ex5113.m

采用Matlab编写的有限元方法

%----------------------------------------------------------------------------
% EX5.11.3.m                                                              
% to solve the transient two-dimensional Laplace's equation             
%   u,t = u,xx + u,yy ,  0 < x < 5, 0 < y < 2                                                                        
% boundary conditions:
%   u(0,y,t) = 100, u(5,y,t) = 100, 
%   u,y(x,0,t) = 0, u,y(x,2,t) = 0
% initial condition:
%   u(x,y,0) = 0 over the domain
% using linear triangular elements and backward difference method
%(see Fig. 5.11.1 for the finite element mesh)
%
% Variable descriptions                                                      
%   k = element matrix for time-independent term (u,xx + u,yy)                                           
%   m = element matrix for time-dependent term (u,t)
%   f = element vector
%   kk = system matrix of k                                            
%   mm = system matrix of m
%   ff = system vector                                                 
%   gcoord = coordinate values of each node
%   nodes = nodal connectivity of each element
%   index = a vector containing system dofs associated with each element     
%   bcdof = a vector containing dofs associated with boundary conditions     
%   bcval = a vector containing boundary condition values associated with    
%           the dofs in 'bcdof'                                              
%----------------------------------------------------------------------------            
clear
%------------------------------------
%  input data for control parameters
%------------------------------------

nel=16;                  % number of elements
nnel=3;                  % number of nodes per element
ndof=1;                  % number of dofs per node
nnode=15;                % total number of nodes in system
sdof=nnode*ndof;         % total system dofs  
deltt=0.4;               % time step size for transient analysis
stime=0.0;               % initial time
ftime=10;                % termination time
ntime=fix((ftime-stime)/deltt); % number of time increment

%---------------------------------------------
%  input data for nodal coordinate values
%  gcoord(i,j) where i->node no. and j->x or y
%---------------------------------------------

gcoord(1,1)=0.0;   gcoord(1,2)=0.0;   gcoord(2,1)=1.25;   gcoord(2,2)=0.0;
gcoord(3,1)=2.5;   gcoord(3,2)=0.0;   gcoord(4,1)=3.75;   gcoord(4,2)=0.0;
gcoord(5,1)=5.0;   gcoord(5,2)=0.0;   gcoord(6,1)=0.0;    gcoord(6,2)=2.5;
gcoord(7,1)=1.25;  gcoord(7,2)=2.5;   gcoord(8,1)=2.5;    gcoord(8,2)=2.5;
gcoord(9,1)=3.75;  gcoord(9,2)=2.5;   gcoord(10,1)=5.0;   gcoord(10,2)=2.5;
gcoord(11,1)=0.0;  gcoord(11,2)=5.0;  gcoord(12,1)=1.25;  gcoord(12,2)=5.0;
gcoord(13,1)=2.5;  gcoord(13,2)=5.0;  gcoord(14,1)=3.75;  gcoord(14,2)=5.0;
gcoord(15,1)=5.0;  gcoord(15,2)=5.0;  

%---------------------------------------------------------
%  input data for nodal connectivity for each element
%  nodes(i,j) where i-> element no. and j-> connected nodes
%---------------------------------------------------------

nodes(1,1)=1;    nodes(1,2)=2;    nodes(1,3)=7;
nodes(2,1)=2;    nodes(2,2)=3;    nodes(2,3)=8; 
nodes(3,1)=3;    nodes(3,2)=4;    nodes(3,3)=9;
nodes(4,1)=4;    nodes(4,2)=5;    nodes(4,3)=10; 
nodes(5,1)=1;    nodes(5,2)=7;    nodes(5,3)=6;
nodes(6,1)=2;    nodes(6,2)=8;    nodes(6,3)=7; 
nodes(7,1)=3;    nodes(7,2)=9;    nodes(7,3)=8;
nodes(8,1)=4;    nodes(8,2)=10;   nodes(8,3)=9; 
nodes(9,1)=6;    nodes(9,2)=7;    nodes(9,3)=12;
nodes(10,1)=7;   nodes(10,2)=8;   nodes(10,3)=13; 
nodes(11,1)=8;   nodes(11,2)=9;   nodes(11,3)=14;
nodes(12,1)=9;   nodes(12,2)=10;  nodes(12,3)=15; 
nodes(13,1)=6;   nodes(13,2)=12;  nodes(13,3)=11;
nodes(14,1)=7;   nodes(14,2)=13;  nodes(14,3)=12; 
nodes(15,1)=8;   nodes(15,2)=14;  nodes(15,3)=13;
nodes(16,1)=9;   nodes(16,2)=15;  nodes(16,3)=14; 

%-------------------------------------
%  input data for boundary conditions
%-------------------------------------

bcdof(1)=1;             % first node is constrained
bcval(1)=100;             % whose described value is 0 
bcdof(2)=5;             % second node is constrained
bcval(2)=100;             % whose described value is 0
bcdof(3)=6;             % third node is constrained
bcval(3)=100;             % whose described value is 0 
bcdof(4)=10;             % 4th node is constrained
bcval(4)=100;             % whose described value is 0
bcdof(5)=11;             % 5th node is constrained
bcval(5)=100;             % whose described value is 0 
bcdof(6)=15;             % 6th node is constrained
bcval(6)=100;             % whose described value is 0

%-----------------------------------------
%  initialization of matrices and vectors
%-----------------------------------------

ff=zeros(sdof,1);       % initialization of system vector
fn=zeros(sdof,1);       % initialization of effective system vector
fsol=zeros(sdof,1);     % initialization of solution vector
kk=zeros(sdof,sdof);    % initialization of system matrix
mm=zeros(sdof,sdof);    % initialization of system matrix
index=zeros(nnel*ndof,1);  % initialization of index vector

%-----------------------------------------------------------------
%  computation of element matrices and vectors and their assembly
%-----------------------------------------------------------------

for iel=1:nel           % loop for the total number of elements

nd(1)=nodes(iel,1); % 1st connected node for (iel)-th element
nd(2)=nodes(iel,2); % 2nd connected node for (iel)-th element
nd(3)=nodes(iel,3); % 3rd connected node for (iel)-th element
x1=gcoord(nd(1),1); y1=gcoord(nd(1),2);% coord values of 1st node
x2=gcoord(nd(2),1); y2=gcoord(nd(2),2);% coord values of 2nd node
x3=gcoord(nd(3),1); y3=gcoord(nd(3),2);% coord values of 3rd node

index=feeldof(nd,nnel,ndof);% extract system dofs associated with element

k=felp2dt3(x1,y1,x2,y2,x3,y3); % compute element matrix
m=felpt2t3(x1,y1,x2,y2,x3,y3); % compute element matrix

kk=feasmbl1(kk,k,index);  % assemble element matrices 
mm=feasmbl1(mm,m,index);  % assemble element matrices 

end

%-----------------------------
%   loop for time integration 
%-----------------------------

for in=1:sdof
fsol(in)=0.0;    % initial condition
end

sol(1,1)=fsol(8);  % sol contains time-history solution of node 8
sol(2,1)=fsol(9);  % sol contains time-history solution of node 9 

kk=mm+deltt*kk; 

for it=1:ntime

fn=deltt*ff+mm*fsol;           % compute effective column vector

[kk,fn]=feaplyc2(kk,fn,bcdof,bcval); % apply boundary condition

fsol=kk\fn;     % solve the matrix equation 

sol(1,it+1)=fsol(8);  % sol contains time-history solution of node 8
sol(2,it+1)=fsol(9);  % sol contains time-history solution of node 9 

end

%------------------------------------
% plot the solution at nodes 8 and 9
%------------------------------------

time=0:deltt:ntime*deltt;
plot(time,sol(1,:),'*',time,sol(2,:),'-');
xlabel('Time')
ylabel('Solution at nodes')

%---------------------------------------------------------------