# Spatially varying mesh quality

Updated version of the 2018 blog post Spatially varying mesh quality. Needs `fmesher` version `0.3.0.9005` or later.

## Fundamentals of triangle mesh refinement

The fmesher package has several features that aren’t widely known. The code started as an implementation of the triangulation method detailed in Hjelle and Dæhlen, “Triangulations and Applications” (2006), which includes methods for spatially varying mesh quality.1 Over time, the interface and feature improvements focused on robustness and calculating useful mesh properties, but the more advanced mesh quality features are still there!

The algorithm first builds a basic mesh, including any points the user specifies, as well as any boundary curves (if no boundary curve is given, a default boundary will be added), connecting all the points into a Delaunay triangulation.

After this initial step, mesh quality is decided by two criteria:

1. Minimum allowed angle in a triangle
2. Maximum allowed triangle edge length

As long as any triangle does not fulfil the criteria, a new mesh point is added in a way that is guaranteed to locally fix the problem, and a new Delaunay triangulation is obtained. This process is repeated until all triangles fulfil the criteria. In perfect arithmetic, the algorithm is guaranteed to converge as long as the minimum angle criterion is at most 21 degrees, and the maximum edge length criterion is strictly positive.

A basic mesh with regular interior triangles can be created as follows:

    library(ggplot2)
    library(fmesher)

    loc <- as.matrix(expand.grid(1:10, 1:10))
    bnd_sf <- fm_nonconvex_hull(loc, convex = 1, concave = 10)
    bnd <- fm_as_segm(bnd_sf)
    loc <- fm_hexagon_lattice(bnd_sf, edge_len = 1)
    mesh1 <- fm_rcdt_2d_inla(
      loc = loc,
      boundary = bnd,
      refine = list(max.edge = Inf)
    )
    ggplot() +
      geom_fm(data = mesh1)

The `fm_nonconvex_hull_inla` function creates a polygon to be used as the boundary of the mesh, extending the domain by a distance `convex` from the given points, while keeping the outside curvature radius to at least `concave`. In this simple example, we used the method `fm_rcdt_2d_inla`, which is the most direct interface to `fmesher`. The `refine = list(max.edge = Inf)` setting makes `fmesher` enforce the default minimum angle criterion (21 degrees) but ignore the edge length criterion. To get smaller triangles, we change the `max.edge` value:

    mesh2 <- fm_rcdt_2d_inla(
      loc = loc,
      boundary = bnd,
      refine = list(max.edge = 0.5)
    )
    ggplot() +
      geom_fm(data = mesh2) +
      coord_sf(default = TRUE)

It is often possible and desirable to increase the minimum allowed angle to achieve smooth transitions between small and large triangles (values as high as 26 or 27, but almost never as high as 35, may be possible). Here we will instead focus on the edge length criterion, and make that spatially varying.

## Spatially varying edge length

We now define a function that computes our desired maximal edge length as a function of location, and feed the output of that to `fm_rcdt_2d_inla` using the `quality.spec` parameter instead of `max.edge`:

    qual_loc <- function(loc) {
      if (inherits(loc, c("sf", "sfc", "sfg"))) {
        loc <- sf::st_coordinates(loc)
      }
      pmax(0.05, (loc[, 1] * 2 + loc[, 2]) / 16)
    }
    mesh3 <- fm_rcdt_2d_inla(
      loc = loc,
      boundary = bnd,
      refine = list(max.edge = Inf),
      quality.spec = list(
        loc = qual_loc(loc),
        segm = qual_loc(bnd$loc)
      )
    )
    ggplot() +
      geom_fm(data = mesh3) +
      coord_sf(default = TRUE)

This gave a smooth transition between large and small triangles!

We can also use different settings on the boundary, e.g. `NA_real_` or `Inf` to make it not care about edge lengths near the boundary:

    qual_bnd <- function(loc) {
      rep(Inf, nrow(loc))
    }
    mesh5 <- fm_rcdt_2d_inla(
      loc = loc,
      boundary = bnd,
      refine = list(max.edge = Inf),
      quality.spec = list(
        loc = qual_loc(loc),
        segm = qual_bnd(bnd$loc)
      )
    )
    ggplot() +
      geom_fm(data = mesh5) +
      coord_sf(default = TRUE)

We can also make more complicated specifications, but beware of asking only for reasonable triangulations; When experimenting, I recommend setting the `max.n.strict` and `max.n` values in the `refine` parameter list, that prohibits adding infinitely many triangles!

    qual_bnd <- function(loc) {
      if (inherits(loc, c("sf", "sfc", "sfg"))) {
        loc <- sf::st_coordinates(loc)
      }
      pmax(0.1, 1 - abs(loc[, 2] / 10)^2)
    }
    mesh5 <- fm_rcdt_2d_inla(
      loc = loc,
      boundary = bnd,
      refine = list(
        max.edge = Inf,
        max.n.strict = 5000
      ),
      quality.spec = list(
        loc = qual_loc(loc),
        segm = qual_bnd(bnd$loc)
      )
    )
    ggplot() +
      geom_fm(data = mesh5) +
      coord_sf(default = TRUE)

## Assessing the mesh quality

The `fm_assess` function may provide some insights into the effect of variable mesh quality settings:

    out <- fm_assess(mesh5,
      spatial.range = 5,
      alpha = 2,
      dims = c(200, 200)
    )
    print(names(out))
    #> [1] "sd"       "sd.dev"   "sd.bound" "edge.len" "geometry"

    ggplot() +
      geom_tile(
        data = out,
        aes(geometry = geometry, fill = edge.len),
        stat = "sf_coordinates"
      ) +
      coord_sf(default = TRUE) +
      scale_fill_distiller(
        type = "seq",
        direction = 1,
        palette = 1,
        na.value = "transparent"
      )

A “good” mesh should have `sd.dev` close to 1; this indicates that the nominal variance specified by the continuous domain model and the variance of the discretise model are similar.

    sd.dev.limits <- 1 + c(-1, 1) * max(abs(range(out$sd.dev, na.rm = TRUE) - 1))
    col.values <- 2 * seq(0, 1, length.out = 100) - 1
    col.values <- (sign(col.values) * abs(col.values)^1.5 + 1) / 2
    ggplot() +
      geom_tile(
        data = out,
        aes(geometry = geometry, fill = sd.dev),
        stat = "sf_coordinates"
      ) +
      coord_sf(default = TRUE) +
      scale_fill_distiller(
        type = "div",
        palette = "RdBu",
        na.value = "transparent",
        limits = sd.dev.limits,
        values = col.values
      )

Here, we see that the standard deviation of the discretised model is smaller in the triangle interiors than at the vertices, but the ratio is close to 1, and more uniform where the triangles are small compared with the spatial correlation length that we set to `5`. The most problematic points seem to be in the rapid transition between large and small triangles. We can improve things by increasing the minimum angle criterion:

    mesh6 <- fm_rcdt_2d_inla(
      loc = loc,
      boundary = bnd,
      refine = list(
        min.angle = 30,
        max.edge = Inf,
        max.n.strict = 5000
      ),
      quality.spec = list(
        loc = qual_loc(loc),
        segm = qual_bnd(bnd$loc)
      )
    )
    out6 <- fm_assess(mesh6,
      spatial.range = 5,
      alpha = 2,
      dims = c(200, 200)
    )

    ggplot() +
      geom_tile(
        data = out6,
        aes(geometry = geometry, fill = sd.dev),
        stat = "sf_coordinates"
      ) +
      coord_sf(default = TRUE) +
      scale_fill_distiller(
        type = "div",
        palette = "RdBu",
        na.value = "transparent",
        limits = sd.dev.limits,
        values = col.values
      )

The `sd.dev` range has clearly decreased (the maximum absolute deviation from 1 decreased from 0.0994 to 0.0989), and there is no longer a band of high values near the upper right corner.

The number of additional vertices needed to accomplish this is small, as seen from the mesh object summaries:

    mesh5
    #> fm_mesh_2d object:
    #>   Manifold:  R2
    #>   V / E / T: 1633 / 4567 / 2935
    #>   Euler char.:   1
    #>   Constraints:   Boundary: 329 boundary edges (1 group: 1), Interior: 0 edges
    #>   Bounding box: (7.471455e-04,1.099925e+01) x (7.471455e-04,1.099925e+01)
    #>   Basis d.o.f.:  1633
    mesh6
    #> fm_mesh_2d object:
    #>   Manifold:  R2
    #>   V / E / T: 2009 / 5692 / 3684
    #>   Euler char.:   1
    #>   Constraints:   Boundary: 332 boundary edges (1 group: 1), Interior: 0 edges
    #>   Bounding box: (7.471455e-04,1.099925e+01) x (7.471455e-04,1.099925e+01)
    #>   Basis d.o.f.:  2009

1. The algorithm is very similar to the one used by `triangle`, available in R via Rtriangle. That only handles planar meshes, and had a potentially problematic non-commercial license clause. Since we needed spherical triangulations for global models, we had to write our own implementation.↩︎