The Finite Element Method

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The Finite Element Method

• General Overview

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General Overview

• widespread use in many engineering applications
• Applications of FEM in Engineering
• Mechanical/Aerospace/Civil/Automobile
  Engineering
• Structure analysis (static/dynamic,linear/nonlin
  ear)
• Thermal/fluid flows
• Electromagnetics
• Geomechanics
• Biomechanics
• ...

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General Overview

• examples
• conduction heat transfer, solve for the
  temperature distribution throughout the body with
  known boundary conditions and material properties
• fluid mechanics problems range from steady
  inviscid incompressible flow to complex viscous
  compressible flow,

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General Overview

• acoustics uses finite element and boundary
  element numerical methods
• electromagnetic solution for magnetic field
  strength provide insight to the design of
  electromagnetic devices
• capabilities extended to include fluid-structure
  interactions, convective heat transfer
• Bio-mechanics-bone structural analysis, blood
  flow in blood vessels

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General Overview

• Finite element method is a numerical method of
  solving a system of governing equations over the
  domain of a continuous physical system
• method applies the many fields of science and
  engineering
• for engineering use, fields of continuum
  mechanics and the theory of elasticity provide
  the governing equations

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Why numerical method
Most engineering problem involve solution of
governing differential equations.
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For heat transfer, torsion of shafts,
irrotational flow, seepage through porous media
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Solution of differential equation is tedious and
some times impossible
Complex geometry, boundary conditions, loading
conditions and material
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General Overview

• Finite element method can be summarized in the
  following steps
• small parts called elements subdivide the domain
  of the solid structure
• elements assemble through interconnections at a
  finite number of points (nodes) on each element
• assembly provides a model of the structure

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General Overview

• within each small domain, we assume a simple
  general solution to the governing equations
• solution for each element is a function of the
  unknown solutions at the nodes

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Fundamental concept of FEM
The fundamental concept of FEM is that continuous
function of a continuum (given domain ?) having
infinite degrees of freedom is replaced by a
discrete model, approximated by a set of
piecewise continuous function having a finite
degree of freedom.
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• since the continuum domain is divided into finite
  elements with nodal values as the unknowns, the
  loads and boundary conditions must be converted
  to nodal quantities
• single forces F apply to the nodes directly
• distributed forces are converted to equivalent
  nodal values
• supports, e.g. grounds are converted to specified
  displacements for the supported nodes

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General Overview

• sources of error
• assumed solution within the element is rarely the
  exact solution
• error between exact and assumed solution
• magnitude depends on the size of the elements
  relative to the solution variation
• in most cases, assumed solution converges to the
  correct as element size decreases

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General Overview

• precision of the algebraic equation solution
• function of computer accuracy, algorithm used,
  number of equations and element size

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General Overview

• all solid structures could be modeled with
  three-dimensional solid elements, but for many
  cases this is overkill
• many structures can be simplified by making some
  assumptions e.g. plane stress and plane strain
  assumptions, simple beam theory

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General Overview

• different types of elements are formulated for
  these simplified cases
• lead to simpler and faster solutions

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General Overview

• elements are categorized as either structural or
  continuum
• structural elements include trusses, beams,
  plates and shells
• formulations are based on same assumptions as in
  their structural theories
• finite element solution is no more accurate than
  a solution using conventional beam or plate theory

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General Overview

• continuum elements are two- and three dimensional
  solid elements
• formulation based on the theory of elasticity
  (provides the governing equations for deformation
  and stress)
• Few closed form or numerical solutions exist for
  these problems

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One-Dimensional Spring System

• Basic finite element method can be simply
  illustrated by a one-dimensional spring system
• the subscripted u values are the nodal
  displacements
• there is an applied force F at node 3
• we want to find the nodal displacements and
  spring forces

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One-Dimensional Spring System

• first formulate a general element
• each spring is identified by the element number
  in the box
• spring elements have a node at each end and
  connect at a common node
• consider a single spring element p with nodes i
  and j

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One-Dimensional Spring System

• assume positive displacements ui at node i and uj
  at node j
• element has spring constant k so nodal forces
  occur when we have displacements
• define fip as the force acting on node i due to
  the nodal displacements of element p
• equilibrium gives

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One-Dimensional Spring System
In the matrix form

• or

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One-Dimensional Spring System

• element 1
• element 2

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One-Dimensional Spring System

• Let the external forces be Fi , where i
  represents the node, so that the equilibrium
  equations are
• at node 1
• at node 2
• at node 3

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One-Dimensional Spring System

• substitute the element equations into these
  (structure assembly process)
• the resulting equation is
• or

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• Solve the matrix equation
• if the applied forces are known, and the nodal
  displacements are unknown, we have 3 simultaneous
  equations to solve
• the stiffness matrix above is singular so we do
  not have a unique solution
• this means the structure is at equilibrium at any
  location in x space i.e. we have rigid body motion

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• a unique solution requires must have some part of
  the structure grounded - i.e. we must apply some
  boundary conditions such as a fixed displacement
  of one of the nodes
• If we apply an external force F at node 3 and we
  attach the spring to ground at node 1, we can set
  the displacement at node 1 to zero.

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One-Dimensional Spring System

• If the external force on node 2 is 0 then
• this effectively zeros the first column of the
  stiffness matrix, and gives three equations in
  two unknowns
• If the value of the reaction force at node 1 is
  unknown, we can also skip the first equation, and
  consider only the last two
• Now we can solve this for the unknown us

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Using a Computer Program

• 3 stages
• preprocessing
• processor
• postprocessing

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Using a Computer Program

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Using a Computer Program

• Engineer
• responsible for interpreting results
• must ensure results are valid

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Analysis Step-by-Step

• procedure includes
• initial planning
• decide if FEA is needed
• doing the analysis
• presenting the results

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Analysis Step-by-Step
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Using a Computer Program

• preprocessing
• create model
• nodal point locations
• element selection
• nodal connectivities
• material properties
• displacement boundary conditions
• loads and load cases
• preprocessor assembles data into a format for
  execution

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Using a Computer Program

• processor
• code that solves the system equations
• generates element stiffness matrices
• stores data in files
• assembles the structure stiffness matrix
• must provide enough displacement boundary
  conditions to prevent rigid body motion
• solution gives nodal displacements
• with element information, get strain and stress

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Using a Computer Program

• postprocessing
• numeric output data difficult to use
• reduces data to graphic displays (contour plots,
  graphs)
• magnifies nodal displacements
• nodal displacements are single valued
• stress at a node can be multivalued if multiple
  elements are attached to the node
• (stress is found from within each element)

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Analysis Step-by-Step

• Start by having some conceptual design or some
  existing design failure for analysis
• define the analysis problem
• what type of analysis (static, dynamic, etc.)
• 2-d or 3-d solution
• criteria for analysis
• important variables (maximum stress, average
  stress, strain, deformation, fracture load, yield
  load, critical stress location, etc.)
• most critical guide the modeling and presentation
  of results

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Analysis Step-by-Step

• time allowed for analysis determines whether an
  adequate analysis can be done
• tight time constraints may cause a sacrifice in
  accuracy
• analyses may only show qualitative aspects of
  behavior
• required solution accuracy helps estimate the
  number of analysis cycles

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Analysis Step-by-Step

• approximate engineering analysis
• determine the load conditions
• estimate the accuracy of loading
• get material property data and its statistical
  variation
• (may need to check for sensitivity to material
  property variations)

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Analysis Step-by-Step

• use approximate structure made up of similar
  simpler structures with known solutions (beams,
  cylinders, simple plates, etc.)
• use approximate solutions to estimate value and
  locations of critical variables
• now decide if you need FEA

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Analysis Step-by-Step

• if Yes
• develop conceptual model
• geometry
• element types
• plan mesh
• look for symmetries that can simplify model
• determine if a 2-d or 3-d solution is needed
• if 3-d, try a 2-d simplified analysis first
• try to lay out mesh based on expected solution

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Analysis Step-by-Step

• choose a computer program to use
• many different programs available for use,
  sometimes you have no choice
• make sure the program has the correct element
  types for the model
• consider its mesh generation and pre- and
  postprocessing capabilities

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Analysis Step-by-Step

• Prepare model
• detailed mesh plan that includes the degree of
  refinement desired at critical locations
• apply boundary conditions
• apply load and load cases

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Analysis Step-by-Step

• Run analysis and check for errors

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Analysis Step-by-Step

• Postprocessing
• look at deformed displacements and check for
  consistency with expected results
• look at stresses and compare to approximate
  solution

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Analysis Step-by-Step

• Refine model by considering the results of the
  first analysis
• high stress and rapid variations Þ reduce element
  size
• low stress Þ increase element size
• Redo analysis and check if results are converging

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Analysis Step-by-Step

• Figure 1-7 is a refined model of 1-6
• note how the maximum stress has increased
• convergence has not yet been achieved
• Serious mistake if only one model is analyzed
• Figure 1-6 is in error by 23, while Figure 1-7
  is in error by 19
• There is no guarantee that results will be
  accurate

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Analysis Step-by-Step

• Use results from the analysis to estimate the
  converged solution
• rate the analysis by estimating the accuracy
  achieved
• determine if important criteria identified
  earlier are satisfied

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Analysis Step-by-Step

• When finished, prepare a report which highlights
  the significant results