Finite Element

Finite Element constitutions are the base for vertex-based deformable geometry in libuipc. The state is given by vertex positions; each constitution adds its own energy (or none) on top of the common kinetic and attribute setup. Constitutions such as Empty, Particle, ARAP, Hookean Spring, Neo Hookean Shell, Discrete Shell Bending, Kirchhoff Rod Bending, and Stable Neo Hookean inherit this base and document only their additional attributes and energy.

State Variable

The state variable for a Finite Element vertex is its position:

\[ \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}. \]

Positions are the degrees of freedom that evolve over time. Each concrete constitution documents only the attributes it introduces and how they map to its energy formula.

Shell thickness convention

For a triangle mesh, the vertex thickness attribute stores the one-sided collision offset (radius) \(r\). The physical slab extends to both sides of the surface, so its full material thickness is \(h=2r\). Shell mass uses \(\rho A_0(2r)\), and shell constitutions that depend on physical thickness must use \(2r\). A zero radius switches the generic vertex measure to triangle area, so mass_density is then interpreted as a surface density.