ex991.m

大量的有限元法求解偏微分方程的实例程序

%----------------------------------------------------------------------------
% Example 9.9.1                                                              
%   axisymmetric analysis of a solid subjected to an internal
%   pressure using linear triangular elements           
%   (see Fig. 9.9.1 for the finite element mesh)
%
% Variable descriptions                                                      
%   k = element matrix                                             
%   f = element vector
%   kk = system matrix                                             
%   ff = system vector                                                 
%   disp = system nodal displacement vector
%   eldisp = element nodal displacement vector
%   stress = matrix containing stresses
%   strain = matrix containing strains
%   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'                                              
%----------------------------------------------------------------------------            

%------------------------------------
%  input data for control parameters
%------------------------------------

clear
nel=10;                  % number of elements
nnel=3;                  % number of nodes per element
ndof=2;                  % number of dofs per node
nnode=12;                % total number of nodes in system
sdof=nnode*ndof;         % total system dofs  
edof=nnel*ndof;          % degrees of freedom per element
emodule=28e6;            % elastic modulus
poisson=0.25;            % Poisson's ratio

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

gcoord=[10.  0.; 10.  1.; 11.  0.; 11.  1.; 12.  0.; 12.  1.; 
13.  0.; 13.  1.; 14.  0.; 14.  1.; 15.  0.; 15.  1.];

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

nodes=[1 3 4; 1 4 2; 3 5 6; 3 6 4; 5 7 8; 
5 8 6;7 9 10; 7 10 8; 9 11 12; 9 12 10];

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

bcdof=[2 4 6 8 10 12 14 16 18 20 22 24];   % axial movement constrained
bcval=[0 0 0 0 0 0 0 0 0 0 0 0];        % whose described values are 0 

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

ff=zeros(sdof,1);       % system force vector
kk=zeros(sdof,sdof);    % system matrix
disp=zeros(sdof,1);     % system displacement vector
eldisp=zeros(edof,1);   % element displacement vector
stress=zeros(nel,4);    % matrix containing stress components
strain=zeros(nel,4);    % matrix containing strain components
index=zeros(edof,1);    % index vector
kinmtax=zeros(4,edof);   % kinematic matrix
matmtx=zeros(4,4);      % constitutive matrix

%----------------------------
%  force vector
%----------------------------
pi=4.0*atan(1);                            % pi=3.141592

ff(1)=2e3*pi*2*10;      % force applied at node 1 in x-axis
ff(3)=2e3*pi*2*10;      % force applied at node 2 in x-axis

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

matmtx=fematiso(3,emodule,poisson);        % compute constitutive matrix

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

%-------------------------------------------------------
%  find the derivatives of shape functions
%-------------------------------------------------------

area=0.5*(x1*y2+x2*y3+x3*y1-x1*y3-x2*y1-x3*y2);  % area of triangule
area2=area*2;
xcenter=(x1+x2+x3)/3;                      % x-centroid of triangle
ycenter=(y1+y2+y3)/3;                      % y-centroid of triangle

shape(1)=((x2*y3-x3*y2)+(y2-y3)*xcenter+(x3-x2)*ycenter)/area2; 
shape(2)=((x3*y1-x1*y3)+(y3-y1)*xcenter+(x1-x3)*ycenter)/area2; 
shape(3)=((x1*y2-x2*y1)+(y1-y2)*xcenter+(x2-x1)*ycenter)/area2;

dhdx=(1/area2)*[(y2-y3) (y3-y1) (y1-y2)];  % derivatives w.r.t. x-axis
dhdy=(1/area2)*[(x3-x2) (x1-x3) (x2-x1)];  % derivatives w.r.t. y-axis

kinmtax=fekineax(nnel,dhdx,dhdy,shape,xcenter);   % kinematic matrix

k=2*pi*xcenter*area*kinmtax'*matmtx*kinmtax;      % element matrix

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

end

%-----------------------------
%   apply boundary conditions
%-----------------------------

[kk,ff]=feaplyc2(kk,ff,bcdof,bcval);

%----------------------------
%  solve the matrix equation
%----------------------------

disp=kk\ff;   

%---------------------------------------
%  element stress computation
%---------------------------------------

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

nd(1)=nodes(ielp,1); % 1st connected node for (iel)-th element
nd(2)=nodes(ielp,2); % 2nd connected node for (iel)-th element
nd(3)=nodes(ielp,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

%-------------------------------------------------------
%  extract element displacement vector
%-------------------------------------------------------

for i=1:edof
eldisp(i)=disp(index(i));
end

area=0.5*(x1*y2+x2*y3+x3*y1-x1*y3-x2*y1-x3*y2);  % area of triangule
area2=area*2;
xcenter=(x1+x2+x3)/3;                      % x-centroid of triangle
ycenter=(y1+y2+y3)/3;                      % y-centroid of triangle

shape(1)=((x2*y3-x3*y2)+(y2-y3)*xcenter+(x3-x2)*ycenter)/area2; 
shape(2)=((x3*y1-x1*y3)+(y3-y1)*xcenter+(x1-x3)*ycenter)/area2; 
shape(3)=((x1*y2-x2*y1)+(y1-y2)*xcenter+(x2-x1)*ycenter)/area2;

dhdx=(1/area2)*[(y2-y3) (y3-y1) (y1-y2)];  % derivatives w.r.t. x-axis
dhdy=(1/area2)*[(x3-x2) (x1-x3) (x2-x1)];  % derivatives w.r.t. y-axis

kinmtax=fekineax(nnel,dhdx,dhdy,shape,xcenter);   % kinematic matrix

estrain=kinmtax*eldisp;             % compute strains
estress=matmtx*estrain;             % compute stresses

for i=1:4
strain(ielp,i)=estrain(i);          % store for each element
stress(ielp,i)=estress(i);          % store for each element          
end

end

%------------------------------------
% print fem solutions
%------------------------------------

num=1:1:sdof;
displace=[num' disp]                         % print nodal displacements

for i=1:nel
stresses=[i stress(i,:)]                     % print stresses
end

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