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 (x_i,j,y_i,j) are matrices for the coordinates of the curvilinear grid points, the function u = f(x,y) is defined as the matrix u_i,j = f(x_i,j,y_i,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 (x_i,j,y_i,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 x_i,j and y_i,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 u_i,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 = u_max.
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 (x_i,j,y_i,j) for coordinates of the triangular grid.
5. Compute the matrix u_i,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

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

u = (1 + y² – x²)/(1 + x² + y²)²