 # Finite Element Analysis and Its Application in Oral Surgery Research

Mercedes Gallas (University of Santiago de Compostela, Spain)
DOI: 10.4018/978-1-60566-733-1.ch010

## Abstract

The Finite Element Method (FEM) is a widely applied mathematical model that permit us to know the biomechanical behavior of the human mandible in various clinical situations under physiological and standardized trauma conditions. The three-dimensional FEM provides to simulate force systems applied and allows analysis of the response of the jawbone to the loads in 3D space. Clinical extrapolations from FEM may not give absolute values but they will provide detailed description of biomechanical pattern and a prediction of regional stresses distribution. This virtual modeling is useful to choose the most efficient localization and design of miniplate osteosynthesis and to test new biomaterials.
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## Introduction

The Finite Element Analysis (FEA) is an accepted theoretical technique used in solution of engineering problems specially (used) for problems involving complicated geometries. It can be used to calculate deflection, stress, vibration, buckling behavior and to analyze deflection under loading or applied displaced. It can be employed as well to analyze elastic deformation or “permanently bent out of shape” plastic deformation.

FEA is a computer-numerical method to obtain a solution to a complex mechanical problem by dividing the problem domain into a collection of much smaller and simpler domains (elements) in which the field variables can be interpolated with the use of shape functions. An overall approximated solution to the original problem is determined based on variational principles. The behavior of an individual element can be described with a relatively simple set of equations. Just as the set of elements would be joined together to build the whole structure, the equations describing the behaviors of the individual elements are joined into an extremely large set of equations that describe the behavior of the whole structure. Therefore in FEA, a structure is modeled with discrete-element mathematical representation by subdividing it into simpler geometric shapes or elements whose apices meet to form nodes. This complex system of points called nodes make a grid called mesh. Each element of the mesh can adopt a specific geometric shape: triangle, square, tetrahedron, … ; depending on the type of analysis such as two-dimensional and three-dimensional. Each element is defined by a specific internal function. This mesh is programmed to contain the material characteristics and structural properties that defined how the structure will react to certain loading conditions. The elastic constants, Poisson´s ratio (ν), Young´s modulus (E) and the material density (ρ) are specified for the modeled material, which each element retaining the material characteristics of the original structure. The assignment of proper material properties to a Finite Element Model (FEM) is the essential step to ensure predictive accuracy. The primary objective of the FEM is to realistically replicate the important parameters and features of the real model. The first step in the creation of a FEM is to describe its geometry in the computer by appropriate software. In general, a FEM is defined by a mesh net-work, which is made up of the geometric arrangement of elements and nodes. Depending on the problem to investigate the numerical representation of the structure under analysis can be achieved either in a two- (2D) or three-dimensionally (3D) ways. As a loading or force is applied, the interconnected elements of mesh start to move. The displacement of nodes is transformed, through calculations, to stress and strain values. The displacements, von Mises stresses and shear stress in different planes could be studied under different and determined situations. Stress distribution was analyzed based on color differences in von Mises equivalent stress.

This process of dividing the whole domain into elements is indispensable in FEA to formulate the solution functions for each finite element and combines them properly to obtain the solution to the whole body. The process of creating the mesh, elements, their respective nodes and defining boundary conditions is denominated “discretization” of the problem domain. Stress and strain distribution patterns in a structure depend on its geometry, the boundary conditions and its material properties. The finite element modeling technique offers the advantage of being able to model structures with intricate shapes and indirectly quantify their complex mechanical behavior at any theoretical point. Because the FEM uses the theories of elasticity and static equilibrium, the effects of multiple external forces acting on a system can be assessed as physical events in terms of deformation, stress or strains.

The information needed to calculate stress and displacements in FEM is: the total number of nodal point and element, the elastic modulus and Poisson´s ratio for the materials associated with each element, the coordinates of each nodal point, the type of boundary strains and the evaluation of the forces applied to the external nodes with a FEA software and computer. Also FEA provides the possibility to simulate clinical loadings forces (for example: traumatic impact, orthodontic force systems, physiological muscle forces) applied and allows analysis of the response of the bone to these loadings conditions in the three-dimensional space.

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