|Year : 2010 | Volume
| Issue : 3 | Page : 425-432
|Infinite to finite: An overview of finite element analysis
A Srirekha, Kusum Bashetty
Department of Conservative Dentistry & Endodontics, The Oxford Dental College & Hospital, Hosur Road, Bangalore, Karnataka, India
Click here for correspondence address and email
|Date of Submission||11-Nov-2009|
|Date of Decision||08-Jan-2010|
|Date of Acceptance||21-May-2010|
|Date of Web Publication||29-Sep-2010|
| Abstract|| |
The method of finite elements was developed at perfectly right times; growing computer capacities, growing human skills and industry demands for ever faster and cost effective product development providing unlimited possibilities for the researching community. This paper reviews the basic concept, current status, advances, advantages, limitations and applications of finite element method (FEM) in restorative dentistry and endodontics. Finite element method is able to reveal the otherwise inaccessible stress distribution within the tooth-restoration complex and it has proven to be a useful tool in the thinking process for the understanding of tooth biomechanics and the biomimetic approach in restorative dentistry. Further improvement of the non-linear FEM solutions should be encouraged to widen the range of applications in dental and oral health science.
Keywords: Anisotropic, finite element method, meshing, modulus of elasticity, nodes, non-linear, Poisson′s ratio, stresses
|How to cite this article:|
Srirekha A, Bashetty K. Infinite to finite: An overview of finite element analysis. Indian J Dent Res 2010;21:425-32
Stress analysis of dental structures has been a topic of interest in recent years with an objective of determining stresses in the dental structures and improvement of the mechanical strength of these structures. Stresses in dental structures have been studied by various techniques, such as brittle coating analysis, strain gauges, holography, 2-dimensional (2D) and 3-dimensional (3D) photo elasticity, finite element analysis (FEA), digital moirι interferometric investigation and other numerical methods. Most of the stress analysis of dental structures was carried out using the photoelastic technique. The advantages of using photoelastic study are that it can quantify stresses throughout a 3D-structure and determine stress gradients. However, it requires a birefringent material and is more difficult with complex geometries. 
|How to cite this URL:|
Srirekha A, Bashetty K. Infinite to finite: An overview of finite element analysis. Indian J Dent Res [serial online] 2010 [cited 2019 Jun 24];21:425-32. Available from: http://www.ijdr.in/text.asp?2010/21/3/425/70813
A more recent method of stress analysis, generally developed in 1956 in the aircraft industry was the finite element method (FEM). Initially, this technique was used widely only in aerospace engineering, but slowly due to the flexibility of the method to model any complex geometries and provide instant results, it made its presence felt in dentistry.  It was first used in dentistry in the 1970's to replace photo elasticity tests.
FEA is a popular numerical method in stress analysis. This method involves a series of computational procedures to calculate the stress and strain in each element, which performs a model solution. FEM and FEA are one and the same. FEA is more popular in industries and FEM at universities. FEM circumvents many of the problems of material analysis by allowing one to calculate physical measurements of stress with in a structure. 
The purpose of this article is to give an insight into the FEA, which has totally overshadowed other experimental analysis due to its ability to model even the most complex of geometries, which is immensely flexible and of adaptable nature.
| Basic Concept of FEM|| |
There are 3 methods to solve any engineering problem: analytical method, numerical method, and experimental method.  The FEM is a numerical procedure used for analyzing structures and consists of a computer model of a material or design that is stressed and analyzed for specific results. FEM uses a complex system of points (nodes) and elements, which make a grid called as mesh. This mesh is programmed to contain the material and structural properties (elastic modulus, Poisson's ratio, and yield strength), which define how the structure will react to certain loading conditions. The mesh acts like a spider web, in that, from each node there extends a mesh element to each of the adjacent nodes. The basic theme is to make calculations at only limited (finite) number of points and then interpolate the results for the entire domain (surface or volume). Any continuous object has infinite degree of freedom (dofs) and it is just not possible to solve the problem in this format. FEM reduces the dofs from infinite to finite with the help of meshing (nodes and elements) and all the calculations are made at limited number of nodes  [Figure 1]. Using these functions and the actual geometry of the element, the equilibrium equations between the external forces acting on the elements and the displacements occurring on its nodes can be determined.
Theoretically, the shape and the size of elements determine the results. Finite elements started with triangular elements and these elements being stiffer, resulted in less stress and displacement. Later, quadrilateral elements were used for accuracy of results. Polyhedral mesh work is in research at the moment. Increasing the number of calculation points (nodes and elements) improves accuracy. For example, increasing the number of lines reduces error margin in finding out the area of a circle [Figure 2]. The number of straight lines are equivalent to the number of elements in FEM.
Generally, 2 types of analysis are used in the industry, which are 2-D modeling and 3-D modeling. 2-D modeling is comparatively simple and it allows the analysis to be run on a relatively normal computer, but it also sometimes tends to yield less accurate results. 3-D modeling produces more accurate results, but it can run only on the fastest computers effectively. For 2-D analysis, the elements are triangular or quadrilateral and in 3-D analysis 10, 12, or 14 faces are used. In some situations mass, spring, damper, and gap elements are also used [Figure 3].
|Figure 3: Other types of elements. (a) mass; (b) spring; (c) damper; and (d) gap|
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FEM is performed with material properties that can be isotropic (same properties) or anisotropic (different properties) [Table 1].  All real-life materials are anisotropic, but it is simplified into isotropic properties or orthotropic properties (different properties along 3 axes, namely- x, y, and z). Elastic modulus, Poisson's ratio (strain in the lateral direction to that in the axial direction when an object is subjected to tensile loading),  and yield strength for the materials are applied. The analysis is performed as linear static analysis or non-linear analysis depending on the allocation of appropriate physical characteristics to the different parts of the tooth. Linear systems are less complex and effective in determining elastic deformation. Many of the non-linear systems are capable of testing a material all the way to fracture and they do account for plastic-deformation [Table 2]. The eventual result of any FEM is the normal and shearing stress values of the structure upon loading. The failure criteria is measured by Von-Mises stresses.  The reason for selecting Von-Mises criteria, which apparently results in a tensile type normal stress, lies in the fact that the brittle materials, which the tooth is a member of, fail primarily because of tensile type of stress. In practice, an FEM usually consists of 3 principle steps [Figure 4]:
- Pre-processing: It includes CAD (computer-aided designing) data, meshing, and boundary conditions.
- Processing or solution: This is the step in which the computer software does the job of calculation. Internally, the software carries out matrix formulations, inversion, multiplication, and solution.
- Post-processing: This step includes viewing results, verifications, conclusions, and thinking about what steps would be taken to improve the design.
|Figure 4: Steps in finite element method: (a) 3D-model; (b) meshing; and (c) resultant stresses|
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| Advantages and Limitations of the Finite Element Method|| |
FEM is a basic research tool that is widely being used in dentistry. When finite element modeling is compared with laboratory testing, it offers several advantages. The variables can be changed easily, simulation can be performed without the need for human material and it offers maximum standardization. FEM helps to visualize the point of maximum stress and displacement, but it is not easy to predict failure in materials with complex geometric shapes made of different materials, complex loading varying with relation to time and point of application further complicated by residual stresses. If tools, such as CAD and CAE (computer-aided engineering), are modeled in an appropriate fashion, one can easily get stress contour plots clearly indicating the locations of high stress and displacement. Designing with CAD/CAM (computer-aided machining) helps to transfer the basic data with advance information and the CAE engineer runs the analysis after meshing. Hence the work gets completed in a shorter duration of time and is also cost effective. FEM can minimize laboratory testing requirement, but it will be wrong to assume that it will totally replace testing. FEM provides faster solutions with logical and reasonable accuracy in an era where the industry prefers faster solutions. FEM may give results with a reasonable degree of accuracy, but this approach has certain limitations,  such as it's inability to simulate accurately the biological dynamics of the tooth and its supporting structure. For example, in non-carious cervical lesions, the structure of the dentin undergoes changes as it is exposed to the oral environment. It is very difficult to develop a predictive model for the complex structure of tertiary dentin, which is formed in response to a stimulus, such as tooth wear. 
Different studies have highlighted the anisotropy in the material properties of enamel and dentin. ,, However, numerical modeling efforts have been made to understand the stress distribution in teeth that commonly led to simplifying assumptions that the elastic modulus of teeth is isotropic. Conclusions drawn from such studies, although useful as a first step, often tend to be misleading. Another shortcoming of the FEM studies is the absence of rigorous experimental validation.
| Advances in Finite Element Analysis|| |
Many studies have been carried out by using 2-D and 3-D modeling. Numerical results obtained by 2-D modeling have some shortcomings. First, human teeth are irregular and the structure has no symmetry, so they cannot be represented in 2-D volumes.  3-D FEM is preferred to obtain an optical realistic analysis with detailed tooth anatomy and computational process. Previous attempts to obtain 3-D models resulted in coarser meshes due to limitations of geometry acquisition method of the tooth model, increased memory requirements for 3-D models, which did not allow fine representation of the geometry. Inner anatomical detail was obtained by some authors with the use of digitized plaster models and by using tooth morphology literature data (pulp, enamel, and dentin volumes). For 3-D approach, maximum anatomic detail of the tooth and other structures are obtained by micro-scale computed tomography data acquisition technique. 
For reducing the meshing time apart from conventional types of elements, research is also going on about polyhedral meshing and mesh-less (or mesh-free) analyses. Advantages of polyhedral meshing could be described as less meshing time with higher accuracy and that too at less number of dofs. Some of the software's have already started providing polyhedral meshing option, but at the moment it would be too early to answer whether polyhedral mesh would be a regular feature of structural as well as other types of analysis also. Hybrid meshing (hex-pyram-tetra) is a very special option and not supported by all the softwares. The basic concept is to use hex (linear) elements in critical areas (high stress locations) and tetra meshing (parabolic) in general areas (areas away from critical areas). If meshing is carried using pure hex elements, then it will take lots of time, hybrid mesh saves meshing time substantially without compromising the quality of results.
One of the problems associated with early finite element models is the difficulty of allocating appropriate physical characteristics to the different constituent parts of the tooth. The properties allocated to the materials under investigation are critical to the validity of FEAs, since each element is assigned specific values that affect the results.
Different researchers have assumed markedly different physical properties of dental tissues, such as enamel.  Some have considered enamel to be an isotropic material in which properties are similar in all directions. , However, Spears  provided evidence that enamel should be considered as anisotropic, rather than isotropic, as it is suspected of having different physical properties in different directions. This is an important issue because when enamel is considered to be anisotropic, the tooth seems to be better able to cope with loading. Not only are the resultant stresses of lower magnitude, but they are also preferably transferred into dentin, which tolerates tensile stress better than the enamel.
Linear static models have been employed extensively in FEM studies. A constant elastic modulus that represents the linear stress-strain relationship of a material is input into a programmer. However, the validity of a linear analysis is questionable as the intra-oral environment gives rise to non-linearities, such as material non-linearities, changing interrelation of objects, and geometric non-linearities. Material nonlinearities cause the stiffness of a structure to change with different load levels. 
The dynamic behavior of the periodontal ligament (PDL) and oral soft tissues is another aspect to be considered.  PDL is a highly specialized connective tissue and is the most deformative tissue in the periodontal system allowing for tooth movement under functional loads. The non-linear simulation of periodontal properties provides precision and reliability of the calculated stress and strain with a wide range of tooth movements, where as linear or bilinear properties are used as a simple solution with limited accuracy.  Qian et al.,  conducted a study by means of a combined experimental and numerical approach, to investigate the full-field distributions of displacement, stress and strain, and their evolution with loading in the entire fresh periodontium under an externally applied force. They concluded that the non-linear and time-dependent viscoelasticity of the PDL enables the acquisition of a full picture of detailed, realistic stress/strain fields, and deformation patterns of the entire fresh periodontium, being of essence in orthodontics and dentistry.
Non-linear FEM calculations also include the transient and residual stresses in dental materials. The determination of elastic, plastic, and viscoelastic properties of a target material often requires mechanical testing prior to FEM. Residual stresses in metals and ceramic restorations,  contraction stresses in composites,  and the prediction of permanent deformation of materials , are some of the nonlinearities to be applied and investigated.
Sliding and friction phenomena critically affect stress and strain created on the contact surfaces between teeth. This non-linear property of tooth is solved by contact analysis depending on the region of contact, load, material, and environmental factors, which are highly unpredictable. , Second, frictional response is dependent on the pair of surfaces in contact, temperature, and humidity.  Contact and friction play a major role in the mechanical behavior of the implant and prosthesis also.  These non-linearities in FEM determine the stresses generated and provide reliable results.
Interfacial stress in restorations is another aspect to be considered. In linear analysis, interfacial surface between tooth and restoration share the same node representing perfect bond. This conventional approach occasionally leads to erroneous interpretation of FEM results. In reality, even when a good bond is obtained, micro cracks or debonding can develop at sites where inter-facial stress exceeds bond strength. This can lead to change in the stress distribution and interfacial failure. Many commercial finite element programs overcome this problem by using an average nodal stress approach. The use of average nodal stresses to interprete interfacial phenomena is unrealistic, because the results are too heavily biased by the values of the materials modulus forming the interface. To predict failure risk of a bonded tooth-restoration interface, it is essential to use non-linear simulation. ,
The non-linear FEM has become an increasingly powerful approach to predict stress and strain within the structures in a realistic situation that cannot be solved by conventional linear static models. This method in dental applications was reported in the literature recently; however, the validity and reliability have not been sufficiently established.
| General Applications of FEM|| |
FEM is widely applied in commercial fields because of its incredible precision in obtaining results. Its applications have also increased because of decreasing cost and increasing technical upgradation of the computer.
- FEM is used for the description of form changes in biological structures (morphometrics), particularly in the area of growth and development.
- FEM and other related morphometric methods, such as the macro-element or the boundary integral equation method, are useful for the assessment of complex shape changes.
- The knowledge of physiologic values of alveolar stresses provides a guideline reference for the design of dental implants and it is also important for the understanding of stress-related bone remodeling.
- FEM is useful with structures containing potentially complicated shapes, such as dental implants and inherent homogenous material.
- It is useful for the analysis of stresses produced in the PDL when subjected to orthodontic forces.
- It is also useful to study stress distribution in tooth in relation to different designs.
- It is used in the area of optimization of the design of dental restorations.
- It is used for investigation of stress distribution in tooth with cavity preparation.
- The type of predictive computer model described may be used to study the biomechanics of tooth movement, even though accurately assessing the effect of new appliance systems and materials without the need to go to animal or other less representative models.
- FEM technique is widely used in structural engineering.
- It is also used to predict and estimate the damages in the electrical fields.
- It is also used in optimization of sheet metal blanking process.
| Applications in Restorative Dentistry and Endodontics|| |
Many studies have been carried out to assess the behavior of tooth under occlusal load, using FEM.
Evidence to the formation of abfraction lesions has been derived from finite element studies.  In 1991, Goel et al.  investigated the stresses arising at the dentino-enamel junction (DEJ) during function and noted that the shape of the DEJ was different under working cusps than non-working cusps. The results of this study showed that tensile stresses were elevated toward cervical enamel and also that mechanical inter-locking between enamel and dentin is weaker in the cervical region than in other areas of the tooth making it susceptible to crack, which could contribute to cervical caries.
Rees and co-authors ,, used this concept to estimate the effect of repeated loading on the restoration of cervical cavities. The hypothesis suggests that continual occlusal loading produced displacements and stresses under the buccal cervical enamel and dentin, increasing crack initiation and encouraging loss of restoration. This suggests that lingual walls of teeth should be equally susceptible to cervical wear as are buccal walls, but this is not supported by clinical findings where lingual surface lesion are comparatively rare. Further FEM studies by these authors showed that exposed dentin could be eroded by acid undermining enamel causing more breakdown and increased wear. 
Ichim and co-authors  have investigated the influence of lesion shape, depth, and occlusal loading on cervical glass ionomer cement restorations. The results of these studies indicate that lesion depth and shape have no significance on restoration, and for better retention of restoration, the authors suggested occlusal readjustment of tooth contacts. Another study  by the same authors using nonlinear technique for crack propagation predicted the mechanical failure of biomaterials in clinical situations. Further studies by these authors on elastic modulus of materials concluded that more flexible materials (elastic modulus of 1 GPa) should be used for cervical restorations. 
Magne and Oganesyan  conducted one study to measure cuspal flexure of intact and restored maxillary premolars with different restorative materials (mesio-occlusal-distal porcelain, and composite-inlay restorations) and occlusal contacts (in enamel, at restoration margin, or in restorative material). They concluded that a relatively small cuspal deformation was observed in all the models. There is an increased cusp-stabilizing effect of ceramic inlays compared with composite ones.
Ausiello et al,  conducted a study using 3D-FEA to identify the adhesive lining thickness and flexibility. They concluded that application of a thin layer of more flexible adhesive (low-elastic modulus) leads to the same stress relief as thick layers of less flexible adhesive (high-elastic modulus). Coelho et al.,  conducted another study to test the hypothesis that micro-tensile bond strength values are inversely proportional to dentin-to-composite adhesive layer thickness through laboratory mechanical testing and FEA. They concluded that the hypothesis was accepted for single bond and rejected for a clearfill self-etch adhesive system.
Zarone et al.,  evaluated the influence of tooth preparation design on the stress distribution and localization of critical sites in maxillary central incisor restored by means of alumina porcelain veneers under functional loading. They concluded that when restoring a tooth by means of porcelain veneers, the chamfer with palatal overlap preparation better restores the natural stress distribution under load than the window technique.
Ausiello et al.,  investigated the effect of differences in the resin-cement elastic modulus on stress-transmission to ceramic or resin-based composite inlay-restored class-II MOD cavities during vertical occlusal loading. They concluded that indirect composite resin-inlays performed better in terms of stress dissipation. Glass-ceramic inlays transferred stresses to the dentinal walls and depending on its rigidity, to the resin-cement and the adhesive layers. For high cement layer modulus values, the ceramic restorations were not able to redistribute the stresses properly into the cavity. However, stress-redistribution did occur with the composite resin-inlays.
Silva et al.,  evaluated stress distribution on endodontically treated maxillary central incisors that have been restored with different prefabricated posts: 4 metallic posts and 1fiberglass post. They concluded that fiber posts show more homogenous stress distribution than metallic posts. The post material seems to be more relevant for the stress distribution in endodontically treated teeth than the post's external configuration. One of the drawbacks of their study was that, they considered all the materials and structures as elastic, isotropic, homogenous, and linear except the fiberglass post, which was considered orthotropic.
Zhou et al.,  evaluated the stress distribution of mandibular second premolar restored with fiber post-core of different shapes and diameters when loads were applied in the vertical and lateral directions. They concluded that irrespective of the shapes, there was no significant change in the stress distribution with increase in post diameter. Under lateral load, the cone post and trapezium post created the least increased range of maximum stress than that under the vertical load. Cone and trapezium fiber posts are the ideal designs to restore the residual crowns and roots. Necchi et al.,  conducted a computational study to check the interaction between the rotating instrument and differently shaped root canals during the insertion and removal procedure using FEA. They concluded that the radius and the position of the canal curvature are the most critical parameters that determined the stress in the instrument with higher stress levels being produced by decreasing the radius and moving from the apical to the mid-root position. The most demanding working conditions were observed in canals with sharp curves, especially in areas where the instruments had larger diameters. To prevent possible damage to instruments and fracture, it is advised that the instruments should be discarded following their use in such canals.
Subramaniam et al.,  analyzed bending and torsional stresses in 2 simulated models of nickel-titanium rotary instruments (Protaper and Profile) using the finite element model. They found that under equal loads, the Protaper model showed uniform stress distribution and less elasticity compared with Profile model. Few shortcomings of their study were that they ignored the variations in taper of the Protaper and Profile instruments and also non-linear mechanical behavior of the Ni-Ti material was not considered.
Kim et al.,  compared the stress distribution during simulated root canal shaping and estimated the residual stress thereafter for some nickel-titanium rotary instruments using a 3D finite-element package, taking into account the non-linear mechanical behavior of the nickel-titanium material. They concluded that the original Protaper design showed the greatest pull in the apical direction and the highest reaction torque from the root canal wall, whereas Profile showed the least. In Protaper, stresses were concentrated at the cutting edge, and the residual stress reached a level close to the critical stress for phase transformation of the material. The residual stress was highest in Protaper followed by Protaper Universal and Profile.
Hong et al.,  analyzed the stress variations of root canal wall that resulted from vertical and lateral condensation. Stress of root canal wall caused by vertical condensation was higher than that by lateral condensation on the same loading condition. They concluded that lateral condensation will not bring on vertical root fracture directly, but over-force and improper operation are both dangerous that give rise to vertical root fracture with any method.
Er et al., conducted a study to determine the distribution and level of temperature, in a model of a maxillary canine, the surrounding periodontal tissues, and the bones, during a System B heat obturation technique simulation using a 3D-FEA. They concluded that the simulation of System B technique created no potentially harmful levels of temperature throughout the maxillary canine model.
| Conclusion|| |
FEM has proved to be less time-consuming process in experimental research. Even though small differences may remain between reality and FEM, the numerical system is able to reveal the otherwise inaccessible stress distribution with-in the tooth-restoration complex. Further development of non-linear FEM solutions is encouraged to gain a wide range of mechanical solutions that would be beneficial for dental and oral health science.
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Department of Conservative Dentistry & Endodontics, The Oxford Dental College & Hospital, Hosur Road, Bangalore, Karnataka
Source of Support: None, Conflict of Interest: None
[Figure 1], [Figure 2], [Figure 3], [Figure 4]
[Table 1], [Table 2]
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