Three-Dimensional Problems

When modeling a system, we try to reduce the problem to a two-, and if possible, to a one-dimensional model. However, often it is not possible to reduce the dimensionality of a problem, forcing us to solve a full three-dimensional model. In principle, solving a problem in 3D using a finite element formulation work the same way as in 1D or 2D. However, set-up effort, and therefore engineering time, as well as computational costs go up drastically when a problem is solved using a full three dimensional model. Most developments as described for ID and 2D finite element formulations are also valid for 3D. In this section, we will present several finite elements and formulations. 

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Components of the governing equations of the process can be decoupled to develop a solution scheme for a three-dimensional problem by combining one- and two-dimensional analyses. 

The hydrogen atom is a three-dimensional problem in which the attractive force of the nucleus has spherical symmetr7. Therefore, it is advantageous to set up and solve the problem in spherical polar coordinates r, 0, and three parts, one a function of r only, one a function of 0 only, and one a function of [Pg.171]

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

We have been discussing a class of penetration problems that are accurately modeled by two-dimensional calculations. There are many three-dimensional problems, however, that can be well approximated by two-dimensional analyses, and the greatly reduced computer memory and time requirements for such calculations make them attractive alternatives for scoping studies, or for parameter sensitivity studies. Although good quantitative predictions may not be obtained with such approximations, the calculations can be expected to reveal trends and qualitative results that will carry over to the full three-dimensional problem. 

Computational methods have played an exceedingly important role in understanding the fundamental aspects of shock compression and in solving complex shock-wave problems. Major advances in the numerical algorithms used for solving dynamic problems, coupled with the tremendous increase in computational capabilities, have made many problems tractable that only a few years ago could not have been solved. It is now possible to perform two-dimensional molecular dynamics simulations with a high degree of accuracy, and three-dimensional problems can also be solved with moderate accuracy. 

Forces are vector quantities and the potential energy t/ is a scalar quantity. For a three-dimensional problem, the link between the force F and the potential U can be found exactly as above. We have... 

Thus, the three-dimensional problem has been reduced to three one-dimensional problems. 

Van Regenmortel, M. H. V. andPellequer.J. L. (1994), Predicting antigenic determinants in proteins looking for unidimensional solutions to a three-dimensional problem Peptide Res., 7, 224 - 228. 

In the finite element method, Petrov-Galerkin methods are used to minimize the unphysical oscillations. The Petrov-Galerkin method essentially adds a small amount of diffusion in the flow direction to smooth the unphysical oscillations. The amount of diffusion is usually proportional to Ax so that it becomes negligible as the mesh size is reduced. The value of the Petrov-Galerkin method lies in being able to obtain a smooth solution when the mesh size is large, so that the computation is feasible. This is not so crucial in one-dimensional problems, but it is essential in two- and three-dimensional problems and purely hyperbolic problems. 

By separation of variables, we immediately obtain the eigenvalues and eigenfunctions of the three-dimensional problem (A.49). 

The indices a, a go from 1 to n + 1, where n is the number of spatial degrees of freedom.3 Thus, for one-dimensional quantum-mechanical problems, n = 2, for two-dimensional problems, n = 3, and for three-dimensional problems, n-4. 

The most general Hamiltonian with dynamic symmetry (II) again has a form similar to Eq. (2.43), with both linear and quadratic terms. This is a peculiar feature of one-dimensional problems. In order to simplify the discussion of three-dimensional problems, we prefer to consider a Hamiltonian with only quadratic terms... 

Dynamical symmetries for three-dimensional problems can be studied by the usual method of considering all the possible subalgebras of U(4). In the present case, since one wants states to have good angular momentum quantum numbers, one must always include the rotation algebra, 0(3), as a subalgebra. One can show then that there are only two possibilities, corresponding to the chains... 

As mentioned before in connection with one-dimensional problems, the states (2.101) or (2.96) provide bases in which all algebraic calculations can be done. These bases are orthogonal bases for three-dimensional problems. They can be converted one into the other by unitary transformations that have been (Frank and Lemus, 1986) written down explicitly. 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

Dunham-like expansion for three dimensional problems... 

The algebraic description of three-dimensional problems is in terms of U(4). The coset space for this case is a complex three-dimensional vector, We denote its complex conjugate by q. One can then introduce the canonical position and momentum variables q and p by the transformation... 

The situation for three-dimensional problems is similar to that discussed in Section 7.15. One begins by constructing the ground state. The coordinate is now a vector, a, and the intrinsic state is... 

For three-dimensional problems, there are in total four boson operators. There must be three more boson operators in addition to Eq. (7.136). They are given by (Leviatan and Kirson, 1988)... 

We limit ourselves to a discussion of coupled three-dimensional problems, since the one-dimensional problems can be obtained from the three-dimensional ones by deleting some terms. The study of coupled systems is done precisely in the same way as that of single systems. We consider in particular the case of linear triatomic molecules described by the algebra... 

The numerical solutions necessary to solve the practical three-dimensional problems agree well with the closed-form analytical solutions for simpler one- and two-dimensional cases with constant material properties. The resin pressure gradient in the thickness (vertical) direction for a well-dammed laminate (no horizontal flow) is nonlinear. 

In the three-dimensional problem, it will be noticed from (71) that in X oJ2 the density of states and the diffusion coefficient occur in the denominator, as they do also in the expression given by Kawabata (1981). If the disorder broadens the band, as will occur in the Anderson model if V0 > B9 then (75) should be modified to... 

Unlike the two-dimensional z-r and r-6 planes, where there are many practical problems are posed and solved, it is hard to think of a two-dimensional problem that is posed on a 6-z surface. Nevertheless, the strain components are certainly required in three-dimensional problems. The derivation follows the same procedure as we have just followed. However, we will not work through the derivations here but simply state the important results. 

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