History of finite analysis

Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration systems.

By the early 70's, FEA was limited to expensive mainframe computers generally owned by the aeronautics, automotive, defense, and nuclear industries. Since the rapid decline in the cost of computers and the phenomenal increase in computing power, FEA has been developed to an incredible precision. Present day supercomputers are now able to produce accurate results for all kinds of parameters.

The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with co-workers at the University of Swansea, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with co-workers at Cornell University. Further impetus was provided in these years by available open source finite element software programs. NASA sponsored the original version of NASTRAN, and UC Berkeley made the finite element program SAP IV widely available. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix. The method has since been generalized for the numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism, heat transfer, and fluid dynamics. Newton also worked in the finite analysis

Quadratic forms

Given a vector space V, we will say a function φ: V->r is a quadractic form if:

φ (x) = μ belonging to R

 φ (ax)= a^2 x

φ (x+y) + φ (x-y) = 2φ(x) + 2φ(y)

The matrix form is formed by φ (x)= a11 (x1)^2 + a22 (x2)^2 + a33 (x3)^2 + 2a12 x1x2 + a13 x1x3 + 2a23 x2x3. The vector x is placed in a row, then the matrix is formed by placed a11, a22 y a33 in the diagonal and the rest of them is dividing a12, a13 and a23 by two and put the in their symmetric place; finally, the vector x is placed again as a column matrix.

The classification os a quadratic form is:

• Positive: When φ(x) > 0
• Semidefined positive: When φ(x) ≥ 0
• Semidefined negative: When φ(x) ≤ 0
• Defined negative: When φ(x) < 0
• Undefined: When φ(x) = ?

Basically, when you calculate the diagonal matrix of a quadratic form, you see the values of the eigenvalues. If all of them are positive, it's positive. When they are positive and at least one of them is zero, semidefined positive. The same thing with the negative and semidefined but with negative values. If none of these condictions aren't showed, then it's undefined. The quadratic forms are the best way to calculate the positive property of a scalar product, where you need to have a defined positive.

Bilineal forms

Given a vector space V, we will say a function of  f: VxV-> R is bilineal if

f(x,y) = μ  belonging to R and being x,y two vectors.

In order to be a bilineal form, it has to be linear in both positions.

Lineal to the left : f(ax+by,z) = af(x,z) + bf(y,z)
Lineal to the right : f(x, ay+bz)=af(x,y) + b(x,z)

There are two types of bilineal forms:

• Symmetrical: f(x,y) = f(y,x)
• Antisymmetrical: f(x,y) = -f(y,x)

Bilinear forms are used to calculate a real number using the image of two vectors. A particular case of this operation is when the image equals the two vectors multiplied by one (eigenvalue = 1), meaning that we have an euclidean space and therefor, a scalar product. The matrix form works similar to the scalar product. You multiply the x vector as a row, the matrix is the image of the combination of the base (e1,e1; e2,e1; e3,e1; e2,e2; e2,e3; e3,e3) and finally the vector is placed as a column. This matrix form of the bilineal function is called "Gramm Matrix".
In this link you can find more information about bilineal forms.

Video: Eigenvalues and eigenvectors

Morphism

A morphism in Algebra is bascailly changing the coordinates of a vector using the same space or going from one space to another. Here is a list of some morphism:

• Monomorphism: f: X -> Y is a monomorphism if f ∘ g1 = f ∘ g2 .It implies that g1=g2 and X ->Z
• Epimorphism: f:X -> Y is a epimorphism if f ∘ g1 = g ∘ g2 implying that g1 = g2 and Y -> Z

       A monomorphism and an epimorphism together form a bimorphism.

• Isomorphism: f: X -> Y is an isomorphism if a morphism g : Y → X such that f ∘ g = id_Y and g ∘f = id_X. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f.
• Endomorphism:  f : X → X is an endomorphism of X. A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h ∘ g with g ∘ h = id. 

       An isomorphism and an endomorphism from a automorphism.

Interview

Here we have an interview with Alonso Martinéz Cuenca, ex-worker in General Electric.

So what did you study at university and where?

I studied Electronical Engineering at Universidad Complutense, Spain.

In how many companies have your worked?

I've been working in two companies. Both of them related with the same activity: Electronic devices.

Can you tell me some of your tasks in both companies?

The first one was "Compagnie General du Electronique"(CGE). It was a french company that manifactured electronical devices for diagnostic imagin, computes, etc, from year 1974 to 1982. Along these eight years, I was working in the technical service in charge of the maintenance  devices of this company as well as ultrasound  systems.

The other company was "General Electric" (GE). In 1984 CGE joined to GEH and I was in charge of technical support for Computed Microchips (CM). After six years I started to do electronical applications, helping to the Engineering department to develop tools for circuits in CM. Eight years later, I started to work in the Marketing and Sales department, finishing my laboral activity as a MR sales specialist.

So, your career's knowledge was very useful in here...

Of course. As I started in a technical department, all my career was very useful and I had to apply all my knowledge when I was working in the field, reparing the different devices as well as developing electrical and electronical instalations.

Did you have to travel in the development of you laboral duties and how often?

Yes, I had to travel along Iberia mainly, but I had to travel to some european countries and Africa to do clinical applications and to the USA to perform some training twice per year.

Did the market keep seeling electronical devices despite the crisis in our country?

The crisis has affected to this kind of companies specially because the social security that is the biggest buyer for this kind of devices has reduced dramatrically the adquisition of this imaging diagnostic devices.

Do you think the market will rise again in a few years?

Fortunately, in the middle of 2015 the market will start to increase their activity and the operating planning has been risen a 4 % over the expectation. 

Introduction

This is the final part of this blog, taking the Algebra subject. I'll make six posts about the subject, similar to the first part of the blog (calculus).

My opinion about "Communication Skills"

So here is the last post involving the "Communication Skill" subject. So I'm going to give my opinion about this.

Basically, a very lost potential. It started okay with a lot of activities to do with huge variety and learning something that people from a science career never learn too well. I was excited about how this subject was going to be, but before the first month passed, my intereset was completely lost.

In the first classes, we had debates and parcipitaded in several videos we see in class and I really liked it, but that was left to do the same thing over and over, so overall, it went really boring. The formula of the subject was: The secretary reviews his blog/activities/wikia, approve last's minute and then go to his chair. Over and over again.

Also, that's my least favourite thing about this: The secretary. I just don't get it. Why a classmate must stand infront of everyone telling stuff that the teacher is not only more capable to handle? The teacher must be the one bringing the blog, wikia and activity and review it, otherwise nobody will every pay attention. This ends with a complete mess with the classmates not knowing everything we have to do and then most of the activities are wrong. Every single student need feedback on their individual activities, not some general tips that you may already have right, because that's what professors are for, correcting all your work on their subjects.

The blog and the wikia are actually a unique way of developing extra work without tending to be linear with the activities. Maybe the one I liked more was the blog instead of the wikia, just that's because I already had to do posts in the Calculus subject.

As for the activities, they are all acceptable and you learn a lot from them. But again, the same problem as before, if you have no personal feedback, you won't be learning anything. Okay, yes, there are a lot of students and a lot of activities and the teacher is not only teaching this specific subject. Not only students have to study other subjects and also so other activities and projects but also, the whole activities have, in general, some procedures of making them, so it's easy to find little mistakes, and those little mistakes are, for example, the lenght of the introduction or the conclusion, how to redact a document as if you were the teacher, etc.

What I'm trying to say is that this is not the way to handle such a subject. In my opinion, it needs more participation but not with the secretary mechanic, like more debates or conferences, because with that, the subject can be way more entertained.