The finite element method
In simple terms, the finite element method is a numerical technique that uses computational power to approximate solutions for complex real-world problems. Although physical problems can be expressed through a mathematical governing equation, these governing equations can range from simple algebraic equations to complex differential equations. 1 And most of the time, these intricate differential equations cannot be solved using conventional methods; therefore, engineers use the finite element method to transform these differential equations into systems of linear equations, whose result can be approximated through the use of computational power. 2 With that being said, the apparent difficulty of solving systems of linear systems is left in the disposal of computation, but the assembling of the equations is no easy task. Generally, the manual process of finite element analysis can be split into largely eight steps: discretization from the continuum, determining element types for the model, corresponding shape functions for each element, forming the governing equations (usually in partial differential equations), transforming these governing equations into the weak form, assembling a global matrix that represents the global domain of the problem, applying boundary conditions, and finally, letting the computers do the work. Much as any engineer would tackle an intricate problem, simplifying a complex model into different components is the optimum approach. Likewise, given its name, ‘ the finite element method ’ , the method discretizes the complex model into finite elements, to which each element is connected at nodes. 3 This collection of finite elements that represents the whole structure is called mesh. 4 The discretization ’s function is to transfer a continuous system, which possesses infinite degrees of freedom, to a discrete model, which has finite degrees of freedom, so that numerical methods can be implemented to approximate solutions. This transformation also turns partial differential equations into ordinary differential equations, simplifying the problem. Along with discretization, initial conditions are sometimes also introduced into the method, depending on the analysis. 5 1 The Finite Element Method (FEM ). https://www.comsol.com/multiphysics/finite-element- method?parent=physics-pdes-numerical-042-62. Consulted: 31/7/2024. See also Nikishkov, G. (2001) ‘Introduction to the Finite Element Method’, UCLA at http://nliebeaux.free.fr/ressources/introfem.pdf . Consulted: 31/7/2024. 2 What is Finite Element Analysis (FEA)? https://www.ansys.com/simulation-topics/what-is-finite-element- analysis#:~:text=FEA%20is%20commonly%20used%20in,bridges%2C%20buildings%2C%20and%20dams. Consulted: 31/7/2024. 3 Murad, J. The Finite Element Method (FEM) – A Beginner's Guide. https://www.jousefmurad.com/fem/the-finite-element-method-beginners- guide/#:~:text=Generally%20speaking%2C%20boundary%20conditions%20(BCs,set%20of%20conditions%20 is%20known. Consulted: 1/8/2024. 4 Jiacheng (JC) Sun. Basic Finite Element Mesh Explained. https://www.midasbridge.com/en/blog/1d-2d-3d- element-comparison-in- fem#:~:text=Meshing%20is%20the%20process%20of,connectivity%20at%20the%20element%20boundaries. Consulted: 3/8/2024. 5 Study Smarter. Discretization . https://www.studysmarter.co.uk/explanations/engineering/solid- mechanics/discretization/#:~:text=Discretization%20in%20the%20finite%20element,approximate%20solutio
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