1. 14 - finite element method
volume growth - theory
14 - finite element method

2. finite element method from continuous problem…
• temporal discretization
implicit euler backward
• spatial discretization
finite element method
• staggered solution
newton raphson iteration
• linearization
gateaux derivative
… to linearized discrete initial boundary value problem
finite element method

3. integration point based
• discrete residual
compare integration point based density growth!
• discrete residual
difference only in stress int point!
discrete residual
finite element method

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

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

6. integration point based
• stiffness matrix / iteration matrix
compare integration point based density growth!
• linearization of residual wrt nodal dofs
difference only in stress int point!
linearization
finite element method

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

8. integration point based constitutive equations
• constitutive equations - given calculate
with
and
with
• evolution of internal variable
with
and
constitutive equations
finite element method

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

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

11. integration point based constitutive equations
• residual
• linearization of residual
• with
constitutive equations
finite element method

12. stress driven volume growth
finite element method

13. integration point based constitutive equations
• discrete volume update
check in matlab!
• residual of biological equilibrium / volume growth
constitutive equations
finite element method

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

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

16. 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