Notes of SIGGRAPH 2012 FEM course
Xutong Jin
2023-02-06
FEM Continuous Discretization Undeformed configuration Deformed location Deformation gradient tensor Deformation Hyperelastic strain energy Strain energy per unit undeformed volume Total energy Elastic force: intuitive understanding Force per unit undeformed volume Aggregate force Force per unit undeformed area Aggregate force Stress tensor 1st Piola-Kirchhoff stress tensor (3*3 matrix) For hyperelastic material Boundary and Internal Inconsistency Intuitive way: Strain measures Green strain tensor Discard rotation as Linear approximation Linear elasticity strain energy density Young's modulus Poisson's ratio Piola stress St. Venant-Kirchhoff model Corotated linear elasticity Isotropic materials hyperelastic constitutive model rotationally invariant and isotropic Isotropic materials and invariants isotropic invariant Neohookean elasticity Energy and force discretization Discrete state Interpolated deformation map Strain energy Discrete energy Discrete elastic forces For individual elements energy force Linear tetrahedral elements In every tetrahedron constant deformation gradient Determine Volume Nonsingular for nonzero volume Deformation gradient Strain energy Force Force differentials Expensive expression (stiffness matrix) Directly Constitutive model related Stress differentials for St. Venant-Kirchhoff materials Stress differentials for Neohookean materials Model reduction Linear modal analysis Nonlinear model reduction cubic polynomial quadrature points Choice of basis Basis from simulation data