Introduction to Finite Element MethodJayadeep U. B.M.E.D., NIT Calicut

Department of Mechanical Engineering, National Institute of Technology Calicut

IntroductionLecture - 01The finite element method is a numerical procedure for obtaining approximate solutions for problems in continuous systems.

The solution means finding the distribution (functions) of system parameters (variables).

In general these distributions should satisfy various conditions, like the equilibrium equations in solid mechanics, which become the system governing equations.

A fundamental step in FEM is the discretization of the domain (body) into simpler shapes (like rectangles in 2D), which are called the finite elements.

The whole method can be summarized as conversion of the system governing equations (Differential Equations) into a set of algebraic relations and solving them.

Department of Mechanical Engineering, National Institute of Technology Calicut

Why Finite Element Method (FEM)?Equilibrium Equations:

An example: The generalized elasticity problemLecture - 01We need to find the stresses, strains and displacements in the engine block shown, under the action of various surface and body loads:

Department of Mechanical Engineering, National Institute of Technology Calicut

Generalized elasticity problem contd.Lecture - 01The stress-strain relations:

Department of Mechanical Engineering, National Institute of Technology Calicut

Generalized elasticity problem contd.Lecture - 01The strain-displacement relations:

The boundary conditions:

Department of Mechanical Engineering, National Institute of Technology Calicut

Generalized elasticity problem contd.Lecture - 01Hence there are 15 equations (3 equilibrium equations, 6 stress-strain relations & 6 strain-displacement relations), which can be solved (at least in theory!!!) for the 15 unknowns (6 stresses, 6 strains & 3 displacements).

The solution means finding the stress, strain and displacement distributions within the body, satisfying the above differential equations at all interior points and boundary conditions on the surface points.

Therefore, even with the assumptions like linear elasticity, isotropy etc., this problem can NOT be solved exactly, except for some special, simplified cases.

This necessitates the use of numerical procedures for getting approximate solutions, and FEM is one of the most powerful methods available in todays world.

Department of Mechanical Engineering, National Institute of Technology Calicut

The Finite Element MethodInfinitesimalFiniteInfiniteWhy the term Finite Element?Leads to Differential Equations (Recall the derivation of equm eqns in elasticity)Used to create elements for problems with infinite / semi-infinite domains (e.g.: underground explosions) Could be a topic for term paper!!!Leads to System of algebraic equations Can be solved using computers (No elaborations here The complete course is about it!!!).Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

The Finite Element Method contd. Problem at hand, say P, is not solvable Replace it with another problem Pn, with solution Sn.

The Fundamental Idea (Common to many Numerical Methods):Solution Sn should be close to the exact solution S within acceptable limits (Concepts of Error & Tolerance).

It would be great, if we can obtain Sn with varying levels of accuracy may be at different costs of time, resources (Convergence Studies).

In FEM, we achieve this by two main steps: Discretize (mesh) the domain into geometrically simple shapes the Finite Elements. Piece-wise approximation of the variable of interest (say, Temperature Profile) over these finite elements.

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

The Fundamental Idea - DiscretizationFind the second moment of area of the I beam cross-section below:

You have already used Discretization?!!By Symmetry the Centroidal axis is known.

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Discretization contd. By Parallel Axis Theorem:

Net Second Moment of Area:

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

Discretization contd. Problem: Find the second moment of area of the I beam cross-section.

A Closer look at the Procedure:Assumptions: By Symmetry the Centroidal axis is known.

Discretize the geometry into geometrically simple shapes finite elements?

Perform element level calculations.

Co-ordinate Transformation.

Assembly into the Global System. (In this case, the required results are obtained at this step itself).

Lecture - 01

Department of Mechanical Engineering, National Institute of Technology Calicut

What is FEM?One of the numerical methods to obtain Approximate Solutions to Physical/Engg. Problems.

The domain of the problem is (hypothetically) divided into sub-domains (called Finite Elements).

Elements are inter-connected at points (called Nodes).

Over each finite element, the variable of interest (Primary Variable) is approximated using Simple Functions (generally Polynomials, but not necessarily!).

Governing equations are written for each element leading to algebraic equations in terms of Nodal Variables.

Assembly process to get the Global Equations.

Boundary conditions (b.c.s) are applied and Solved for Nodal Variables.

Secondary Variables (if any!) are calculated.

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Why is it so Widely Used?Versatile A lot of problems solved by same methodology.

Advantages of FEM:Convergence Studies are possible by increasing the number of sob-domains or elements Mesh Refinement Studies or by increasing the order of functions used in approximation.

Strong mathematical foundation Aids in Error Analysis.

Very much linked to computers became much cheaper with improvement in computational resources.

The complete process needs to be repeated for small changes in the domain.

Mistakes are easily made Like applying wrong boundary conditions.

Disadvantages of FEM:Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Origins of FEMClosely related to many Numerical Methods No precise date (or time period) of origin.

Discretization was used by Ancient Greeks and Chinese for problems in geometry. E.g., finding the area / perimeter of a circle by approximating using regular polygons.

By AD480, a Chinese Engineer called Tsu Chung Chik, determined to lie between 3.1415926 & 3.1415927 (My calculator gives: 3.141592654!!!).

Archimedes used finite elements(!) for determining the volume of solids Almost near the invention of Calculus.

Development of numerical procedures like Weighted Residual Methods, Finite Difference methods, Variational Calculus

The (now!) famous paper by Richard Courant in 1943 He suggested use of triangular mesh to solve torsion problem.

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

The Family Tree of FEMRef.: Zienkiewicz, O. C., The Finite Element Method, TMH Edition.Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Origins of FEMFinite Difference and Variational Finite Difference Methods.

Contributions by Mathematicians:Variational Methods, Weighted Residuals, Piecewise Continuous trial functions

Mainly interested in devising ways to solve the Partial Differential Equations, convergence studies

Interested in actual problems in design specifically related to the structural design of aircrafts.

Matrix methods in Structural Analysis of Discrete Systems.

Approximation of continuous systems as discrete systems.

Contributions by Engineers:The stiffness method and continuum elements.

Lecture - 02

Department of Mechanical Engineering, National Institute of Technology Calicut

Concluding RemarksFEM is a method for obtaining approximate solution for problems with continuous systems, i.e., field problems.

Continuous systems lead to differential equations Mathematically speaking, FEM can be thought as a numerical procedure for solving differential equations.

The basic idea is to convert the differential equations into a system of algebraic equations, which can be solved for getting an approximation for variables (functions) involved.

Lecture - 02This conversion is achieved by the discretization of the domain into simpler finite elements, and using an approximation for the functions over these sub-domains.

The method aims at the solution of the D.E. only Hence, FEM can be used for a large variety of physical problems.

We will study the analysis of discrete systems, which directly lead to algebraic equations, before delving into the real FEM.