Mastering Finite Element Methods & Computational Simulations using Python

Mastering Finite Element Methods & Computational Simulations using Python
(In Association with iHUB Divyasampark IIT Roorkee)

About the Course:

Course Objectives:

• Build a strong foundation in Python programming for numerical computing and scientific visualization.

• Implement and analyse iterative linear solvers (Jacobi, Gauss–Seidel and SOR) and understand their convergence behaviour.

• Apply key Numerical discretization schemes and evaluate their relative accuracy and stability.

• To develop a strong foundation in the Finite Element Method (FEM) through the mathematical formulation and numerical implementation of 1-D engineering problems.

• To develop the ability to formulate, analyse, and implement linear two-dimensional finite element models for structural and thermal engineering problems.

• To develop a comprehensive understanding of nonlinear Finite Element Analysis of 2D structures by studying geometric and material nonlinearities, nonlinear solution techniques, elastoplastic constitutive models, yield criteria, hardening laws, & stress integration algorithms.

• To introduce learners to the fundamental principles of structural dynamics through the modelling and analysis of single-degree-of-freedom mass–spring–damper systems.

• Perform error estimation and mesh-convergence studies to assess numerical accuracy and solution reliability in FEM.

Prerequisites and eligibility:

Course Overview:

      Module 1: Python Programming Fundamentals

• Installation of Anaconda Environment, working with Jupyter notebook.
• Basics of Python Language; Python objects with details of shell/numbers/variables etc.
• Introduction to NumPy Library.
• Arithmetic and Numerical operations using NumPy
• Data Visualization and plotting using Matplotlib.

      Module 2: Numerical Schemes and Solver

• Forward, backward and central schemes, order of accuracy.
• Assessment of various schemes through python coding.
• Introduction to Iterative Solvers
• Implementation of Jacobi, Gauss-Seidel and SOR solvers using Python and study on their convergence behavior.

      Module 3: 1D Finite Element Method Fundamentals

• Governing differential equations, strong form vs. weak form, and the Principle of Virtual Work / Galerkin formulation. 
• Formulation and implementation of Dirichlet (essential) and Neumann (natural) boundary conditions.
• 1D linear and quadratic isoparametric elements; shape functions, strain-displacement matrices, and interpolation errors.
• Isoparametric mapping, Gaussian quadrature rules, and numerical integration accuracy.
• Python Implementations: 1D Axial bar under static and distributed loads, 1D Steady-state heat conduction

      Module 4:  Linear 2D Finite Element Methods

• 3-node triangular elements (CST), 4-node linear quadrilaterals (Q4), and 8-node quadratic quadrilaterals (Q8). 
• 2D Poisson’s equation formulation; Plane Stress and Plane Strain elasticity fundamentals.
• Stress and flux computation; post-processing and visualization
• Python Implementations: Plane stress analysis of a plate with a central hole (stress concentration factor evaluation). 2D Transient heat conduction with a moving heat source (e.g., welding process simulation).

      Module 5: 2D Non-Linear Finite Element Analysis

• Geometric (large deformations/strains), material (plasticity, hyperelasticity)
• Full and modified Newton-Raphson methods.
• 1D/2D Von Mises yield criteria, isotropic/kinematic hardening rules, and return-mapping stress integration algorithms (Radial Return Algorithm).
• Total Lagrangian vs. Updated Lagrangian formulations, Green-Lagrange strain tensor, and Second Piola-Kirchhoff stress tensors.
• Python Implementations: Non-linear elastoplastic analysis of a 2D cantilever beam/plate subject to cyclic loading.

      Module 6: Introduction to Dynamics

• Single-degree-of-freedom systems: mass–spring–damper models and equations of motion.
• Numerical time-integration methods for dynamic systems; introduction to explicit and implicit schemes.
• Python Implementations: Simulation of free and forced vibrations of a mass–spring–damper system and study of dynamic response under different loading conditions.

      Module 7: Error Analysis & Convergence in FEM

• Sources of FEM error, error norms
• Mesh convergence studies (h-convergence)
• Patch test & consistency check  
• Python implementation in the Stress convergence in bar problem

Contact Person: Dr. Subrat Kotoky

Email: [email protected] / [email protected]

Phone: 9085317465 / 8473874389

Expert Profiles:1.  Dr. Chaitanya Vundru (FEM)

       


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