The finite element method
The discretized elements can take a range of shapes and dimensions. From a beam element (1D), to a rectangle (2D), to finally a tetrahedron (3D), different element shapes will have their own corresponding shape functions. These shape functions are used to ‘ interpolate the solution between the discrete evaluated nodal values. ’ After determining the element used and its shape function, we scrutinize each individual node. 6 Depending on the types of analysis that is conducted using this model, the governing equation will be different for different analyses. For example, in a static problem, the governing equations are to be derived from equilibrium and material behaviour conditions, which form a stiffness matrix and its load vectors; on the other hand, for problems related to fluid, heat, etc., deriving the respective element equation is very different from a static scenario. 7 By definition, governing equations are mathematical expressions or equations that represent the most detailed and fundamental model of physics principles in a given system, and these equations usually take the form of partial differential equations. 8 As partial differential equations, the governing equations are usually in their strong form, and sometimes they are discontinuous functions, for which taking a derivative of a discontinuous function is not possible. However, through the use of weak formulation — this is usually achieved by multiplying the original PDE with a test function, which is differentiable — the PDE is transformed into an integral. While reducing the order of the derivative, the weak formulation can introduce boundary conditions that accurately represent the domain of analysis and allow the governing equation to be more suitable for numerical approximation. 9 To be specific, weak formulation is a process that transforms equations to hold solutions with respect only to the test function, rather than conditions being met at every point. This allows using the concept of linear algebra to tackle partial derivative, significantly simplifying the equations for approximation. For example, the governing equation, Euler-Bernoulli equation, of the deflection of a beam is given:
𝑑 2 𝑑𝑥 2
𝑑 2 𝑦 𝑑𝑥 2
(𝐸𝐼
)= 𝑞(𝑥)
However, solving the ‘ strong form ’ of the equation is only limited to simple elements. To generalize the solution, weak formulation is used; to be specific, the Galerkin method, an approximation that uses the
ns%20to%20complex%20problems. Consulted: 3/8/2024; see also Radhakrishnan, H. How to Mesh and Simulate Welds with Ansys Mechanical . https://www.ansys.com/blog/how-to-simulate-welds. Consulted: 3/8/2024. 6 What Are Shape Functions In FEA – And How Are They Derived? https://www.fidelisfea.com/post/what-are-shape-functions-in-fea-and-how-are-they- derived#:~:text=Shape%20functions%20are%20the%20backbone,of%20computer%20solvable%20algebraic% 20equations. Consulted: 3/8/2024. 7 The Efficient Engineer. Understanding the Finite Element Method. https://www.youtube.com/watch?v=GHjopp47vvQ&t=445s. Consulted: 3/8/2024. See also Murad (n. 3). 8 Byjus. Partial Differential Equation, https://byjus.com/maths/partial-differential-equation/. Consulted: 3/8/2024. 9 Chien Liu. A Brief Introduction to the Weak Form. https://www.comsol.com/blogs/brief-introduction-weak- form#:~:text=The%20weak%20formulation%20turns%20a,specifying%20fluxes%20at%20the%20boundaries. Consulted: 5/8/2024.
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