8/13/2019 FEA basics
1/43
Indo German Winter Academy 2009
Vinay Prashanth SubbiahTutor Prof. S. Mittal
Indian Institute of Technology, Madras
8/13/2019 FEA basics
2/43
Introduction to FEM FEM Process Fundamentals - FEM
Interpolation Functions Variational Method Rayleigh-Ritz Method - Example Method of Weighted Residuals
IIT Madras
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
Vinay P Subbiah
8/13/2019 FEA basics
3/43
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
IIT Madras
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
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
4/43
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
IIT Madras
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
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
5/43
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
Ref 4 Pg. 40, Fig 3.1
8/13/2019 FEA basics
6/43
Differential Equation Boundary Conditions
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
7/43
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
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
Ref 4 Pg. 40, Fig 3.2
Ref 4 Pg. 42, Fig 3.3
8/13/2019 FEA basics
8/43
Nm are independent trial functions
Properties:1. Nm is chosen such that u u as m
IIT Madras
omp e eness requ remen
2. Nm depends only on geometry andno. of nodes
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
9/43
Piecewise defined trial functions:one dimensional linear
Varies linearly across each element
Properties:1. Nm = 1 at node m
IIT Madras
=
n
i ik
i
xx
xx
1
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
. 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
8/13/2019 FEA basics
10/43
Simplex element
Simplest geometric shape used to approximate an irregular surface
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
Solving for 1, 2, 3 in terms of nodal coordinates, nodal values & rewritingexpression for T(x,y)
Ref 4 Pg. 49, Fig 3.7
8/13/2019 FEA basics
11/43
where
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
and interpolation functions
A Area of triangle
8/13/2019 FEA basics
12/43
Similarly quadratic triangular elements are also possible for betteraccuracy
Higher orders are also possible
IIT Madras
2 D Mesh: Quadrilateral Element
In simplest form rectangular element
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
Ref 4 Pg. 58, Fig 3.13
8/13/2019 FEA basics
13/43
where:[K] stiffness matrix;
{ T } Vector of unknowns (like temperature);{ f } forcing or loading vector
Exam le Heat Transfer
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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.
8/13/2019 FEA basics
14/43
u is solution to continuum problemF,E are differential operators
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
15/43
Since above holds true for an a
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
16/43
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
IIT Madras
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
8/13/2019 FEA basics
17/43
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
IIT Madras
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
8/13/2019 FEA basics
18/43
Example -
Using the governing equation as theEuler-Lagrange equation
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
- where I is the Variational Integral
Ref 4 Pg. 75, Fig 3.24
8/13/2019 FEA basics
19/43
Integrating by parts
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
And applying boundary conditions
Variational Integral is =
Note: order of
derivativein integrand
8/13/2019 FEA basics
20/43
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
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
K a = f
Now, checking for stationarity of thevariational integral by differentiating with unknown
parameters
Set of linear algebraic equations
8/13/2019 FEA basics
21/43
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
IIT Madras
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
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
22/43
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
IIT Madras
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
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
23/43
1) Point Collocation: Dirac Delta Function
Equivalent to making the residual R equal to zero at a number of chosen points
IIT Madras
2) Sub-Domain:
Integrated error over N sub-domains should each be zero
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
24/43
3) Galerkin:
Advantages include
i. Better accuracy in many cases
IIT Madras
ii. Coefficient Matrix is symmetric which makes computations easier
4) Least Squares: Attempt to minimize sum of squares of residual at eachpoint in domain
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
R
8/13/2019 FEA basics
25/43
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
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
26/43
Now consider the integral of this form
Here, the weighted residual integrated overdomain
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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.
8/13/2019 FEA basics
27/43
Governing Equation
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
28/43
Galerkin method requiresSince weighting function = shape function
Integrating by parts andapplying boundary conditions
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
Algebraic EquationEquation of the elements
8/13/2019 FEA basics
29/43
Assembling elementstogether and solving forUnknown parameters =
nodal temperatures
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
30/43
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
IIT Madras
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
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
31/43
Assembly of two triangular elements
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
Ref 4 Pg. 324, Fig C.1
8/13/2019 FEA basics
32/43
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
33/43
Poisson Equation
In 3-d Cartesian coordinates
Laplace Equation
-
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
34/43
Poisson Equation
Boundary Conditions
= 0
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
Finiding the Variational Integral
Integrating by parts &Using relation
8/13/2019 FEA basics
35/43
Approximating integral to achieve algebraicequations using shape functions
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
and accuracy required in analysis
8/13/2019 FEA basics
36/43
On differentiating with respect to individual parameters ai
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
37/43
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
IIT Madras
This simple summation of terms of the integrand evaluated at certain pointsand weighted with certain functions serves as the approximation
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
38/43
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
IIT Madras
At n = 1 Trapezoidal Rule
At n = 2 Simpsons Rule
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
Ref 1 Pg. 218, Fig 9.12a
8/13/2019 FEA basics
39/43
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
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
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
8/13/2019 FEA basics
40/43
Ref 1 Pg. 83, Table 3.2
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
41/43
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
IIT Madras
Method of Weighted Residuals : Galerkin Method Laplace & Poisson Equations
FEM Discretization Poisson Equation
Numerical Integration simplify complex integrals
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
42/43
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
IIT Madras
5) The Method of Weighted Residuals and Variational Principles, 1972 B.
Finlayson
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah
8/13/2019 FEA basics
43/43
IIT Madras
Fundamentals of Finite Element Methods: Variational methods for the Laplace and Poisson Equations
Vinay P Subbiah