Equivalence constraint

It is illustrative to compare Figure 5-58 with the equivalent result of the ALS analysis in Figure 5-55. The most striking difference is the number of iterations required to reach the outcome. RFA arrives at the optimal resolution, within the constraints, in 10 iterations. ALS, using equivalent constraints results in acceptable matrices C and A but even after 100 iterations the optimum clearly has not been reached. 

Energy expressions given by Eqs. (44)-(48) are possible because of the assumed shell structure. This equivalence constraint may be expressed as... 

Geometry SCF optimized with no symmetry or atom equivalence constraints. bIn the 6-31G basis optimized geometry. 

Geometry SCF optimized with no symmetry or atom equivalence constraints in the 6-31 + G basis set. bElectron affinity (in eV) for radical (Table 4)- anion (this table) using the MP2/6-31 + G energies. c Relative to the molecular centre of mass. 

The Equivalent Constraint Model (ECM) was introduced by Fan and Zhang (1993) with the aim to analyse the in-situ constraint effects on damage evolution in a particular lamina within a multidirectional laminate. In this model, all the laminae below and above the chosen lamina are replaced with homogeneous layers having the equivalent constraining effect. It is assumed that the in-plane stiffness properties of the equivalent constraint layers can be calculated from the classical lamination theory, provided stresses and strains in them are known. 

The purpose of the analysis of an ECM laminate is to determine the reduced stiffness properties of the explicitly damaged layer. Since the reduced stiffness properties of the equivalent constraint layer(s) used in the analysis of an ECM laminate, are assumed to be determined from the analysis of the other ECM laminate, problems for ECMl and ECM2 laminates are coupled. The advantage of the ECM-based approach is that it avoids cumbersome consideration of a repeated laminate element defined by the intersecting pairs of macrocracks used by other researchers (Hashin, 1987 Tsai and Daniel, 1992 Henaff-Gardin et al, 1996a,b). 

Figure 2. Equivalent Constraint Model of a cross ply laminate with multilayer damage.

Here is the in-plane stiffness matrix of layer of the ECM/r laminate, and is the modified in-plane stiffness matrix of the equivalent constraint layer (k layer). Crack surfaces are assumed to be stress-free, so that... 

In this paper, the Equivalent Constraint Model (ECM) of the damaged lamina was applied to predict theoretically stiffness-damage behaviour of cross-ply [0m/90 ] ... 

Kashtalyan, M., and Soutis, C., 1999, Application of the Equivalent Constraint Model to investigate stiffness properties of transversally cracked and split FRP laminates. Advanced Composites Letters 8 205. 

The selection of an appropriate liquid phase for an analysis depends on several factors but principally on the compounds to be separated. Phases which may be suitable for the separation of a small selected group of organic acids may not be suitable when other acids are present. Clearly, non-polar phases separate compounds with respect to boiling point or molecular weight and will only permit elution of compounds which are volatile within the thermal stability range of the phase. Equivalent constraints apply to polar phases. Prior to the introduction of the modified silicone oil liquid phases, the phases most commonly in use were, for non-polar conditions, silicone grease or Apiezone L and for polar columns the polyglycols, polyesters or polyethers. 

The gaseous tracer method yields the equivalent piston flow linear velocity of the gas flow in the pipe without any constraints regarding flow regime under the conditions prevailing for flare gas flow. 

The characteristic of a relational database model is the organization of data in different tables that have relationships with each other. A table is a two-dimensional consti uction of rows and columns. All the entries in one column have an equivalent meaning (c.g., name, molecular weight, etc. and represent a particular attribute of the objects (records) of the table (file) (Figure 5-9). The sequence of rows and columns in the tabic is irrelevant. Different tables (e.g., different objects with different attributes) in the same database can be related through at least one common attribute. Thus, it is possible to relate objects within tables indirectly by using a key. The range of values of an attribute is called the domain, which is defined by constraints. Schemas define and store the metadata of the database and the tables. 

The resulting overall energy balance for the plant at nominal load conditions is shown in Table 3. The primary combustor operates at 760 kPa (7.5 atm) pressure the equivalence ratio is 0.9 the heat loss is about 3.5%. The channel operates in the subsonic mode, in a peak magnetic field of 6 T. AH critical electrical and gas dynamic operating parameters of the channel are within prescribed constraints the magnetic field and electrical loading are tailored to limit the maximum axial electrical field to 2 kV/m, the transverse current density to 0.9 A/cm , and the Hall parameter to 4. The diffuser pressure recovery factor is 0.6. 

Again we point out the rather remarkable fact that the coupled Eqs. (4.17) are exact and are equivalent to the full HEM if no otho- constraints, other than those of Eq. (4.18), are placed on v, 3 , nor S. These coupled equations describe... 

The ionization of anilinium ion in water at 25° was adopted as the basis set for this type. This choice was dictated by the fact that extensive reliable data for both m- and p-substituents are available which meet the minimal substituent requirement in both series. Two methods were found which give essentially the same set of parameters. First, the or(ba) parameters for the substituents CH3, F, Cl, and Br were assumed to apply to the data for m- and p-substituted anilinium ions. This constraint was sufficient to give a set essentially equivalent to the values given in Table V. Second, the data include results for the... 

In addition to the standard constraints introduced previously, structural constraints obtainable from the effects of the paramagnetic center(s) on the NMR properties of the nuclei of the protein can be used (24, 103). In iron-sulfur proteins, both nuclear relaxation rates and hyperfine shifts can be employed for this purpose. The paramagnetic enhancement of nuclear relaxation rates [Eqs. (1) and (2)] depends on the sixth power of the nucleus-metal distance (note that this is analogous to the case of NOEs, where there is a dependence on the sixth power of the nucleus-nucleus distance). It is thus possible to estimate such distances from nuclear relaxation rate measurements, which can be converted into upper (and lower) distance limits. When there is more than one metal ion, the individual contributions of all metal ions must be summed up (101, 104-108). If all the metal ions are equivalent (as in reduced HiPIPs), the global paramagnetic contribution to the 7th nuclear relaxation rate is given by... 

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