LAB 6: THREE-DIMENSIONAL GRAPHICS AND CURVILINEAR GRIDS

 

Mathematics:

A function of two variables u = f(x,y) is plotted as a surface in a space of three dimensions (x,y,u). When a rectangular domain of (x,y) is divided into sets of rectangles at the grid points in x and in y , the function u = f(x,y) is defined on the rectangular (Cartesian) grid. Other curvilinear grids are also used in various problems due to symmetries of the function f(x,y), symmetries of the domain of (x,y), or for a better design of numerical algorithms. This project is to study plotting of three-dimensional surfaces on the circular and triangular (curvilinear) grids.

 

MATLAB three-dimensional graphics can be used for functions defined at the curvilinear grids. If (xi,j,yi,j) are matrices for the coordinates of the curvilinear grid points, the function u = f(x,y) is defined as the matrix ui,j = f(xi,j,yi,j). Provided that the sizes of these three matrices agree, contour plots, mesh surfaces and painted surfaces with lighting can be constructed for functions defined at the curvilinear grids.

 

Objectives:

         Understand methods of building grid matrices (xi,j,yi,j) from coordinates of the curvilinear grid points.

         Exploit visualization of surfaces with circular symmetry.

         Exploit visualization of surfaces defined at the triangular grids.

 

MATLAB script for plotting functions on a circular grid:

 

When the function u = f(x,y) has a circular symmetry and is defined in a disk or in an annulus, it is convenient to locate the grid points at circles of radius r. The function u = f(x,y) can be rewritten as the function u = u(r,) in polar coordinates:

x = r*cos(), y = r*sin(),

where r is radius from (x,y) to the origin (0,0) and is the angle between the vector (x,y) and the positive x axis. For example, the torus is defined by the function:

u = (R2 (r-a)2)1/2 for a-R r a+R,

where a > R. The circular grid for the torus and the surface in the space (x,y,u) are shown here:

 

 

Steps in writing the script:

  1. Define the grid for radius vector r between [a-R,a+R] for R = 1 and a = 2.
  2. Define the grid for angle vector theta between [0,2*pi].
  3. Compute the matrices xi,j and yi,j in polar coordinates. Use the outer products of vectors r and .
  4. Open window #1 and plot the circular grid alone.
  5. Compute the matrix ui,j for each grid point on the circular grid.
  6. Open window #2 and plot the contour plots of the torus between u = 0 and u = umax.
  7. Open window #3 and plot the surface for the torus. Display positive and negative values of u at the same graph. Draw the surface and the circular grid at the same graph.

 

Exploiting the MATLAB script:

  1. Use surface with lighting for the following colormaps: hot, cold, jet, grey. Observe the difference in visualization of the surface.
  2. Use surface with lighting for the following shadings: flat, interp. Observe the difference in visualization of the surface.
  3. Change view of the tour by rotating the graph with view.

 

MATLAB script for plotting functions on a triangular grid:

 

When the function u = f(x,y) is computed as solution of a boundary value problem based on finite element methods, the function is usually defined on a triangular grid, that consists of triangular elements. An example of a triangular grid consisting of 64 triangles and 41 vertices (nodal points) is shown here:

Suppose the function u = f(x,y) = cos(x)*sin(y) is computed in the vertices of the triangular grid (e.g. with the use of the finite-element method). When the numerical approximation of the function u = f(x,y) is known at the vertices, the contour plot and the surface can be computed in the given domain of (x,y). The contour plot of the function looks like:

 

 

Steps in writing the script:

  1. Define a structure for two coordinates of each vertex (nodal point) on the triangular grid.
  2. Define a structure for three vertex point numbers of each triangular element of the triangular grid.
  3. Open window #1 and plot the triangular grid alone.
  4. Build the matrices (xi,j,yi,j) for coordinates of the triangular grid.
  5. Compute the matrix ui,j for each grid point at the triangular grid.
  6. Open window #2 and plot 6 contour plots of the function u = f(x,y). Draw the boundary of the domain at the same graph.
  7. Open window #3 and plot the surface for the function u = f(x,y). Display positive and negative values of u at the same graph. Draw the surface and the grid at the same graph.

 

Exploiting the MATLAB script:

  1. Construct the same function u = f(x,y) = cos(x)*sin(y) on a rectangular grid between 4 x 4 and -4 y 4 with step-size 1.
  2. Draw the contour plot and the surface as in steps (6)-(7). Compare the approximation of the function on rectangular and triangular grids.

 

QUIZ:

 

  1. Plot the sausaged torus defined in polar coordinates:

u = (R2 (r-a)2)1/2*cos(3) for a-R r a+R

 

  1. Plot the lump soliton in the disk of radius 4 by using polar coordinates:

u = (1 + y2 x2)/(1 + x2 + y2)2