1 08 - finite element method 08 - finite element method - density growth - theory QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
1 08 - finite element method 08 - finite element method - density growth - theory.
Jan 29, 2016
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
108 - finite element method
08 - finite element method -
density growth - theory
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
2finite element method
from continuous problem…
… to linearized discrete initial boundary value problem
• temporal discretization
• linearization
• spatial discretization• staggered/simultaneous
implicit euler backwardfinite element methodnewton raphson iterationgateaux derivative
3finite element method
sequential solution - element based
staggered solution - integration point based
simultaneous solution - node point 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]
weinans, huiskes & grootenboer [1992], harrigan & hamilton [1992],,[1994], jacobs, levenston, beaupré,simo & carter [1995]
jacobs, levenston, beaupré,simo & carter [1995], fischer, jacobs,levenston & carter [1997], nackenhorst [1997], levenston [1997]]
4finite element method
from strong form … to weak form (1d)• strong / differential form
• weak / integral form - nonsymmetric
• strong form / residual format
• integration by parts• integral theorem & neumann bc‘s• weak form / integral form - symmetric
5finite element method
integration point based
strong form
• cast it into its residual format
• with residual
• start with nonlinear mechanical equilibrium equation
6finite element method
integration point based
boundary conditions
• dirichlet / essential boundary conditions
• neumann / natural boundary conditions
• strong / differential form
7finite element method
integration point based
weak form
• mulitplication with test function & integration
• weak form / nonsymmetric
• strong / differential form
8finite element method
integration point based
weak form
• gauss theorem & boundary conditions
• integration by partsQuickTime™ and a
TIFF (Uncompressed) decompressorare needed to see this picture.
• weak form / symmetric
9finite element method
integration point based
spatial discretization
• interpolation of test functions
• interpolation of trial functions
• discretization
10finite element method
integration point based
discrete residual
• discrete residual format
• discrete residual
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.• discrete weak form
11finite element method
integration point based
discrete residual
• discrete residual
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
• residual of mechanical equilibrium/balance of momentum
check in matlab!
12finite element method
integration point based
linearization
• incremental residual
• system of equations
• linearization / newton raphson scheme
• incremental iterative update
13finite element method
integration point based
linearization
• stiffness matrix / iteration matrix
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
• linearization of residual wrt nodal dofs
14finite element method
integration point based
linearization
• stiffness matrix / iteration matrix
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
• linearization of residual wrt nodal dofs
check in matlab!
15finite element method
integration point based
constitutive equations
with
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
• constitutive equations - given calculate
and from
16finite element method
integration point based
constitutive equations
• stress calculation @ integration point level
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
• constitutive equations - given calculate
check in matlab!
17finite element method
integration point based
constitutive equations
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.• constitutive equations - given calculate
• temporal discretization - euler implizit
with
with
• local newton iteration
18finite element method
integration point based
constitutive equations
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
check in matlab!
• residual of biological equilibrium / balance of mass
• discrete density update
19finite element method
integration point based
constitutive equations
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
depends on time discretization
with
• constitutive equations - given calculate
20finite element method
integration point based
constitutive equations
• tangent operator / constitutive moduli
QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
• linearization of stress wrt deformation gradient
check in matlab!
21finite element method
integration point based
staggered solution
loop over all time stepsglobal newton iteration
loop over all elementsloop over all quadrature pointslocal newton iteration to determine
determine element residual & partial derivativedetermine global residual and iterational matrixdetermine
determine state of biological equilibrium
22finite element method
integration point based
staggered solution
loop over all time stepsglobal newton iteration
loop over all elementsloop over all quadrature pointslocal newton iteration to determine
determine element residual & partial derivativedetermine global residual and iterational matrixdetermine
determine state of biological equilibrium
Related Documents
