Differential elements and finite elements

Classical analysis employs differential elements, e.g. dx dy dz for bodies in right handed global coordinates X Y Z , or dr rd, , ϕdz in semi-polar coordinates , ,r ϕ , or – in z the simplest configuration – ds for 1D beams with the coordinates s as the position of a cross-section measured from the chosen origin of the beam. The characteristic feature is that these are limit shapes, i.e. it is assumed that differentials dx dy etc. approach zero without , any limits, which is often expressed – inaccurately – in a popular way that the elements are infinitely small. What is in fact substantial for these elements is the variability in the vicinity of zero. No equation from the classical analysis could be derived if we imagine an element 0 0 0× × or ε ε ε× × with constant ε . Even the fundamental term of the classical analysis, derivative, is defined by a limit transition, e.g.

0

The above-mentioned formulas express the values of derivatives through limits of differences of values that are valid beyond the point x , i.e. for x+ε , and values that are valid in the point x . Therefore, these are forward derivatives. The formulas can be written also for backward derivatives – then we have to use differences f x( )− f x( −ε), or as middle derivative using a symmetrical form (f x+ε 2)− f x( −ε 2) etc. Engineering faculties devote quite a considerable part of mathematics curriculum to derivatives and therefore, here we only refer to the appropriate textbooks, books or guides and popular monographs [57], [58]. For points of the vicinity approach point B from any side or direction.

A similar definition applies to an arbitrary derivative f . If we use an illustrative concept of derivative f x as a tangent (tan) of the angle between (i) the tangent to the graph of f and ( ) (ii) the x -axis, then the conception of a common tangent on the left- or right-hand side (alternatively limit middle secant) leads to the popular definition of continuity of the first derivative: the graph of the function is not allowed to have a peak in point B. This applies also to two variables if we consider sections y=const or x=const across the graph ( , )f x y . If we admit the conception of n -dimensioned graphs embedded into (n+1) dimensional space, it is possible to handle the term peak even in these graphs that cannot be displayed by means of any technical measures in a real 3D model of the space (Euclid, Newton). Similarly for the second derivative, it is possible under certain assumptions approach to the conception

4.1 Geometric properties of elements

of graph curvature (1 variable) or three components of the curvature of a planar graph (2 variables) that must have – in the case of satisfied continuity – the same limit from all sides, etc.

4.1.1.1 How many differential elements are there?

This provocative question can be perceived in different ways. If it is to mean how semi-polar, curved, etc.). Consequently, we get an unlimited number of shapes of differential elements. However, the question can be understood in different way: We have a given 1D, 2D, or 3D body, we define in it just one fixed system of coordinates and we divide the body into differential elements – all of the same shape. How many of these elements are there? This formulation of the question has a closer relation to the following text dealing with FEM and it is useful to think about it a bit more. Every FEM user, including students, would give the correct answer that their number is unlimited, as their dimension approaches, in limit, zero, or as students sometimes put it: they are infinitely small and therefore there are infinitely

If all the elements are identical and their number is n , then:

n e

∞0. The subscript 0 indicates the lowest possible cardinality of the set of the infinite number of elements, which is termed aleph-null. If we admit, within the same conception, that differential element Ω_e has a zero measure, then the meaning of expression (4.1.4) is just a sum of countable (even though infinite) number of nulls (zeros), i.e. Ω =0! Apparently, something is wrong. And painstaking students who had good teachers of mathematics report the same problem.

A continuum of an arbitrary dimension (1D, 2D, etc.) and size (from the smallest possible domains to meta-galactic dimensions) is defined by unlimited divisibility. Each of its parts has again properties of a continuum. More illustrative example of 1D continuum: the interval between arbitrarily close points x , x+h is again a continuum, it thus have the same

number of points as the continuum between −∞ and +∞. The same must be understood in the meaning of the possibility to assign the elements of one set to the elements of a second set. In the case of infinite sets we cannot speak about a concrete number since the number is unlimited, but about cardinality which is identical for the two sets, if such assignment can be made and no elements remain unassigned in either set. What is not correct in the result Ω =0 is apparently the wrong conception of a continuum that sticks to the common sense of a practical engineer who does not accept some strange higher infinity. It is sufficient to get over this useless backwardness and simply admit that the number of points in an arbitrary continuum is greater that countable infinity, i.e. that they cannot be assigned to natural numbers 1,2,3,...

If we had an unlimited amount of time and started to assign natural numbers 1,2,3,... to the points of any howsoever small (e.g. ( ,x x+h)) or large (e.g. (0,∞)) continuum, which would represent a never-ending process, so that some infinitely long-lived creatures would have to carry on with it after the extinction of the universe, they would see with some astonishment that the continuum has not decreased at all, that it still has the same cardinality.

After billions of years of assigning, e.g. with the speed of 1 point / second, still almost no point of the continuum would be marked, which is the mathematical formulation of the continual measure of a countable, even though infinite, set of points.

We can imagine a differential element around each point of a continuum, so that in limit dx→0, dy→0, etc., we get a set of differential elements which is not only infinite, but which has much larger cardinality than countable infinity. Thus the number of differential elements can be marked e.g. ∞₁, with the subscript 1 indicating a greater cardinality aleph one. The extremely interesting mathematical considerations, disputes and proofs about whether there is something between ∞₀ and ∞₁, the alternative up-to-date definition of the measure, etc., are not interesting for FEM users. On the other hand, it is very useful for the needs of modelling of inputs and interpretation of outputs of FEM programs to consider the principal capabilities of FEM in general, with no regard to advertising slogans printed on the glossy paper of world-renowned companies’ colourful marketing materials. These capabilities are ultimately limited by a simple factor:

The number of finite elements in FEM is always finite, as it follows from the name of the method and the nature of the matter: division of body (4.1.2) to elements of the same dimension (FEM) and similarly for the boundary of the body (BEM) or partially discretised problems (FLM and strip method). The number of parameters d, i.e. geometric quantities in the deformation variant, or statical quantities in the force variant, or both in the mixed

ee) Replacement of differential equations by the variational principles in which we handle countable sets of free fields (art. 3.3.2.).

4.1 Geometric properties of elements

ff) Reduction of the system of ∞₀ equations to systems of N equations, which can be processed by the current computers.