Finite Element Method: Variational Methods to Computer Programming

Week 1:Part1: Variational Methods:Functional and Minimization of Functional; Derivation of Euler Lagrange equation: (a) First variation of Functional, (b) Delta operator Functional with (a) several dependent variables, (b) higher order derivatives; Variational statement Weak Form); Variational statement to Minimization problemRelation between Strong form, Variational statement and Minimization problem;Different approximation methods with Computer Programming: Galerkin, method, Weighted Residual method; Rayleigh Ritz methodWeek 2:Part 2. One dimensional Finite Element Analysis:Gauss Quadrature integration rules with Computer Programming; Steps involved in Finite Element Analysis; Discrete system with linear springs;Continuous systems: Finite element equation for a given differential equation Linear Element: Explaining Assembly, Solution, Post- processing with Computer Programming Quadratic element with Computer Programming: Finite element equation, Assembly, Solution, Post-processing; Comparison of Linear and Quadratic elementWeek 3:Part 3. Structural Elements in One dimensional FEM:Bar Element with Computer Programming: Variational statement from governing differential equation; Finite element equation, Element matrices, Assembly, Solution, Post-processing; Numerical example of conical bar under self-weight and axial point loads.Truss Element with Computer Programming: Orthogonal matrix, Element matrices, Assembly, Solution, Post-processing; Numerical exampleWeek 4:Beam Formulation: Variational statement from governing differential equation; Boundary terms; Hermite shape functions for beam element Beam Element with Computer Programming: Finite element equation, Element matrices, Assembly, Solution, Post-processing, Implementing arbitrary distributive load; Numerical exampleWeek 5:Frame Element with Computer Programming: Orthogonal matrix, Finite element equation; Element matrices, Assembly, Solution, Post- processing; Numerical examplePart 4. Generalized 1D Finite Element code in Computer Programming:Step by step generalization for any no. of elements, nodes, any order Gaussian quadrature;Generalization of Assembly using connectivity data; Generalization of loading and imposition of boundary condition; Generalization of Post-processing using connectivity data;Week 6:Part 5. Brief background of Tensor calculus:Indicial Notation: Summation convention, Kronecker delta and permutation symbol, epsilon-delta identity; Gradient, Divergence, Curl, Laplacian; Gauss-divergence theorem: different formsWeek 7 & 8 :Part 6. Two dimensional Scalar field problems:2D Steady State Heat Conduction Problem, obtaining weakform, introduction to triangular and quadrilateral elements, deriving element stiffness matrix and force vector, incorporating different boundary conditions, numerical example.Computer implementation: obtaining connectivity and coordinate matrix, implementing numerical integration, obtaining global stiffness matrix and global force vector, incorporating boundary conditions and finally post-processing.Week 8 & 9:Part 7. Two dimensional Vector field problems:2D elasticity problem, obtaining weak form, introduction to triangular and quadrilateral elements, deriving element stiffness matrix and force vector, incorporating different boundary conditions, numerical example.Iso-parametric, sub-parametric and super-parametric elements Computer implementation: a vivid layout of a generic code will be discussed Convergence, Adaptive meshing, Hanging nodes, Post- processing, Extension to three dimensional problems Axisymmetric Problems: Formulation and numerical examplesWeek 10:Part 8. Eigen value problemsAxial vibration of rod (1D), formulation and implementation Transverse vibration of beams (2D), formulation and implementationWeek 11:Part 9. Transient problem in 1D & 2D Scalar Valued ProblemsTransient heat transfer problems, discretization in time : method of lines and Rothe method, Formulation and Computer implementationsWeek 12: Choice of solvers: Direct and iterative solvers

Thanks to the support from MathWorks, enrolled students have access to MATLAB for the duration of the course.