3DFEM

Python 3D finite element code

This python code allows for solving 3D structural problems using the finite element method. New features will be added over time.

This code has NOT been validated on reference cases yet.

Requirements and dependencies

• Python 3

• numpy, scipy

• Paraview is suggested for visualizing exported VTK files, especially since animations are generated as a series of VTK files as handled by this software.

Current features:

Meshes

• Tetrahedral mesh generation from a set of points using scipy.spatial.Delaunay
• 4-node tetrahedral (Tet4), 6-node prism (Prism6), 8-node brick (Brick8) elements support
• Support for meshes containing different types of elements, possibly of different orders

Materials

• Linear isotropic elastic materials

Structural matrices

• Consistent mass matrix
• Linear elastic stiffness matrix
• Rayleigh damping matrix (linear combination of mass and stiffness matrices)
• Support only for zero-Dirichlet boundary conditions
• Full Gauss quadrature scheme

Reduced-Order Modeling

• Projection on linear elastic modes

Forces

• Nodal forces
• Support for modulation of the initial force vector by a given function over time steps or frequency steps in dynamical analyses

Solvers

• Modal analysis
• Linear static analysis
• Linear time-domain dynamics using the Newmark scheme and a reduced-order model based on elastic modes

Post-processing

• Export of a deformed mesh to VTK format (PolyData legacy format)
• Export of a series of deformed meshes to VTK format for animations (for visualizing elastic modes, or deformations in time-domain dynamical analyses)

Examples of solutions

Modal analysis

First linear elastic mode of a clamped-clamped beam displayed in Paraview

Time-domain dynamics

Linear elastic cantilever beam with Rayleigh damping pulled up by a constant force on its free end, then released

Linear elastic plate with Rayleigh damping excited by a nodal force, on the center of the upper face, varying with a bell curve envelope

Intended future features:

Meshes

• Support for Tet10, Prism18, Brick27 elements

Materials

• All anisotropy classes for elastic materials
• Linear viscoelastic materials using the Generalized Maxwell Model

Structural matrices

• Arbitrary Dirichlet boundary conditions

Reduced-Order Modeling

• Proper Orthogonal Decomposition (POD) for nonlinear problems

Forces

• Body forces
• Surface forces

Solvers

• Frequency-domain dynamical analysis
• Newton-Raphson method for geometrically nonlinear elastostatics and elastodynamics
• Arc-length method for geometrically nonlinear elastostatics and elastodynamics with strong nonlinearities (e.g. post-buckling analysis)

Post-processing

• Plotting of Frequency-Response Functions (FRF)
• Support for including element stress and strain components in the VTK files

Uncertainty Quantification

• Typical Maximum Entropy-probability distributions for elastic coefficients (Young's modulus, Poisson's ratio, anisotropic coefficients) for parametric uncertainties
• Typical Maximum Entropy-probability distributions for mass, linear elastic stiffness, Rayleigh damping and linear viscoelastic damping matrices for nonparametric uncertainties
• Monte Carlo method for uncertainty propagation with plotting of confidence intervals
• Gaussian Kernel Density Estimation (GKDE) for the estimation of probability density functions of observable quantities
• Causal nonparametric probabilistic models (linear viscoelasticity)