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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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