Motivos centrales

3.3.1 Position of variational principles in mechanics

The elementary explanation of articles 1 and 2 is sufficient for a general idea about how the finite element method provides for the substitution of unknown functions for their approximate shapes – a linear combination of base functions – that are closely related to the division of the analysed body Ω (FEM) or Γ (BEM) into finite elements. To sum up: how the discretisation of the project and the related finite number of unknown parameters is carried out in FEM. The solution of the systems of algebraic equations is the most effective procedure the state-of-the-art computers can offer. Non-linear problems must be solved by the repetitive application of the same procedure too. The given explanation focused on the most often used variant called deformation variant of FEM, in which the unknown parameters d have the nature of geometrical quantities: components of displacement and small rotations, or their derivatives. In article 1.3 we showed that the application of the first-rate FEM apparatus is possible in all the situations in which the functional (whose extreme (minimum or maximum) is being sought) is of additive nature. We also explained that formulas (1.3.2) - (1.3.4) clearly show how such a functional can be created in problems where we lack any physical definition for it – through the principle of weighed residua (1.3.4). This can be done also in non-mechanical, even non-physical, purely mathematical problems. It is clear that for engineers especially mechanical problems are of great importance. For these, the development of FEM has offered not only the deformation, but also force and mixed

variants – for problems of varying content and extent. Some FEM systems contain them in the set of implemented finite elements. Therefore, it is useful to be familiar with the variational principles that represent the basis of the particular solution in order to be able to evaluate their practical advantages and disadvantages. There exist many illustrious original resources dealing with these issues, e.g. the literature cited in [3-9]. For the first acquaintance with the subject let us mention certain “sine qua non” for the engineers.

The principles of mechanics belong to the oldest principles of physics. This includes such statements that can be used to derive the equations of motion or under special conditions of statics the equations of mechanical systems, both in arbitrary coordinates. Depending on the mathematical formulation and mechanical meaning we can distinguish two types of principles:

Differential principles examine an arbitrary instantaneous state of the system and compare it with the nearby state.

Integral principles are in general variational principles that examine such states which the system passes during a limited time interval and compare them with certain nearby states.

Some principles can exist in both formulations and are usually closely related to each other.

The differential principles include e.g. Gauss’s principle of least constraint and Hertz’s principle of least curvature. The typical integral principles include Hamilton’s, Euler-Maupertuis’s and Jacobi’s principles – called together the principle of least effect (effect = energy × time) – which differ from each other in the assignment of the comparative trajectories. D’Alembert’s principle and the principal of virtual work are not in fact extremal principles, but also represent statements from which the equations of motion or, as the case may be, equilibrium, can be derived.

When dealing with problems in the field of statics of elastic bodies, we can start either with the system of differential equations of with energetic reasoning, which leads either to the classical boundary problem or to the variational problem from the theory of elasticity. There is no general answer to the question which formulation of the problem should be considered the primary one, unless we take into account some additional, e.g. methodological, positions.

Each basic principle is actually an unprovable axiom and can be replaced only by another axiom (or derived from it). The basis for the finite element is however formed by the fundamental variational (integral) principles.

The oldest principles of mechanics focus only on what is termed mechanical systems of mass points or rigid bodies that have finite number of degrees of freedom and such connections whose reactions do no work for any virtual displacement. On the other hand, the virtual work of stresses appears in elastic bodies.

The most general principle is Gauss’s principle of least constraint: the real motion of the system is such that in every time instant the resistance Z is minimal:

2

on condition that we take into account all possible motion states of the system with the same positions and speeds but different accelerations a (_i i=1, 2,K ) that meet the given ,N conditions. P means external forces and m_{i i} the mass of N points of the system. Under special circumstances, this principle is equivalent to d'Alembert’s principle, but makes it

3.3 Variational principles of mechanical problems of FEM

possible to take into account more general (non-linear, non-homogeneous) conditions. It is based on the method of the least sum of squares and for a free motion without forces (P_i =0) it can be harmonised with the Hertz’s principle of least curvature.

The most important principle for the field of statics is the principle of virtual works, which can be traced back to Aristotle (383 – 322 B.C.) and which was generally formulated by Bernoulli in 1717 in a letter to Varignon who published it in 1725. In the original form it is applicable just to the systems of rigid bodies, non-compressible liquids without friction and non-expansible ropes and reads: a mechanical system is in equilibrium only if the virtual work V of given forces P (_i i=1, 2,K ) is for any virtual displacement ,N δ (ri i=1, 2,K ) equal ,N

A virtual displacement is an arbitrary infinitesimal change in the position of the mechanical system that is assumed in a certain time instant and that complies with the given geometrical connections. Forces P (contrary to the reactions that are subsequently determined by the _i analysis) must be defined in advance. They are of physical origin and, in general, depend on physical constants (gravitation, friction, electrical or magnetic forces, etc.). They can act both outside and inside of the system. The principle of virtual works can be extended to elastic continuum only axiomatically. All the mechanical (statical) conditions of equilibrium in the body and on its boundary are then based on it. On the other hand, if we accept these conditions as an axiom, the principle of virtual work can be derived from them. The statements are actually equivalent.

In order to understand the wider context, let us add the following facts. We deal only with what is termed scleronomous mechanical systems with scleronomic constraints in which time t is not explicitly included (contrary to rheonomic constraints). In addition, we take into account only such forces P whose potential is independent on time or – more exactly – that do not follow the deformation of the bodies and keep their direction, so that their potential energy Π is unequivocally defined by the deformation of the body. We term such system shortly as conservative. The following statement applies to them: A scleronomous mechanical system with given conservative forces moves in such a way that the sum of kinetic ( K ) and potential (Π) energy remains constant during the full movement (k is the energetic constant).

K+ Π =k (3.3.3)

It must be stated that in technical practice we may encounter also rheonomic constraints and, in particular, given forces that are not conservative and their potential energy is not independent on time, e.g. external forces follow the deformation of the structure (fluid pressure according to Pascal’s law), are influenced by its vibration (various aerodynamic effects), etc. Similarly, internal forces may follow more complex constitutive law in the case of ideally elastic bodies. The system is then non-conservative (dissipative) and the mechanical energy is partially transformed into other forms of energy, e.g. thermal, etc.

3.3.2 Scalar, vector and tensor field in FEM inputs and outputs

Texts dealing with FEM often use the term field as the opposite to the term single value. Users sometimes misunderstand it and consider it identical to the term function, which is known to be a relation that relates one set if independent variables to another set of dependant variables, regardless of whether the relation is analytical (formulas), numerical (tables) or graphical (graphs). A field is just a description of the state in all points figures (3D), and occasionally even 4D figures in what is called space-time elements in space ( , , , )x y z t . Such elements were developed for certain dynamic problems and were applied in practice. In general, however, another discretisation of time factor t has proved useful. In free vibration problems it is the harmonic analysis, in forced vibration it is either modal analysis (decomposition into the free vibration shapes) or what is termed direct integration in time t , e.g. over reliably small differences ∆t, which allows for a good estimate of the response of the analysed system to the given dynamic excitation. Consequently, we stick to three dimensions N =1, 2, 3 (as the configuration N =1 and 2 can always be obtained from the basic configuration N =3), because this is the way they are defined by means of various static and in particular geometric restrictions concerning the nature of quantities acting in real 3D bodies (other bodies do not exist in physical terms). Therefore, the explanation can be limited to 3D body Ω and its points x=( ,x x x_{1 2}, ₃)=( , , )x y z with the illustrative notation of coordinate axes x i_i( =1, 2, 3) using letters , ,x y z .

The subscript notation i (or j if two different axes play a certain role) is in FEM common also for other quantities than coordinates. For example, displacement components

( 1, 2, 3) the directions specified by the subscripts. The fraction 1 2 has been introduced to ensure that quantity ε has the character of tensor, i.e. that it transforms during the rotation of the coordinate trihedral following the same rules as any other tensor, e.g. σ . In addition, this coefficient is important for the unification of the notation of the geometrical link between displacements ui and components of strain ε . The following notation is introduced in which _ij the subscript following the comma denotes the partial derivative with respect to this index:

1

3.3 Variational principles of mechanical problems of FEM

Components of rotation that were earlier neglected for many reasons were implemented into modern FEM programs (approximately from 1980 onwards). Formally, they differ from the components of strain only by the minus sign in the general formula that is similar to (3.3.5):

(

)

The notation with one subscript denotes the axis of rotation, the notation with two subscripts the plane of rotation. It represents the real physical rotation. If it is small enough, it is exactly satisfied for the limit approaching zero, then the rotation can be considered a vector and the three quantities (3.3.7) its components in the x –, y –, and z –axis. As a preliminary point it should be observed that they can be accepted as parameters of deformation in nodes of finite elements, which makes it possible to concentrate all the parameters into their vertices.

This eliminates the intermediate side parameters that worsen the good numerical conditionality of the equation systems for the solution. They increase the band width BW or occupy more elements in matrix K .

A scalar field is defined in each point of body Ω by a single value (e.g. by specific weight or density ρ , temperature T etc.) as a function of coordinate ⁰ x , i.e. the position of the _i point. Whether we know the values (e.g. definition of the density of the body) or not (e.g.

temperature T ) plays no role in the nature of the field. On the other hand, a scalar field (i) ⁰ can vary over time t , which is a common situation towards the rheological process, or (ii) can be constant over time t , e.g. in the case of stationary (stable) thermal flow through the body each point has permanently constant temperature.

A vector field is in each point of body Ω by one vector, which is a quantity representable by an arrow the origin of which is in the analysed point, the length expresses the magnitude and the arrow defines the direction and orientation of the vector. The way of input is once again unimportant. A useful aid is an idea of a dense system of arrows each of which starts in one point of a body.