FEA Course Lecture I

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• FEA Course Lecture I Outline
• 10/02/03 - UCSD
• Formal Definition of FEA
• An approximate mathematical analysis tool to
  study the behavior of a continua (or a system) to
  an external influence such as stress, heat,
  pressure, magnetic filed etc. This involves
  generating a mathematical formulation of the
  physical process followed by a numerical solution
  of the mathematics model.
• History of FEA
• Greek Mathematicians were the first to use the
  basic principles of FE to solve a physical
  problem (i.e., finding the area of a circle, or
  find the value of Pi)
• A ? R
• C 2?R (More on this later).

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• History of FEA (continued)
• Archimedes had used a concept of splitting a
  domain and re-assembling it to calculate the
  volume of a wedge by breaking it into a series of
  triangles.
• Modern FEA as we know it.
• 1941 Hrenikoff Framework method for Plane
  elastic medium represented as collection of bars
  and beams
• 1943 Courant solved a St. Venants Torsion
  Problem through an assemblage of triangular
  elements
• 1956 Turner, Cough, Martin and Topp UC
  Berkeley/Aerospace
• 1960 Clough was the first to use the formal
  name of Finite Elements.

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Basic Concept Division of a given domain into a
set of simple sub-domains called finite elements
accompanied with polynomial approximations of
solution over each element in terms of nodal
values. Assembly of element equation with
inter-element continuity of solution and balance
of force considered. What are Finite Elements?
Any geometric shape that allows computation of
solutions (with approximation) or provides
necessary relations among the values of solution
at selected points (called nodes) of the
sub-domain.

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B) Basic Illustration Approximation of
Circumference of a Circle

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• FE Descitization
• Each line segment is an element.
• Collection of these line segments is called a
  mesh.
• Elements are connected at nodes.
• Element Equations
• He 2R sin (q/2)
• Assembly of Equations and Solution
• Pn Sigma He (n1, N)
• For q 2?/n, He 2R sin (?/n), Pn
  n2Rsin(?/n)
• Convergence
• As n approaches infinity, P 2?R
• if x 1/n
• Pn 2Rsin(?x)/x
• As n approaches infinity, x-gt0,
• Limit (2Rsin(?x)/x) as x-gt0 limit
  (2?Rcos(?x)/1) 2?
• Error Estimation
• Error, Ee Se He
• R2?/n 2Sin(?/n)
• Total Error nEe

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C) Some Examples of the Second Order Equations in
1- Dimension, -d/dx(adu/dx) q for 0 lt x lt L
Field Primary Variable u Constant a Source term q Secondary Variable Qo
Transverse Deflection of a Cable Transverse Deflection Tension in Cable Distributed Transverse Load Axial Force
Axial Deformation of a bar Longitudinal Displacement EA (E Young's Modulus, A Cross Sectional Area) Friction or contact force on surface of bar Axial Force
Heat Transfer Temperature   Thermal Conductivity Heat Source Heat
Flow Through Pipes Hydrostatic Pressure PD4/128m (D- Diameter, m - viscosity) Flow Source (Generally Zero) Flow Rate
Laminar Incompressible Flow through a Channel under Constant Pressure Gradient Velocity Viscosity Pressure Gradient Pressure
Flow Through Porous Media Fluid Head Coefficient of Permeability Fluid Flux Flow (seepage)
Electrostatics Electrostatic Potential Dielectric Constant Charge Density Electric Flux
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D) Some Examples of the Poisson Equation ?.
(k?u) f
Field of Application Primary Variable u Material Constant K Source Variable f Secondary Variables d, du/dx, du/dy
Heat Transfer Temperature T Conductivity k Heat Source Q Heat Flow q comes from conduction k ?T/?n and convection h(T-T?)
Irrotational Flow of an Ideal Fluid Stream Function y Velocity Potential f Density r Density r Mass Production s (normally zero) Mass Production s (normally zero) Velocities ? y /?x -v ? y /?y u ? f /?x -v ? f /?y u
Groundwater Flow Piezometric Head f Permeability K Recharge Q Seepage q k? f/dn Velocities u -k? f/dx , v -k? f/dy
Torsion of Members with Constant Cross-Section Stress Function Y k 1 G Shear Modulus f 2 q angle of twist per unit length Gqdf/dx -syz Gqdf/dx -sxz
Electrostatics Scalar Potential f Dielectric Constant e Charge Density r Displacement Flux density Dn
Magnetostatics Magnetic Potential f Permeability m Charge density r Magnetic Flux density Bn
Transverse Deflection of Elastic Membranes Transverse deflection u Tension T in membrane Transversely distributed Load Normal force q
Both tables taken from J. N. Reddy's Book
"Introduction to the Finite Element Method", J.N.
Reddy, McGraw Hill Publishers, 2nd Edition, Page
71
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• E) Some Examples of Coupled Systems
• Plane Elasticity
• dsx/dx dsxy/dy fx rd2u/dt2
• dsxy/dx dsy/dy fy rd2v/dt2
• Flow of Viscous Incompressible Fluids Navier
  Stokes Equations
• Conservation of Linear Momentum
• rdu/dt - d/dx(2mdu/dx) - d/dym(du/dy dv/dx )
  dP/dx fx 0
• rdv/dt - d/dxm(dv/dy du/dx ) -
  d/dy(2mdu/dy) dP/dy fy 0
• Conservation of Mass
• (du/dx dv/dy) 0

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System Level Modeling

• System Level Modeling Reduced-order macro
  models are converted into simulation templates
  where the physically correct result can be
  further optimized with system level trade-offs.

Sample elector-mechanical library elements
Courtesy Coventor
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• SOFTWARE-Specific Session
• Intro to ANSYS. Basic file operations. Simple
  plate problem.
• Intro to FEMLAB. Fluid mechanics problem.
  Critical look at results.
• Intro to software-specific issues. h-elements,
  p-Elements
• Homework 1 and Reading Assignments.