Introduction to Finite Element Method

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Introduction to Finite Element Method

• José Luis Gómez-Muñoz

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Finite Element Method

• Finite Element Method (FEM, FEA) is a collection
  of techniques used to obtain approximated
  solutions to partial differential equations that
  appear in Engineering and Physics
• The problem domain is subdivided in small
  regions, called elements

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Origins

• FEM was conceived by engineers in 1950s to
  analyze structural systems in airplanes.
• First papers about FEM Turner et al. (1956),
  Clough (1960) and Argyris (1963)
• Theoretical basis was found later, based on
  Variational Calculus (Rayleigh 1877, Ritz 1909)
  and Galerkin method (1905).

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Nowadays

• FEM is a powerful tool in Engineering. Its main
  advantage is that it can handle domains of
  complex geometry
• There are many computational packages that use
  FEM, among them we have Ansys, Cosmos and Algor
• FEM can be used in Mathematica with the IMTEK
  Mathematica Supplemtent http//www.imtek.de/simu
  lation/mathematica/IMSweb/

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1D Example Nodes

• Five nodes have being placed in the domain 1ltxlt5.1

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1D Example Base Functions (1)

• Every node has a base function, such that it
  takes the value of 1 at the node, descends
  linearly to 0 at the two adjacent nodes, and is
  zero in the rest of the domain

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1D Example Base Functions (2)

• Base function at the 4th node

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1D Example Base Functions (3)

• Base functions

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Example 1D Function that we want to interpolate

• Imagine we have a function that we want to
  interpolate between the nodes

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Example 1D Function values at the nodes

• We will use the function values at every node

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Example 1D Function values at the nodes (2)

• We will use the function values at every node

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Example 1D Multiply each base function by the
node value

• We multiply each base function by the
  corresponding node value.

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Example 1D Linear Combination

• We multiply each base function by the
  corresponding node value and we add all of them.
  The result is the interpolating function

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Example 1D Interpolation Function

• We multiply each base function by the
  corresponding node value and we add all of them.
  The result is the interpolating function

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Ejemplo 1D Lagrange Interpolation

• This is called Lagrange Interpolation.

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Lagrange Interpolation vs. FEM

• In the Lagrange Interpolation, the values of the
  function at the nodes are known
• On the other hand, in FEM we only know that the
  function must be the solution of a differential
  equation (with boundary conditions)
• In other words, with FEM we interpolate an
  unknown function!

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How do we interpolate an unknown function?

• How do we interpolate an unknown function?
• We do know that the function is the solution of a
  differential equation with boundary conditions
• The interpolation function does Not solve exactly
  the differential equation, there is an error
• Minimizing (in some sense) the error produces a
  linear system of equations, where the unknowns
  are the values of the function at the nodes.
• When we solve that linear system, the
  interpolation is obtained

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Example 1D Problem

• Imagine that in this domain we want to solve the
  problem y-20, y(0)0, y(5.1)0

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Example 1D Elements

• The interval between two nodes is an element
• We have four elements in this example

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Example 1D Base functions and elements

• Every element has two halves of base function

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Example 1D Base functions and elements (2)

• Every element has two halves of base function
• Each half of base function is called shape
  function

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Example 1D Base functions and elements (3)

• Every element has two halves of base function
• Each half of base function is called shape
  function

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Example 1D Base functions and elements (4)

• From all the possible linear combinations of
  shape functions, FEM finds the one that minimizes
  the error when substituted in the differential
  equation (Galerkin method)

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Example 1D Linear System

• Minimizing the error produces a linear system of
  equations, where the unknowns are the values of
  the function at the nodes
• That linear system is solved, and the
  interpolation is obtained

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Example 1D Interpolation of the unknown solution

• This is the interpolation of the (unknown)
  function that solves the differential equation
• It is the linear combination of base functions
  that, in some sense (Galerkin), minimizes the
  error when substituted in the differential
  equation

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Example 2D Nodes

• Nodes in a 2D domain

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Example 2D Elements

• Elements and Nodes in a 2D domain
• Base functions look like pyramids
• Every element has different pieces of the base
  functions

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Example 2D A linear combination

• A linear combination of base functions. It can be
  regarded as the interpolation of a function of
  two variables