is of a harmonic type, so that d(t) = d exp( — io t), the dipolar moment al orl and radiates monochromatic light with a frequency Oq. This means that the optical line is described by a Lorentzian with a vanishing half-width i.e. by a 5-function. By using the integral representation for 5-fimction, we can write for the optical line shape function the following expression ... [Pg.126]

The representation in Eq. 12 is called the stochastic harmonic function representation (SHE) and is of great flexibility by choosing different PDEs for the randomized frequencies (Chen et al. 2013). Particularly, if the PDE of ojj takes the shape of PSD, i.e.. [Pg.1991]

The mathematical formulation for the representation of the near-fault ground velocity pulses proposed by Mavroeidis and Papageorgiou (2003) is the product of a harmonic oscillation with a bell-shaped function. That is ... [Pg.2526]

First, we fit the experimental G-E data with a simple step function or stair-step curve, as shown in Fig. 8.B.3, for example. This step function representation of the s-shaped G-E curve has the properties ... [Pg.180]

Fig. 25. Schematic representation of imprinting (a) cross-linking polymerization ia the presence of a template (T) to obtain cavities of specific shape and a defined spatial arrangement of functional groups (binding sites. A—C) (b) cross-linked polymer prepared from the template monomer and ethylene... Fig. 25. Schematic representation of imprinting (a) cross-linking polymerization ia the presence of a template (T) to obtain cavities of specific shape and a defined spatial arrangement of functional groups (binding sites. A—C) (b) cross-linked polymer prepared from the template monomer and ethylene...
The subunits which can be defined can include atoms or collections of atoms corresponding to functional groups. The subunits can be represented as regions of space defined by electon density. These representations correspond well with the qualitative concepts that arise from valence bond stmctures. The mathematical evaluation can assign shape and charge density to atoms. Table 1.17 gives the C and H charge densities in some... [Pg.58]

FIGURE 1119 The lysozyme molecule is a typical enzyme molecule. Lysozyme is present in a number of places in the body, including tears and the mucus in the nose. One of its functions is to attack the cell walls of bacteria and destroy them. This "ribbon" representation shows only the general arrangement of the atoms to emphasize the overall shape of the molecule the ribbon actually consists of amino acids linked together (Section 19.13). [Pg.688]

Figure 16. Families of steady-state shapes for System III os a function of incre2is-ing growth rate P, as computed in a Xe/2 sample size, including the transition to deep cells. The amplitude of the cellular shapes is denoted by A, as defined by (10). Continuation of (A/4)-family computed with the mixed cylindri-cal/cartesian representation is shown 2ls a dotted (...) curve. Figure 16. Families of steady-state shapes for System III os a function of incre2is-ing growth rate P, as computed in a Xe/2 sample size, including the transition to deep cells. The amplitude of the cellular shapes is denoted by A, as defined by (10). Continuation of (A/4)-family computed with the mixed cylindri-cal/cartesian representation is shown 2ls a dotted (...) curve.
Because of the previously mentioned inadequacy of the function a —l/a, a different value for the parameter %i is required for the set of points (Fig. 135) at each elongation a. These values are —0.90, — 0.73, and —0.56 for a = 1.4, 2.0, and 3.0, respectively. If the function a — l/a were replaced by an empirical representation of the shape of the stress-strain curve, a single value of xi would suffice to represent all of the data within experimental error. This limitation of Eq. (41) relates to an unexplained feature of the stress-strain curve and is... [Pg.581]

Since these formal bases, which are supposed to describe the true continuum background, will be represented upon finite sets, all the qnantities which must be interpolated from these representations (i.e. matrix elements and phaseshifts) must be smooth functions of the energy index this reqnires a snitable redefinition of the channel hamiltonian Hp if this supports narrow shape resonances. [Pg.369]

Figure 1. (a) Schematic representation of the evolution by radioactive decay of the daughter-parent (N2/N1) activity ratio as a function of time t after an initial fractionation at time 0. The initial (N2/Ni)o activity ratio is arbitrarily set at 2. Time t is reported as t/T2, where T2 is the half-life of the daughter nuclide. Radioactive equilibrium is nearly reached after about 5 T2. (b) Evolution of (N2/N1) activity ratios for various parent-daughter pairs as a function of time since fractionation (after Williams 1987). Note that the different shape of the curves in (a) and (b) is a consequence of the logarithmic scale on the x axis in (b). [Pg.127]


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See also in sourсe #XX -- [ Pg.272 ]




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