FEM

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FEM???? 2005/09/12
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Agenda

• Variational Principles in Static and Dynamics
• One-dimensional Beam Extension
• One-dimensional Finite Element Method
• Extension to Two-dimensional FEM
• Inelastic Deformation

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Variational Principle in Statics
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Internal energy
equilibrium
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Cartesian coordinates
Geometric constraint
Variational principle in statics
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Introduce Lagrange multiplier
equilibrium
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Variational Principle in Dynamics
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Lagrangean
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Largange equation of motion
Equation of rotation
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Cartesian coordinates
Geometric constraint
Lagrangean under geometric constraint
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Largange equation of motion
magnitude of constraint force
direction of constraint force
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ODE solver
constant time step
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adaptive time step
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Constraint Stabilization Method
Simple pendulum with Cartesian coordinates
Equations of motion under geometric constraint
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Algebraic equation
Differential equation
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1D Beam Extension
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Static Modeling
Elastic Potential Energy
Work done by External Force
Geometric constraint
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Variational Principle in Statics
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Finite Element Method

• Divide an integral (U or T) over a region into
  integrals over small regions.
• Approximate a function in each small region by a
  simple (e.g. linear) function.
• Calculate the each divided integrals.
• Sum up all calculated integrals to derive the
  original integral.

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Piecewise Linear Approximation
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Piecewise Linear Approximation
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Optimization w.r.t. function
Optimization w.r.t. vector
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Stiffness matrix
External force
Geometric constraint
geo
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Variational principle in statics
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Static extension of uniform beam
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Static extension of irregular beam
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Dynamic extension of beam
Lagrangean under geometric constraint
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Lagrange equation of motion
elastic force
external force
constraint force
inertial force
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Constraint Stabilization Method
algebraic equation
differential equation
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Dynamic equations
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Viscoelastic deformation
Connection matrix
material
object
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Extension to 2D Deformation
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Piecewise linear approximation
point Pi
point Pj
point Pk
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Stress-strain relationship
Stress (pseudo) vector
Strain (pseudo) vector
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Isotropic elastic deformation
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Strain-displacement relationship
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Elastic potential energy
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Displacement in DPiPjPk
Stress in DPiPjPk
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Potential energy in DPiPjPk
connection matrices (geometric)
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Variational principle in statics
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Kinetic energy
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Inertia matrix
Lumped Inertia Matrix
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Lagrangean under geometric constraints
elastic forces
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Computation of elastic forceswithout
constructing total matrix K
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Computation of elastic forces
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Physical parameters
E Youngs modulus (elastic modulus) n Poisson
ratio
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Viscoplastic Deformation
relaxation function
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2D/3D isotropic viscoplasticity
1D material
2D/3D material
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Viscoplastic forces
2D/3D material
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Dyn. eqs. viscoplatic deformation
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Computation of viscoplastic forces
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Physical parameters
E Youngs modulus (elastic modulus) nela
Poisson ratio for elasticity
lela, mela
c viscous modulus nvis Poisson
ratio for viscosity
lvis, mvis
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Rheological deformation
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Dyn. eqs. rheological deformation
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Computation of rheological forces
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for example
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Physical parameters
E Youngs modulus (elastic modulus) nela
Poisson ratio for elasticity
lela, mela
c1, c2 viscous moduli nvis1 ,nvis1
Poisson ratios for viscosity
lvis1, mvis1 , lvis2, mvis2
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Summary

• Continua model provides a foundation of object
  deformation.
• Finite element approximation comes from a nature
  of integrals an integral over a region can be
  divided into integrals over small regions that
  cover the original region.
• Inelastic deformation including viscoplastic and
  rheological deformation can be described by a set
  of differential equations.

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