Therefore an obvious procedure for the generation of the shape functions of the element shown in Figure 2.8 is to obtain the products of linear interpolation functions in the x and y directions. The four-noded rectangular element constructed in this way is called a bi-linear element. Higher order members of this family are also readily generated using the tensor products of higher order one-dimensional interpolation functions. For example, the second member of this group is the nine-noded bi-quadratic rectangular element, shown in Figure 2.9, whose shape functions are formulated as the products of quadratic Lagrange polynomials in the x and y directions.  [c.26]

Figure 2.9 Bi-quadratic rectangular element Figure 2.9 Bi-quadratic rectangular element
The standard technique for improving the accuracy of finite element approximations is to refine the computational grid in order to use a denser mesh consisting of smaller size elements. This also provides a practical method for testing the convergence in the solution of non-linear problems through the comparison of the results obtained on successively refined meshes. In the h-version of the finite element method the element selected for the domain discretization remains unchanged while the number and size of the elements vary with each level of mesh refinement. Alternatively, the accuracy of the finite element discretizations can be enhanced using higher-order elements whilst the basic mesh design is kept constant. For example, after obtaining a solution for a problem on a mesh consisting of bi-Iinear elements another solution is generated via bi-quadratic elements while keeping the number, size and shape of the elements in the mesh unchanged. In this case the number of the nodes and, consequently, the node-to-element ratio in the mesh will increase and a better accuracy will be obtained. This approach is commonly called the p-version of the finite element method.  [c.40]

Rectangular Taylor-Hood Bi-quadratic Bi-linear Comers, mid-sides and centre Corners  [c.73]

Figure 5.2 shows the finite element mesh corresponding to the configuration shown in Figure 5.1. This mesh consists of 225 nine-node bi-quadratic elements and its utihzation in the present model is based on the application of isoparametric mapping, described in Chapter 2. Figure 5.2 shows the finite element mesh corresponding to the configuration shown in Figure 5.1. This mesh consists of 225 nine-node bi-quadratic elements and its utihzation in the present model is based on the application of isoparametric mapping, described in Chapter 2.
Figure 5.2 The mesh used in this example consisting of 225 nine-node bi-quadratic elements Figure 5.2 The mesh used in this example consisting of 225 nine-node bi-quadratic elements
It has been long established that during the flow of long-chain polymers stresses within the fluid may rise significantly (overshoot) at certain locations in the domain. This phenomenon is more clearly observed in contracting flows where the polymer moves from wider to narrower sections of the domain. Let us consider the flow of a polymeric fluid in a two-dimensional gap between the cross-sectional plane of a symmetric. screw and a cylindrical outer tube. Initially the entire domain is discretized into a mesh consisting of 512 bi-quadratic finite elements, as shown in Figure 5.11. The flow inside this domain is generated by rotation of the screw in the anticlockwise direction. Using the simulation results obtained on this mesh, the boundary conditions for a representative section of the domain confined between two flights are defined and evaluated. A more refined mesh consisting of 256 bi-quadratic elements for this section is constructed, Considering the symmetry of the domain, simulation of stress field in this section should provide an insight for the entire domain. Predicted normal  [c.156]

SHAPE. Gives the shape functions in terms of local coordinates for bi-linear or bi-quadratic quadrilateral elements.  [c.211]


More accurately, as the inverse problem process computes a quadratic error with every point of a local area around a flaw, we shall limit the sensor surface so that the quadratic error induced by the integration lets us separate two close flaws and remains negligible in comparison with other noises or errors. An inevitable noise is the electronic noise due to the coil resistance, that we can estimate from geometrical and physical properties of the sensor. Here are the main conclusions  [c.358]

We try to estimate the function H(u), noted H, by minimization of the quadratic residual error  [c.747]

More correctly, the regression problem involves means instead of averages in (1). Furthermore, when the criterion function is quadratic, the general (usually nonlinear) optimal solution is given by y = [p u ], i.e., the conditional mean of y given the observation u .  [c.888]

Gas mixtures are subject to the same degree of non-ideality as the one-component ( pure ) gases that were discussed in the previous section. In particular, the second virial coefficient for a gas mixture can be written as a quadratic average  [c.359]

There are many physical systems which are modelled by Hamiltonians, which can be transfonned tln-ough a canonical transfomiation to a quadratic fomi  [c.392]

Van der Waals (1890) extended his theory to mixtures of components A and B by introducing mole-fraction-dependent parameters a and b defined as quadratic averages  [c.622]

In figure A3.3.9 the early-time results of the interface fonnation are shown for = 0.48. The classical spinodal corresponds to 0.58. Interface motion can be simply monitored by defining the domain boundary as the location where i = 0. Surface tension smooths the domain boundaries as time increases. Large interconnected clusters begin to break apart into small circular droplets around t = 160. This is because the quadratic nonlinearity eventually outpaces the cubic one when off-criticality is large, as is the case here.  [c.743]

Research over the past decade has demonstrated that a multidimensional TST approach can also be used to calculate an even more accurate transmission coefficient than for systems that can be described by the fiill GLE with a non-quadratic PMF. This approach has allowed for variational TST improvements [21] of the Grote-Hynes theory in cases where the nonlinearity of the PMF is important and/or for systems which have general nonlinear couplmgs between the reaction coordinate and the bath force fluctuations. The Kramers turnover problem has also been successfiilly treated within the context of the GLE and the multidimensional TST picture [22]. A multidimensional TST approach has even been applied [H] to a realistic model of an Sj 2 reaction and may prove to be a promising way to elaborate the explicit microscopic origins of solvent friction. Wliile there has been great progress toward an understanding and quantification of the dynamical corrections to the TST rate constant in the condensed phase, there are several quite significant issues that remain largely open at the present time. For example, even if the GLE were a valid model for calculating the dynamical corrections, it remains unclear how an accurate and predictive microscopic theory can be developed for the friction kernel q(t) so that one does not have to resort to a molecular dynamics simulation [17] to calculate this quantity. Indeed, if one could compute the solvent friction along the reaction coordinate in such a maimer, one could instead just calculate the exact rate  [c.890]

It is not difficult to show that, for a constant potential, equation (A3.11.218) and equation (A3.11.219) can be solved to give the free particle wavepacket in equation (A3.11.7). More generally, one can solve equation (A3.11.218) and equation (A3.11.219) numerically for any potential, even potentials that are not quadratic, but the solution obtained will be exact only for potentials that are constant, linear or quadratic. The deviation between the exact and Gaussian wavepacket solutions for other potentials depends on how close they are to bemg locally quadratic, which means  [c.1002]

I - FH) r on a quadratic surface. Here I is the unit matrix and denotes the nth cycle. The ultimate convergence rate is governed by the magnitude of the largest eigenvalue of the matrix (I - FH). This will be  [c.2335]

WEIGHTED RESIDUAL FINIl E ELEMENT METHODS - AN OUTLINE Nine-node bi-quadratic element  [c.30]

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 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.  [c.83]

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.  [c.84]

A general procedure is developed for the computation of the total energies of molecules at their equilibrium geometries. Ab Initio molecular orbital theory is used to calculate electronic energies by a composite method, utilizing large basis sets (including diffuse-sp, double-d and f-polarization functions) and treating electron correlation by the Mqller Plesset perturbation theory and by quadratic configuration interaction. The theory is also used to compute zero-point vibrational energy corrections. Total atomization energies for a set of 31 molecules are found to agree with experimental thermochemical data to an accuracy of greater than 2kcal mol in most cases. Similar agreement is achieved for ionization energies, electron and proton affinities. Residual errors are assessed for the total energies of neutral atoms.  [c.322]

It has be shown that the minimization of (1) having B(/t) in the form (9) yields to the procedure of the by a weighted matching of the values in the voxels having the data p . In (9) 5(p ) is not a quadratic functional of p, so equation (1) becomes nonlinear. One of the problems on the way to its numerical solution is the appearance of negative values of p during the iteration procedure by taking the logarithm of these values. The next is, that being insensitive to the small bright points in the image, this functional blurs large defects, forcing their matching with the background. Its main advantage, the support of the total number of absorbed quanta in the whole image and providing the maximum possible level of the entropy, is very important for image restoration because it proceeds the most unbiased prior about the image of the object under testing. This makes, that the support by (9) is nonaltemative for any structure from one side, and a weak one from the other.  [c.115]

Simulations of the adaptive reconstruction have been performed for a single slice of a porosity in ferritic weld as shown in Fig. 2a [11]. The image matrix has the dimensions 230x120 pixels. The number of beams in each projection is M=131. The total number of projections K was chosen to be 50. For the projections the usual CT setup was used restricted to angels between 0° and 180° with the uniform step size of about 3.7°. The diagonal form of the quadratic criteria F(a,a) and f(a,a) were used for the reconstruction algorithms (5) and (6).  [c.124]

One of the remarkable features of time evolution of wavepackets is tire close coimection they exhibit with the motion of a classical particle. Specifically, Elirenfest s theorem indicates that for potentials up to quadratic, the average value of position and momentum of the quantum wavepacket as a function of time is exactly the same as that of a classical particle on the same potential that begins with the corresponding initial conditions in position and momentum. This classical-like behaviour is illustrated in figure Al.6.1 for a displaced Gaussian wavepacket in a hannonic potential. For the case shown, the initial width is the same as the ground-state width, a coherent state , and hence the Gaussian moves without spreadmg. By way of contrast, if the initial Gaussian has a different width parameter, the centre of the Gaussian still satisfies the classical equations of motion however, the width will spread and contract periodically in time, twice per period.  [c.227]

It should be noted that the friction kernel is not in general independent of the reaction coordinate motion [17], i.e. a nonlinear response, so the GLE may have a limited range of validity [18,19 and 20]. Furtliemrore, even if the equation is valid, the strengdi of the friction might be so great that the second and third temis on the right-hand side of (A3.8.9) could dominate the dynamics much more so than the force generated by the PMF. It should also be noted drat, even though the friction in (A3.8.9) may be adequately approximated to be dynamically mdependent of the value of tlie reaction coordinate, the equation is still in general nonlinear, depending on the nature of the PMF. For non-quadratic  [c.889]

The bromate-ferroin reaction has a quadratic autocatalytic sequence, but in this case the induction period is detennined primarily by the time required for the concentration of the hiliibitor bromide ion to fall to a critical low value tlirough the reactions  [c.1097]

Figure B3.2.11. Total energy versus lattice constant of gallium arsenide from a VMC calculation including 256 valence electrons [118] the curve is a quadratic fit. The error bars reflect the uncertainties of individual values. The experimental lattice constant is 10.68 au, the QMC result is 10.69 (+ 0.1) an (Figure by Professor W Schattke). Figure B3.2.11. Total energy versus lattice constant of gallium arsenide from a VMC calculation including 256 valence electrons [118] the curve is a quadratic fit. The error bars reflect the uncertainties of individual values. The experimental lattice constant is 10.68 au, the QMC result is 10.69 (+ 0.1) an (Figure by Professor W Schattke).
Both defects of the Newton method can be eliminated by replacing the exact inverse Hessian by a (fixed) positive definite approximation to it, F. This method is known as simple relaxation. In both geometry and wavefiinction optunization, it is usually possible to construct a fairly good approximate Hessian. For geometry optimization, this can be based on the molecular coimectivity and transferability of potential parameters, or on previous low-level calculations. For wavefiinction optimization, a guess based on orbital energy differences is often reasonably accurate. Far from the minimum, approximate Hessian methods using positive definite matrices are preferable to the Newton method, as they have the descent property, i.e., the energy decreases for sufficiently small steps. However, they lack the quadratic tenninal convergence rate of the Newton method. Instead, the residual error vector (the distance from the accurate minimum) is given by r  [c.2335]

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods.  [c.2336]

See pages that mention the term Bi-quadratic : [c.35]    [c.1190]    [c.3062]    [c.28]    [c.73]    [c.84]    [c.63]    [c.248]    [c.380]    [c.607]    [c.741]    [c.1002]    [c.1236]    [c.1973]    [c.1973]    [c.1973]    [c.2333]    [c.2334]    [c.2335]   
Practical aspects of finite element modelling of polymer processing (2002) -- [ c.0 ]