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Ex1291.m
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%----------------------------------------------------------------------------
% Example 12.9.1
% to solve the ordinary differential equation given as
% u'' = 1, 0 < x < 1
% u(0) = 0 and u(1) = 0
% and to compute the mean-square norm for different element sizes.
%
% Variable descriptions
% k = element matrix
% f = element vector
% kk = system matrix
% ff = system vector
% 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'
%----------------------------------------------------------------------------
%------------------------------------
% input data for control parameters
%------------------------------------
clear
for int=1:7 % loop for the different element numbers
nel=2^int; % number of elements
nnel=2; % number of nodes per element
ndof=1; % number of dofs per node
nnode=nel+1; % total number of nodes in system
sdof=nnode*ndof; % total system dofs
%-----------------------------------------
% input data for nodal coordinate values
%-----------------------------------------
elemsize(int)=1.0/nel;
for i=1:nnode
gcoord(i)=elemsize(int)*(i-1);
end
%-----------------------------------------------------
% input data for nodal connectivity for each element
%-----------------------------------------------------
for i=1:nel
nodes(i,1)=i;
nodes(i,2)=i+1;
end
%-----------------------------------------
% input data for coefficients of the ODE
%-----------------------------------------
acoef=1; % coefficient 'a' of the diff eqn
bcoef=0; % coefficient 'b' of the diff eqn
ccoef=0; % coefficient 'c' of the diff eqn
%-------------------------------------
% input data for boundary conditions
%-------------------------------------
bcdof(1)=1; % first node is constrained
bcval(1)=0; % whose described value is 0
bcdof(2)=nnode; % 4th node is constrained
bcval(2)=0; % whose described value is 0
%-----------------------------------------
% initialization of matrices and vectors
%-----------------------------------------
ff=zeros(sdof,1); % initialization of system force vector
kk=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
nl=nodes(iel,1); nr=nodes(iel,2); % extract nodes for (iel)-th element
xl=gcoord(nl); xr=gcoord(nr);% extract nodal coord values for the element
eleng=xr-xl; % element length
index=feeldof1(iel,nnel,ndof);% extract system dofs associated with element
k=feode2l(acoef,bcoef,ccoef,eleng); % compute element matrix
f=fef1l(xl,xr); % compute element vector
[kk,ff]=feasmbl2(kk,ff,k,f,index); % assemble element matrices and vectors
end % end of loop for elments
%-----------------------------
% apply boundary conditions
%-----------------------------
[kk,ff]=feaplyc2(kk,ff,bcdof,bcval);
%----------------------------
% solve the matrix equation
%----------------------------
fsol=kk\ff;
%-----------------------------
% compute mean-square error
%-----------------------------
error(int)=0.0;
for iel=1:nel
nl=nodes(iel,1); nr=nodes(iel,2); % extract nodes for (iel)-th element
xl=gcoord(nl); xr=gcoord(nr);% extract nodal coord values for the element
eleng=xr-xl; % element length
soll=fsol(nl); solr=fsol(nr); % extract fem solution
co1=(soll-solr)/eleng-0.5;
co2=(xl*solr-xr*soll)/eleng;
error(int)=error(int)+(xr^5-xl^5)/20.0+co1*(xr^4-xl^4)/4.0 ...
+(co1^2+co2)*(xr^3-xl^3)/3.0+co1*co2*(xr^2-xl^2)+co2^2*eleng;
end
error(int)=sqrt(error(int));
end % end of loop of different element numbers
%------------------------------------------------------------
% log-log plot of the mean-square error vs element size
%------------------------------------------------------------
loglog(elemsize,error);
xlabel('Element Size')
ylabel('Mean-Square Error')