1 08 - finite element method 08 - finite element method - density growth - theory

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Text of 1 08 - finite element method 08 - finite element method - density growth - theory

  • 08 - finite element method08 - finite element method -density growth - theory

  • finite element methodfrom continuous problem to linearized discrete initial boundary value problem temporal discretization linearization spatial discretization staggered/simultaneousimplicit euler backwardfinite element methodnewton raphson iterationgateaux derivative

  • finite element methodsequential solution - element basedstaggered solution - integration point basedsimultaneous solution - node point basedhuiskes, weinans, grootenboer, dalstra, fudala & slooff [1987],carter, orr, fhyrie [1989], beaupr, orr & carter [1990], weinans,huiskes & grootenboer [1992], [1994], jacobs, levenston, beaupr,simo & carter [1995], huiskes [2000], carter & beaupr [2001]weinans, huiskes & grootenboer [1992], harrigan & hamilton [1992],,[1994], jacobs, levenston, beaupr,simo & carter [1995]jacobs, levenston, beaupr,simo & carter [1995], fischer, jacobs,levenston & carter [1997], nackenhorst [1997], levenston [1997]]

  • finite element methodfrom strong form to weak form (1d) strong / differential form weak / integral form - nonsymmetric strong form / residual format integration by parts integral theorem & neumann bcs weak form / integral form - symmetric

  • finite element methodintegration point basedstrong form cast it into its residual format with residual start with nonlinear mechanical equilibrium equation

  • finite element methodintegration point basedboundary conditions dirichlet / essential boundary conditions neumann / natural boundary conditions strong / differential form

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

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

  • finite element methodintegration point basedspatial discretization interpolation of test functions interpolation of trial functions discretization

  • finite element methodintegration point baseddiscrete residual discrete residual format discrete residual discrete weak form

  • finite element methodintegration point baseddiscrete residual discrete residual residual of mechanical equilibrium/balance of momentumcheck in matlab!

  • finite element methodintegration point basedlinearization incremental residual system of equations linearization / newton raphson scheme incremental iterative update

  • finite element methodintegration point basedlinearization stiffness matrix / iteration matrix linearization of residual wrt nodal dofs

  • finite element methodintegration point basedlinearization stiffness matrix / iteration matrix linearization of residual wrt nodal dofscheck in matlab!

  • finite element methodintegration point basedconstitutive equationswith constitutive equations - given calculateandfrom

  • finite element methodintegration point basedconstitutive equations stress calculation @ integration point level constitutive equations - given calculatecheck in matlab!

  • finite element methodintegration point basedconstitutive equations constitutive equations - given calculate temporal discretization - euler implizit with with local newton iteration

  • finite element methodintegration point basedconstitutive equationscheck in matlab! residual of biological equilibrium / balance of mass discrete density update

  • finite element methodintegration point basedconstitutive equationsdepends on time discretizationwith constitutive equations - given calculate

  • finite element methodintegration point basedconstitutive equations tangent operator / constitutive moduli linearization of stress wrt deformation gradientcheck in matlab!

  • finite element methodintegration point basedstaggered solutionloop over all time stepsglobal newton iterationloop over all elementsloop over all quadrature pointslocal newton iteration to determinedetermine element residual & partial derivativedetermine global residual and iterational matrixdeterminedetermine state of biological equilibrium

  • finite element methodintegration point basedstaggered solutionloop over all time stepsglobal newton iterationloop over all elementsloop over all quadrature pointslocal newton iteration to determinedetermine element residual & partial derivativedetermine global residual and iterational matrixdeterminedetermine state of biological equilibrium