Finite element interpolation

One example of non-IRC trajectory was reported for the photoisomerization of cA-stilbene.36,37 In this study trajectory calculations were started at stilbene in its first excited state. The initial stilbene structure was obtained at CIS/6-31G, and 2744 argon atoms were used as a model solvent with periodic boundary conditions. In order to save computational time, finite element interpolation method was used, in which all degrees of freedom were frozen except the central ethylenic torsional angle and the two adjacent phenyl torsional angles. The solvent was equilibrated around a fully rigid m-stilbene for 20 ps, and initial configurations were taken every 1 ps intervals from subsequent equilibration. The results of 800 trajectories revealed that, because of the excessive internal potential energy, the trajectories did not cross the barrier at the saddle point. Thus, the prerequisites for common concepts of reaction dynamics such TST or RRKM theory were not satisfied. 

Fig. 3.5. Illustration of the finite element interpolated approximate wave functions (a) ground state wave function for the M = 2 approximation (h) ground state and first excited state wave functions for the N = 3 approximation.

From a geometric perspective, the physics of degree of freedom elimination is based upon the kinematic slavery that emanates from the use of the finite element as the central numerical engine of the method. An alternative view is that of constraint. By virtue of the use of finite element interpolation, vast numbers of the atomic-level degrees of freedom are constrained. The main point is that some subset of the full atomic set of degrees of freedom is targeted as the representative set of atoms, as shown in fig. 12.11. These atoms form the nodes in a finite element mesh, and the positions of any of the remaining atoms are found by finite element interpolation via... 

All these considerations lead to the need of a new B-bar operator if the fundamental conservation laws of energy and momentum are to be preserved. The new operator needs to account not only for the discrete finite element interpolations in space, but also the discrete structure in time of the EDMC time-stepping algorithms, as presented in this paper. 

Ch. D. Berweger, W. F. van Gunsteren and F. Miiller-Plathe Finite element interpolation for combined classical/quantum mechanical molecular dynamics simulations, J. Comp. Chem. 18, 1484-1495 (1997). 

We next introduce finite element interpolations for vf and wf using the shape functions 4>r ... 

Lastly, computational efficiency needs to be discussed. However complete the formulation of the coarse-grained alternative to MD methodology is up to here, additional approximations are required to make it computationally efficient. To begin with, it is assumed that the thermally averaged positions of the constrained atoms can be expressed as a finite-element interpolation of the positions of the representative atoms, i.e., using finite-element shape functions. This is analogous to the procedure followed in the standard QC method to determine the instantaneous positions of the nonrepresentative atoms. Moreover, the computation of Vcg is noticibly expedited when both the local harmonic approximation and the Cauchy-Born rule are taken into account. Under such circumstances, Vcg becomes... 

The moving or deforming FEM method has been demonstrated successfully for modeling the solidification of pure materials by Lynch and O Neill [79],and Zabaras and Ruan [80, 81]. In this method, time dependent finite element interpolation functions are introduced and finite element nodes move with time to adapt to the motion of the solid/liquid interface. Because the interface is continuously traced. 

The main consequence of the discretization of a problem domain into finite elements is that within each element, unknown functions can be approximated using interpolation procedures. 

WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE 2.1.1 Interpolation models... 

Let be a well-defined finite element, i.e. its shape, size and the number and locations of its nodes are known. We seek to define the variations of a real valued continuous function, such as/, over this element in terms of appropriate geometrical functions. If it can be assumed that the values of /on the nodes of Oj, are known, then in any other point within this element we can find an approximate value for/using an interpolation method. For example, consider a one-dimensional two-node (linear) element of length I with its nodes located at points A(xa = 0) and B(a b = /) as is shown in Figure 2.2. 

Inherent in the development of approximations by the described interpolation models is to assign polynomial variations for function expansions over finite elements. Therefore the shape functions in a given finite element correspond to a... 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). 

The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

Hughes, T. J. R., Franca, L. P. and Balestra, M., 1986. A new finite-element formulation for computational fluid dynamics. 5. Circumventing the Babuska-Brezzi condition - a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolations. Cornput. Methods Appl. Meek Eng. 59, 85-99. 

The Galerkin finite element method results when the Galerkin method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-80) to provide the Galerkin finite element equations. For example, with the grid shown in Fig. 3-48, a linear interpolation would be used between points x, and, vI+1. 

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

---