General Issues

When the finite element method is used, a continuous area Ω occupied by a structure and possessing an infinite number of the degrees of freedom is approximated by a discrete model consisting from a set of subareas (finite elements) that have a finite number of the degrees of freedom and interact with each other only at the nodal points.

Each finite element (FE) Ω_e (e = 1,…, m) is characterized by the following properties:

• a dimensionality of the used space (one-dimensional, two-dimensional, three-dimensional);
• a geometric shape, which is usually one of the simplest geometrical figures (line segment, triangle, rectangle, quadrangle, tetrahedron, etc.);
• a set of nodes placed (as a rule, though not always) on lines (surfaces) that divide the elements and are common for the adjacent ones;
• a set of the used degrees of freedom usually related to the nodes (though not necessarily to the nodes) — displacements, rotations, etc.;
• rules that define the relationship between the displacements of the nodes of the finite element and the nodes of the system. Nodes of the element, for example, can be attached to the nodes of the system rigidly or hingedly;
• a system of approximating functions which are defined within the area Ω_e and which enable to approximately express the components of the displacements in any point of the element in terms of its degrees of freedom;
• a physical law that defines a relationship between the internal forces and displacements;
• a class of problems to which this finite element type can be applied (plane stress state plate FE, Kirchhoff–Love plate FE, FE of the Reissner plate resting on the bi-parametric elastic subgrade, Timoshenko bar for a spatial problem, etc.);
• a set of allowed loads and actions which can be applied directly to the finite element and the method of their specification;
• and last but not least — a list of limitations and recommendations on the application.

One of the most important properties of a finite-element model is the maximum diameter of the elements

\[ h=\max\limits_{e} \left( {\sup\limits_{x,y\in \Omega_{e} } \left| {x-y} \right|} \right) \]

depending on which the errors of the method are often estimated. In other words, h is a minimum diameter of a sphere which can be described around any finite element of the design model. Moreover, it is usually assumed that if a diameter is infinitely decreased, i. e. when h → 0, the following regularity conditions are met — a sphere with a radius ρ ≥ Ch can be embedded into every finite element, where the constant С does not depend on h. This prevents the use of the so-called "needle" elements (too elongated rectangles, triangles with very small angles, etc.).

The approximating functions in the finite element method (FEM) are usually polynomial or piecewise-polynomial (subareas method), although there are elements with rational (the so-called isoparametric elements), trigonometric, logarithmic and other approximations of the displacement field. The selection of the degrees of freedom of the element and the corresponding approximating functions completely defines the rate of convergence and the FEM error estimate.

If we fix all parameters of a finite element design model, except for the size of the finite elements h, we can imagine that by changing this size, we will obtain a sequence of the approximate solutions of the problem u_h . When talking about the convergence of FEM, we mean that this sequence tends to an exact solution of the problem u*, when h → 0.

The above concept of the finite element modeling of a problem is based on what is known in the literature as the application of h-elements. The relative simplicity of this approach enables to solve the engineering problems directly and effectively. In recent years some software systems (for example, NASTRAN or StressCheck) have been using another finite element type, a so-called p-element. Unlike the h-elements, the p-elements are capable of representing the curvature of surfaces and peculiarities of stress fields directly on simpler meshes containing a smaller number of elements. The accuracy of the analysis is controlled by a p-level (power of the polynomial) specified for each element: the higher the p-level, the better the accuracy. When the h-elements are used, the accuracy of the result can be improved by increasing the number of the elements; the computation time will grow accordingly. When the p-elements are used, the same phenomenon will take place when the “p-power” grows.

Despite the many positive characteristics of p-elements, the h-elements have a number of advantages. A global behavior of a structure can be analyzed better with the h-elements. They are also better in the representation of solutions of problems where a stress discontinuity may occur, such as ribbed plates, and in the solution of nonlinear problems. Finally, the h-elements are used in the time-proven computational technology elaborated in every detail. These are the reasons why the SCAD finite element library contains only the h-elements.