finite elements methods

T.Sri.Kalyan

Indian Institute of Technology, Kharagpur

Brief Outlook Introduction Basic Process of FEM Strong vs Weak Formulation Variational methods

• Rayleigh Ritz method

• Method of Weighted Residuals • Galerkin method

Domain Discretization Interpolation Functions Numerical Integration Computer Implementation Application in Fluid flows Summary References

2 T Sri Kalyan Finite element methods

Introduction to FEM

Numerical technique for the development of mathematical models for solving structural systems.

One of the most popular methods for solving the PDE’s.

Converts the PDE’s into set of algebraic equations that are easy to solve.

All the complexities of the problems like varying shapes, boundaries etc. are easily maintained

Gives approximate solutions using numerical methods


FEM vs Classical Methods

Classical Methods Finite Element Method

Exact solutions are obtained for the governing equations

Approximate solutions are obtained

Makes drastic assumptions for complexities in shape, boundary conditions, loading.

Basic assumptions are made without affecting the physics involved.

Difficult to deal with the material and geometrical non-linearities

Can be dealt easily


Basic Process of FEM

The steps to be followed in the analysis are:

Select suitable field variables (like displacement in solid mechanics) and element shapes

Convert the PDE’s into integral form.

Divide the domain into fine elements

Select interpolation functions for respective elements.

Find the element properties

Assemble element properties to get global properties

Impose the boundary conditions

Solve the system equations to get the nodal unknowns


Strong vs Weak Formulation Short Example

Consider a beam subjected to uniform loading as shown in the fig:

The governing equation and the boundary conditions are:

0p=xp

L

0

00

2

=dx

duEA

=u

p=dx

udEA

Lx

02


Strong form

The boundary conditions along with the governing equations forms the strong form

Strong form used directly when the governing equations and boundary conditions are not too complex

Thus our strong form is:

0

00

2

=dx

duEA

=u

p=dx

udEA

Lx

02


Weak form A weak form or variational form is a weighted-integral statement

of a differential equation Developed so as to ease the restrictions on boundary conditions. In our case it is

Strong form

Residual Function (R)

Weak form v= Weight function The solution is given by assuming

0

0

0

0

2

0

2

2

=vdxpdx

ud

=pdx

ud

p=dx

ud

L

2

2

02

0

1

( ) ( )N

N j j

j

u U c x x


Development of Weak form Consider a more complex differential equation

With the following boundary conditions

Steps to be done:

Make the weighted integral form of the equation

w= weight function

for 0<x<L

u(0) = 𝑢0, a(𝜕𝑢

𝑑𝑥) = 𝑄

[ ( ) ] ( )d du

a x q xdx dx


Integrate the first part of the equation

Q represents the secondary variable Care should be taken in identifying the natural and

essential boundary conditions Weaker statement of the problem with weaker continuity

of the dependent variable

Weak form

0

0 0

[ ]

L L

Lwdv vdw wv Basic equation used for by-parts


Rayleigh Ritz method OBJECTIVE: Find an approximate solution in the form of assumed linear combination, undetermined parameters(𝑐𝑗)

Ritz method employs weak form for solution.

Approximate solutions are found by using finite subspace instead of infinite dimensional space

Choice of weight functions is restricted to approximation functions, w=∅𝑗

Requires that the set of approximation functions {∅𝑗} need to be complete

∅𝑗 usually taken to be polynomials satisfying the essential boundary conditions


Consider the variational problem resulting from the weak form of the problem

B(w,u)=l(w)

The approximation of u(x) is now employed in above equation and finally expressed in matrix form

or {K}{c} = {F} (Matrix form)

𝑤ℎ𝑒𝑟𝑒 𝐾𝑖𝑗 = B(∅𝑖 , ∅𝑗); 𝐹𝑖 = l(∅𝑖 ) - B(∅𝑖 , ∅0)

(1)

(2)

0

1

( ) ( )N

N j j

j

u U c x x


EXAMPLE: Determining the deflection in a simply supported beam

Governing Eq: 𝑀

𝐸𝐼=𝜕2𝑦

𝜕𝑥2

Let y(x) = 𝑎𝑖𝑠𝑖𝑛𝑚𝜋𝑥

𝑙𝑎𝑚=1,3

Quadratic functional

𝑎𝑖 ’s obtained by minimizing 𝜋

Thus y=


Weighted Residuals Method Employs weighted-integral form for the solution

Residual function R made zero in a “weighted-residual” sense

R integrated over a set of linearly independent weight functions (𝑊𝑗)

The requirements on ∅𝑖 , ∅0 are more stringent

Different sets of equations produced from different methods

0jW R


Process Consider the operator and approximation equation

A(u) = f(x) A = Differential Operator

Residual function R= A(Un) – f

Parameters𝑐𝑗determined by the following equation

Number of weight functions equal to number of unknown parameters

0

1

( ) ( )N

N j j

j

u U c x x


Least-Squares method

Parameters 𝑐𝑗 are determined by minimizing the

integral of the square of the residual

Can be represented in a matrix form {A}{c} = {F}

Where

Coefficient matrix involves same order of differentiation as in governing equation


Collocation Method:

Residual function made zero at N selected points thus generating N equations

Selection of N specific points is crucial in this method

𝑅(𝑥𝑖 , 𝑐𝑗) = 0 (i = 1,2,…….N)

Sub-domain Method:

Domain is divided into large number of subdomains.

𝑊𝑗 = 1 in ∅𝑗

= 0 else

j

j jW R R


Galerkin Method Weight function taken to be equal to approximation

function (𝜓𝑖 = ∅𝑖) Results in development of symmetric matrices When 𝜓𝑖 ≠ ∅𝑖 it is referred as Petrov–Galerkinmethod

(i = 1,2,……N) In solid mechanics it is the virtual work method Approximation functions are of higher order than those in

Ritz method Ritz and Galerkin are same only in two cases:

When all the boundary conditions are essential Same approximation functions are used

0iR x


Example:

Consider the equation with given boundary conditions

Approximation functions are given by

∅0= x; ∅1 = x(2-x); ∅2 = 𝑥2(1-2x/3)

The final residual function is


Least Square Method:

Taking 𝜓𝑖 = 𝜕𝑅

𝜕𝑐𝑖 we get 2 equations for 𝑐𝑗 ’s

Galerkin Method:

Take 𝜓𝑖 = ∅𝑖 we get

Different solutions are obtained from different methods


Domain Discretization Domain (Ω) of the problem divided into set of

subintervals called finite elements

Collection of these forms the finite element mesh

Discretization is often dictated by geometry, loading and material properties

Approximate solution on these elements need to be continuous at the interfaces

Element type, density and number of elements controls the accuracy of the solution to large extent

Mess refinement to be done to check the convergence of the solution


Element Shapes:

Depending on the problem there can be one, two, three dimensional elements

One dimension: Line elements

Two dimension: Triangle, Quadrilateral

Three dimension: Tetrahedron

Higher order of elements are also used by increasing the number of nodes


Interpolation Functions FEM seeks approximate solution over each element as

opposed to variational methods

Solution found at element nodes called interpolation points or element nodes

The approximate solution over an element is given by

where,

Uje = Value of solution at nodes

𝜓 = Interpolation functions


Lagrange Interpolation: These interpolate only function values but not the derivatives

Hermite Interpolation: Interpolate both the function values and the derivatives

Accuracy of solution can be increased with higher order of approximation


Element Equations Linear Elements:

Function assumes a value at nodes and nullifies at other points

In general the approximate functions are:

( ) 0e e

i jx

( ) 1e e

i jx

if i≠j

if i=j and,

1

( ) 1n

e

i

i

x

1 1

h


Triangular Element:

Most commonly used element in 2 dimensions

The approximation function is given by

The unknowns ci’s are obtained by matrix U = {A}{C}

Where, 𝛼𝑖 =


Rectangular Elements:

Because of four nodes a bilinear term is added in the approximation solution

Interpolation function becomes

Quadratic Elements:

Each side of the element should have three nodes

In this case the solution assumes the form,


Numerical Integration Evaluation of the complicated integral forms or where

the integrand is discontinuous

Can be easily implemented in a computer in form of a program.

BASIC IDEA: To find a function that is a good approximation of integrand and easily integrable.

F(x) is approximated by P4(x)

Thus, 𝐹 𝑥 𝑑𝑥𝑏

𝑎≅ 𝑃4 𝑥 𝑑𝑥

𝑏

𝑎


Newton-Cotes Quadrature Employs values of function at equally spaced points

where,

𝑊𝑖 = Weighting coefficients;

𝑥𝑖= Base points; r = Number of base points

It is to be ensured that 𝑤𝑖𝑟1 = 1

For r =2 it equals the Trapezoidal rule

For r=3 it is Simpsons rule

Odd points results in higher order of accuracy


Computer Implementation Finite element program consists of three basic units:

Preprocessor: This involves reading geometry, material data and boundary conditions of the problem

Processor: Here all the calculations like generation of mesh, element matrices etc are done

Postprocessor: Computing the solution at desired points of the domain

The major steps in processor are:

Generation of element matrices

Assembly of element equations

Imposition of boundary conditions


Formulations for fluid flows: Basic governing equations are derived from

Law of conservation of mass (Continuity equation)

Law of conservation of momentum (Equations of motion)

Law of conservation of energy

Two different viewpoints are used in the analytical description of the equations of a continuous medium

Eulerian Description (Fixed spatial location)

Lagrangian Description (Fixed set of material particles)

The physical properties that are of main interest are viscosity, compressibility and rotationality.


Continuity equation for two dimensional flow is given by

The conservation of momentum for 2-D flows leads to

the famous Navier-Stokes equations.

The equations are simplified by the introduction of either stream function or velocity potential

FEM used in finding the stream function or velocity potential of a given problem.


Summary FEM – Allows accurate representation of complex

geometry and dissimilar material properties

Ritz, Galerkin and least square methods melts down to matrix form for determining the parameters 𝑐𝑗

Difficulty faced in constructing the approximation functions

Triangular element is elementary shape for two dimensional problems

Numerical Integration necessary for computer implementation


References:

J.N Reddy: Introduction to Finite Element method, TMH, 3rd edition, 2005.

S.S Bhavikatti: Finite Element Analysis, 2nd edition, 2010

B. Finlayson: Method of Weighted Residuals and Variational Principles, 1972.


finite elements methods

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