14 finite element method volume growth theory 14

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

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

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

from strong form … to weak form (1 d) • 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 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 element residual & partial derivative determine global residual and iterational matrix determine state of biological equilibrium staggered solution 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 element residual & partial derivative determine global residual and iterational matrix determine state of biological equilibrium staggered solution finite element method 22