The
mechanical analysis of systems having structural function
(buildings, bridges, aircrafts, etc.) often represents
an hard task because of the large complexity of their geometry
and of the loading conditions (dynamic
forces, etc.). These systems can be studied through
basic principles of mechanics by means of a set of differential
equations. Usually these differential equations cannot be solved, apart from very simple
cases in which the simplicity of geometry, boundary conditions
and loading makes it possible a direct integration.
Numerical methods are adopted in order to solve mechanical
problems concerning systems with complex geometry and subject to complex loading conditions, obtaining solutions which are approximate but still suitable for engineering purposes.
The Finite Element Method (FEM) belongs to the class of numerical
methods and nowadays it is the most common method adopted
in structural analysis.
FEM was first proposed in 1943 by R. Courant to obtain approximate solutions for vibrating
systems. Since then the method has significantly evolved and it has found application in different fields, such as civil engineering, mechanical
engineering and more recently biomedical engineering.
Dental biomechanics represents a field
in which the use of FEM is particularly promising. FEM is currently applied with the aim of improving the design of materials, structures and manufacturing procedures,
thus also improving clinical results in implantology.
Finite
Element Method
The aim of a mechanical analysis by means of the Finite Element Analysis is to simulate the behaviour
of a structure subjected to specific boundary conditions, such as loads or displacements restrictions.
The first step in a Finite Element Analysis is to split the real geometry of the structure in discrete portions, which are called finite elements. Each finite element is characterised by a set of points called nodes. Finite elements and nodes form the mesh. The laws of balance, which are required to solve the problem of the equilibrium of a body under specific loading conditions, are written and solved for each element,
thus reducing the number of equations and unknowns from infinite, for the continuum body, to a finite number, for the body subdivided in finite elements. Solving the above system of equations the displacements at the nodes are calculated, then stresses and strains in each finite element are determined from the value of the displacements previously obtained.
In order to carry out a Finite Element Analysis the structure is drawn with a CAD software and the virtual model created is imported
in the pre-processor program. Here the mesh generation takes place. Material properties and element properties are defined.
Loads applied to the structure and boundary conditions are then specified, where boundary conditions are usually in the form of zero displacements
applied to specific points, lines or surfaces in the model.
The analysis solver software is divided into three parts: the pre-solver, the mathematical
engine and the post-solver. The pre-solver formulates the mathematical
model reading data from the pre-processor, the math engine calculates
the solution, while in the post-processing
stage the results are read and interpreted.
Control
Volume Method
The Control
Volume Method is usually applied to solve problems of fluid dynamics where rigorous conservation of mass and/or energy is very important in the simulation of the physical system. Considering an arbitrarily
chosen volume, the Control Volume Method permits to calculate
the values of the field variables (mass, temperature, momentum, etc.) averaged across the
volume. The balance equation for a given variable states that the rate of accumulation of the variable in a control volume is equal to the net rate of flux of the variable across the surface bounding the volume (if no generation process occurs within the volume). The differential form of the balance equation for each variable is integrated over the control volume to form a system of equations that are solved as in the case of the Finite Element Method.
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