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

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

integration point based • discrete residual compare integration point based density growth! • discrete residual difference only in stress calculation @ int point! 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 compare integration point based density growth! • linearization of residual wrt nodal dofs difference only in stress calculation @ int point! 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 with • evolution of internal variable with and 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 • residual • linearization of residual • with constitutive equations finite element method

stress driven volume growth finite element method

integration point based constitutive equations • discrete volume update check in matlab! • residual of biological equilibrium / volume growth 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