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116 lines
4.0 KiB
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116 lines
4.0 KiB
Org Mode
#+TITLE: Validation Examples for 2D Heterogeneous Diffusion
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#+AUTHOR: MDL <delucia@gfz-potsdam.de>
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#+DATE: 2023-08-26
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#+STARTUP: inlineimages
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#+LATEX_CLASS_OPTIONS: [a4paper,9pt]
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#+LATEX_HEADER: \usepackage{fullpage}
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#+LATEX_HEADER: \usepackage{amsmath, systeme}
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#+LATEX_HEADER: \usepackage{graphicx}
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#+LATEX_HEADER: \usepackage{charter}
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#+OPTIONS: toc:nil
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* Simple setup using deSolve/ReacTran
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- Square of side 10
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- Discretization: 11 \times 11 cells
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- All boundaries closed
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- Initial state: 0 everywhere, 1 in the center (6,6)
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- Time step: 1 s, 10 iterations
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The matrix of spatially variable diffusion coefficients \alpha is
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constant in 4 quadrants:
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\alpha_{x,y} =
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| 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |
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| 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |
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| 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |
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| 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |
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| 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |
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| 1 | 1 | 1 | 1 | 1 | 1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
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| 1 | 1 | 1 | 1 | 1 | 1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
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| 1 | 1 | 1 | 1 | 1 | 1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
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| 1 | 1 | 1 | 1 | 1 | 1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
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| 1 | 1 | 1 | 1 | 1 | 1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
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| 1 | 1 | 1 | 1 | 1 | 1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |
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The relevant part of the R script used to produce these results is
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presented in listing 1; the whole script is at [[file:scripts/HetDiff.R]].
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A visualization of the output of the reference simulation is given in
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figure [[fig:1][1]].
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Note: all results from this script are stored in the =outc= matrix by
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the =deSolve= function. I stored a different version into
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[[file:../scripts/gold/HetDiff1.csv]]: this file contains 11 columns (one
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for each time step including initial conditions) and 121 rows, one for
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each domain element, with no headers.
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#+caption: Result of ReacTran/deSolve solution of the above problem at 4
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#+name: fig:1
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[[./images/deSolve_AlphaHet1.png]]
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#+name: lst:1
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#+begin_src R :language R :frame single :caption Listing 1, generate reference simulation using R packages deSolve/ReacTran :captionpos b :label lst:1
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library(ReacTran)
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library(deSolve)
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## harmonic mean
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harm <- function(x,y) {
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if (length(x) != 1 || length(y) != 1)
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stop("x & y have different lengths")
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2/(1/x+1/y)
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}
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N <- 11 # number of grid cells
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ini <- 1 # initial value at x=0
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N2 <- ceiling(N/2)
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L <- 10 # domain side
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## Define diff.coeff per cell, in 4 quadrants
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alphas <- matrix(0, N, N)
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alphas[1:N2, 1:N2] <- 1
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alphas[1:N2, seq(N2+1,N)] <- 0.1
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alphas[seq(N2+1,N), 1:N2] <- 0.01
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alphas[seq(N2+1,N), seq(N2+1,N)] <- 0.001
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cmpharm <- function(x) {
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y <- c(0, x, 0)
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ret <- numeric(length(x)+1)
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for (i in seq(2, length(y))) {
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ret[i-1] <- harm(y[i], y[i-1])
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}
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ret
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}
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## Construction of the 2D grid
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x.grid <- setup.grid.1D(x.up = 0, L = L, N = N)
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y.grid <- setup.grid.1D(x.up = 0, L = L, N = N)
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grid2D <- setup.grid.2D(x.grid, y.grid)
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dx <- dy <- L/N
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D.grid <- list()
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## Diffusion coefs on x-interfaces
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D.grid$x.int <- apply(alphas, 1, cmpharm)
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## Diffusion coefs on y-interfaces
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D.grid$y.int <- t(apply(alphas, 2, cmpharm))
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# The model
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Diff2Dc <- function(t, y, parms) {
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CONC <- matrix(nrow = N, ncol = N, data = y)
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dCONC <- tran.2D(CONC, dx = dx, dy = dy, D.grid = D.grid)$dC
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return(list(dCONC))
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}
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## initial condition: 0 everywhere, except in central point
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y <- matrix(nrow = N, ncol = N, data = 0)
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y[N2, N2] <- ini # initial concentration in the central point...
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## solve for 10 time units
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times <- 0:10
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outc <- ode.2D(y = y, func = Diff2Dc, t = times, parms = NULL,
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dim = c(N, N), lrw = 1860000)
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#+end_src
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