Body fitted coordinates

Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or cuives which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 

Shyy, W., Application of body-fitted coordinates in transport processes, in Advances in Transport Processes, Mujumdar, A. S. and Mashelkar, R. A., Eds., Vol. 9, Elsevier, Amsterdam, 1993. 

These equations were derived for flow over a plane surface. They may be applied to flow over a curved surface provided that the boundary layer thickness remains small compared to the radius of curvature of the surface. When applied to flow over a curved surface, x is measured along the surface and y is measured normal to it at all points as shown in Fig. 2.15, i.e., body-fitted coordinates are used. 

Thompson, J.F., Numerical Solution of Flow Problems Using Body-Fitted Coordinate Systems, Lecture Series 1978-4, von Karman Institute for Fluid Dynamics, Brussels, Belgium, Mar. 1978. 

The COPO fiicility was modelled two-dimensionally using the body fitted coordinate option... 

An important extension to rigid-body fitting is the so-called directed tweak technique [105]. Directed tweak allows for an RMS fit, simultaneously considering the molecular flexibility. By the use of local coordinates for the handling of rotatable bonds, it is possible to formulate analytical derivatives of the objective function. With a gradient-based local optimizer flexible RMS fits are obtained extremely fast. However, no torsional preferences may be introduced. Therefore, directed tweak may result in energetically unfavorable conformations. 

The body-fitted curvilinear coordinate system is obtained by solving the covariant Laplace equations for v and cr considered as functions of and rj ... 

J.F. Thompson, Z.U.A. Warsi, and C.W. Mastin (1974) Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies. J. Comput. Phys. 15, 299-319. J.F. Thompson, Z.U.A. Warsi, and C.W. Mastin (1985) Numerical Grid Generation. North-Holland, New York. 

In the present work, we must carry out transformations of the dipole moment functions analogous to those descrihed for triatomic molecules in Refs. [18,19]. Our approach to this problem is completely different from that made in Refs. [18,19]. We do not transform analytical expressions for the body-fixed dipole moment components (/Zy, fiy, fi ). Instead we obtain, at each calculated ab initio point, discrete values of the dipole moment components fi, fiy, fif) in the xyz axis system, and we fit parameterized, analytical functions of our chosen vibrational coordinates (see below) through these values. This approach has the disadvantage that we must carry out a separate fitting for each isotopomer of a molecule Different isotopomers with the same geometrical structure have different xyz axis systems (because the Eckart and Sayvetz conditions depend on the nuclear masses) and therefore different dipole moment components (/Z, fiy, fij. We resort to the approach of transforming the dipole moment at each ab initio point because the direct transformation of analytical expressions for the body-fixed dipole moment components (/Zy, fiyi, fi i) is not practicable for a four-atomic molecule. The fact that the four-atomic molecule has six vibrational coordinates causes a huge increase in the complexity of the transformations relative to that encountered for the triatomic molecules (with three vibrational coordinates) treated in Refs. [18,19]. 

The continuous region or body is subdivided into a finite number of subregions or elements (Fig. 15.5). The elements may be of variable size and shape, and they are so chosen because they closely fit the body. This is in sharp contrast to finite difference methods, which are characterized by a regular size mesh, describable by the coordinates that describe the boundaries of the body. 

The general numerical approach employs rigid body motions and least-squares fitting. Given two sets of points xt,i = 1,2,..., N and yui= 1,2,..., N (here xt and yt are vectors specifying atomic coordinates), find the best rigid motion of the points y h) such that the sum of the squares of the deviations E xt — Yt 2 is a minimum. 

Here, by is the multi-body parameter for bond-formation energy affected by local atomic arrangement, especially, by the presence of other neighboring atoms (atom k is the effective coordination number which is a function of the angle between Vy and and is fitted to stabilize the tetrahedral structure [20, 21]. The values of parameters are listed in Table 1. 

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