Finite Element Method in Thermal Engineering

Week 1: Introduction to tensors, linear dependence and independence of vectors, strong form of a differential equation, primary and secondary variables in a continuum problem, concept of variational methods, illustration of Rayleigh-Ritz, least squares, collocation, Galerkin and Petrov-Galerkin methods, limitations of variational methods, development of finite element method from the basic variational method
Week 2:The Green-Gauss theorem, weak form of a differential equation, the functional, weight and shape functions, displacements or degrees-of-freedom, types of boundary conditions, domain discretization and generation of nodes, elements, structured and unstructured meshes, bar element, triangular and quadrilateral elements, Pascal’s triangle, element connectivity and assembly using IEN (Identification of Element Nodes), ID (Identification) and LM (Location Matrix) arrays
Week 3:The concepts of local and global coordinates, Galerkin finite-element method, comparison of finite element solutions of one-dimensional elliptic partial differential equations of heat conduction with solutions using alternate methods, such as, finite difference and finite volume
Week 4:Isoparametric elements, natural coordinates, coordinate transformation, numerical integration, computation of element level matrices and vectors through local coordinates and by Gauss quadrature for one and two-dimensional elliptic partial differential equations of heat conduction
Week 5:Implementation of essential, natural and convective boundary conditions for elliptic partial equations of heat conduction in one- and two-dimensions, assembly of element level matrices and vectors and generation of the global/assembled matrix equation system of linear algebraic equations, properties of the assembled matrix equation system,
Week 6:Basic matrix equation solvers for symmetric and asymmetric linear matrix equation systems, solution of the matrix equation system, concepts of preconditioners and matrix-free methods
Week 7:FEM discretization of parabolic partial differential equations of heat conduction corresponding to unsteady equations, the trapezoidal rule: implicit and explicit formulations
Week 8:Introduction to convective transport, the linear convection-diffusion equation, Galerkin formulation of the one-dimensional convection-diffusion equation using linear and quadratic elements, asymmetry in convection-diffusion matrix versus symmetry in conduction (diffusion) matrix, difference in banded structure of the coefficient matrix of convection-diffusion and diffusion matrices, solution of the convection-diffusion matrix equation system
Week 9:Limitations of Galerkin method for flow problems, illustration of odd-even decoupling of solutions in one-dimension for high Peclet number, the need for stabilization of the weak form of flow equations, the streamline-upwind Petrov Galerkin (SUPG) stabilization, Petrov-Galerkin formulation for one-dimensional convection-diffusion equation using linear bar elements
Week 10:Introduction of multiple degrees-of-freedom problems, i.e. the Navier-Stokes (N-S) equations of motion, surface traction vector, stress-divergence form of N-S equations, coupled versus segregated formulation of NavierStokes equations, Q1Q0 and Q1Q1 bilinear quadrilateral elements or staggered and collocated arrangements of primitive variables, connectivity and assembly of element level equations for multiple degrees-of-freedom problems
Week 11:Coupled Galerkin formulation of steady Navier-Stokes equations in two-dimensions using Q1Q0 elements
Week 12:Coupled Petrov-Galerkin formulation of steady Navier-Stokes equations in two-dimensions using Q1Q1 elements