The last part of this series is short. You can check out the Matlab code for implementing FEM today! I encourage you to see how the code demonstrates ideas that we studied. These include pullback to parent domain, Gauss quadrature for numerical integration, and postprocessing for visualization.
The geometries that we considered so far are simple. If we are to analyze a complex, real-world structure, we need a way to accurately model its geometry. Not to worry, we can map a simple element to a complex one using a function.
Today, we will study the isoparametric approach. It tells us how to define the function so that we can evaluate integrals fast and achieve optimal rate of convergence. It is key to making FEM practical and powerful.
Recall the continuum approach to mechanics. We view the body, the structure of our interest, as a continuous region in the vector space of . Then, we can completely describe the structure’s state by finding the set of fields that satisfies the force and moment equilibriums, the stress-strain relation (constitutive equation), and the strain-displacement relation (kinematics).
The problem is, finding the fields exactly is impossible for all but simple structures. Today, we will look at how to make the problem simpler so that we can analyze real-life structures.
In continuum mechanics, we view the body—the structure (solid or fluid) that is of our interest—as a continuous region that lives in the three-dimensional space .
The word continuous means, we assume the body to be made up of infinitely many, infinitely small points, with no space in-between (like the cloud of paint seen above), rather than discrete, finitely small atoms, with an atomistic space in-between.
The space is special in many ways (mathematically, that is), and allows us to set up and solve problems in mechanics in a rigorous manner. To keep things interesting and concise, we will replace some definitions with layman, more intuitive explanations.
Over the next few posts, I will share the writings and programs that I had created for a class in computational structural analysis—how we efficiently analyze a structure using numerical methods. The field is a subset of computational mechanics, which combines the disciplines of mathematics, computer science, and engineering.
The class consisted of juniors and seniors in mechanical or aerospace engineering. In other words, for brevity, I will assume that you have some familiarity with statics, linear algebra, and calculus. If time permits in future, I will upload my class notes to fill gaps and share more drawings.