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Diffusion Equation Example Using Isoparametric Elements

LetD = {(x, y) : 0< x, y 0,

u(x,y, 0) = 0, u(x, 0, t) = 0, u(x, 1, t)/y = x, 0x 1, t > 0 andu(0, y , t) = 0, u(1, y , t)/x =y, 0 y 1, t > 0. The weak form of this problem is

D

vu

tdA+

D

v u dA=

1

0

vu(1, y , t)

x dy+

1

0

vu(x, 1, t)

y dx

Using bilinear elements we takeu=n

j=1uj(t)j(x, y),v = i(x, y), i= 1, . . . , n(We ignore bound-ary conditions for the moment, as usual, these will be imposed near the end of the computation.)This assumption leads to the element mass and stiffness matrices

m=

De

He(He)T dA, k=

De

He

x

(He)T

x +

He

y

(He)T

y

dA

where He = [He1 (x, y), . . . , H e4 (x, y)]

T is the element shape function vector. The two contributionsto the element load vectors are of the form

ye+1ye

xe+1xe

y dy=

ye+1ye

00

He3

(1, y)He

4(1, y)

ydy,

xe+1xe

He(x, 1)x dx=

xe+1xe

0He

2(x, 1)

He3

(x, 1)0

x dx

Because the shape functions are bilinear, the Hek(1, y), Hej (x, 1) are linear functions of their argu-

ments. In this particular problem the boundary contributions to the element load functions areindependent of time in the general case these contributions can depend on time. We can now builda Matlab file to solve this initial boundary value problem based on our Textbook files from Chapters5 and 6. (All files necessary to compile this example are included below with the exception of the

utility files gridgen.m, gridplot.m.)

function heat_eq(nx,ny)

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

% Based on Textbook Example 6.6.2

% to solve the two-dimensional Laplaces equation given as

% u,t-u,xx - u,yy =0, 0 < x < 1, 0 < y < 1

% u(x,0) = 0, u,y(x,1) = x

% u(0,y) = 0, u,x(1,y) = y

% u(x,y,0)=0

% using isoparametric four-node quadrilateral elements

%

% Variable descriptions% k = element matrix

% f = element vector

% kk = system matrix

% ff = system vector

% gcoord = coordinate values of each node

% nodes = nodal connectivity of each element

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% 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

% point2 - integration (or sampling) points

% weight2 - weighting coefficients

% nglx - number of integration points along x-axis% ngly - number of integration points along y-axis

% xcoord - x coordinate values of nodes

% ycoord - y coordinate values of nodes

% jacob2 - jacobian matrix

% shape - four-node quadrilateral shape functions

% dhdr - derivatives of shape functions w.r.t. natural coord. r

% dhds - derivatives of shape functions w.r.t. natural coord. s

% dhdx - derivatives of shape functions w.r.t. physical coord. x

% dhdy - derivatives of shape functions w.r.t. physical coord. y

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

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

% input data for control parameters

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

bl=[0,0]; ur=[1,1];

nx1=nx+1; ny1=ny+1;

nel=nx*ny; % number of elements

nnel=4; % number of nodes per element

ndof=1; % number of dofs per node

nnode=nx1*ny1; % total number of nodes in system

nglx=2; ngly=2; % use 2x2 integration rule

sdof=nnode*ndof; % total system dofs

edof=nnel*ndof; % dofs per element

deltt=0.04; % time step size for transient analysis

stime=0.0; % initial time

ftime=4; % 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

% nodes(i,j) where i-> element no. and j-> connected nodes

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

[gcoord,nodes]=gridgen(bl,ur, nx, ny, 1);

gridplot(nx, ny, gcoord, nodes, 1);

input(press any key to continue);

close all

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

% input data for boundary conditions

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

bcdof=[1:nx1,1+nx1:nx1:nx1*ny1-nx];

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nbcdof=length(bcdof);

bcval=zeros(1,nbcdof);

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

% initialization of matrices and vectors

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

ff=zeros(sdof,1); % initialization of system force vector

kk=zeros(sdof,sdof); % initialization of system matrix

mm=zeros(sdof,sdof); % initialization of system mass matrix

index=zeros(nnel*ndof,1); % initialization of index vector

% Note: The only contribution to ff in this problem is from the boundary fluxes

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

% loop for computation and assembly of element matrices

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

[point2,weight2]=feglqd2(nglx,ngly); % sampling points & weights

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

for i=1:nnel

nd(i)=nodes(iel,i); % extract connected node for (iel)-th element

xcoord(i)=gcoord(nd(i),1); % extract x value of the node

ycoord(i)=gcoord(nd(i),2); % extract y value of the node

end

k=zeros(edof,edof); % initialization of element matrix to zero

m=zeros(edof,edof); % element mass matrix

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

% numerical integration

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

for intx=1:nglx

x=point2(intx,1); % sampling point in x-axis

wtx=weight2(intx,1); % weight in x-axis

for inty=1:ngly

y=point2(inty,2); % sampling point in y-axis

wty=weight2(inty,2) ; % weight in y-axis

[shape,dhdr,dhds]=feisoq4(x,y); % compute shape functions and

% derivatives at sampling point

jacob2=fejacob2(nnel,dhdr,dhds,xcoord,ycoord); % compute Jacobian

detjacob=det(jacob2); % determinant of Jacobian

invjacob=inv(jacob2); % inverse of Jacobian matrix

[dhdx,dhdy]=federiv2(nnel,dhdr,dhds,invjacob); % derivatives w.r.t.

% physical coordinate

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

% compute element matrix

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

for i=1:edof

for j=1:edof

k(i,j)=k(i,j)+(dhdx(i)*dhdx(j)+dhdy(i)*dhdy(j))*wtx*wty*detjacob;m(i,j)=m(i,j)+shape(i)*shape(j)*wtx*wty*detjacob;

end

end

end % y loop

end % x loop

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

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

% assemble element mass and stiffness matrices

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

kk=feasmbl1(kk,k,index);

mm=feasmbl1(mm,m,index);

end % end of element loops

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

% Now compute the flux contributions to the load vector.

% These are independent of time for this problem -- so do not

% have to be computed in the time integration loop.

% The loops below could really be moved into the main assembly loop

% and then the textbook file feasmbl2() could be used to

% assemble kk and ff at the same time.

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

[point1, weight1]=feglqd1(ngly); % get integration points for 1D boundary integrals

for iel=nx:nx:nx*ny % loop over elements with boundary on x=1, 0

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% pt (y=sum_j H_j(1,s)y_j)

jacob1=fejacob1(2,dhdr1,ycoord(2:3));

for i=2:3

f(i)=f(i)+shape(i)*bfnc*wty*jacob1; % only shape(2),shape(3) nonzero

end

end

for i=2:3 ff(nd(i))=ff(nd(i))+f(i); end % assemble load vectorend

[point1, weight1]=feglqd1(nglx);

for iel=nx*ny-nx+1:nx*ny % loop over elements with boundary on y=1,0

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fsol=zeros(sdof,1); %initial value of solution

sol(1,1)=fsol((nnode+1)/2); % store time history solution for middle node (assumes nnode is odd

kn=mm+deltt*kk; % compute effective system matrix (time invariant)

for it=1:ntime % start loop for time integration

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

[kn,fn]=feaplyc2(kn,fn,bcdof,bcval); % apply essential boundary conditions

fsol=kn\fn; % solve the matrix equation

sol(1,it+1)=fsol((nnode+1)/2); %

end

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

% analytical solution at node middle node

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

xypt=gcoord((nnode+1)/2,:);

for it=1:ntime+1

tt=(it-1)*deltt;

esol(it)=xypt(1)*xypt(2)-trans(xypt(1),tt,10)*trans(xypt(2),tt,10);

end

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

% plot the solution at nodes middle node

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

time=0:deltt:ntime*deltt;

plot(time,sol(1,:), time, esol, o);

xlabel(Time)

ylabel(Solution at nodes)

%===================================================================

% end of main routine

%===================================================================

function [point1,weight1]=feglqd1(ngl)

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

% Purpose:

% determine the integration points and weighting coefficients

% of Gauss-Legendre quadrature for one-dimensional integration

%

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% Synopsis:

% [point1,weight1]=feglqd1(ngl)

%

% Variable Description:

% ngl - number of integration points

% point1 - vector containing integration points

% weight1 - vector containing weighting coefficients%-------------------------------------------------------------------

% initialization

point1=zeros(ngl,1);

weight1=zeros(ngl,1);

% find corresponding integration points and weights

if ngl==1 % 1-point quadrature rule

point1(1)=0.0;

weight1(1)=2.0;

elseif ngl==2 % 2-point quadrature rule

point1(1)=-0.577350269189626;

point1(2)=-point1(1);

weight1(1)=1.0;

weight1(2)=weight1(1);


point1(1)=-0.774596669241483;

point1(2)=0.0;


weight1(1)=0.555555555555556;

weight1(2)=0.888888888888889;



point1(1)=-0.861136311594053;

point1(2)=-0.339981043584856;



weight1(1)=0.347854845137454;

weight1(2)=0.652145154862546;



else % 5-point quadrature rule

point1(1)=-0.906179845938664;

point1(2)=-0.538469310105683;

point1(3)=0.0;



weight1(1)=0.236926885056189;

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weight1(2)=0.478628670499366;

weight1(3)=0.568888888888889;



end

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

function [point2,weight2]=feglqd2(nglx,ngly)

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

% Purpose:

% determine the integration points and weighting coefficients

% of Gauss-Legendre quadrature for two-dimensional integration

%

% Synopsis:

% [point2,weight2]=feglqd2(nglx,ngly)

%

% Variable Description:% nglx - number of integration points in the x-axis

% ngly - number of integration points in the y-axis

% point2 - vector containing integration points

% weight2 - vector containing weighting coefficients

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

% determine the largest one between nglx and ngly

if nglx > ngly

ngl=nglx;

else

ngl=ngly;

end

% initialization

point2=zeros(ngl,2);

weight2=zeros(ngl,2);

% find corresponding integration points and weights

[pointx,weightx]=feglqd1(nglx); % quadrature rule for x-axis

[pointy,weighty]=feglqd1(ngly); % quadrature rule for y-axis

% quadrature for two-dimension

for intx=1:nglx % quadrature in x-axis

point2(intx,1)=pointx(intx);

weight2(intx,1)=weightx(intx);

end

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for inty=1:ngly % quadrature in y-axis

point2(inty,2)=pointy(inty);

weight2(inty,2)=weighty(inty);

end

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

function [shapeq4,dhdrq4,dhdsq4]=feisoq4(rvalue,svalue)

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

% Purpose:

% compute isoparametric four-node quadilateral shape functions

% and their derivatves at the selected (integration) point

% in terms of the natural coordinate

%

% Synopsis:

% [shapeq4,dhdrq4,dhdsq4]=feisoq4(rvalue,svalue)

%

% Variable Description:% shapeq4 - shape functions for four-node element

% dhdrq4 - derivatives of the shape functions w.r.t. r

% dhdsq4 - derivatives of the shape functions w.r.t. s

% rvalue - r coordinate value of the selected point

% svalue - s coordinate value of the selected point

%

% Notes:

% 1st node at (-1,-1), 2nd node at (1,-1)

% 3rd node at (1,1), 4th node at (-1,1)

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

% shape functions

shapeq4(1)=0.25*(1-rvalue)*(1-svalue);

shapeq4(2)=0.25*(1+rvalue)*(1-svalue);

shapeq4(3)=0.25*(1+rvalue)*(1+svalue);

shapeq4(4)=0.25*(1-rvalue)*(1+svalue);

% derivatives

dhdrq4(1)=-0.25*(1-svalue);

dhdrq4(2)=0.25*(1-svalue);

dhdrq4(3)=0.25*(1+svalue);

dhdrq4(4)=-0.25*(1+svalue);

dhdsq4(1)=-0.25*(1-rvalue);

dhdsq4(2)=-0.25*(1+rvalue);

dhdsq4(3)=0.25*(1+rvalue);

dhdsq4(4)=0.25*(1-rvalue);

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

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function [dhdx,dhdy]=federiv2(nnel,dhdr,dhds,invjacob)

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

% Purpose:

% determine derivatives of 2-D isoparametric shape functions with% respect to physical coordinate system

%

% Synopsis:

% [dhdx,dhdy]=federiv2(nnel,dhdr,dhds,invjacob)

%


% dhdx - derivative of shape function w.r.t. physical coordinate x

% dhdy - derivative of shape function w.r.t. physical coordinate y

% nnel - number of nodes per element

% dhdr - derivative of shape functions w.r.t. natural coordinate r

% dhds - derivative of shape functions w.r.t. natural coordinate s

% invjacob - inverse of 2-D Jacobian matrix

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

for i=1:nnel

dhdx(i)=invjacob(1,1)*dhdr(i)+invjacob(1,2)*dhds(i);

dhdy(i)=invjacob(2,1)*dhdr(i)+invjacob(2,2)*dhds(i);

end

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

function [jacob2]=fejacob2(nnel,dhdr,dhds,xcoord,ycoord)

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

% Purpose:

% determine the Jacobian for two-dimensional mapping

%

% Synopsis:

% [jacob2]=fejacob2(nnel,dhdr,dhds,xcoord,ycoord)

%


% jacob2 - Jacobian for one-dimension


% dhdr - derivative of shape functions w.r.t. natural coordinate r

% dhds - derivative of shape functions w.r.t. natural coordinate s

% xcoord - x axis coordinate values of nodes

% ycoord - y axis coordinate values of nodes

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

jacob2=zeros(2,2);

for i=1:nnel

jacob2(1,1)=jacob2(1,1)+dhdr(i)*xcoord(i);

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jacob2(1,2)=jacob2(1,2)+dhdr(i)*ycoord(i);

jacob2(2,1)=jacob2(2,1)+dhds(i)*xcoord(i);

jacob2(2,2)=jacob2(2,2)+dhds(i)*ycoord(i);

end

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

function [kk]=feasmbl1(kk,k,index)

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

% Purpose:

% Assembly of element matrices into the system matrix

%

% Synopsis:

% [kk]=feasmbl1(kk,k,index)

%


% kk - system matrix

% k - element matri

% index - d.o.f. vector associated with an element%-----------------------------------------------------------

edof = length(index);

for i=1:edof

ii=index(i);

for j=1:edof

jj=index(j);

kk(ii,jj)=kk(ii,jj)+k(i,j);

end

end

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

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

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

% Purpose:

% Apply constraints to matrix equation [kk]{x}={ff}

%

% Synopsis:

% [kk,ff]=feaplybc(kk,ff,bcdof,bcval)

%


% kk - system matrix before applying constraints

% ff - system vector before applying constraints

% bcdof - a vector containging constrained d.o.f

% bcval - a vector containing contained value

%

% For example, there are constraints at d.o.f=2 and 10

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% and their constrained values are 0.0 and 2.5,

% respectively. Then, bcdof(1)=2 and bcdof(2)=10; and

% bcval(1)=1.0 and bcval(2)=2.5.

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

n=length(bcdof);

sdof=size(kk);

for i=1:n

c=bcdof(i);

for j=1:sdof

kk(c,j)=0;

end

kk(c,c)=1;

ff(c)=bcval(i);

end

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

function [index]=feeldof(nd,nnel,ndof)

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

% Purpose:

% Compute system dofs associated with each element

%

% Synopsis:

% [index]=feeldof(nd,nnel,ndof)

%


% index - system dof vector associated with element "iel"

% iel - element number whose system dofs are to be determined


% ndof - number of dofs per node

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

edof = nnel*ndof;

k=0;

for i=1:nnel

start = (nd(i)-1)*ndof;

for j=1:ndof

k=k+1;

index(k)=start+j;

end

end

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

function [jacob1]=fejacob1(nnel,dhdr,xcoord)

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

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% Purpose:

% determine the Jacobian for one-dimensional mapping

%

% Synopsis:

% [jacob1]=fejacob1(nnel,dhdr,xcoord)

%

% Variable Description:% jacob1 - Jacobian for one-dimension


% dhdr - derivative of shape functions w.r.t. natural coordinate

% xcoord - x axis coordinate values of nodes

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

jacob1=0.0;

for i=1:nnel

jacob1=jacob1+dhdr(i)*xcoord(i);

end

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

function [shape,dhdr]=feisol2(rvalue)

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

%Purpose

% Compute isoparametric 2-node shape functions

% and their derivatives at the selected

% point in terms of the natural coordinate.

%

%Synopsis:

% [shape,dhdr]=kwisol2(rvalue)

%

%Variable Description:

% shape - shape functions for the linear element

% dhdr - derivatives of shape functions

% rvalue - r coordinate valute of the selected point

%

%Notes:

% 1st node at rvalue=-1

% 2nd node at rvalue=1

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

shape(1)=0.5*(1.0-rvalue); % first shape function

shape(2)=0.5*(1.0+rvalue); % second shape function

dhdr(1)=-0.5; % derivative of the first shape function

dhdr(2)=0.5; % derivative of the second shape function

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

function t=trans(z,t,n)

sgn=-1;

sum=0;

for k=1:n

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lambda=(2*k-1)*pi/2; lamsq=lambda*lambda;

sine=sin(lambda*z);

expfnc=exp(-lamsq*t);

sum=sum+sgn*sine*expfnc/lamsq;

sgn=-sgn;

end;

t=2*sum;

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

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