An Outline to Finite Element Method

“We are all connected To each other biologically. To the Earth, chemically. To the rest of the Universe atomically” — Neil deGrasse Tyson

Hi, my besties!Thank You, as always, for ringing by and giving this cutie your love. Now, who wasn't aware of the impact of mathematics in world applications? World basically runs on mathematics-subtle flex ;} And for problems devoid of accurate analytic solutions calls for numerical approximations, a marvel as it finds its application everywhere from terrestrial to galactic realms. Being the first on the southern pole of the Moon gives me extra pride and another reason to flex as an Indian(The list is countless ngl), so today, I'll bring something special for y’all. Now, diving to complex matters, have you thought about how on earth we can solve partial derivative equations? Complex problems with irregular boundaries? So learn this new method then!

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FEM or Finite Element Method is a numerical technique that numerically integrates and further solves partial differential equations or can be formulated as functional minimization. A domain of interest is represented a an assembly of finite elements.

This method is a very general method, hence quite flexible to applications. It is highly systematic so we can easily automate it and structures as well so it can be used in almost all aspects of science and technology.

Before we dive deeper, we need to know that to get a correct solution, the problem should be well-posed.

A problem is well posed if

• A solution exists for the problem
• The solution must be unique
• The solution depends continuously on the data ie small change in data gives a small change in solutions.

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How FEM works?

To summarize in general terms how fem works :

• Discretize the region into finite elements: Break the domain into subdomains.
• Select the interpolation functions: Interpolation functions are basically shape functions.

Interpolation or shape functions are fundamental components of the finite element method (FEM). They are mathematical functions used to approximate the behavior of the unknowns (such as displacement, temperature, pressure, etc.) within each finite element in a FEM analysis. These functions help to bridge the gap between the continuous problem domain and the discrete elements, allowing us to express the behavior of the problem within each element.

Here’s how interpolation functions work:

1. Element Discretization: In FEM, the problem domain is divided into smaller elements (triangles, quadrilaterals, tetrahedra, hexahedra, etc.). These elements are geometric shapes that collectively make up the entire problem domain.
2. Nodal Values: The unknown variables are defined at each element's nodes (or vertices). For example, in a structural analysis, the displacement of a structure might be defined at the nodes.
3. Interpolation Functions: Interpolation functions are chosen based on the type of element used and the desired level of accuracy. These functions are designed to satisfy certain properties:

• They take the value of 1 at the node where the unknown is being interpolated (node i) and 0 at all other nodes.
• They are usually polynomials that can capture the variations in the unknowns across the element.

1. Interpolation Process: The values of the unknowns within an element are expressed as a linear combination of the nodal values and the interpolation functions. This is known as the interpolation process.

Mathematically, if we have nodal values u_i​ at node i, and ϕ_i​ is the interpolation function associated with node i, then the interpolated value u within the element can be expressed as:

Where n is the number of nodes in the element.

Different types of elements (linear, quadratic, cubic, etc.) use different interpolation functions to approximate the behavior of the unknowns. The choice of interpolation functions affects the accuracy of the finite element solution. In practical applications, the accuracy and efficiency of the analysis need to be balanced, and the appropriate type of element with its associated interpolation functions is selected accordingly.

Deriving interpolation functions involves satisfying certain conditions like partitioning unity property (the sum of interpolation functions is 1 over the entire element) and achieving the desired continuity across element boundaries.

Interpolation functions are a key component of the finite element formulation, and they play a crucial role in the accuracy and reliability of the numerical solution obtained from the analysis.

These are also known as the basis function.

• Finding the element properties: The matrix equation for the finite elements should be established relating to the nodal values of the unknown fn to other parameters. The most convenient method used for this are the variational method and the Galerkin method
• Assemble the element eqs
• Solve the global equation system
• compute additional results\

Now the question comes: What are the Finite elements we are mentioning everywhere?

Finite Elements (FE) are the blocks we break into while solving a problem using FEM. Here, we solve the mathematical problems by breaking them into smaller, more manageable parts. It is potent in solving partial differential equations (PDEs), which arise in various scientific and engineering fields, including structural mechanics, heat transfer, fluid dynamics, electromagnetics, and more.
The core idea behind finite elements is to approximate a continuous domain (or system) by discretizing it into minor, more superficial elements. These elements are usually triangles or quadrilaterals in 2D and tetrahedra or hexahedra in 3D. The problem domain is divided into these finite elements, and within each element, the behavior of the problem is approximated using a set of mathematical functions (usually polynomials). The original PDE problem is transformed into a system of algebraic equations based on the behavior of these approximations within each element.

There are many other parts of FEM that I will discuss in the next blog, stay tuned ❤ FEM is extensively used in elliptic PDE and finds its application in various complex problems in heat transfer, fluids, electromagnetic, solid mechanics, satellite problems, and many more.