Introduction to Finite Elements

An introduction to FEM V.S.S.Srinivas

Plan Day 1 A brief introduction to Finite Difference Method (FDM) using a simple example Introduction to the concept of Finite Element Method (FEM) using a top-down approach Weighted residuals—illustration using an example Interpolation functions—illustration using an example Comparison of the numerical solutions obtained using FEM and FDM against the analytical solution—discussion Applying boundary conditions—Natural boundary conditions Concept of element Day 2 Revisiting FEM using a bottom-up approach—The Standard Procedure Element shape functions Natural coordinates—Geometric coordinates Coordinate transformation—Jacobian Numerical integration—Gauss quadrature Nodal connectivity—assembly of element matrices—global matrix Applying boundary conditions—Essential boundary conditions Questions session

A glance at Finite Difference Method Consider a steady one dimensional heat conduction case Approximate the governing equations and boundary conditions with algebraic equivalents Impose the equivalent conditions at select locations q

Illustration by example 1 2 3 4 q

Summary Derivatives are approximated using Taylor series The resultant difference (algebraic) equations are imposed at nodes The set of linear algebraic equations are solved A solution is obtained for the approximated system of equations At the outset, Finite Element Method differs from FDM in the above aspect

Finite Element Method We approximate the solution Interpolation functions Let l=1

Finite Element Method-Galerkin Weighted Residuals Analytical solution is the exact solution for a system of differential equations We seek approximate solution when there is no exact one How do we go about it Can we satisfy the equations in an average sense? How can we improve upon the solution we are seeking

FEM-Galerkin Weighted Residuals

Analytical solution and comparison Analytical solution Comparison of all the three solutions

Other Weighted Residual Methods Least squares method Point collocation method Subdomain collocation method

Concept of assembly The total integral can be considered as the sum of integrals over a set of sub-domains In finite element terminology, they are called elements 1 2 3 1 2 3 4

Concept of Assembly Assembled matrix

Contd.. Boundary term can also be decomposed into sum of integrals over each subdomain If you notice, except at the end points, the integral cancels at every other point or node in the domain. Essentially, this is to say that the whole integral can be seen as the sum of integral over each subdomain Till now, we dissected the whole integral and saw the details. We depart at this point and resume FEM by assembling the integrals of every subdomain (element) In this process, we will visit the standard procedure of finite element method

Revisiting interpolation functions Element point of view Non-zero functions in element 1: N1, N2 Non-zero functions in element 2: N2, N3 Non-zero functions in element 3: N3, N4 For every element, the components of interpolation functions are presenting a common picture It is easy to obtain the matrix for every element and then assemble them to obtain the global matrix

Interpolation functions from an element point of view

FEM-Standard Procedure Reconsider the example discussed before, resuming from the last point of departure The integrand in the equation cannot be always analytically integrated For example, if Or k can also be a function of Temperature. What is the way out?

Element shape functions Most of the times, the integrand is not numerically integrable We resort to numerical integration then

FEM Standard Procedure- Coordinate Transformation Numerical integration, popularly known as gauss quadrature This rule is for a generic element Limits of the integration are from -1 to 1 instead of x e1 and x e2 Necessitates a coordinate transformation Old coordinates ‒G eometric coordinates New non-dimensional coordinates ‒N atural coordinates The coordinate transformation brings in a scaling factor named Jacobian

Pictorial representation-coordinate transformation Notion of isoparametric formulation Jacobian

Assembly of element matrices nodal connectivity-1D 1 2 3 4 1 2 3 1 2 Local node no. Global node no. 1 - 3, 2 – 4 3 1 - 2, 2 – 3 2 1 - 1, 2 - 2 1 Local to global Element

Nodal Connectivity-2D Global node no. Local node no. (i,j) entry in every element conductivity matrix goes to (I,J) entry in global conductivity matrix (i,j)—local node nos, (I,J)—Global node nos. 1 2 3 4 1 5 6 7 8 9 2 3 4 1 2 3 4 1 – 4, 2 – 5,3 – 8, 4 – 7 3 1 – 5, 2 – 6, 3 – 9, 4 – 8 3 2 – 2, 2 – 3,3 – 6,4 – 5, 2 1 – 1,2– 2, 3 – 5, 4 – 4 1 Local to global Element

Applying Boundary Conditions Natural or neuman boundary conditions are applied in the integral form Number of ways to impose essential (dirichlet) conditions Revisiting the example,T 4 is known, T 1 , T 2 , T 3 have to be solved Considering the assembled system of equations

Contd.. We can take any set of three equations Consider the first three equations Subtract the term associated with T 4 from both sides Solve for the unknowns

Contd.. Subtract the fourth column multiplied by T 4 from the right hand side Remove the fourth row and column Remove the fourth entry from the right hand side Solve for T 1 , T 2 , T 3 using the resulting set of linear equations Other popular methods are lagrange multiplier, penalty etc.

Summary Considered a steady state heat conduction problem as the example problem to illustrate the concepts of FDM and FEM To lay a platform for the comparison of FDM and FEM, the problem is solved using FDM Next, obtained solution using FEM. In the process, explained the important concepts Weighted residuals Integral form Interpolation functions Imposition of natural boundary conditions Notion of element Compared the FDM and FEM solutions against the analytical solution Finite element method is explained by using a dissection approach Next, the standard approach of assembly starting from the element stiffness matrices is explained Natural or intrinsic coordinates, spatial coordinates are explained Local-global nodal connectivity, gauss quadrature, applying essential boundary conditions are explained The concepts of Jacobian and Gauss Quadrature are introduced

References An Introduction to Finite Element Method, J.N.Reddy, McGraw-Hill Science Engineering Introduction to Finite Elements in Engineering (3rd Edition) by Tirupathi R. Chandrupatla and Ashok D. Belegundu, Differential equations with exact solutions: http://eqworld.ipmnet.ru

Introduction to Finite Elements

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Introduction to Finite Elements