FEM Elasticity

FEM Elasticity#

The following tutorial assumes that a viable volumetric mesh has been generated with segmented regions to apply boundary conditions.

Formulation#

Elasticity FEM is a 3-DOF model, whereby spatial integration is performed using a Continuous Galerkin technique (constructing the solution piecewise using shapefunctions in a transformed basis). Such elasticity problems are solved using the three components of Displacement as the degrees-of-freedom (DOF).

Materials#

Elasticity FEM requires the following material properties:

  • Mass|Density

  • YoungsModulus

  • Poisson|Ratio

If materials are not defined for a volume region, properties default to the physical constants.

Initial Conditions#

For transient problems, an initial state must be defined. Transient functionality for elasticity FEM is a work-in-progress.

Boundary Conditions#

For Elasticity FEM, it is not required to set boundary conditions on all surfaces.

  • Use NAN (not-a-number) to skip over one or more components of a boundary condition (leaving that specific DOF free).

  • Ensure that the problem is sufficiently constrained in the XYZ directions, else the solver will not converge!

Dirichlet (state)#

Displacement asserts XYZ strain onto a surface in linear length units of meters.

Neumann (gradient)#

Stress asserts the magnitude of Cauchy Stress via a local strain gradient normal to the surface multiplied by the Young’s Modulus, yielding units in N/m2 aka Pa.

Symmetry asserts zero local strain gradient normal to the surfaces.

Robin (composite)#

Traction asserts a strain reference and stiffness parameter in units of meters and N/m, respectively.

Solvers#

Once the physics and conditions are defined, an adjacency matrix A is assembled to represented the behavior of the system. An interleaved residual vector b is also defined. In preparation for solution v, linear operations are performed to manipulate the matrix and residual vector into a solvable system-of equations of form A*v = b.

The BiCGSTAB (Biconjugate Gradient Stabilized) method solves such multi-DOF formulations on the available GPU within a tolerance defined by Convergence in compute.cfg.

Steady-State#

Linear Steady State converges upon the solution for t->infinity.

Transient#

Linear Euler converges upon the solution iteratively in time with timestep defined in the solver settings.