AS 5850 Finite Element AnalysisAS 5850 Finite Element Analysis

Prof. K. V. N. GopalDepartment of Aerospace Engineering

IIT Madras

INTRODUCTION

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Goal of engineering computations

Perform analysis and design of physical systems and processes subjected to imposed conditions (or loads) and help in engineering decision making

Analysis:Determination of the behavioral response exhibited by a particular structural configuration under specific loads.

(input, system) -> determine output

Design:Process of altering dimensions, shapes, and materials to find the best structural configuration to perform a specific function or give a desired response.

(input, output) -> determine the right system

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Engineering Decision Process

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Mathematical Formulation

• Identify the dependent and independent variables of the problem

• Obtain governing equations involving the dependent variables in terms of the independent variables

- algebraic, differential or integral equations

• Identify primary and secondary variables and associated essential and natural boundary conditions

Boundary/Initial Value ProblemsBoundary/Initial Value Problems

Partial differential equations model the behaviour of a wide range of problems in engineering.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Solution Methods

Analytical solutions for most of the realistic PDE models are either too complex or non-existent.

Numerical methods can be used to give approximatesolutions.

A general mathematically validated, computational efficient(programmable) method can give reasonably accurateapproximate solutions for complex problems.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Numerical Methods

Finite Difference Method (FDM)Point-wise approximation of the differential equation using array of grid points

Finite Element Method (FEM)Integral approximation to the differential equation using an assembly of finite elements and satisfying boundary conditions exactly

Boundary Element Method (BEM)Integral formulation of the equations satisfied exactly on the boundaryand satisfying boundary conditions approximately.

Finite Volume Method (FVM), spectral element method,meshless methods etc.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Advantages of FEM

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Basic Idea of FEM

1. Divide whole into simple parts (finite element mesh)

2. Set up the `problem’ over a typical part of the domain. i.e. derive a set of relationships between primary and secondary variables

3. Assemble the parts to obtain the solution to the whole

Divide and ConquerDivide and Conquer

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Illustrating the idea of FEM

Centre of Mass of a 3-D machine Component

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Approximating a domain

Total Area Under a Curve

( )b

a

I F x dx

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Example – 2 (Contd)

Approximation of the Curve

1 1 2

2 2 2 3

3 3 3

, ( ) , x

, x

a b x a x xF x a b x x x

a b x x b

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Example -2 (contd.)Total Area Under the Curve

1 1 2

2 2 2 3

3 3 3

, ( ) , x

, x

a b x a x xF x a b x x x

a b x x b

1 2 3I I I I

1 1

( ) ( )i i

i i

x x

i i i ix x

I F x dx a b x dx

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Finite Element Terminology

Element A geometric sub-domain of the region being simulated, with the property that it allows a unique derivation of the approximation (interpolation) functions.

NodeA geometric location in the element which plays a role in the derivation of the interpolation functions and it is the point atwhich solution is sought.

Mesh A collection of elements (or nodes) that replaces the actual domain.

Weak FormAn integral statement equivalent to the governing equations and natural boundary conditions.

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Finite Element Discretization

Introduction Department of Aerospace Engineering

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K. V. N. Gopal

Steps in Finite Element Modeling

Begin with the governing equations of the problem

Develop its weak form (weighted-integral statement) over a typical element

Approximate the solution over each finite element

Obtain relations among the quantities ofinterest over each finite element

Assemble these relations over the entire domain

Solve the resultant equations

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Applications of FEA

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

Development of FEMHistorical LandmarksHistorical Landmarks

1943R. Courant, Variational methods for the solution of problems of equilibrium and vibration, Bull. Amer. Math. Soc. 49 (1943), 1-23, MR 4, 200.“ Thirty years ago, Courant gave a remarkable lecture to this Society….

To begin with, his idea was forgotten. Perhaps you have forgotten it too ; it had to do with approximation by piecewise polynomials……Certainly there was an idea whose time was coming. When it finally came, fifteen years after Courant's lecture, it developed into what is now the most powerful technique for solving a large class of partial differential equations—the finite element, method. The only sad part is that virtually the whole development took place as if Courant had never existed.” – G. Strang, American Mathematical Society, Apr 1973

M.J. Turner, R.W. Clough, H.C. Martin and L.J. Topp , Stiffness and deflection analysis of complex structures. J Aero Sci 23 (1956), pp. 805–823.

1956

This paper presented the idea of dividing the real continuum directly into elements of arbitrary shape and directly establishing their stiffness. This became known as the method of Finite Elements only in

1960, following a paper presented by Ray Clough.

Ray Clough

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

History of FEM - Timeline

• 1943 – R. Courant proposed a torsion solution using triangular subregions (not implemented since no computers were available)

• 1950s – Digital computers which solved large systems of equations became available. Aerospace companies began solving structures problems. Classic Paper by Clough and others in 1956

• 1960 – term “Finite Element Method” first coined by R.W.Clough

• Early and mid 1960’s – new elements formulated, method became accepted, extended to heat transfer and fluid flow problems

• Late 1960’s, early 1970’s – Commercial availability of large general purpose software (NASTRAN, ANSYS, SAP)

• 1980s – Inexpensive PC versions, interactive graphics

• 90’s and beyond – Widespread application in numerous industries, Integration with CAD packages.

• Most recent emphasis – Simulation of multi-physics problems

Introduction Department of Aerospace Engineering

AS5850 FEA

K. V. N. Gopal

References

1. J. N. Reddy An Introduction to the Finite Element Method

2. O.C. Zienkiewicz et. al., Finite Element Method (3 vols)

3. K. J. Bathe Finite Element Procedures

4. T. J. R. Hughes The Finite Element Method – Linear, Static and Dynamic Finite Element Analysis

5. S. S. Rao ‘The Finite Element Method in Engineering’

6. R. D. Cook, D. S. Malkus, M. E. Pleisha ‘Concepts and Applications of Finite Element Analysis’