Finite Element Method: Variational Methods to Computer Programming (English)

   Thursday, March 28, 2024   

Starting a new Lecture Notes Series on Finite Element Method: Variational Methods to Computer Programming

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Lecture 1: Finite Element Method: Variational Methods to Computer Programming

Lecture 2: Lec 1: Functional, First variation, Euler Lagrange equation; Several Dependent variables

Lecture 3: Lec 2: Functional with higher order derivatives; Variational statement

Lecture 4: Lec 3: Differential equation, Variational statement and Minimization problem; Rayleigh-Ritz method

Lecture 5: Lec 4: FEM steps: Explained with discrete linear springs; Gaussian Quadrature rule for integration

Lecture 6: Lec 5: Solving one Ordinary Differential Equation using Linear Finite Element

Lecture 7: Lec 6: Solving one Ordinary Differential Equation using Quadratic Finite Element

Lecture 8: Lec 7: Bar Element: Elemental equation; Matlab Implementation with Example

Lecture 9: Lec 8: Bar Element: Postprocessing; Comparison with Analytical Solution; Bar with linear springs

Lecture 10: Lec 9: Truss Element: Elemental equation; Matlab Implementation with Example

Lecture 11: Lec 10: Beam Element: Variational statement; Hermite shape function

Lecture 12: Lec 11: Beam Element: Elemental equation; Matlab implementation with Example

Lecture 13: Lec 12: Beam Element: Matlab implementation for the example with Non-uniform distributed load

Lecture 14: Lec 13: Frame Element: Derivation of elemental equation in global reference frame

Lecture 15: Lec 14: Frame Element: Matlab implementation with one Example

Lecture 16: Lec 15: Generalization of Geometry data; Stiffness matrix, Load vector formation at element level

Lecture 17: Lec 16: Generalization of Assembly, Imposition of Boundary condition and Load information

Lecture 18: Lec 17: Indicial Notation: Summation convention, Kronecker delta, Permutation symbol

Lecture 19: Lec 18: Second order tensor; Gradient, Divergence, Curl and Laplacian in Indicial notation

Lecture 20: Lec 19: Gauss Divergence theorem and its application in Heat transfer and Structural analysis

Lecture 21: Lec 20: Derivation of weak form of 2D steady-state heat conduction problem

Lecture 22: Lec 21: Triangular element, calculating element stiffness and element force vector

Lecture 23: Lec 22: Numerical example, assembly, mapping

Lecture 24: Lec 23: Numerical integration, Neumann boundary, and higher order shape functions

Lecture 25: Lec 24: Quadrilateral element, Lagrange shape functions, Serendipity elements

Lecture 26: Lec 25: Development of a MATLAB code for solving 2D steady-state heat conduction problem

Lecture 27: Lec 26: Demonstration of the MATLAB code

Lecture 28: Lec 27: Elasticity problems in two dimension and obtaining the weak form

Lecture 29: Lec 28: Deriving element stiffness matrix and element force vector, numerical example

Lecture 30: Lec 29: Development of a MATLAB code for solving planar elasticity problems

Lecture 31: Lec 30: Superconvergent Patch Recovery, error estimator, adaptive refinement

Lecture 32: Lec 31: Solving eigenvalue problem in bar and beam, writing FEM code in MATLAB

Lecture 33: Lec 32: Solving eigenvalue problem of membrane, writing FEM code in MATLAB

Lecture 34: Lec 33:Solving transient problems (parabolic type)

Lecture 35: Lec 34: Solving transient problems (hyperbolic type)

Lecture 36: Lec 35: Solving elasticity problems in 3D using FEM, Solvers