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Code covered by the BSD License  

Highlights from
Biohydrodynamics Toolbox

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from Biohydrodynamics Toolbox by Alexandre Munnier
Tool to simulate easily the motion of moving solids or swimming robots in a potential fluid flow.

bht_link2(t,der,varargin) 
function y = bht_link2(t,der,varargin)
% BHT_LINK2
% 
% Boundary describing functions
% 
% Syntax
% 
%     y = BHT_LINK2(t,der,settings)
% 
% Description
% 
%     Thies function defines a boundary as a parameterized line over
%     [0,2*pi[. The input variable t (the parameter) must be a vector with
%     all of its elements within [0,2*pi[. The function returns, depending
%     on the flag der:
% 
%     * der = 0: the coordinates of the points corresponding to the
%       parameters t. 
%     * der = 1: the coordinates of the tangent vectors (the
%       firsts derivatives of the parameterization). 
%     * der = 2: the second derivatives of the parameterization.
% 
%     The inpute variable settings is a row vector of eight elements. 
%     It is used to set the boundary's shape. 
%
% See also BHT_BOUNDARY_CHECK

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                                         %
%                      BIOHYDRODYNAMICS MATLAB TOOLBOX                    %
%                                                                         %
%                         A. MUNNIER and B. PINCON                        %
%                                                                         %
% alexandre.munnier@iecn.u-nancy.fr        bruno.pincon@iecn.u-nancy.fr   %
% http://www.iecn.u-nancy.fr/~munnier  http://www.iecn.u-nancy.fr/~pincon %
%                                                                         %
%                                                                         %
%                      INSTITUT ELIE CARTAN, NANCY 1                      %
%                       http://www.iecn.u-nancy.fr/                       %
%                                                                         %
%                      INRIA Lorraine, Projet CORIDA                      %
%                    http://www.iecn.u-nancy.fr/~corida/                  %
%                                                                         %  
%                                                                         %  
%                                                                         %
%                                                                         %
%                            August 15th 2008                             %
%                                                                         %
%                                               GNU GPL v3 licence        %
%                                                                         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%##########################################################################
% Reading settings
if isempty(varargin)
    error('Boundary''s parameters (row vector of eigth elements) are missing.')
else
parameters = varargin{1};
end

if length(parameters)~=8
    error('Boundary''s parameter vector must have eight elements');
end
%==========================================================================
% boundary's settings
a = parameters(1:4);
b = parameters(5:8);
%==========================================================================
n = length(t); % number of points
t = reshape(t,1,n); 
%==============================

y = zeros(2,n);

%==========================================================================
if der == 0
    
y(1,:) = (a(1)/6+a(2)/3+a(3)/3+a(4)/6)...
        +3/pi^2*(a(1)+a(2)-a(3)-a(4))*cos(t)...
        +9/4/pi^2*(a(1)-a(2)-a(3)+a(4))*cos(2*t)...
        +4/3/pi^2*(a(1)-2*a(2)+2*a(3)-a(4))*cos(3*t)...
        +9/16/pi^2*(a(1)-a(2)-a(3)+a(4))*cos(4*t)...
        +3/25/pi^2*(a(1)+a(2)-a(3)-a(4))*cos(5*t);

y(2,:) = sum(b)/6 ...
        +3/2/pi^2*(b(1)-b(2)-b(3)+b(4))*cos(t)...
        -9/8/pi^2*sum(b)*cos(2*t)...
        -4/3/pi^2*(b(1)-b(2)-b(3)+b(4))*cos(3*t)...
        -9/32/pi^2*sum(b)*cos(4*t)...
        +3/50/pi^2*(b(1)-b(2)-b(3)+b(4))*cos(5*t)...
        +3*sqrt(3)/2/pi^2*(b(1)+b(2)-b(3)-b(4))*sin(t)...
        +9*sqrt(3)/8/pi^2*(b(1)-b(2)+b(3)-b(4))*sin(2*t)...
        -9*sqrt(3)/32/pi^2*(b(1)-b(2)+b(3)-b(4))*sin(4*t)...
        -3*sqrt(3)/50/pi^2*(b(1)+b(2)-b(3)-b(4))*sin(5*t);
    
elseif der == 1
    
    y(1,:) = ...
        -3/pi^2*(a(1)+a(2)-a(3)-a(4))*sin(t)...
        -2*9/4/pi^2*(a(1)-a(2)-a(3)+a(4))*sin(2*t)...
        -3*4/3/pi^2*(a(1)-2*a(2)+2*a(3)-a(4))*sin(3*t)...
        -4*9/16/pi^2*(a(1)-a(2)-a(3)+a(4))*sin(4*t)...
        -5*3/25/pi^2*(a(1)+a(2)-a(3)-a(4))*sin(5*t);

y(2,:) = ...
        -3/2/pi^2*(b(1)-b(2)-b(3)+b(4))*sin(t)...
        +2*9/8/pi^2*sum(b)*sin(2*t)...
        +3*4/3/pi^2*(b(1)-b(2)-b(3)+b(4))*sin(3*t)...
        +4*9/32/pi^2*sum(b)*sin(4*t)...
        -5*3/50/pi^2*(b(1)-b(2)-b(3)+b(4))*sin(5*t)...
        +3*sqrt(3)/2/pi^2*(b(1)+b(2)-b(3)-b(4))*cos(t)...
        +2*9*sqrt(3)/8/pi^2*(b(1)-b(2)+b(3)-b(4))*cos(2*t)...
        -4*9*sqrt(3)/32/pi^2*(b(1)-b(2)+b(3)-b(4))*cos(4*t)...
        -5*3*sqrt(3)/50/pi^2*(b(1)+b(2)-b(3)-b(4))*cos(5*t);
    
    
elseif der == 2
    
    y(1,:) = ...
        -3/pi^2*(a(1)+a(2)-a(3)-a(4))*cos(t)...
        -4*9/4/pi^2*(a(1)-a(2)-a(3)+a(4))*cos(2*t)...
        -9*4/3/pi^2*(a(1)-2*a(2)+2*a(3)-a(4))*cos(3*t)...
        -16*9/16/pi^2*(a(1)-a(2)-a(3)+a(4))*cos(4*t)...
        -25*3/25/pi^2*(a(1)+a(2)-a(3)-a(4))*cos(5*t);

y(2,:) = ...
        -3/2/pi^2*(b(1)-b(2)-b(3)+b(4))*cos(t)...
        +4*9/8/pi^2*sum(b)*cos(2*t)...
        +9*4/3/pi^2*(b(1)-b(2)-b(3)+b(4))*cos(3*t)...
        +16*9/32/pi^2*sum(b)*cos(4*t)...
        -25*3/50/pi^2*(b(1)-b(2)-b(3)+b(4))*cos(5*t)...
        -3*sqrt(3)/2/pi^2*(b(1)+b(2)-b(3)-b(4))*sin(t)...
        -4*9*sqrt(3)/8/pi^2*(b(1)-b(2)+b(3)-b(4))*sin(2*t)...
        +16*9*sqrt(3)/32/pi^2*(b(1)-b(2)+b(3)-b(4))*sin(4*t)...
        +25*3*sqrt(3)/50/pi^2*(b(1)+b(2)-b(3)-b(4))*sin(5*t);
    
end

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