Standard boundary representation

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A B REP is an entity that has a scope. A B REP may have a material property associated with it. The B REP is a self-contained entity in the sense that no entity in the B REP may refer to an entity outside the scope of the B REP. All referenced entities must be within the B REP scope itself. The scope of a B REP contains both topological and geometrical entities. Geometry is represented by lists of the entities POINT, DIRECTION, EDGE CURVE and FACE SURFACE which are referenced by the topological entities defined subsequently. Topology is represented by lists of the entities, VERTEX, EDGE, LOOP, FACE and SHELL in that order so that no entity is referenced before it is defined. 

V Niomber of vertices E Number of edges F Number of faces L Number of loops S Number of shells G Genus of the solid object i.e. number of through holes 

This formula still allows the same object. 

The objectives for creating a topological structure for a given object on the basis of the Euler formula are, therefore, 

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ProcGen generates a scaled 3D model of the test specimen geometry, in the form of a faceted boundary representation. This model is made available for use by other software tasks in the system. The STEP file format (the ISO standard for product data exchange) was chosen to provide future compatibility with CAD models produced externally. In particular part 204 (faceted b-rep) of this standard is used. 

An alternative to the standard-form representation is the differential-algebraic equation (DAE) representation, which is stated in a general form as g(r, y, y). The lower portion of Fig. 7.3 illustrates how the heat equation is cast into the DAE form. The boundary conditions can now appear as algebraic constraints (i.e., they have no time derivatives). For a problems as simple as the heat equations, this residual representation of the boundary conditions is not necessary. However, recall that implicit boundary-condition specification is an important aspect of solving boundary-layer equations. 

The key feature to making a semiclassical approach practical is to avoid having to deal explicitly with the double-ended boundary conditions in Eq. (3.3) [16-20]. (The initial condition x(X, p( 0) = X is obviously easy to deal with.) To do this, one uses the standard coordinate space representation of Eq. (2.5),... 

This chapter has reviewed existing results in addressing the analysis and control of multiple-time-scale systems, modeled by singularly perturbed systems of ODEs. Several important concepts were introduced, amongst which the classification of perturbations to ODE systems into regular and singular, with the latter subdivided into standard and nonstandard forms. In each case, we discussed the derivation of reduced-order representations for the fast dynamics (in a newly defined stretched time scale, or boundary layer) and the corresponding equilibrium manifold, and for the slow dynamics. Illustrative examples were provided in each case. 

We begin with a powerful solution method that can be applied for general 3D flows whenever the boundaries of the domain can be expressed as a coordinate surface for some orthogonal coordinate system. In this case, we can use an invariant vector representation of the velocity and pressure fields to simultaneously represent (solve) the solutions for a complete class of related problems by using so-called vector harmonic functions, rather than solving one specific problem at a time, as is necessary when we are using standard eigenfunction expansion techniques. 

Example 6.26. Let us calculate the fundamental group of the projective plane RP using Van Kampen s theorem. Consider a standard representation of KP as a unit disk in the plane with the boundary self-identified by the antipodal map, i.e., x —x. Let A be the disk centered at the origin with radius 1/2, and let B be obtained from MP by removal of the interior of A. We have Au B = RP, and it is easy to equip RP with a CW structure such that A and B are CW subcomplexes (alternatively, one could thicken A and B a little bit so as to obtain a topologically identical open covering of RP ). 

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