Finite Element Method Explained in 3 Levels of Difficulty

The video introduces the Finite Element Method (FEM) as a popular numerical technique for solving differential equations. It establishes a specific 1D problem: finding a function U(x) whose second derivative is a given function F(x), with defined boundary conditions. An analytical solution for a simple case is presented to illustrate the problem before discussing the need for computational methods like FEM for more complex scenarios.

Computers operate on finite-dimensional problems, unlike functions which are infinite-dimensional. FEM addresses this by discretizing the unknown function U(x) into a finite number of 'nodes' and 'elements'. Linear 'shape functions' are used to approximate the function between nodes, reducing the problem to a finite number of unknowns (nodal values). The strong formulation of the problem is shown to be incompatible with these simple shape functions, leading to the introduction of the 'weak formulation' through integration and partial integration. This weak formulation allows the use of finite element approximations by reducing the differentiability requirements.

The video demonstrates how to substitute the finite element approximation for U(x) and test functions V(x) into the weak formulation. By using vector notation for nodal values and shape functions, the integral equation is transformed into a system of linear equations: K*U = F, where K is the system matrix, U is the vector of unknown nodal values, and F is the system vector. The calculation of components for K and F is shown, highlighting properties like symmetry and sparsity for computational efficiency. The resulting FEM solution is compared to the analytical solution for the 1D problem.

To automate integral calculations, the video introduces element-wise defined shape functions. Instead of global shape functions, each element has its own set of shape functions. This allows integrals over the entire domain to be expressed as a sum of integrals over individual elements. The concept of 'element matrices' (Ke) and 'element vectors' (Fe) is introduced. The process of 'assembly' is then explained, showing how these element-level components are combined to form the global system matrix K and system vector F, which will be identical to those obtained in Level 1.

The final level introduces the 'reference element' – a standardized element defined on a coordinate axis between -1 and 1. This reference element serves as a blueprint for all actual elements. A mapping function is defined to transform coordinates between the actual and reference elements, based on the 'isoparametric concept'. This transformation allows integrals over actual elements to be converted into integrals over the reference element, which can then be efficiently computed using numerical methods like 'Gaussian quadrature'. The video illustrates how Gaussian quadrature works and applies it to calculate entries of element matrices and vectors, demonstrating how it automates and simplifies the computation compared to manual integration.