FEM Elasticity

FEM Elasticity#

This tutorial assumes a sufficient-quality volumetric mesh has been generated with segmented regions ready for boundary conditions.

Formulation#

Elasticity uses a 3-DOF small-displacement model, currently optimized for isotropic ductile materials. A continuous Galerkin approach constructs the spatial solution using piecewise shape functions. The elliptic PDE is discretized into a linear matrix weak form using Displacement components as degrees-of-freedom (DOF).

Local Strain is solved using a hardware-accelerated stabilized biconjugate gradient solver. A gradient operator then resolves Strain|Gradient, enabling calculation of the CauchyStress tensor and von MisesStress scalar.

Material Properties#

Elasticity FEM requires:

  • Mass|Density

  • YoungsModulus

  • Poisson|Ratio

Properties can be assigned to specific volumes. Unassigned volumes default to system physical constants.

Initial Conditions#

Transient problems require an initial state definition. Transient elasticity functionality is currently under development.

Boundary Conditions#

Not all surfaces require boundary conditions. Use NAN to leave specific DOFs unconstrained.

Warning

Ensure the problem is sufficiently constrained in XYZ directions. Under-constrained systems will fail to converge.

Dirichlet (Value)#

  • Displacement: Asserts XYZ position in meters (or millimeters if geometry uses mm).

  • Symmetry: Enforced by setting displacement to 0 in the symmetric direction(s). Example for Y-symmetry: Displacement={NAN, 0, NAN}

Neumann (Gradient)#

  • Stress: Load as strain gradient vector scaled by Young’s Modulus. Units: Pa (N/m²)

  • Pressure: Scalar static load normal to surface, scaled by Young’s Modulus. Units: Pa

Robin (Composite)#

  • Traction: Load with external finite-stiffness material grip. Units: m (or mm) and N/m

Solvers#

Once physics and conditions are defined, an adjacency matrix A and residual vector b are assembled. Linear operations transform the system into A*v = b.

The BiCGSTAB solver resolves multi-DOF formulations on the GPU to maximum double-precision tolerance (~1e-16).

Steady-State#

Linear Steady State converges to the solution as t → ∞.

Transient#

Linear Newmark provides time-iterative solutions. Note: Transient integration for elasticity is a work-in-progress and not yet supported.