# 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: 880 / 2598 / 1719
    #>   Euler char.:   1
    #>   Constraints:   39 boundary edges (1 group: 1), 111 interior edges (1 group: 1)
    #>   Bounding box: (-11.963110,  8.246925) x (-5.522181, 5.499657) x (0,0)
    #>   Basis d.o.f.:  880
    plot(mesh2, axes = TRUE)

    (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 it's triangular Barycentric coordinates
    bary <- fm_bary(mesh2, loc = pts)
    # How many points are outside the mesh?
    sum(is.na(bary$t))
    #> [1] 2
    head(bary$bary)
    #>            [,1]      [,2]      [,3]
    #> [1,] 0.04720217 0.5813611 0.3714367
    #> [2,] 0.13930150 0.7219260 0.1387725
    #> [3,] 0.43189448 0.1384910 0.4296145
    #> [4,] 0.20127814 0.3716721 0.4270498
    #> [5,] 0.21259133 0.3686084 0.4188003
    #> [6,] 0.44390951 0.1739205 0.3821700

    # Construct an evaluator object
    evaluator <- fm_evaluator(mesh2, loc = pts)
    sum(!evaluator$proj$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] 5.438358e-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$t))
    #> [1] 0
    head(bary1$bary)
    #>          [,1]      [,2]
    #> [1,] 2.000000 -1.000000
    #> [2,] 1.992993 -0.992993
    #> [3,] 1.985986 -0.985986
    #> [4,] 1.978979 -0.978979
    #> [5,] 1.971972 -0.971972
    #> [6,] 1.964965 -0.964965

    # Construct an evaluator object.
    evaluator1 <- fm_evaluator(mesh1, loc = pts1)
    # mesh_1d basis functions are defined everywhere
    sum(!evaluator1$proj$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")

## Plotting

### Base graphics

    plot(mesh2)

### `ggplot` graphics

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

    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))

## 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)