08 - finite element method

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08 - finite element method
08 - finite element method - density growth -
theory
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finite element method
from continuous problem
temporal discretization
implicit euler backward
spatial discretization
finite element method
staggered/simultaneous
newton raphson iteration
linearization
gateaux derivative
to linearized discrete initial boundary value
problem
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finite element method
sequential solution - element based
huiskes, weinans, grootenboer, dalstra, fudala
slooff 1987, carter, orr, fhyrie 1989,
beaupré, orr carter 1990, weinans, huiskes
grootenboer 1992, 1994, jacobs, levenston,
beaupré, simo carter 1995, huiskes 2000,
carter beaupré 2001
staggered solution - integration point based
weinans, huiskes grootenboer 1992, harrigan
hamilton 1992,, 1994, jacobs, levenston,
beaupré,simo carter 1995
simultaneous solution - node point based
jacobs, levenston, beaupré,simo carter 1995,
fischer, jacobs, levenston carter 1997,
nackenhorst 1997, levenston 1997
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finite element method
from strong form to weak form (1d)
strong / differential form
strong form / residual format
weak / integral form - nonsymmetric
integration by parts
integral theorem neumann bcs
weak form / integral form - symmetric
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finite element method
integration point based
start with nonlinear mechanical equilibrium
equation
cast it into its residual format
with residual
strong form
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finite element method
integration point based
strong / differential form
dirichlet / essential boundary conditions
neumann / natural boundary conditions
boundary conditions
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finite element method
integration point based
strong / differential form
mulitplication with test function integration
weak form / nonsymmetric
weak form
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finite element method
integration point based
integration by parts
gauss theorem boundary conditions
weak form / symmetric
weak form
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finite element method
integration point based
discretization
interpolation of test functions
interpolation of trial functions
spatial discretization
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finite element method
integration point based
discrete weak form
discrete residual format
discrete residual
discrete residual
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finite element method
integration point based
discrete residual
check in matlab!
residual of mechanical equilibrium/balance of
momentum
discrete residual
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finite element method
integration point based
linearization / newton raphson scheme
incremental residual
system of equations
incremental iterative update
linearization
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finite element method
integration point based
stiffness matrix / iteration matrix
linearization of residual wrt nodal dofs
linearization
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finite element method
integration point based
stiffness matrix / iteration matrix
check in matlab!
linearization of residual wrt nodal dofs
linearization
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finite element method
integration point based
constitutive equations - given calculate
with
and
from
constitutive equations
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finite element method
integration point based
constitutive equations - given calculate
check in matlab!
stress calculation _at_ integration point level
constitutive equations
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finite element method
integration point based
constitutive equations - given calculate
temporal discretization - euler implizit
local newton iteration
with
with
constitutive equations
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finite element method
integration point based
discrete density update
check in matlab!
residual of biological equilibrium / balance of
mass
constitutive equations
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finite element method
integration point based
constitutive equations - given calculate
with
depends on time discretization
constitutive equations
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finite element method
integration point based
tangent operator / constitutive moduli
check in matlab!
linearization of stress wrt deformation gradient
constitutive equations
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finite element method
integration point based
loop over all time steps
global newton iteration
loop over all elements
loop over all quadrature points
local newton iteration to determine
determine element residual partial derivative
determine global residual and iterational matrix
determine
determine state of biological equilibrium
staggered solution
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finite element method
integration point based
loop over all time steps
global newton iteration
loop over all elements
loop over all quadrature points
local newton iteration to determine
determine element residual partial derivative
determine global residual and iterational matrix
determine
determine state of biological equilibrium
staggered solution