The finite element method
sum of a number of trial functions of unknown coefficients to estimate the solution, is used. The Galerkin method is something much more complicated and uses other complex mathematical tools such as Lagrange elements, related to shape functions, and Hermite elements, related to interpolation. 10 After deriving the governing equations for specific elements, these equations are to be assembled into a global matrix that represents the behaviour of the whole system with respect to independent variables. The assembly of the global matrix takes into account all formulated equations at each node. It is typical to first assemble the matrix for each individual element and then combine each element’s matrix into a global matrix of the system. Depending on the number of nodes within each element and the number of degrees of freedom each individual element possesses, each matrix for a corresponding element varies in size. 11 In static situations, using the stiffness matrix of a structure to solve for its individual nodal displacement is a highly effective demonstration. Typically, the number of rows and columns for a stiffness matrix is dependent on the number of degrees of freedom that the element possesses. 12 For instance, a two-nodal beam element in two dimensions has six degrees of freedom, and its position and motion can be defined by three equilibrium equations: two for translation (denoted as X and Y for their corresponding vector directions) and one for rotation (denoted as M for the point moment along its central axis). Hence, the matrix of the whole beam element can be split into the axial stiffness matrix (2x2) and the transverse/rotational stiffness matrix (4x4). 13 The matrices can be formed by enforcing the principle of equilibrium on the beam as it is a static problem. Its equilibrium equations given as
∑𝐹 𝑥 =0,
∑𝐹 𝑦 =0,
∑𝑀 =0
By treating solids like a spring that follows Hooke’s law, an equation is formed for each node:
{𝐹} = [𝑘]{𝑥}
In which the [k] matrix represents the stiffness matrix and is formulated through the equation of young’s modulus, given as:
𝐹𝐿 𝐴𝑥
𝐸 =
Rearranging gives:
𝐸𝐴 𝐿
𝐹 =
𝑥
10 Kelly. The (Galerkin) Finite Element Method. https://pkel015.connect.amazon.auckland.ac.nz/SolidMechanicsBooks/FEM/One_Dimensional/02_FE_Method .pdf. Consulted: 5/8/2024. 11 The Efficient Engineer (n. 7). 12 Dean, J. ‘Introduction to the Finite Element Method (FEM), Lecture 1 The Direct Stiffness Method and the Global Stiffness Matrix.’ Cambridge. https://www.ccg.msm.cam.ac.uk/system/files/documents/FEMOR_Lecture_1.pdf. Consulted: 10/8/2024. 13 The Efficient Engineer (n. 7).
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