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EX5115.m
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%----------------------------------------------------------------------------
% EX5.11.5.m
% to solve the two-dimensional Laplace's equation given as
% u,xx + u,yy =0, 0 < x < 5, 0 < y < 10
% u(x,0) = 0, u(x,10) = 100sin(pi*x/10),
% u(0,y) = 0, u,x(5,y) = 0
% using linear triangular elements
%(see Fig. 5.9.1 for the finite element mesh)
%
% Variable descriptions
% k = element matrix
% f = element vector
% kk = system matrix
% ff = system vector
% fn = effective system vector
% kn = effective system matrix
% fsol = solution vector
% sol = time-history solution of selected nodes
% 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=4; % number of nodes per element
ndof=1; % number of dofs per node
nnode=25; % total number of nodes in system
sdof=nnode*ndof; % total system dofs
deltt=0.04; % time step size for transient analysis
stime=0.0; % initial time
ftime=2; % termination time
ntime=fix((ftime-stime)/deltt); % number of time increment
a=0.04;
%---------------------------------------------
% 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;
gcoord(16,1)=0.0; gcoord(16,2)=7.5;
gcoord(17,1)=1.25; gcoord(17,2)=7.5;
gcoord(18,1)=2.5; gcoord(18,2)=7.5;
gcoord(19,1)=3.75; gcoord(19,2)=7.5;
gcoord(20,1)=5.0; gcoord(20,2)=7.5;
gcoord(21,1)=0.0; gcoord(21,2)=10.;
gcoord(22,1)=1.25; gcoord(22,2)=10.;
gcoord(23,1)=2.5; gcoord(23,2)=10.;
gcoord(24,1)=3.75; gcoord(24,2)=10.;
gcoord(25,1)=5.0; gcoord(25,2)=10.;
%---------------------------------------------------------
% 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(1,4)=6;
nodes(2,1)=2; nodes(2,2)=3; nodes(2,3)=8; nodes(2,4)=7;
nodes(3,1)=3; nodes(3,2)=4; nodes(3,3)=9; nodes(3,4)=8;
nodes(4,1)=4; nodes(4,2)=5; nodes(4,3)=10; nodes(4,4)=9;
nodes(5,1)=6; nodes(5,2)=7; nodes(5,3)=12; nodes(5,4)=11;
nodes(6,1)=7; nodes(6,2)=8; nodes(6,3)=13; nodes(6,4)=12;
nodes(7,1)=8; nodes(7,2)=9; nodes(7,3)=14; nodes(7,4)=13;
nodes(8,1)=9; nodes(8,2)=10; nodes(8,3)=15; nodes(8,4)=14;
nodes(9,1)=11; nodes(9,2)=12; nodes(9,3)=17; nodes(9,4)=16;
nodes(10,1)=12; nodes(10,2)=13; nodes(10,3)=18; nodes(10,4)=17;
nodes(11,1)=13; nodes(11,2)=14; nodes(11,3)=19; nodes(11,4)=18;
nodes(12,1)=14; nodes(12,2)=15; nodes(12,3)=20; nodes(12,4)=19;
nodes(13,1)=16; nodes(13,2)=17; nodes(13,3)=22; nodes(13,4)=21;
nodes(14,1)=17; nodes(14,2)=18; nodes(14,3)=23; nodes(14,4)=22;
nodes(15,1)=18; nodes(15,2)=19; nodes(15,3)=24; nodes(15,4)=23;
nodes(16,1)=19; nodes(16,2)=20; nodes(16,3)=25; nodes(16,4)=24;
%-------------------------------------
% input data for boundary conditions
%-------------------------------------
bcdof(1)=1; % first node is constrained
bcval(1)=0; % whose described value is 0
bcdof(2)=2; % second node is constrained
bcval(2)=0; % whose described value is 0
bcdof(3)=3; % third node is constrained
bcval(3)=0; % whose described value is 0
bcdof(4)=4; % 4th node is constrained
bcval(4)=0; % whose described value is 0
bcdof(5)=5; % 5th node is constrained
bcval(5)=0; % whose described value is 0
bcdof(6)=6; % 6th node is constrained
bcval(6)=0; % whose described value is 0
bcdof(7)=11; % 11th node is constrained
bcval(7)=0; % whose described value is 0
bcdof(8)=16; % 16th node is constrained
bcval(8)=0; % whose described value is 0
bcdof(9)=21; % 21st node is constrained
bcval(9)=0; % whose described value is 0
bcdof(10)=22; % second node is constrained
bcval(10)=38.2683; % whose described value is 38.2683
bcdof(11)=23; % third node is constrained
bcval(11)=70.7107; % whose described value is 70.7107
bcdof(12)=24; % 4th node is constrained
bcval(12)=92.3880; % whose described value is 92.3880
bcdof(13)=25; % 5th node is constrained
bcval(13)=100; % whose described value is 100
%-----------------------------------------
% initialization of matrices and vectors
%-----------------------------------------
ff=zeros(sdof,1); % system vector
fn=zeros(sdof,1); % effective system vector
fsol=zeros(sdof,1); % solution vector
sol=zeros(1,ntime+1); % time-history solution
kk=zeros(sdof,sdof); % initialization of system matrix
mm=zeros(sdof,sdof); % initialization of system matrix
kn=zeros(sdof,sdof); % effective 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
nd(4)=nodes(iel,4); % 4th 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
x4=gcoord(nd(4),1); y4=gcoord(nd(4),2);% coord values of 4th node
xleng=x2-x1; % element size in x-axis
yleng=y4-y1; % element size in y-axis
index=feeldof(nd,nnel,ndof);% extract system dofs associated with element
k=felp2dr4(xleng,yleng); % compute element matrix
m=a*felpt2r4(xleng,yleng); % 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)=100.0; % initial condition
end
sol(1)=fsol(13); % sol contains time-history solution at node 13
kn=2*mm+deltt*kk; % compute effective system matrix
for it=1:ntime
fn=deltt*ff+(2*mm-deltt*kk)*fsol; % compute effective column vector
[kn,fn]=feaplyc2(kn,fn,bcdof,bcval); % apply boundary condition
fsol=kn\fn; % solve the matrix equation
sol(it+1)=fsol(13); % sol contains time-history solution at node 13
end
%------------------------------------
% plot the solution at node 13
%------------------------------------
time=0:deltt:ntime*deltt;
plot(time,sol);
xlabel('Time')
ylabel('Solution at the center')
%---------------------------------------------------------------