Geometry

Geometry#

Engineering geometries must facilitate:

  • Accurate physical representation

  • Element discretization and parallel computation

  • Region segmentation for conditions & couplings

  • Spatial transformations and instancing

Introduction#

Structured vs Unstructured#

  • Unstructured: Traditional declarative meshes storing nodal positions and topology as discrete elements.

  • Structured: Procedural grids/trees computing positions and topology on-demand. Used for SDF and background spatial computations.

Implicit Shapes#

Level sets express shapes without discrete topology. Fields map coordinates x to scalar values φ(x). The implicit boundary is defined where φ(x) = C (default C=0).

Signed Distance Field (SDF)#

When |∇φ(x)| = 1, the field is a distance field. With C=0, interior values are negative, exterior positive, satisfying SDF criteria.

Analytic vs Sampled#

  • Analytic: Primitive SDFs combined via operators for unlimited resolution. Requires gradient error handling.

  • Sampled: Discrete SDF on background grids (1M–1B nodes) enabling constant-time parallel sampling. Exact distances solved for narrow-band shells; winding numbers determine sign robustly.

Computable Topology#

Physics simulation requires shape discretization into computable elements. Nodes are points in space-time connected by elements to facilitate information flow.

Element Types#

Intrinsic attributes:

  • type (NODE1, LINE2, TRI3, QUAD4, TET4, HEX8, …)

  • id (0 default, positive for users)

  • indices (defining vertices)

Extrinsic functions: center, domain, distance, contains, decompose, quality, neighbors

Nodes (0D)#

  • Supported: NODE1

  • Functions: volume (based on median dual fragment)

Edges (1D)#

  • Supported: LINE2, DUAL2

  • Functions: length, area (interface vector for DUAL2)

Faces (2D)#

  • Supported: TRI3, QUAD4

  • Functions: area (normal vector)

Cells (3D)#

  • Supported: TET4, HEX8

  • Functions: volume (scalar)

Median Duals#

The primary mesh is what users see/save. Downstream numerics may interpret it differently:

  • FEM: Uses primary mesh directly with Gauss-Legendre quadrature.

  • FVM: Interprets primary mesh as Voronoi inverse, yielding median-dual control volumes for flux accuracy.

Regions of Interest#

Regions assign IDs to element subsets for boundary conditions and coupling.

Dimension

Region Type

Element Type

Visibility Default

0D

Groups

Nodes

groups=false

1D

Loops

Edges

loops=false

2D

Surfaces

Faces

surfaces=true

3D

Volumes

Cells

volumes=true

Elements and regions are complementary: each element has an id, and region id corresponds to elements with matching id.