Transcript finite element method

14 - finite element method volume growth - theory
14 - finite element method
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from continuous problem…
• temporal
discretization
• spatial discretization
implicit euler
backward
finite element method
•
newton raphson
staggered/simultaneou
iteration
• linearization
gateaux derivative
s
… to linearized discrete initial boundary
value problem
finite element method
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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]]
finite element method
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from strong form … to weak form (1d)
• strong / differential form
• strong form / residual format
• weak / integral form - nonsymmetric
• integration by parts
• integral theorem & neumann
bc‘s
• weak form / integral form symmetric
finite element method
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integration point based
• start with nonlinear mechanical equilibrium
equation
• cast it into its residual format
• with residual
strong form
finite element method
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integration point based
• strong / differential form
• dirichlet / essential boundary
conditions
• neumann / natural boundary
conditions
boundary conditions
finite element method
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integration point based
• strong / differential form
• mulitplication with test function & integration
• weak form / nonsymmetric
weak form
finite element method
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integration point based
• integration by parts
Quic kT ime™ and a
T IFF (Uncompres sed) decompres sor
are needed to s ee this picture.
• gauss theorem & boundary conditions
• weak form / symmetric
weak form
finite element method
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integration point based
• discretization
• interpolation of test
functions
• interpolation of trial functions
spatial discretization
finite element method
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integration point based
• discrete weak form
Qui
TIFF (Uncomp
are needed
• discrete residual format
• discrete residual
discrete residual
finite element method
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integration point based
Qui
TIFF (Uncomp
are needed
• discrete residual
check in matlab!
• residual of mechanical equilibrium/balance of
momentum
discrete residual
finite element method
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integration point based
• linearization / newton raphson
scheme
• incremental residual
• system of equations
• incremental iterative update
linearization
finite element method
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integration point based
Qu ickT ime ™ a nd a
TIF F (U nco mpre ssed ) de com pres sor
are nee ded to s ee th is pi cture .
• stiffness matrix / iteration matrix
• linearization of residual wrt nodal
dofs
linearization
finite element method
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integration point based
Qu ickT ime ™ a nd a
TIF F (U nco mpre ssed ) de com pres sor
are nee ded to s ee th is pi cture .
• stiffness matrix / iteration matrix
check in matlab!
• linearization of residual wrt nodal
dofs
linearization
finite element method
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integration point based
• constitutive equations - given
Qui
TIFF (Uncomp
are needed
calculate
with
and
from
constitutive equations
finite element method
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integration point based
• constitutive equations - given
Qui
TIFF (Uncomp
are needed
calculate
check in matlab!
• stress calculation @ integration point level
constitutive equations
finite element method
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integration point based
• constitutive equations - given
calculate
Qui
TIFF (Uncomp
are needed
• temporal discretization - euler implizit
• local newton iteration
with
with
constitutive equations
finite element method
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integration point based
Qui
TIFF (Uncomp
are needed
• discrete density update
check in matlab!
• residual of biological equilibrium / balance of
mass
constitutive equations
finite element method
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integration point based
• constitutive equations - given
calculate
Qu ickT ime ™ a nd a
TIF F (U nco mpre ssed ) de com pres sor
are nee ded to s ee th is pi cture .
with
depends on time
discretization
constitutive equations
finite element method
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integration point based
Qu ickT ime ™ a nd a
TIF F (U nco mpre ssed ) de com pres sor
are nee ded to s ee th is pi cture .
• tangent operator / constitutive moduli
check in matlab!
• linearization of stress wrt deformation
gradient
constitutive equations
finite element method
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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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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
finite element method
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