fea basics

8/13/2019 FEA basics

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Indo German Winter Academy 2009

Vinay Prashanth SubbiahTutor Prof. S. Mittal

Indian Institute of Technology, Madras


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Introduction to FEM FEM Process Fundamentals - FEM

Interpolation Functions Variational Method Rayleigh-Ritz Method - Example Method of Weighted Residuals

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Galerkin FEM Assembly of Element Equations Laplace & Poisson Equations Discretization of Poisson Equation Variational Method

Numerical Integration Summary References

Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations

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Numerical solution of complex problems in Fluid Dynamics, StructuralMechanics

General discretization procedure of continuum problems posed bymathematically defined statements differential equations

Difference in approach between Mathematician & Engineer

Mathematical approaches

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n te erences2) Method of Weighted Residuals

3) Variational Formulations

Engineering approaches

Create analogy between finite portions of a continuum domain and realdiscrete elements

Example: Replace finite elements in an elastic continuum domain bysimple elastic bars or equivalent properties


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Finite Element Process based on following two conditions

1) Finite number of parameters determine behavior of finite number of

elements that completely make up continuum domain2) Solution of the complete system is equivalent to the assembly of the

individual elements

The process of solving governing equations using FEM

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1) Define problem in terms of governing equations (differential equations)2) Choose type and order of finite elements and discretize domain

3) Define Mesh for the problem / Form element equations

4) Assemble element arrays

5) Solve resulting set of linear algebraic equations for unknown

6) Output results for nodal/element variables


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Ref 4 Pg. 40, Fig 3.1


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Differential Equation Boundary Conditions

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Being an approximate process will seek solution of formwhere: Ni - shape functions or trialfunctions

ai unknowns or parameters tobe obtained to ensure good fit

Defined: Ni = 0 at boundary of domain


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Domain is broken up into number of non-overlapping elements

Functions used to represent nature of solution in elements trial / shape /

basis / interpolation functions These serve to form a relation between the differential equation and

elements of domain

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Ref 4 Pg. 40, Fig 3.2

Ref 4 Pg. 42, Fig 3.3


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Nm are independent trial functions

Properties:1. Nm is chosen such that u u as m

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omp e eness requ remen

2. Nm depends only on geometry andno. of nodes


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Piecewise defined trial functions:one dimensional linear

Varies linearly across each element

Properties:1. Nm = 1 at node m

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=

n

i ik

i

xx

xx

1


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. m = a a o er no es3. Nm

e = 1 for element e4. Number of nodes = number of

functions5. If Nm is a polynomial of order n-1,

then

Nke = , node k, element e,

k i

Ref 4 Pg. 44, Fig 3.4


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Simplex element

Simplest geometric shape used to approximate an irregular surface

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Solving for 1, 2, 3 in terms of nodal coordinates, nodal values & rewritingexpression for T(x,y)

Ref 4 Pg. 49, Fig 3.7


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where

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and interpolation functions

A Area of triangle


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Similarly quadratic triangular elements are also possible for betteraccuracy

Higher orders are also possible

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2 D Mesh: Quadrilateral Element

In simplest form rectangular element


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Ref 4 Pg. 58, Fig 3.13


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where:[K] stiffness matrix;

{ T } Vector of unknowns (like temperature);{ f } forcing or loading vector

Exam le Heat Transfer

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Qi Heat flux through node i;e element;l length of element;k thermal conductivity;A Area.Now, { T }e is the unknown temperature at either node

Consider a single element on a one dimensionaldomain with nodes i, j.


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u is solution to continuum problemF,E are differential operators

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Now, u is exact solution if for any arbitrary u

ie. if variational integral is made stationary

Now, the approximate solution can be found by substitutingtrial function expansion


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Since above holds true for an a

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Parameters ai are thus found from above equations

Note:

The presence of symmetric coefficient matrices for above equations is one of theprimary merits of this approach


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Natural: Variational forms which arise from physical aspects of problem itself

Example Min. potential energy equilibrium in mechanical systems

However, not all continuum problems are governed by differential

equations where variational forms arise naturally from physical aspects of

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Contrived:1. Lagrange Multipliers: extending number of unknowns by addition of

variables

2. Least square problems: Procedures imposing higher degree ofcontinuity requirements

Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson EquationsVinay P Subbiah


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Method depends on theorem from theory of the calculus of variations

The function T (x) that extremises the variational integral corresponding to thegoverning differential equation (called Euler or EulerLagrange equation) is thesolution of the original governing differential equation and boundary conditions

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Process

1) Derive Variational Integral from governing differential equation

2) Vary the solution function until Variational Integral is made stationary

with respect to all unknown parameters (ai) in the approximation

Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson EquationsVinay P Subbiah


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Example -

Using the governing equation as theEuler-Lagrange equation

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- where I is the Variational Integral

Ref 4 Pg. 75, Fig 3.24


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Integrating by parts

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And applying boundary conditions

Variational Integral is =

Note: order of

derivativein integrand


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Let I =

Substitute the approximation into integral and forcing Ito be stationary with respect to unknown parameters

yields set of linear equations to be solved for theunknown parameters

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K a = f

Now, checking for stationarity of thevariational integral by differentiating with unknown

parameters

Set of linear algebraic equations


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One can observe from the variational integral that despite the governingequation being a second derivative equation, the integrand is only of thefirst derivative

In cases where the second derivative tends to infinity or does not exist, thisformulation is very useful as it does not require the second derivative

Example Change in rate of change of temperature where two different

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Hence, the variational formulation of a problem is often called the Weakformulation

However, this formulation may not be possible for all differential equations

An alternative approach is - Method of Weighted Residuals


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Residual = A(T) A(T) where: T exact; T approximate; A GoverningEquation

Since, A(T) = 0 Residual ( R ) = A(T)

Method of weighted residuals requires that ai be found by satisfyingfollowing equation -

K a = f

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where: wi(x) are n arbitrary weighting functions

Essentially, the average Residual is minimized Hence, minimizing error inapproximation. R 0 when n

weighting functions can take any values

However, depending on the weighting functions certain special cases aredefined and commonly used


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1) Point Collocation: Dirac Delta Function

Equivalent to making the residual R equal to zero at a number of chosen points

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2) Sub-Domain:

Integrated error over N sub-domains should each be zero


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3) Galerkin:

Advantages include

i. Better accuracy in many cases

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ii. Coefficient Matrix is symmetric which makes computations easier

4) Least Squares: Attempt to minimize sum of squares of residual at eachpoint in domain


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R


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One of the most important methods in using Finite Element Analysis. Here, the weighting function is the same as the trial function at each of theelements nodes

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Interestingly, the solution obtained via this method is exactly the same as thatobtained by the variational methodIt can further be shown that, if a physical problem has a natural variationalprinciple attached to it or in other words if a governing equation can be writtenas a variational integral, then the Galerkin and Variational methods are identical

and thus provide same solution


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Now consider the integral of this form

Here, the weighted residual integrated overdomain

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Now, on integration by parts + Boundary Terms

C & D are operators with lower order of differentiation as compared to A,Hence lower orders of continuity are demanded from the trial functions.


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Governing Equation

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Consider domain to consist of 5 linear elements & 6 nodes

Approximate solution from Elements

Ni, Nj are interpolation functions across node ii , j are nodal unknowns

Ref 4 Pg. 75, Fig 3.24


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Galerkin method requiresSince weighting function = shape function

Integrating by parts andapplying boundary conditions

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Algebraic EquationEquation of the elements


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Assembling elementstogether and solving forUnknown parameters =

nodal temperatures

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Interestingly, this method provides better results when compared toapproximate methods that use a

function profile that satisfies BC and is

assumed before hand MWR or Rayleigh-Ritz


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Based on two criteria for interpolation functions:1. Compatibility

2. Completeness

Compatibility: Field Variable and any of its partial derivatives up to one order less

than hi hest in variational inte ral should be continous

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Example Continuity of Temperature with reference to heatconduction governing equation

Completeness: Within each element continuity must exist up to order of highest

derivative in variational integral Essentially, as number of nodes , Residual 0


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Assembly of two triangular elements

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Ref 4 Pg. 324, Fig C.1


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Poisson Equation

In 3-d Cartesian coordinates

Laplace Equation

-

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Essentially, Laplace equation is a special case of the Poisson Equationwhere the RHS = 0

A physical example is the Steady-State Heat Conduction equation;

Others include fluid mechanics, electrostatics etc.

Using the Steady-state heat equation, FEM discretization is carried out inthe following slides


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Poisson Equation

Boundary Conditions

= 0

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Finiding the Variational Integral

Integrating by parts &Using relation


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Approximating integral to achieve algebraicequations using shape functions

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and accuracy required in analysis


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On differentiating with respect to individual parameters ai

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Note:Galerkin and Variational procedures must give the same answer for caseswhere natural variational principles exist example is the case above

We get system of linear algebraic equations to be solvedfor unknown parameters


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As elements increase in order, the integrals required to solve for theelement matrices become complex and algebraically tedious

Hence, methods of numerical integration are used especially incomputational techniques like finite element

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This simple summation of terms of the integrand evaluated at certain pointsand weighted with certain functions serves as the approximation


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Here, a polynomial of degree n-1 is used

to approximate the integrand by sampling

at n points on the integrand curve

Resulting polynomial is integrated instead

to approximate value of the integral

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At n = 1 Trapezoidal Rule

At n = 2 Simpsons Rule


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Ref 1 Pg. 218, Fig 9.12a


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Here, sampling points are not assigned but are

rather evaluated; thus, we get exact value ofintegral when integrand is of order p ( n),

also to be determined

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Consider polynomial of order p; Use abovesummation approximation to evaluate integralwith this polynomial as integrand and comparingcoefficients with exact value we get p + 1 = 2 (n+1)

Thus, after sampling at n + 1 points order of accuracy = 2n + 1;In earlier approach, order of accuracy = n only achieved

Ref 1 Pg. 218, Fig 9.12b


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Ref 1 Pg. 83, Table 3.2

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FEM Numerical solution to complex continuum problems

Breaks down domain into discrete elements which are then

piecewise approximated and assembled back to get completesolution

Differential Equation Integral Form Linear Algebraic Equations

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Method of Weighted Residuals : Galerkin Method Laplace & Poisson Equations

FEM Discretization Poisson Equation

Numerical Integration simplify complex integrals


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1) The Finite Element Method Its Basis and Fundamentals, 2005, SixthEdition O.C. Zienkiewicz, R.L. Taylor & J.Z. Zhu

2) Finite Elements and Approximation, 1983 - O.C. Zienkiewicz, K. Morgan

3) Finite Element Procedures in Engineering Analysis, 2001 K. Bathe

4) Fundamentals of the Finite Element Method for Heat and Fluid Flow,2004 R.W. Lewis, P. Nithiarasu, K.N. Seetharamu

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5) The Method of Weighted Residuals and Variational Principles, 1972 B.

Finlayson


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