5.3.1. Basic Concept of Finite Element Methods

The domain is divided into smaller regions called elements. These elements are often triangles or quadrilaterals in 2D problems and tetrahedra or hexahedra in 3D problems. The collection of all elements is called the mesh.

In the Finite Element Method (FEM), the first step involves breaking down a complex physical domain into smaller, manageable sections known as 'elements'. For 2D problems, these elements can take the shape of triangles or quadrilaterals, while in 3D, they can be tetrahedra or hexahedra. Collectively, all these elements make up what is referred to as the 'mesh'. This discretization is essential because it transforms a continuous problem into a finite one, allowing for numerical solutions.

Think of a large, complicated puzzle (the continuous domain) that you want to solve. To make it easier, you cut the puzzle into smaller pieces (elements). Each piece can be solved individually, making the overall task more manageable. In the same way, FEM helps to simplify complex problems by dividing them into smaller parts.

Within each element, the solution is approximated using basis functions or shape functions. These functions are typically polynomials that are defined over the elements.

Once the domain has been discretized into elements, the next step is to approximate the solution within each element. This is done using what are known as 'basis functions' or 'shape functions', which are often polynomials. These functions serve as building blocks to express the solution across the element, enabling complex geometrical shapes to be represented and solved mathematically. By using these functions, we can interpolate the solution between the nodes of the elements.

Imagine you have a piece of clay (the element) that you can mold into different shapes. To create a specific design, you can use different tools (the basis functions) to shape the clay effectively. Each tool represents a different polynomial that helps you achieve the desired shape in your clay modeling, akin to how shape functions help form the solution within each element.

The governing PDE is converted into its weak form using integration by parts. This step reduces the order of the derivatives, making it easier to solve.

After choosing approximation functions, the governing Partial Differential Equation (PDE) is converted into its 'weak form' through a process called integration by parts. This transformation serves to lower the order of the derivatives in the equation, which facilitates easier computation. The 'weak form' essentially represents the same physical principles encapsulated by the PDE but in a form that can be more readily manipulated numerically. This step is crucial because it allows FEM to handle cases where classical derivatives might not exist or be easily calculated.

Consider trying to solve a Rubik's Cube. Initially, the cube's puzzles can seem complicated (like the original PDE). By breaking down the process into steps (the integration by parts), you simplify your approach, making it easier to solve and manipulate each piece to eventually finish the cube. Similarly, FEM simplifies the complexity of PDEs to make numerical solutions feasible.

The weak form of the PDE is then discretized by integrating over the elements, leading to a system of algebraic equations that can be solved for the unknown values of the function at the nodes of the mesh.

Following the formulation of the weak form, the next phase involves discretization over the elements. This is achieved by integrating the weak form across each element, generating a system of algebraic equations. These equations represent relationships among the values of the function at the mesh nodes, allowing for the solution to be computed iteratively. This assembling process ensures that each piece of the mesh contributes to the overall solution, leading to precise numerical results.

Imagine building a network of roads (the mesh) connecting different cities (the nodes). Each connection (equation) explains how to travel from one city to another, forming a comprehensive map of routes. By determining each connection, you can navigate from one point to any other efficiently, just as FEM allows understanding of the solution across the entire domain through its assembled equations.

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