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 Table of Contents  
Year : 2022  |  Volume : 22  |  Issue : 3  |  Page : 211-214

Finite element analysis – Concepts for knowledge and implementation in dental research

Department of Prosthodontics, Faculty of Dental Sciences, Sri Ramachandra Institute of Higher Education and Research, Chennai, Tamil Nadu, India

Date of Submission17-Jun-2022
Date of Decision28-Jun-2022
Date of Acceptance01-Jul-2022
Date of Web Publication18-Jul-2022

Correspondence Address:
Anand Kumar Vaidyanathan
Department of Prosthodontics, Faculty of Dental Sciences, Sri Ramachandra Institute of Higher Education and Research, Chennai, Tamil Nadu
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Source of Support: None, Conflict of Interest: None

DOI: 10.4103/jips.jips_299_22

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How to cite this article:
Vaidyanathan AK, Banu R F. Finite element analysis – Concepts for knowledge and implementation in dental research. J Indian Prosthodont Soc 2022;22:211-4

How to cite this URL:
Vaidyanathan AK, Banu R F. Finite element analysis – Concepts for knowledge and implementation in dental research. J Indian Prosthodont Soc [serial online] 2022 [cited 2022 Aug 11];22:211-4. Available from: https://www.j-ips.org/text.asp?2022/22/3/211/351279

Finite element analysis (FEA) helps to visualize the location, direction, and magnitude of the applied force, and the stress evolved the three-dimensional anatomic structure, which may not be otherwise feasible in a clinical scenario.[1] The advantage is that the physical property of the material is unchanged by the applied force, and the tests can be done multiple times until there is no skewing of the results from the mean values. FEA is frequently used in dentistry by undergraduates and postgraduates for their short-term research, and the outcome solely depends on the analyst's perspective of understanding the clinical situation. Errors could occur during the stages, from designing a virtual model to the analysis of the results, and the researcher compiles this erroneous data for submission of the outcome. Literature in dentistry has defined the application of FEA, while this editorial will enable the researcher to provide clinicians' insight into finite element research to minimize errors for higher quality results.

At the preprocessing stage, based on the clinical scenario, the researcher could choose a one-dimensional (1D) element for a lengthy and thin structure such as wires, a two-dimensional (2D) element for plate or shell-like structures, and a three-dimension (3D) element for a structure that is solid and has complex geometry that cannot be simplified for analysis [Figure 1].[2] The researcher also needs to understand that analyzing the property of an implant in 1D or 2D could reveal a false outcome as the vertical impact of force at a point could create shear stress on another point of a solid body. Similarly, assessing a maxillary major connector in three dimensions could add an extra element to the dimension that may hamper the speed of processing. However, the false outcome will be minimal in a three-dimensional model design when real-time data are captured through cone-beam computed tomography. Planning to analyze a 2D element in 3D space gives a better in-depth output, provided the design requires the third space. Especially when assumptions are made for idealized geometry that would enable us to understand the material better at each cross sections.
Figure 1: (a) 1D first order; (b) 1D second order; (c) 2D tri 3 (first order); (d) 2D tri 6 (second order); (e) 2D quad 4 (first order); (f) 2D quad 8 (second order); (g) 3D tet 4 (first order); (h) 3D tet 10 (second order); (i) 3D hex 8 (first order); (j) 3D hex 20 (second order), 1D: One dimensional, 2D: Two dimensional, 3D: Three dimensional

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Elastic modulus and Poisson's ratio of an element are available in the literature for isotropic and anisotropic models based on the complex anatomy of the human body or the device. Isotropic mineral crystals have the same and consistent characteristics throughout the material due to their uniform composition, and they are not direction-dimension dependent. Whereas, human tissues are anisotropic, with the mineral crystal related to the compositional differences having varying properties in different orientations of the mineral surface. These are direction-dimension dependent and hence, the researcher should assign appropriate values accordingly to the element.

An element can be meshed into triangular elements (TRI/TET), or more robust quadratic elements (QUAD/HEX) based on the first order or second order or third order. A 2D or 3D model can be a linear element, also called first order, whereas a quadratic element is referred to as second order.[3] The first order has the nodes only at the ends of a line [Figure 1]a, whereas the second order and third order have additional nodes in between the lines to help in capturing the deformation in detail during an analysis [Figure 1]b. The presence of nodes at the center in second-order or third-order elements helps to model either a concave or convex shape and can show deformation on mapping to curvilinear geometry with ease. The second-order or third-order elements are advantageous when handling nonlinear elastoplastic or hyperelastic materials. However, the computational effort and duration taken for the output are higher than when using first-order elements. Most commonly, first-order elements are used for meshing, but high precision is obtained only with higher orders, especially when bending movement needs to be analyzed.

Nodes decide the mesh density that confirms the accuracy to yield a better output at minimal time consumption. A 2D mesh could be triangular with three nodes (first order) [Figure 1]c, or six nodes (second order) [Figure 1]d, or quadrilateral with four nodes (first order) [Figure 1]e or eight nodes (second order) [Figure 1]f.[4],[5] The triangular nodes are not better choices compared to the latter as they may give less detailed output, and a quadrilateral mesh is preferred, especially the eight nodes when modeling a 2D element for better strain output. A 3D mesh could be tetrahedral with four nodes (first order) [Figure 1]g, 10 nodes (second order) [Figure 1]h, or hexahedron with eight nodes (first order) [Figure 1]i, 20 nodes (second order) [Figure 1]j.[4],[5] Similar to the 2D element, the second-order element is better for a 3D mesh, but computing a hex with a 20-node model would consume time. Hybrid meshing (hex-pyram-tetra) is a very special option where different mesh densities are used simultaneously in one single model, and not all software supports its application.[6] It is better to begin with a coarse mesh to monitor the time and accuracy, and later the mesh fineness can be improved. The mesh refinement methods include the h-method (MesH), which refers to reducing the size of the mesh; the p-method, which relates to an increase of the polynomial order in the element that is good for regions with a low-stress gradient; and the r-method, which relocates the position of a node.[7] The combinations of the methods, for example, “hr” is good for regions with a large stress gradient, “hp” for a low-stress gradient.[7]

The processing stage of biological structures is affected by the dynamic nature, and the models should be fixed by setting the boundary conditions of the environment to avoid inconsistent results. The researcher needs to decide whether to analyze the 2D element in 2D space or in 3D space or the 3D element in 3D space [Figure 2], as each inclusion of space adds extra data to the dimension. To avoid or minimize discretion errors in this process, the researchers can use higher-order shape functions or smaller elements. A 2D object should be fixed and stable in 3D space with the addition of a support element to the model [Figure 2]c.[8] Evaluating the 2D object in 2D space would decrease the added data and fastens the outcome as the degree of freedom is reduced. However, the deformation (or buckling) of a 2D element out of space would not be considered [Figure 3]. For example, when an endodontic post material is evaluated at the point of impact for vertical loading in 2D space, the oblique loading beyond the point of impact may not be measurable. Hence, a 3D space is required that would increase the number of elements, and delay the process. Moreover, misinterpretation of multiple numerical data would also occur if the elements are not defined precisely.
Figure 2: (a) 2D structure in 2D space; (b) 3D structure in 3D space; (c) 2D structure in 3D space, 2D: Two dimension, 3D: Three dimension

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Figure 3: The buckling effect was seen in load application on 2D structure in 3D space, 1D: One dimension, 2D: Two dimension, 3D: Three dimension

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Load applications in FEA are determined based on specific events such as a thermal cycle, shock from a drop, vibration, or static flexure.[9] The researchers should not simplify complex loads or reduce the number of loads from the optimum required for a test. Similarly, if the load is applied at two different regions which are at sufficiently large distances from the point of measurement, the effect of the load becomes insignificantly small at the area of analysis and may not represent the clinical situation. A static load would be time independent, whereas a dynamic load is time dependent. In a static load, the board-level displacement and elastic stresses/strains are analyzed and not the creep strains/energies, which is time-dependent plastic deformation. It is also essential to capture the inertial effect of an object in a dynamic load since the counteracting force that inhibits the motion of the object would alter the output. Hence, for a dynamic load, as in the investigation of solder fatigue, creep properties must be included, which are modeled by simulated cycles.

The researcher should perform a patch or single element test to determine the response to different loads before final processing and analyze different states of stress and strain to check the validity and inconsistency of the designed model.[8] The researcher should choose plane strain to simulate the interior of a very thick component loaded in a single plane [Figure 4], and plane stress to simulate a very thin component and also the surface behavior of thick plates.[10] The presence of too small or large element aspect ratios (width and length ratio in the FEA model) and corner angles, warped elements, large Poisson's ratio, and increased curved shell element spans need to be analyzed before receiving the final output.[7] The researcher should ask the analyst to verify the coincident nodes (i.e., two nodes meeting at the same point when two elements are joined) and a proper mesh application along with the material property. An inbuilt automated adaptive solution in the software proceeds by refining the mesh until the maximum error is below the optimal limit.[8] The current research on polyhedral meshing and mesh-less (or mesh-free) analysis for reducing the meshing time is being experimented.[6] They are also highly accurate, with fewer degrees of freedom.
Figure 4: Plane strain for thick components assessed at x and y plane

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Postprocessing stage to reduce the numerical error, a modified nonlinear iterative can be formulated; and to reduce the computing time, an adaptive time step size algorithm can be undertaken.[11] Integration errors caused by Gauss integration lead to numerical instabilities. To counteract, increasing the Gauss points is an option, but it is expensive.[7] Similarly, rounding off numerical data (such as 1.1567–1.2) for one element leads to the cumulative effect of appreciable error throughout the model. Hence, rounding should be done with caution.[7] The principle stresses should be zero at an unloaded boundary and the displayed stress, for example, principle stress versus shear stress versus Von Mises stress, should be closer to the standard model in literature or less skewed during the repetition of loading conditions. There should not be large displacements to cause a change in the force directions, especially when soft-tissue or periodontal ligament models are used, and this could be prevented by considering a nonlinear analysis. Nonlinear analysis helps in processing the dynamic behavior of the model like tooth movement and transient and residual stresses in dental materials.[6]

Innovative proposals that cannot be done through in vivo and in vitro studies make us rely on FEA, which has its own limitations in medical research to simulate a biomechanical environment. The limitations can be limited only if the researcher has a greater role in understanding the FEA model. Furthermore, the outcome of FEA should be accompanied by in vitro and/or in vivo models to confirm that the published design is effective.

  References Top

Trivedi S. Finite element analysis: A boon to dentistry. J Oral Biol Craniofac Res 2014;4:200-3.  Back to cited text no. 1
Wai CM, Rivai A, Bapokutty O. Modelling optimization involving different types of elements in finite element analysis. IOP Conf Ser Mater Sci Eng 2013;50:012036.  Back to cited text no. 2
Skotny L. 28 February 2019, What are the Types of Elements Used in FEA? Meshing. Available from: https://enterfea.com/what-are-the-types-of-elements-used-in-fea/. [Last accessed on 2022 Jun 01].  Back to cited text no. 3
Berlioz A, Trompette P. Solid Mechanics Using The Finite Element Method. USA: John Wiley & Sons Inc.; 2010.  Back to cited text no. 4
Rugarli P. Structural Analysis with Finite Elements. UK: Thomas Telford Limited; 2010.  Back to cited text no. 5
Bandela V, Kanaparthi S. Finite element analysis and its applications in dentistry. In: Baccouch M, editor. Finite Element Methods and Their Applications. London: IntechOpen; 2020. Available from: https://www.intechopen.com/chapters/74006. [Last accessed on 2022 Jun 17]. [doi: 10.5772/intechopen. 94064].  Back to cited text no. 6
Murad J. Errors in FEA and Understanding Singularities (Beginners' Guide). FEA; June 14, 2021. Available from: https://www.simscale.com/blog/2016/06/errors-in-fea-and-singularities/. [Last accessed on 2021 Jun 14].  Back to cited text no. 7
Cook RD, Malkus DS, Plesha ME, Witt RJ. Concepts and Applications of Finite Element Analysis. 4th ed., Hoboken, New Jersey, U.S.: Wiley; 2001.  Back to cited text no. 8
Cyprien. Loads; July 03, 2017. Available from: https://feaforall.com/glossary/loads/#:~:text=Applying%20the%20right%20loads%20to,(4)%20the%20 time%20dependency. [Last accessed on 2022 Apr 01].  Back to cited text no. 9
Hurlston R. Plane Stress versus Plane Strain – What are They and Which Should You Choose? July 15, 2021, Available from: https://www.fidelisfea.com/post/plane-stress-versus-plane-strain-what-are-they-and-which-should-you-choose. [Last accessed on 2022 Apr 01].  Back to cited text no. 10
Ho SL, Niu S, Fu WN. Reduction of numerical errors of time-stepping finite element analysis for dynamic simulation of electric machines. IEEE Trans Appl Supercond 2010;20:1864-8.  Back to cited text no. 11


  [Figure 1], [Figure 2], [Figure 3], [Figure 4]


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