Global coordinate

In the figure operation (M) represents a one-to-one transformation between the local and global coordinate systems. This in general can be shown as... 

Isoparametric transformation functions between a global coordinate system and local coordinates are, in general, written as... 

Differentiation of locally defined shape functions appearing in Equation (2.34) is a trivial matter, in addition, in isoparametric elements members of the Jacobian matrix are given in terms of locally defined derivatives and known global coordinates of the nodes (Equation 2.27). Consequently, computation of the inverse of the Jacobian matrix shown in Equation (2.34) is usually straightforward. 

Although the elemental stiffness Equation (2.55) has a common form for all of the elements in the mesh, its utilization based on the shape functions defined in the global coordinate system is not convenient. Tliis is readily ascertained considering that shape functions defined in the global system have different coefficients in each element. For example... 

The unknowns in this equation are the local coordinates of the foot (i.e. and 7]). After insertion of the global coordinates of the foot found at step 6 in the left-hand side, and the global coordinates of the nodal points in a given element in the right-hand side of this equation, it is solved using the Newton-Raphson method. If the foot is actually inside the selected element then for a quadrilateral element its local coordinates must be between -1 and +1 (a suitable criteria should be used in other types of elements). If the search is not successful then another element is selected and the procedure is repeated. 

First of all, one needs to choose the local coordinate frame of a molecule and position it in space. Figure 2a shows the global coordinate frame xyz and the local frame x y z bound with the molecule. The origin of the local frame coincides with the first atom. Its three Cartesian coordinates are included in the whole set and are varied directly by integrators and minimizers, like any other independent variable. The angular orientation of the local frame is determined by a quaternion. The principles of application of quaternions in mechanics are beyond this book they are explained in detail in well-known standard texts... 

When we look in the local segment coordinate system, the symmetry of the equation seen in the global coordinate system is lost, and we will see azimuthal variations. We wish to express the equation for the segment surface in its local coordinate system... 

Sets of local coordinate systems describing certain local features of complicated objects are often advantageous when compared to a single, global coordinate system. Within a topological framework, the general theory of sets of local coordinate systems is called manifold theory. Often, the local coordinate systems are interrelated, and these relations can be expressed by continuous, and in the case of differentiable manifolds, by differentiable mappings, called homeomorphisms (see Equation (15)), and diffeomorphisms, respectively. 

Coordinates of molecules may be represented in a global or in an internal coordinate system. In a global coordinate system each atom is defined with a triplet of numbers. These might be the three distances x,, y,-, z, in a crystal coordinate system defined by the three vectors a, b, c and the three angles a, / , y or by a, b, c, a, P, y with dimensions of 1,1,1,90°, 90°, 90° in a cartesian, i. e. an orthonormalized coordinate system. Other common global coordinate systems are cylindrical coordinates (Fig. 3.1) with the coordinate triples r, 6, z and spherical coordinates (Fig. 3.2) with the triples p, 9, . 

The International Council of Chemical Associations—made up of the American Chemistry Council, the European Chemical Industry Council, and the Japan Chemical Industry Association—is the global coordinator of the Long-Range Research Initiative (LRI), a research program that funds research in the effects of chemicals on human health and the environment (LRI 2001). 

A system of N spherical particles in an electrolyte solution with permittivity em is considered. Particle radii are denoted as ak, and their permittivities are denoted as ek (k = 1, 2,TV). We link the local polar spherical coordinates (rk, 0k, (pk) with the particle centers (rk is a polar radius, 6k is an azimuth angle, q>k is a polar angle). The arrangement of two arbitrarily chosen particles from the ensemble is shown in Figure 1 with corresponding coordinates indicated. Global coordinates (x,y,z) of an observation point P(x,y,z) are determined by vectors rk, r. in the local coordinates, and a distance between centers of the spheres is Rkj (Figure 1). 

Distances and angles. Structures can be presented in an internal coordinate system (symmetry adapted coordinates used in spectroscopy or Z-matrices, that is interatomic distances, three center angles and four center angles) instead of a global coordinate system (coordinate triples, for example cartesian, crystal, cylindrical or spherical coordinates). 

The second symmetry requirement that the expression for the inter-molecular potential has to meet is that it must be invariant under any rotation of the global coordinate frame. The transformation properties of the symmetry-adapted functions Gj Hw) under such a rotation are easily obtained from Eqs. (10) and (5) ... 

Here we have partitioned the sums over all atoms a and /3 in the molecules P and P in the following manner. First, we sum over equivalent atoms within the same class a E a and (3 E b, which have the same chemical nature X = Xa and Xp = X and the same distance da = da and dp = db to the respective molecular center of mass. Next, we sum over classes a E P and b E P. The orientations da and dp of the position vectors of the atoms d and dp, relative to the molecular centers of mass, are still given with respect to the global coordinate frame. If we denote the polar angles of da and dp in the molecule fixed frames by d°a and dp and remember that the molecular frames are related to the global frame by rotations through the Euler angles o)P and to/., respectively, we find that... 

An IMPROVE subproject was responsible for the case study, which was modeled in WOMS. Thus, this subproject served as a global coordinator which integrated the contributions of individual subprojects into a coherent design process. 

Within an organization, AHEAD serves as the central instance for the planning of the overall process, e.g. it is used for its global coordination. The composition of process fragments into a coherent overall process is realized using djmamic task nets, so that the djmamic character of the design process is adequately supported by AHEAD. 

FIGURE 28.4 Global coordination (FTEs, full-time equivalents). 

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