finite elements methods
DESCRIPTION
Elementos finitosTRANSCRIPT
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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=
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 =
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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,
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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
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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
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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
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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.
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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.
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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
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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.
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