# R/FEMobjects.R In fdaPDE: Statistical Analysis of Functional and Spatial Data, Based on Regression with PDE Regularization

#### Documented in create.FEM.basisFEM

#' Create a FEM basis
#'
#' @param mesh A \code{mesh.2D} or \code{mesh.2.5D} object representing the domain triangulation.
#' @return A \code{FEMbasis} object. This contains the \code{mesh}, along with some additional quantities:
#' \itemize{
#' 	\item{\code{order}}{Either "1" or "2". Order of the Finite Element basis.}
#' 	\item{\code{nbasis}}{Scalar. The number of basis.}
#' 	\item{\code{transf_coord}}{It takes value only in the 2D case. It is a list of 4 vectors: diff1x, diff1y, diff2x and diff2y.
#' 	Each vector has length #triangles and encodes the information for the tranformation matrix that transforms the
#' 	nodes of the reference triangle to the nodes of the i-th triangle.
#' 	The tranformation matrix for the i-th triangle has the form [diff1x[i] diff2x[i]; diff1y[i] diff2y[i]].}
#' 	\item{\code{detJ}}{It takes value only in the 2D case. A vector of length #triangles. The ith element contains
#' 	the determinant of the transformation from the reference triangle to the nodes of the i-th triangle.
#' 	Its value is also the double of the area of each triangle of the basis.}
#' }
#' @description Sets up a Finite Element basis. It requires a \code{mesh.2D} or \code{mesh.2.5D} object,
#' as input.
#' The basis' functions are globally continuos functions, that are polynomials once restricted to a triangle in the mesh.
#' The current implementation includes linear finite elements (when \code{order = 1} in the input \code{mesh}) and
#' quadratic finite elements (when \code{order = 2} in the input \code{mesh}).
#' @usage create.FEM.basis(mesh)
#' @examples
#' ## Upload the quasicircle2D data
#' data(quasicircle2D)
#' boundary_nodes = quasicircle2D$boundary_nodes #' boundary_segments = quasicircle2D$boundary_segments
#' locations = quasicircle2D$locations #' data = quasicircle2D$data
#'
#' ## Create the 2D mesh
#' mesh = create.mesh.2D(nodes = rbind(boundary_nodes, locations), segments = boundary_segments)
#' ## Plot it
#' plot(mesh)
#' ## Create the basis
#' FEMbasis = create.FEM.basis(mesh)
#' ## Upload the hub2.5D data
#' data(hub2.5D)
#' hub2.5D.nodes = hub2.5D$hub2.5D.nodes #' hub2.5D.triangles = hub2.5D$hub2.5D.triangles
#'
#' ## Create the 2.5D mesh
#' mesh = create.mesh.2.5D(nodes = hub2.5D.nodes, triangles = hub2.5D.triangles)
#' ## Plot it
#' plot(mesh)
#' ## Create the basis
#' FEMbasis = create.FEM.basis(mesh)
#' @export

create.FEM.basis = function(mesh)
{
if(class(mesh)!='mesh.2D' & class(mesh)!='mesh.2.5D' & class(mesh)!='mesh.3D')
stop("Unknown mesh class")

if (class(mesh)=="mesh.2D"){

#  The number of basis functions corresponds to the number of vertices
#  for order = 1, and to vertices plus edge midpoints for order = 2

nbasis = dim(mesh$nodes)[] eleProp = R_elementProperties(mesh) #eleProp = NULL #if(CPP_CODE == FALSE) #{ # eleProp = R_elementProperties(mesh) #} FEMbasis = list(mesh = mesh, order = as.integer(mesh$order), nbasis = nbasis, detJ=eleProp$detJ, transf_coord = eleProp$transf_coord)
class(FEMbasis) = "FEMbasis"

FEMbasis
} else if (class(mesh) == "mesh.2.5D" || class(mesh) == "mesh.3D"){

FEMbasis = list(mesh = mesh, order = as.integer(mesh$order),nbasis = mesh$nnodes)
class(FEMbasis) = "FEMbasis"
FEMbasis
}
}
#' Define a surface or spatial field by a Finite Element basis expansion
#'
#' @param coeff A vector or a matrix containing the coefficients for the Finite Element basis expansion. The number of rows
#' (or the vector's length) corresponds to the number of basis in \code{FEMbasis}.
#' The number of columns corresponds to the number of functions.
#' @param FEMbasis A \code{FEMbasis} object defining the Finite Element basis, created by \link{create.FEM.basis}.
#' @description This function defines a FEM object.
#' @usage FEM(coeff,FEMbasis)
#' @return An \code{FEM} object. This contains a list with components \code{coeff} and \code{FEMbasis}.
#' @examples
#' library(fdaPDE)
#' ## Upload the horseshoe2D data
#' data(horseshoe2D)
#' boundary_nodes = horseshoe2D$boundary_nodes #' boundary_segments = horseshoe2D$boundary_segments
#' locations = horseshoe2D$locations #' #' ## Create the 2D mesh #' mesh = create.mesh.2D(nodes = rbind(boundary_nodes, locations), segments = boundary_segments) #' ## Create the FEM basis #' FEMbasis = create.FEM.basis(mesh) #' ## Compute the coeff vector evaluating the desired function at the mesh nodes #' ## In this case we consider the fs.test() function introduced by Wood et al. 2008 #' coeff = fs.test(mesh$nodes[,1], mesh$nodes[,2]) #' ## Create the FEM object #' FEMfunction = FEM(coeff, FEMbasis) #' ## Plot it #' plot(FEMfunction) #' @export FEM<-function(coeff,FEMbasis) { if (is.null(coeff)) stop("coeff required; is NULL.") if (is.null(FEMbasis)) stop("FEMbasis required; is NULL.") if(class(FEMbasis) != "FEMbasis") stop("FEMbasis not of class 'FEMbasis'") coeff = as.matrix(coeff) if(nrow(coeff) != FEMbasis$nbasis)
stop("Number of row of 'coeff' different from number of basis")

fclass = NULL
fclass = list(coeff=coeff, FEMbasis=FEMbasis)
class(fclass)<-"FEM"
return(fclass)
}

R_elementProperties=function(mesh)
{
nele = dim(mesh$triangles)[] nodes = mesh$nodes
triangles = mesh$triangles #detJ = matrix(0,nele,1) # vector of determinant of transformations #metric = array(0,c(nele,2,2)) # 3-d array of metric matrices #transf = array(0,c(nele,2,2)) transf_coord = NULL transf_coord$diff1x = nodes[triangles[,2],1] - nodes[triangles[,1],1]
transf_coord$diff1y = nodes[triangles[,2],2] - nodes[triangles[,1],2] transf_coord$diff2x = nodes[triangles[,3],1] - nodes[triangles[,1],1]
transf_coord$diff2y = nodes[triangles[,3],2] - nodes[triangles[,1],2] # Jacobian or double of the area of triangle detJ = transf_coord$diff1x*transf_coord$diff2y - transf_coord$diff2x*transf_coord\$diff1y

#Too slow, computed only for stiff from diff1x,diff1y,..
# for (i in 1:nele)
# {
#   #transf[i,,] = rbind(cbind(diff1x,diff2x),c(diff1y,diff2y))
#   #  Compute controvariant transformation matrix OSS: This is (tranf)^(-T)
#   Ael = matrix(c(diff2y, -diff1y, -diff2x,  diff1x),nrow=2,ncol=2,byrow=T)/detJ[i]
#
#   #  Compute metric matrix
#   metric[i,,] = t(Ael)%*%Ael
# }

#FEStruct <- list(detJ=detJ, metric=metric, transf=transf)
FEStruct <- list(detJ=detJ, transf_coord=transf_coord)
return(FEStruct)
}


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fdaPDE documentation built on July 2, 2020, 2:22 a.m.