Lanza, Alessandro

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Almae Matris Studiorum Campus

UNIVERSITY OF BOLOGNA – FACULTY OF ENGINEERINGINTERNATIONAL MASTER CORSE IN CIVIL ENGINEERING   2011/2012INTRODUCTION TO NUMERICAL METHODSLAB1b:   MATLAB - 3D GRAPHICSMatlab provides several functions that allow to create different types of three-dimensional plots:• Line plots• Surface plots• Contour plotsLine plotsLines in three-dimensional space can be plotted by using the Matlab function plot3, whose syntax is:plot3(x,y,z)Example: t = [0:0.1:10*pi] ;x = exp(-t/20) .* cos(t);   y = exp(-t/20) .* sin(t);   z = t;plot3(x,y,z);xlabel('x');  ylabel('y');  zlabel('z');Surface plotsA generic function of two variables z = f(x,y) represents a surface when plotted in three-dimensional space. The Matlab functions mesh and surf allow to generate, respectively, a wireframe and a solid surface plot. Before being used, both functions require a grid of points in the  xy plane (function  f domain) to be created by means of the Matlab function meshgrid, so that the z = f(x,y) function is then calculated in correspondence of the grid points. The sintax of the function meshgrid is:[X,Y] = meshgrid(x,y)After the grid has been generated, the surface plot can be created by the mesh (wireframe surface) or the surf (solid surface) function. The sintax is:mesh(X,Y,Z)                  surf(X,Y,Z)Examples:a) using the Matlab function mesh to visualize the two variables function z = f(x,y) = sin(x)cos(y):x = linspace(0,2*pi,50);y = linspace(0,pi,50);[X,Y] = meshgrid(x,y);Z = sin(X).*cos(Y);mesh(X,Y,Z);xlabel('x'); ylabel('y'); zlabel('z');grid;b) using the Matlab function surf to visualize the two variables function z = f(x,y) = exp(-(x2+y2) / 3):[X,Y] = meshgrid(-3:.2:3 , -2:.2:2);Z = exp( -(X.^2+Y.^2)/3 );surf(X,Y,Z);Pag. 1 di 3UNIVERSITY OF BOLOGNA – FACULTY OF ENGINEERINGINTERNATIONAL MASTER CORSE IN CIVIL ENGINEERING   2011/2012INTRODUCTION TO NUMERICAL METHODSxlabel('x'); ylabel('y'); zlabel('z');grid;Contour plotsIn topography,  three-dimensional  earth surface is  generally represented by  two-dimensional  maps, called countour maps. These maps contain contour lines (or isolines), that are lines joining points of equal elevation above a given level, such as the mean sea level. More generally,  the shape of two variables functions can be better understood by means of countour plots.In Matlab, these plots can be created by using the function contour, whose sintax is:contour(X,Y,Z)The usage of this function is similar to that of mesh and surf, in the sense that a grid of points has to be preliminarly created by means of the Matlab function meshgrid.Example: [X,Y] = meshgrid(-3:.2:3 , -2:.2:2);Z = exp( -(X.^2+Y.^2)/3 );contour(X,Y,Z);xlabel('x'); ylabel('y'); zlabel('z');grid;Solve the following exercises within the Matlab environment.1. Consider the following function of two variables:221),(yxyxyxfz+++==Represent the function, that is the surface (hyperbolic paraboloid), in the domain –1 ≤ x ≤ 1 , -1 ≤ y ≤ 1, using the Matlab commands seen before (mesh, surf, meshc, surfc, contour…) and trying different graphics options (colormap, shading, axis…).2. Create the contour and surface plots for the following two variables function:z = f(x,y) = – x2+2xy+ 3y2This  function  has  a  critical  point  at  (x=0  ,  y=0),  but  the  point  corresponds  neither  to  a minimum nor to a maximum (inflectional point): which isolines correspond to this point?3. A square metal plate is heated at  90°C in correspondence of the corner  (x=L , y=L), with  L denoting  the  length  of  the  square’s  side.  Temperature  distribution  within  the  square  plate [0 , L] × [0 , L] is given by the following function of two variables:22 )(3)(90),( LyLx eeyxTT −−−− ⋅⋅==Create the contour and surface plots for the temperature.Pag. 2 di 3Refer to the Matlab Help to gather further information on functions that generate three-dimensional graphs ( help graph3d )UNIVERSITY OF BOLOGNA – FACULTY OF ENGINEERINGINTERNATIONAL MASTER CORSE IN CIVIL ENGINEERING   2011/2012INTRODUCTION TO NUMERICAL METHODSCreation of moviesMatlab offers the possibility to store a series of plots and then to show them one after the other so as to obtain a real animation. The basic idea is to store tidily the plots within the columns of a proper  matricial data structure. The Matlab function movie, then, allows to show the animation.Firstly, the plots to be shown have to be generated (usually through a for loop) and stored, by means of the Matlab command getframe, in the columns of the matrix. It might be useful to set the axes limits through the command axis([xmin xmax ymin ymax]).Example:    x=linspace(0,pi);for k=1:5    plot(x,sin(k*x));    grid;    title('Sin(kx), k=1,2,3,4,5');    axis([0,pi,0,1]);    xlabel('x');    ylabel('y');    M(:,k)=getframe; % the current kth plot is stored in the kth column endThen, the movie is simply shown by the command:movie(M)It is worth noticing (in the Workspace panel) the large amount of memory required to store the movie (matrix M)!Solve the following exercises within the Matlab environment.1. Create a movie for the one variable function:axaxaxfy −== 2);(with  a = 0,1,2,3,4,5,6,7,8,9  in the interval ]5.1,5.0[−∈x .2. Create a movie for the two variables function:)cos()sin();,( ykxkkyxfz pipi==with  k = 0,1,2,3,4,5  in the interval ]1,0[]1,0[),( ×∈yx .Pag. 3 di 3Refer to the Matlab Help to gather further information on functions that generate movies

INM_LAB1_EXb

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INM_LAB1_EXb

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Almae Matris Studiorum Campus

Almae Matris Studiorum Campus