Mesh Types

Mesh Types

Comprehensive guide for selecting mesh types in numerical simulations.

Structured Meshes

Cartesian (Regular)

Uniform spacing in each direction.

Grid points at: (i×dx, j×dy, k×dz)
where i, j, k are integers

Properties:

Best for:

• Phase-field simulations
• Spectral methods
• Regular domains (boxes)
• Prototyping and testing

Rectilinear (Non-uniform Cartesian)

Variable spacing, still axis-aligned.

Grid points at: (x_i, y_j, z_k)
where x_i, y_j, z_k are 1D arrays

Properties:

Use for:

• Boundary layer refinement
• Interface refinement
• Graded meshes

Stretching functions:

Geometric: dx_i = dx_0 × r^i
Hyperbolic tangent: concentrated at boundaries
Polynomial: smooth variation

Curvilinear (Body-Fitted)

General structured mesh, not axis-aligned.

Physical: (x, y, z)
Computational: (ξ, η, ζ) on [0,1]³

Mapping: x = x(ξ,η,ζ), etc.

Properties:

Metric terms:

∂/∂x = (1/J)[∂ξ/∂x × ∂/∂ξ + ∂η/∂x × ∂/∂η + ∂ζ/∂x × ∂/∂ζ]
where J = Jacobian of mapping

Common transformations:

• Polar/cylindrical
• Elliptic smoothing
• Algebraic stretching

Block-Structured

Multiple structured blocks patched together.

Block 1: (i,j,k) ∈ [0,N1] × [0,M1] × [0,P1]
Block 2: (i,j,k) ∈ [0,N2] × [0,M2] × [0,P2]
...
Interface: matching or non-matching

Properties:

Unstructured Meshes

Triangular (2D) / Tetrahedral (3D)

Elements: triangles (2D) or tetrahedra (3D).

Properties:

Data structures:

Nodes: [(x_0, y_0), (x_1, y_1), ...]
Elements: [(n_0, n_1, n_2), ...]  # node indices
Edges: derived from elements

Quality metrics:

• Aspect ratio
• Minimum angle
• Circumradius/inradius ratio

Quadrilateral (2D) / Hexahedral (3D)

Elements: quads (2D) or hexahedra (3D).

Properties:

Advantages over tri/tet:

• Fewer elements for same accuracy
• Better alignment with flow/field directions
• Lower numerical diffusion for advection

Mixed/Hybrid

Combine different element types.

Near walls: structured quad/hex (boundary layer)
Interior: unstructured tri/tet (flexibility)

Common patterns:

• Prism layers near walls + tets in bulk
• Quad faces on boundaries + tet interior
• Hanging nodes with transitions

Special Mesh Types

Octree/Quadtree

Hierarchical refinement by recursive subdivision.

Root cell covers domain
Subdivide cells based on criterion
Continue until resolution satisfied

Properties:

Refinement criteria:

• Gradient magnitude
• Error estimator
• Geometric features
• Distance to interface

Voronoi/Polyhedral

Cells are general polygons/polyhedra.

Properties:

Overset (Chimera)

Multiple overlapping meshes.

Background mesh: covers domain
Body-fitted mesh: around objects
Interpolation: at overlap boundaries

Use for:

• Moving bodies
• Multiple components
• Complex geometry with relative motion

Mesh Selection Guide

By Geometry Complexity

By Physics

By Method

Mesh Refinement Strategies

H-Refinement

Subdivide cells/elements.

Original: element of size h
Refined: 4 elements (2D) or 8 elements (3D) of size h/2

P-Refinement

Increase polynomial order within elements.

Original: linear elements (p=1)
Refined: quadratic elements (p=2)

R-Refinement (Mesh Movement)

Move existing nodes without changing connectivity.

Nodes move toward regions needing resolution
Total node count unchanged

HP-Refinement

Combine h and p adaptively.

Smooth regions: increase p
Non-smooth regions: decrease h

Mesh Generation Considerations

Input Requirements

Output Quality Checks

• No inverted elements
• Aspect ratio within bounds
• Skewness within bounds
• Minimum angle acceptable
• Smooth size transitions
• Boundary conformity

Common Tools

Quick Reference

Mesh Type Trade-offs