1. 08 - finite element method
density growth - theory
08 - finite element method

2. 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
finite element method

3. 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]]
finite element method

4. 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

5. integration point based
• start with nonlinear mechanical equilibrium equation
• cast it into its residual format
• with residual
strong form
finite element method

6. integration point based
• strong / differential form
• dirichlet / essential boundary conditions
• neumann / natural boundary conditions
boundary conditions
finite element method

7. integration point based
• strong / differential form
• mulitplication with test function & integration
• weak form / nonsymmetric
weak form
finite element method

8. integration point based
• integration by parts
• gauss theorem & boundary conditions
• weak form / symmetric
weak form
finite element method

9. integration point based spatial discretization
• interpolation of test functions
• interpolation of trial functions
spatial discretization
finite element method

10. integration point based
• discrete weak form
• discrete residual format
• discrete residual
discrete residual
finite element method

11. integration point based
• discrete residual
check in matlab!
• residual of mechanical equilibrium/balance of momentum
discrete residual
finite element method

12. integration point based
• linearization / newton raphson scheme
• incremental residual
• system of equations
• incremental iterative update
linearization
finite element method

13. integration point based
• stiffness matrix / iteration matrix
• linearization of residual wrt nodal dofs
linearization
finite element method

14. integration point based
• stiffness matrix / iteration matrix
check in matlab!
• linearization of residual wrt nodal dofs
linearization
finite element method

15. integration point based constitutive equations
• constitutive equations - given calculate
with
and
from
constitutive equations
finite element method

16. integration point based constitutive equations
• constitutive equations - given calculate
check in matlab!
• stress integration point level
constitutive equations
finite element method

17. integration point based constitutive equations
• constitutive equations - given calculate
• temporal discretization - euler implizit
• local newton iteration
with
constitutive equations
finite element method

18. integration point based constitutive equations
• discrete density update
check in matlab!
• residual of biological equilibrium / balance of mass
constitutive equations
finite element method

19. integration point based constitutive equations
• constitutive equations - given calculate
with
depends on time discretization
constitutive equations
finite element method

20. integration point based constitutive equations
• tangent operator / constitutive moduli
check in matlab!
• linearization of stress wrt deformation gradient
constitutive equations
finite element method

21. 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

22. 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