Direct solution methods

The most important direct solution algorithms used in finite element computations are based on the Gaussian elimination method. 

To describe the basic concept of the Gaussian elimination method we consider the following system of simultaneous algebraic equations 

Let us suppose that we can convert the n x coefficient matrix in equation system (6.5) into an upper triangular form as 

The Gaussian elimination method provides a systematic approach for implementation of the described forward reduction and back substitution processes for large systems of algebraic equations. 

---

The direct-solution method of Akers and Wade [1] is among several which attempt to reduce the amount of trial-and-error solutions. This has been accomplished and has proven quite versatile in application. The adaptation outlined modifies the symbols and rearranges some terms for convenient use by the designer [3]. Dew point and bubble point compositions and the plate temperatures can be determined directly. Constant molal overflow is assumed, and relative volatility is held constant over sections of the column. 

Iterative solution methods are more effective for problems arising in solid mechanics and are not a common feature of the finite element modelling of polymer processes. However, under certain conditions they may provide better computer economy than direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing and hence are potentially more suitable for three-dimensional flow modelling. In this chapter we focus on the direct methods commonly used in flow simulation models. 

Three basic approaches have been used to solve the equations of motion. For relatively simple configurations, direct solution is possible. For complex configurations, numerical methods can be employed. For many practical situations, particularly three-dimensional or one-of-a-kind configurations, scale modeling is employed and the results are interpreted in terms of dimensionless groups. This section outlines the procedures employed and the limitations of these approaches (see Computer-aided engineering (CAE)). 

Electrochemical Impedance Spectroscopy (EIS) and AC Impedance Many direct-current test techniques assess the overall corrosion process occurring at a metal surface, but treat the metal/ solution interface as if it were a pure resistor. Problems of accuracy and reproducibility frequently encountered in the application of direct-current methods have led to increasing use of electrochemical impedance spectroscopy (EIS). 

Direct mechanical methods can be used to determine the swelling pressure of hydrogels, e.g., by means of devices in the form of a cylindrical chamber equipped with a piston in which the gel contacts the solution through a porous membrane. This technique allows measuring very low pressure (of the order of 0.1-10 kPa) and makes it possible to analyze the SAH with swelling up to 700 ml g-1 [102, 103]. Among others, the method of osmotic deswelling is to be mentioned [104]. 

Slower introduction of the dinitrogen tetroxide, by either dropwise addition of an ether solution or entrainment in a slow stream of nitrogen, gave similar results. The direct injection method was found to be easiest. The checkers distilled dinitrogen tetroxide at atmospheric pressure (b.p. 21°) prior to use. 

Owing to the constraints, no direct solution exists and we must use iterative methods to obtain the solution. It is possible to use bound constrained version of optimization algorithms such as conjugate gradients or limited memory variable metric methods (Schwartz and Polak, 1997 Thiebaut, 2002) but multiplicative methods have also been derived to enforce non-negativity and deserve particular mention because they are widely used RLA (Richardson, 1972 Lucy, 1974) for Poissonian noise and ISRA (Daube-Witherspoon and Muehllehner, 1986) for Gaussian noise. 

Numerical solution of elliptic grid equations necessitates, in view of their peculiarities, creating special economical algorithms, because direct economical methods are applicable only in some narrow classes of grid equations. We will elaborate on this later. 

The difference in rates of release of BCNU from wafers produced by the trituration or solution methods is also seen in vivo (11,14), as is shown in Fig. 6. Wafers of PCPP-SA 20 80 were prepared by either the solution or trituration methods, as described above, and were implanted into the brains of rabbits. The animals were sacrificed at various times after implantation and the brains were removed, fixed, and processed for quantitative autoradiography. To quantitate the percentage of the brain exposed to BCNU released from these wafers, the following calculation was performed. The percentage of the brain in which the radioactivity from the tritiated BCNU released from the wafers exceeded the background counts by at least two standard deviation units was plotted as a function of time following implantation in Fig. 6. A control set of rabbits had a solution of BCNU injected directly into the same location in the... 

The advantage of the direct dissolution method is its simplicity. Systems that form vesicles under this method are easily dissolved in aqueous solution, unlike many other systems that form irregular aggregates due to strong hydrophobic interactions. Also, since the aqueous environment is relatively mild, this method... 

On the other hand, in case of the direct (3-glucosidation using 4-equivalents of 4-hydroxyphenylethyl alcohol and cinnamyl alcohol congeners in 90% tert-butanol/HjO solution (method II), chemical yields of (3-D-glucopyranosides were not always satisfactory and should be improved (Table 4, entries 5-11). 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

This hybrid approach can significantly extend the domain of applicability of the AIMS method. The use of interpolation significantly reduces the computational effort associated with the dynamics over most of the timescale of interest, while regions where the PESs are difficult to interpolate are treated by direct solution of the electronic Schrodinger equation during the dynamics. The applicability and accuracy of the method was tested using a triatomic model collisional quenching of Li(p) by H2 [125], which is discussed in Section III.A below. 

Fig. 2-2. The recovery of atmospheric carbon dioxide calculated by the direct Euler method. The solid line is the analytical solution, and the lines with markers...

---