Cartesian coordinates Discretization

It is always possible to convert internal to Cartesian coordinates and vice versa. However, one coordinate system is usually preferred for a given application. Internal coordinates can usefully describe the relationship between the atoms in a single molecule, but Cartesian coordinates may be more appropriate when describing a collection of discrete molecules. Internal coordinates are commonly used as input to quantum mechanics programs, whereas calculations using molecular mechanics are usually done in Cartesian coordinates. The total number of coordinates that must be specified in the internal coordinate system is six fewer... 

Figure 7.1. Discretized flow and mass transport i+1, j, k+1 domain in Cartesian coordinates.

From a classical point of view the behavior of a system of discrete particles is uniquely determined by Newton s laws of motion and the laws of force acting between the particles. We can write for each particle in the system three second-order differential equations which determine the values of the three cartesian coordinates of the particle as functions of time. 

In Cartesian coordinates the Reynolds shear stress pv Vy represents a flux of rr-momentum in the direction of y. Prandtl assumed that this momentum was transported by discrete lumps of fluid, which moved in the y direction over a distance I without interaction conserving the momentum and then mixed with the existing fluid at the new location. The mixing length, /, is supposed to be a variable analogous to the mean free path of kinetic theory in this process. 

In order to study the principles of spatial discretization, we consider the steady state diffusion of a property in a one dimensional domain as sketched in Fig 12.3. In Cartesian coordinates the process is governed by ... 

The model derivation is outlined in Cartesian coordinates. The governing equations are then more conveniently written in vector notation (vector symbolism). For practical applications and simulations these vector equations are converted into cylindrical coordinates and finally reduced to the 2D axi-symmetric bubble column problem. The axi-symmetric model is discretized by use of the IPSA-SIMPLEC solution algorithm in sect C.4.1. 

The data are sampled on a discrete mesh in k space, which is defined on cylindrical or spherical coordinates. Discrete Fourier transformation requires Cartesian coordinates. 

The auxiliary conditions which must be satisfied by a solution of the amplitude equation in order that it be an acceptable wave function are given in Section 9c. These conditions must hold throughout configuration space, that is, for all values between — oo and + oo for each of the ZN Cartesian coordinates of the system. Just as for the one-dimensional case, it is found that acceptable solutions exist only for certain values of the energy parameter W. These values may form a discrete set, a continuous set, or both. 

Solving discrete phase particle track under particle effect balance equation of lagrangian coordinates. The form of particle effect balance equation under cartesian coordinate system (x direction) is (Morsi, S.A. Alexander, A.J. 1972) ... 

Beckermann and coworkers [42,43] discretized (10.29) using a control volume-based FDM, in which the transient term is treated by a fully implicit scheme. The resulting algebraic equation in cartesian coordinates can be expressed as ... 

Consider a discrete dislocation-loop which moves at a uniform velocity v. I et (y) be a moving Cartesian coordinate system which is connected to the dislocation-loop so that y.=x.-v.t. Now the disloc-... 

Although the Verlet scheme shown in equation (16) was derived from a discrete time representation of Hamilton s equations, it has the form of Newton s equation of motion, which is a second-order differential equation expressed, in terms of the Cartesian coordinates, as... 

All governing equations are all solved using a finite volume discretization, see [7]. All vectors quantities, e.g. position vector, velocity and moment of momentum, are expressed in Cartesian coordinates. Non-staggered variable arrangement is used to define dependent variables all physical quantities are stored and computed at cell centers. An interpolation practice of second order accuracy is adopted to calculate the physical quantities at cell-face center [8]. The deferred correction approach [9] is used to compute the convection term appearing in the governing equations by blending the upwind difference and the centi difference scheme. 

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

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