CODEN (USA): IJCRGG, ISSN: 0974-4290, ISSN(Online):2455-9555 Vol.10 No.10, pp 248-256, 2017

Abstract : In this paper, we present a few selected applications of Numerical methods to other parts of mathematics and to various other fields in general. Numerical methods are rapidly moving into the mainstream of mathematics mainly because of its applications in diverse fields which include chemistry, electrical engineering, operation research. The wide scope of these and other applications has been well documented. Keywords: Numerical Methods, Engineering Applications.

Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions. Whether the goal is integration or solution of complex differential equations, there are many tools available to reduce the solution of what can be sometimes quite difficult analytical math to simple algebra.

2. PDEs Contains Two Main Aspects

Both the mathematical analysis of the PDEs and the numerical analysis of methods rely heavily on the strong tools of functional analysis. Numerical approximation of PDEs is a cornerstone of the mathematical modeling since almost all modeled real world problems fail to have analytic solutions or they are not known in the scope of pure mathematics because of their complexity1. The history of numerical solution of PDEs is much younger than that of analytic methods, but the development of high speed computers nowadays makes the advent of numerical methods very fast and productive. On the other hand, the numerical approximation of PDEs often demands knowledge of several aspects of the problem, in order to understand and interpret the behaviour of expected solutions, or the algorithmic aspects concerned with the choice of the numerical method and the accuracy that can be achieved2. The aim of this study is to discuss some modeling problems and provide the knowledge of Finite Element techniques for the numerical approximation of the model equations. Especially the theory and application of finite element methods is a very nice combination of mathematical theory with aspects of functioning, analysing, and applications.

3. Multiple Scales in the Modelling of Real World Problems

Most of the phenomena in nature are concerned with the behaviour of a big number of individual objects which are always in a close interaction with each other. On the other hand, the important features are visible on a much coarser macro scale, where a mean behaviour of objects is observable. Let us consider a very short list of fields where such behaviour arises in real life problems and where the mathematical investigation is needed to answer the questions stated by the problem.

It deals with the interaction of air (oxygen, nitrogen, ozone, etc.) and water molecules, as those exchanges are responsible for the behaviour of macroscopic variables like temperature, pressure humidity, and wind. The main objective of meteorological stations is to develop a system which permits reliable monitoring of climate changes3. The monitoring is of high importance for like airports, offshore wind parks, etc.

Fig.1 Air molecules in atmosphere

In many aspects of our life a huge amount of different materials are used. Glass, wood, metals, concrete, which are directly use almost every minute in our everyday life. Thus, the modification of materials and prediction of their properties are very important objectives for the manufacturers. In order to produce high quality materials the engineers in industry, among other problems, are very much interested in the elastic behaviour or loading capacity of the material4. While it is known that the bonding forces between the atoms of the material are responsible for their physical and chemical properties. So, to manufacture a new product with higher quality, a detailed investigation of the material on the atomic level is not required in most cases. A mathematical model is needed for the quantitative description of the change of material properties under external influences. The concepts of differential equations come to help us as an excellent tool for the development of such a model.

Fig. 2. Crystal Lattice

People spend several hours on their way back home because of the traffic jams on the roads after their hectic work. During the driving process every driver has one’s own behaviour which depends on the objectives of being fast and avoiding accidents. So, in this way a driver interacts with other cars. At all times drivers drive unthinkingly in such a manner as to avoid the traffic jams on the roads5. Again we need the help of mathematical model which can provide the understanding necessary to make the life of drivers more pleasant. The developed model can serve as ancient model also for other application problems. For example, the traffic jam model is similar to gas flow models which allow for the appearance of shock waves. In aircraft traffic, analogous problems cause noise pollution near airports.

Fig.4. Flux

5.Pdes as a Modeling Tool

If the left end of the rod is at a higher temperature, then heat energy will be transferred from left to right across the rod toward the colder part. In conduction process motion of the material as a whole is not considered7. Thermal energy is conserved in the conduction process. The content of energy depends on density and specific heat cp depends on the temperature T via The heat flux tries to equi-distribute heat over a piece of material. Thus, heat flows from hotter to cooler part. Fourier's law governs the conduction process, in a homogeneous medium the rate of heat flow is directly proportional to the temperature difference along the path of heat flow,

Where k > 0 is a material parameter and is called the thermal conductivity. The governing equation for one-dimensional heat conduction is given by,

6. Galerkin Method

Galerkin methods are used for converting a continuous operator problem to a distinct problem. It is the method of applying the variation of parameters to a function space, by transforming the equation to a weak formulation.

Let us introduce Galerkin's method with an abstract problem posed as a weak formulation on a Hilbert space namely,

Choose a subspace

of dimension n and solve the projected problem:Find

such that for all .The equation has not changed and the spaces have changed, converting the problem to a finite-dimensional vector subspace allows us to numerically compute

as a finite linear combination of the basis vectors in 8.

The key property of the Galerkin approach is that the error is orthogonal to the chosen subspaces. Since , we can use as a test vector in the equation. We get the Galerkin orthogonality relation for the error, which is the error between the solution of the original problem, , and the solution of the Galerkin equation,In Galerkin's method the production of a linear system of equations is needed, we build its matrix form, which can be used to compute the solution by a computer program.

. Then, it is sufficient to use these in turn for testing the Galerkin

We expand

with respect to this basis, and insert it into the equation,This previous equation is actually a linear system of equations

, whereThe matrix of the Galerkin equation is symmetric if the bilinear form

is symmetric.As per symmetric bilinear forms,

The application of the standard Galerkin method becomes much easier. Petrov–Galerkin method may be required in the non-symmetric case9. The analysis of these methods proceeds in various steps. It is to be showed that the Galerkin equation is a well-posed problem in the sense of Hadamard.

The analysis will mostly rest on two properties of the bilinear form, namely

Boundedness: for all

for some constantEllipticity: for all

holds for some constantThe norms in the following sections will be norms for which the above inequalities hold (these norms are often called an energy norm).

Since

, boundedness and ellipticity of the bilinear form apply to . Therefore, the wellposedness of the Galerkin problem is actually inherited from the well-posedness of the original problem.7 Quasi-Best Approximation (Céa's Lemma)

The error

between the original and the Galerkin solution admits the estimate.This means, that up to the constant

, the Galerkin solution is as close to the original solution as any other vector in . In particular, it will be sufficient to study approximation by spaces, completely forgetting about the equation being solved.

Proof

Since the proof is very simple and the basic principle behind all Galerkin methods, by boundedness of the bilinear form (inequalities) and Galerkin orthogonality (equals sign in the middle), we have for arbitrary

Dividing by

and taking the infimum over all possible yields the lemma.Here, we consider a beam which is simply supported at both ends, as shown in Fig. The overall length of the beam is L10.

Fig.5. Load distribution

Here z, we assume: -Material is isotropic. -Normal uniform load of intensity q0 is applied over the length of the beam. For such a system, the governing equation for normal deflection is: EI(d4 w/dx 4) ‐q0 = 0. The accuracy of Galerkin solution for the above eq. with exact solution. In such a case, the boundary conditions are:

Thus, we assume: w o(x) = Acos (πx/L)

• Such a displacement function satisfies thedisplacement BC at both ends of beam. Substituting this function in governing equation gives us error in the force. The relation for this error is:

E[w(x)] = AEI∙(π/L)4∙ cos (πx/L) ‐q0

• Thevirtual work doneby this error force as defined in the above eq , when integrated over thelength of beam

is equal to zero. So, ∫ΩE[w(x)]∙ εw1(x)dx = 0, where integration limits are ‐L/2 and L/2. or, ∫Ω [AEI∙(π/L)4∙ cos (πx/L) ‐q0 ]∙ εw1(x)dx = 0.

wGalerkin(0) = 0.01309[q0L4/(EI)]

wo(x) = A1 cos (πx/L) + A3 cos (3πx/L)

• Such an assumed expression for w(x) satisfied the kinematic boundary conditions at both ends of the simply supported beam. At this stage, we also assume an expression for virtual displacement.

w1(x) = B1 cos (πx/L) + B3 cos (3πx/L)

• The equation for w(x), the equation for virtual displacement, w1(x), also satisfies kinematic boundary conditions.

• At this stage, we develop an expression for virtual work over the entire domain, and equate it to zero. Thus:

But we know that B1, and B2 can have arbitrary magnitudes. Hence, integral of virtual work over domain can only be zero, if both terms in [ ] are individually zero.

Thus, we get two parallel equations in A1, and A3. Solving for these equations gives us:

Thus,

w(x) = [4q0L4/(π5EI)] [cos (πx/L) ‐(1/35) cos (3πx/L)].

• At the center, i.e. when x = 0, the value of displacement of thebeam is:

wGalerkin(0) = 0.01309[q0L4/(EI)] using 1‐term solution

wGalerkin(0) = 0.01302[q0L4/(EI)] using 2‐term solution

wExact(0) = 0.1302 [q0L4/(EI)] exact solution.

Thus, 2‐term solution brings us remarkably close to the exact solution.

Based on this analysis, we make following observations:

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