Constrained Delaunay Triangulation Meshes for Spatial 'SPDE' Models.

tulpaMesh

Spatial meshes and FEM matrices for SPDE models, from point clouds to precision matrices in three lines.

Quick Start

library(tulpaMesh)

coords <- cbind(runif(200), runif(200))
mesh <- tulpa_mesh(coords, max_edge = 0.1)
fem  <- fem_matrices(mesh, obs_coords = coords)
# fem$C (mass), fem$G (stiffness), fem$A (projection) ready for SPDE

Statement of Need

Spatial models based on the SPDE approach (Lindgren, Rue & Lindstrom 2011) need a triangulated mesh and finite element matrices. tulpaMesh handles mesh generation, quality refinement, and FEM matrix assembly with no external dependencies beyond Rcpp and Matrix.

Common applications:

• Species distribution models with spatial random effects
• Geostatistical models with Matern covariance via SPDE
• Occupancy models with spatially structured detection
• Any Bayesian hierarchical model needing a spatial mesh

Features

Mesh Generation

• tulpa_mesh() - Constrained Delaunay from points, formulas, or sf boundaries
• tulpa_mesh_sphere() - Icosahedral geodesic meshes for global-scale models
• tulpa_mesh_1d() - Temporal meshes for space-time SPDE
• tulpa_mesh_graph() - Network meshes for river/road SPDE

FEM Assembly

• fem_matrices() - Mass (C), stiffness (G), and projection (A) matrices
• fem_matrices(lumped = TRUE) - Diagonal mass for SPDE Q-builder
• fem_matrices(parallel = TRUE) - Multithreaded assembly via TBB
• fem_matrices(barrier = ...) - Barrier models (Bakka et al. 2019)
• fem_matrices_p2() - Quadratic (P2) elements for higher accuracy
• fem_matrices_nonstationary() - Spatially varying range and variance

Quality & Diagnostics

• mesh_quality() - Per-triangle angles, aspect ratio, area
• mesh_summary() - Min/median/max quality statistics
• plot() - Mesh visualization with optional quality coloring
• Ruppert refinement via min_angle and max_area parameters

Mesh Operations

• subdivide_mesh() - Uniform 1-to-4 triangle subdivision
• subset_mesh() - Extract submesh from triangle indices
• mesh_components() - Find disconnected components
• refine_mesh() - Adaptive refinement from error indicators
• as_tulpa_mesh() - Convert fmesher/INLA meshes
• barrier_triangles() - Identify barrier regions from polygons

Installation

Install from CRAN:

install.packages("tulpaMesh")

Development version:

# install.packages("pak")
pak::pak("gcol33/tulpaMesh")

Usage Examples

Mesh with Ruppert Refinement

library(tulpaMesh)

set.seed(42)
coords <- cbind(runif(100), runif(100))
mesh <- tulpa_mesh(coords, min_angle = 25, max_edge = 0.15)
mesh_summary(mesh)
plot(mesh, color = "quality", vertex_col = "black")

Barrier Model (Coastline)

library(sf)

# Barrier polygon (e.g., land mass that field can't cross)
barrier <- st_polygon(list(rbind(c(3,3), c(7,3), c(7,7), c(3,7), c(3,3))))
bt <- barrier_triangles(mesh, st_sfc(barrier))
fem <- fem_matrices(mesh, barrier = bt)
# fem$G has zero stiffness across barrier triangles

Spherical Mesh for Global Models

globe <- tulpa_mesh_sphere(subdivisions = 4)
globe
#> tulpa_mesh (sphere, radius = 1):
#>   Vertices:   2562
#>   Triangles:  5120
#>   Edges:      7680
fem <- fem_matrices(globe, lumped = TRUE)
# Total surface area approximates 4*pi
sum(fem$va)

Migrate from fmesher

library(fmesher)
fm <- fm_mesh_2d(loc = coords, max.edge = c(0.3, 0.6))
tm <- as_tulpa_mesh(fm)  # direct conversion
fem <- fem_matrices(tm)  # same C, G, A as fmesher

Documentation

• Quick Start
• Spatial Workflows — boundaries, barriers, sf integration
• Spherical and Temporal Meshes — globe, space-time, graphs, P2
• Full Reference

Support

"Software is like sex: it's better when it's free." — Linus Torvalds

I'm a PhD student who builds R packages in my free time because I believe good tools should be free and open. I started these projects for my own work and figured others might find them useful too.

If this package saved you some time, buying me a coffee is a nice way to say thanks. It helps with my coffee addiction.

License

MIT (see the LICENSE.md file)

Citation

@software{tulpaMesh,
  author = {Colling, Gilles},
  title = {tulpaMesh: Constrained Delaunay Triangulation Meshes for Spatial SPDE Models},
  year = {2026},
  url = {https://CRAN.R-project.org/package=tulpaMesh},
  doi = {10.32614/CRAN.package.tulpaMesh}
}