1. 12 - finite element method
density growth - alternative formulation
12 - finite element method

2. 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

3. finite element method node point based strong form
• strong form / residual format
• residuals
strong form
finite element method

4. finite element method node point based boundary conditions
• dirichlet / essential boundary conditions
• neumann / natural boundary conditions
boundary conditions
finite element method

5. finite element method node point based weak form • weak form
• weak form expressions
weak form
finite element method

6. finite element method node point based weak form
• integration by parts
• gauss theorem & boundary conditions
• weak form
weak form
finite element method

7. temporal discretization
node point based
• discretization
• time discrete weak form
• interpolation of material time derivatives
euler backward
temporal discretization
finite element method

8. spatial discretization
node point based
• discretization
• interpolation of test functions
• interpolation of trial functions
spatial discretization
finite element method

9. finite element method node point based discrete residuals
• discrete residual format
• discrete residuals
discrete residuals
finite element method

10. finite element method node point based linearization
• linearization / newton raphson scheme
• incremental residual
• system of equations
linearization
finite element method

11. finite element method node point based linearization
• iteration matrices
linearization
finite element method

12. simultaneous solution
node point based
loop over all time steps
global newton iteration
loop over all elements
loop over all quadrature points
evaulate balance of mass and momentum
determine element residuals & partial derivatives
determine global residuals and iterational matrices
determine and
determine state of biological equilibrium
simultaneous solution
finite element method

13. 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

14. integration vs node point based discontinuous solution
finite element method

15. integration vs node point based
continuous solution
finite element method

16. integration vs node point based
continuous solution
finite element method

17. example - adaptation in bone
different load cases
[1]
midstance phase of gait
2317 N
24˚
703 N
28˚
[2]
extreme range of abduction
1158 N
-15˚
351 N
-8 ˚
[3]
extreme range of adduction
1548 N
56˚
468 N
35˚
Carter & Beaupré [2001]
example - adaptation in bone

18. example - adaptation in bone
node point based - h-refinement
example - adaptation in bone

19. example - adaptation in bone
node point based - p-refinement
example - adaptation in bone

20. example - adaptation in bone
integration point based - p-refinement
example - adaptation in bone