Author: Finn Lindgren

# Basic fmesher use

#### Finn Lindgren

    suppressPackageStartupMessages(library(fmesher))
    set.seed(1234L)

## Mesh construction

    domain <- cbind(rnorm(4, sd = 3), rnorm(4))
    (mesh2 <- fm_mesh_2d(
      boundary = fm_extensions(domain, c(2.5, 5)),
      max.edge = c(0.5, 2)
    ))
    #> fm_mesh_2d object:
    #>   Manifold:  R2
    #>   V / E / T: 911 / 2685 / 1775
    #>   Euler char.:   1
    #>   Constraints:   Boundary: 45 boundary edges (1 group: 1), Interior: 127 interior edges (1 group: 1)
    #>   Bounding box: (-11.976144,  8.248637) x (-5.528888, 5.506054)
    #>   Basis d.o.f.:  911

    (mesh1 <- fm_mesh_1d(
      c(0, 2, 4, 7, 10),
      boundary = "free", # c("neumann", "dirichlet"),
      degree = 2
    ))
    #> fm_mesh_1d object:
    #>   Manifold:  R1
    #>   #{knots}:  5
    #>   Interval:  ( 0, 10)
    #>   Boundary:  (free, free)
    #>   B-spline degree:   2
    #>   Basis d.o.f.:  6

## Point lookup and evaluation

    pts <- cbind(rnorm(400, sd = 3), rnorm(400))

    # Find what triangle each point is in, and its triangular Barycentric
    # coordinates
    bary <- fm_bary(mesh2, loc = pts)
    head(bary)
    #> # A tibble: 6 × 2
    #>   index where[,1]  [,2]   [,3]
    #>   <int>     <dbl> <dbl>  <dbl>
    #> 1  1369     0.515 0.124 0.361
    #> 2   491     0.818 0.157 0.0254
    #> 3  1521     0.679 0.234 0.0876
    #> 4  1275     0.465 0.144 0.392
    #> 5  1054     0.785 0.158 0.0569
    #> 6  1532     0.191 0.230 0.579
    # How many points are outside the mesh?
    sum(is.na(bary$index))
    #> [1] 2
    bary$where[is.na(bary$index), ]
    #>      [,1] [,2] [,3]
    #> [1,]   NA   NA   NA
    #> [2,]   NA   NA   NA

    # Evaluate basis functions
    basis <- fm_basis(mesh2, loc = pts) # Raw SparseMatrix
    basis_object <- fm_basis(mesh2, loc = pts, full = TRUE) # fm_basis object
    sum(!basis_object$ok)
    #> [1] 2

    # Construct an evaluator object
    evaluator <- fm_evaluator(mesh2, loc = pts)
    sum(!fm_basis(evaluator, full = TRUE)$ok)
    #> [1] 2

    # Values for the basis function weights; for ordinary 2d meshes this coincides
    # with the resulting values at the vertices, but this is not true for e.g.
    # 2nd order B-splines on 1d meshes.
    field <- mesh2$loc[, 1]
    value <- fm_evaluate(evaluator, field = field)
    sum(abs(pts[, 1] - value), na.rm = TRUE)
    #> [1] 4.604303e-14

    pts1 <- seq(-2, 12, length.out = 1000)

    # Find what segment, and its interval Barycentric coordinates
    bary1 <- fm_bary(mesh1, loc = pts1)
    # Points outside the interval are treated differently depending on the
    # boundary conditions:
    sum(is.na(bary1$index))
    #> [1] 0
    head(bary1)
    #> # A tibble: 6 × 2
    #>   index where[,1]   [,2]
    #>   <int>     <dbl>  <dbl>
    #> 1     1      2    -1
    #> 2     1      1.99 -0.993
    #> 3     1      1.99 -0.986
    #> 4     1      1.98 -0.979
    #> 5     1      1.97 -0.972
    #> 6     1      1.96 -0.965

    # Evaluate basis functions
    basis1 <- fm_basis(mesh1, loc = pts1) # Raw SparseMatrix
    basis1_object <- fm_basis(mesh1, loc = pts1, full = TRUE) # fm_basis object
    sum(!basis1_object$ok)
    #> [1] 0

    # Construct an evaluator object.
    evaluator1 <- fm_evaluator(mesh1, loc = pts1)
    # mesh_1d basis functions are defined everywhere
    sum(!fm_basis(evaluator1, full = TRUE)$ok)
    #> [1] 0

    # Values for the basis function weights; for ordinary 2d meshes this coincides
    # with the resulting values at the vertices, but this is not true for e.g.
    # 2nd order B-splines on 1d meshes.
    field1 <- rnorm(fm_dof(mesh1))
    value1 <- fm_evaluate(evaluator1, field = field1)
    plot(pts1, value1, type = "l")

Evaluated 1D function

## Plotting

### Base graphics

    plot(mesh2)

2D triangulation mesh (base graphics version)

### `ggplot` graphics

    suppressPackageStartupMessages(library(ggplot2))
    ggplot() +
      geom_fm(data = mesh2)

2D triangulation mesh (ggplot version)

    ggplot() +
      geom_fm(data = mesh1, weights = field1 + 2, xlim = c(-2, 12)) +
      geom_fm(data = mesh1, linetype = 2, alpha = 0.5, xlim = c(-2, 12))

1D B-spline function space basis functions with evaluated function (ggplot version)

## Finite element calculations

    fem1 <- fm_fem(mesh1, order = 2)
    names(fem1)
    #> [1] "c0"  "c1"  "g1"  "g2"  "g01" "g02" "g12"
    fem2 <- fm_fem(mesh2, order = 2)
    names(fem2)
    #> [1] "b1" "c0" "c1" "g1" "g2" "k1" "k2" "ta" "va"

## Stochastic process simulation

    samp <- fm_matern_sample(mesh2, alpha = 2, rho = 4, sigma = 1)[, 1]
    evaluator <- fm_evaluator(
      mesh2,
      lattice = fm_evaluator_lattice(mesh2, dims = c(150, 50))
    )
    image(evaluator$x, evaluator$y, fm_evaluate(evaluator, field = samp), asp = 1)

Simulated 2D Matérn field