A Comparative Analysis of IGAFEM and Traditional FEM in Structural Mechanics

Getting Started with IGAFEM: A Guide to Isogeometric Analysis in MATLAB

Isogeometric Analysis (IGA) bridges the gap between Computer-Aided Design (CAD) and Finite Element Analysis (FEA). By using the same mathematical basis—typically Non-Uniform Rational B-Splines (NURBS)—for both geometric modeling and analysis, IGA eliminates the need for time-consuming mesh generation.

For researchers and students looking to understand the mechanics of IGA, IGAFEM is a widely used, open-source MATLAB code repository. This guide will walk you through the fundamentals of IGA and show you how to get started with the IGAFEM toolbox. What is IGAFEM?

IGAFEM is an educational MATLAB implementation of Isogeometric Analysis originally developed by researchers to demystify IGA concepts. Unlike complex commercial software, IGAFEM provides a transparent, script-based environment where you can see exactly how NURBS basis functions, knot vectors, and control points are translated into stiffness matrices and displacement vectors. Key Concepts of IGA

Before diving into the MATLAB code, it is essential to understand the three core pillars of IGA:

Knot Vectors: A sequence of non-decreasing numbers that divide the parametric space into elements.

Control Points: Points that dictate the shape of the geometry, acting similarly to nodes in traditional FEA but not necessarily lying on the physical mesh.

Basis Functions: B-spline or NURBS functions used to interpolate both the CAD geometry and the unknown field variables (like displacements). Setting Up IGAFEM in MATLAB 1. Download the Code

You can download the IGAFEM source code from its official repository (often hosted on GitHub or open-source scientific repositories). 2. Configure the MATLAB Path

Extract the downloaded folder to your preferred directory. Open MATLAB and add the folder (and its subfolders) to your search path: addpath(genpath(‘C:\path\to\IGAFEM’)); savepath; Use code with caution. Step-by-Step Workflow of an IGAFEM Script

A typical IGAFEM simulation follows a structured pipeline. Below is the breakdown of how a 1D or 2D elasticity problem is solved. Step 1: Define the Geometry (NURBS Data)

Instead of an element connectivity matrix, you define the knot vectors and control points.

% Example: 1D Knot vector Xi = [0 0 0 0.5 1 1 1]; % Control points (Coordinates and weights) controlPoints = [0 0 1; 1 0 1; 2 1 1; 3 0 1]; p = 2; % Polynomial degree Use code with caution. Step 2: Initialize Numerical Integration (Gauss Quadrature)

IGAFEM uses Gauss-Legendre quadrature to integrate the stiffness matrix over the elements defined by the knot spans. [gaussWeights, gaussPoints] = gaussQuadrature(p + 1); Use code with caution. Step 3: Assemble the Stiffness Matrix

The code loops over each knot span (element), evaluates the NURBS basis functions and their derivatives, and builds the global stiffness matrix

% Conceptual loop inside IGAFEM for e = 1:numberOfElements % Compute Jacobian and basis function derivatives % Perform numerical integration K(sctr, sctr) = K(sctr, sctr) + B’C * B * jDet * w; end Use code with caution. Step 4: Apply Boundary Conditions and Solve

Essential boundary conditions (Dirichlet) and loads (Neumann) are applied directly to the control points or boundaries. The linear system

is then solved for the unknown control point displacements ( Step 5: Post-Processing and Visualization

IGAFEM includes built-in plotting functions to visualize the exact CAD geometry, the deformed mesh, and stress distributions. plotStress2D(elementData, controlPoints, u); Use code with caution. Tips for Beginners

Start with 1D: Begin by running the 1D bars or beams examples in the repository. They are easier to debug and understand line-by-line.

Track the “Sctr” Matrix: In IGAFEM, the sctr (scatter) vector maps element degrees of freedom to global degrees of freedom. Understanding this mapping is crucial to mastering the code. Modify the Degree (

): One of the greatest advantages of IGA is easy high-order refinement. Try changing

in the sample scripts to observe how accuracy improves without changing the CAD mesh. Conclusion

IGAFEM is an invaluable tool for transitioning from traditional FEA to the world of Isogeometric Analysis. By using MATLAB’s intuitive matrix environment, it strips away the complexity of commercial CAD/CAE software, allowing you to focus purely on the elegant mathematics behind IGA.