The purpose of deriving displacement functions in FEM (Finite Element Method) homework is to determine the mathematical relationship between the unknown nodal displacements and the known nodal coordinates. This is essential for solving the system of equations in FEM and obtaining the solution for the problem at hand.
Shape functions in FEM are mathematical functions that are used to interpolate the values of the physical quantities (such as displacement, stress, strain) within an element. They are typically polynomials that are defined over an element and are used to approximate the actual solution of the problem.
Displacement functions in FEM are derived by applying the principle of minimum potential energy to the FEM model. This involves setting up the governing equations for the problem, applying the boundary conditions, and then solving for the unknown nodal displacements using the shape functions and the finite element method.
Some common displacement functions used in FEM are linear, quadratic, and cubic shape functions. These functions are typically used in 1D, 2D, and 3D problems respectively. In addition, there are also higher-order shape functions that can be used for more accurate solutions.
It is important to understand how to derive displacement functions in FEM because it is a fundamental concept in the FEM method. By understanding how to derive these functions, one can gain a better understanding of the underlying principles of FEM and how it can be applied to solve complex engineering problems.