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Elastodynamics - Motivation

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! The discrete elastodynamics equations can be derived from eitherHamiltonian, Lagrangian or principle of virtual work for (dAlembertsprinciple)

! with initial conditions

! Discretization with finite elements

! Element mass matrix

! The stiffness matrix and the load vector are the same as for the static case

Elastodynamics -1-

(build-in boundaries)

element mass matrix

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! Semi-discrete equation of motion

! Mass matrix

! Stiffness matrix

! External force vector

! Initial conditions

! Semi-discrete because it is discretized in space but continuous in time

! Viscous damping

! Rayleigh damping

Elastodynamics -2-

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! Kinetic virtual work

! Rotationary inertia (very small for thin beams)

Timoshenko Beam - Virtual Kinetic Work

reference

configuration

deformed

configuration

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! Discretization with linear shape functions

! Lumped mass matrix (lumping by row-sum technique)

! In practice the rotational contribution can mostly be neglected

! For the equivalent Reissner-Mindlin plate, the components of the mass matrix aresimply the total element mass divided by four

Timoshenko Beam - Mass Matrix

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! Semi-discrete equation of motion

! Discretization in time (or integration in time)! Assume displacements, velocities, and accelerations

for t!tn are known

! Central difference formula for the velocity

! Central difference formula for the acceleration

! Discrete equilibrium at t=tn

! Displacements at t=tn+1 follow from these equations as

Explicit Time Integration -1-

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! Provided that the mass matrix is diagonal the update of displacements andvelocities can be accomplished without solving any equations

! Explicit time integration is very easy to implement. The disadvantage isconditional stability. If the time step exceeds a critical value the solution willgrow unboundedly! Critical time step size

! Longitudinal wave speed in solids

Explicit Time Integration -2-

exact solution

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! Semi-discrete heat equation

! is the temperature vector and its time derivative

! is the heat capacity matrix

! is the heat conductivity matrix

! is the heat supply vector

! Initial conditions

! Family of time integrators

Semi-discrete Heat Equation -1-

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! Common names for the resulting methods

! forward differences; forward Euler

! trapezoidal rule; midpoint rule; Crank-Nicholson

! backward differences; backward Euler

! Explicit vs. implicit methods

! For the method is explicit

! For the method is implicit

! Implementation: Predictor-corrector form

Semi-discrete Heat Equation -2-

known

substituting in

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! For elastodynamics most widely used family of time integration schemes

! Assume that the displacements, velocities, and accelerations for t!tn are

known

! Unconditionally stable and undamped for

! Implementation: a-form (according to Hughes)

! Compute predictors

The Newmark Method -1-

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! To compute the accelerations at n+1 following equation needs to be solved

The Newmark Method -2-