function [gMat,gVec] = AxiStressEquationQuadM(Mesh,EFunc,nuFunc,fFunc,gDFunc,gNFunc)
%%  [gMat,gVec] = AxiStressEquationQuadM(Mesh,EFunc,nuFunc,fFunc,gDFunc,gNFunc)
%%
%%  setup the equation for an axisymmetric problem with quadratic elements

%% evaluate the functions
nElem = size(Mesh.elem,1); nGP = size(Mesh.GP,1);
nDOF = Mesh.nDOF;

if ischar(EFunc)
  EV = reshape(feval(EFunc,Mesh.GP,Mesh.GPT),nGP/nElem,nElem);
elseif isscalar(EFunc)
  EV = EFunc*ones(nGP/nElem,nElem);
else
  EV = reshape(EFunc,nGP/nElem,nElem);
endif

if ischar(nuFunc)
  nuV = reshape(feval(nuFunc,Mesh.GP,Mesh.GPT),nGP/nElem,nElem);
elseif isscalar(nuFunc)
  nuV = nuFunc*ones(nGP/nElem,nElem);
else
  nuV = reshape(nuFunc,nGP/nElem,nElem);
endif

w1 = (155 - sqrt(15))/2400;  w2 = (155 + sqrt(15))/2400; w3 = 0.1125;
w = 2*[w1,w1,w1,w2,w2,w2,w3]';  DW = diag(w);
rV = reshape(Mesh.GP(:,1),nGP/nElem,nElem);  %% radii at Gauss points
den = 1./((1+nuV).*(1-2*nuV));  %% a common denominator
a1 = DW*(rV.*EV.*(1-nuV).*den);
a2 = DW*(EV.*(1-nuV)./rV.*den);
a3 = DW*(rV.*EV.*nuV.*den);
a4 = DW*(EV.*nuV.*den);
a5 = DW*(rV.*EV./(2*(1+nuV)));

if ischar(fFunc{1})
  f1V = reshape(feval(fFunc{1},Mesh.GP,Mesh.GPT),nGP/nElem,nElem);
elseif isscalar(fFunc{1})
  f1V = fFunc{1}*ones(nGP/nElem,nElem);
else
  f1V = reshape(fFunc{1},nGP/nElem,nElem);
endif
f1V = f1V.*rV;  %% multiply the values of f with the radii

if ischar(fFunc{2})
  f2V = reshape(feval(fFunc{2},Mesh.GP,Mesh.GPT),nGP/nElem,nElem);
elseif isscalar(fFunc{2})
  f2V = fFunc{2}*ones(nGP/nElem,nElem);
else
  f2V = reshape(fFunc{2},nGP/nElem,nElem);
endif
f2V = f2V.*rV;  %% multiply the values of f with the radii

%% create memory for the sparse matrix and the RHS vector
Si   = zeros(4*36*nElem,1); Sj = Si; Sval = Si; %% maximal number of contributions
gVec = zeros(sum(nDOF),1);

l1 = (12 - 2*sqrt(15))/21;   l2 = (12 + 2*sqrt(15))/21;

xi = [l1/2, 1-l1, l1/2, l2/2, 1-l2, l2/2, 1/3]';
nu = [l1/2, l1/2, 1-l1, l2/2, l2/2, 1-l2, 1/3]';

%% the interpolation matrices for the function and its partial derivatives
M = [(1-xi-nu).*(1-2*xi-2*nu) xi.*(2*xi-1) nu.*(2*nu-1) 4*xi.*nu 4*nu.*(1-xi-nu) 4*xi.*(1-xi-nu)];
Mxi = [-3+4*(xi+nu) 4*xi-1 0*xi 4*nu -4*nu  4-8*xi-4*nu];
Mnu = [-3+4*(xi+nu) 0*xi 4*nu-1 4*xi  4-4*xi-8*nu -4*xi];

% insert the element matrices and vectors into the global matrix
ptrDOF = 1;       %% counter for the DOF we are working on
for k = 1:nElem   %%for each element
  cor = Mesh.nodes(Mesh.elem(k,:),:);  % coordinates of the nodes
 %% compute element stiffness matrix and vector
  area = Mesh.elemArea(k);  % area = 0.5*det(T)
  Gr = [(cor(3,2)-cor(1,2))*Mxi + (cor(1,2)-cor(2,2))*Mnu]/(2*area);
  Gz = [(cor(1,1)-cor(3,1))*Mxi + (cor(2,1)-cor(1,1))*Mnu]/(2*area);
  mat1 = area*[Gr'*diag(a1(:,k))*Gr+M'*diag(a2(:,k))*M + Gr'*diag(a4(:,k))*M...
       + M'*diag(a4(:,k))*Gr + Gz'*diag(a5(:,k))*Gz,...
      Gr'*diag(a3(:,k))*Gz + M'*diag(a4(:,k))*Gz + Gz'*diag(a5(:,k))*Gr];
  mat2 = area*...
	 [Gz'*diag(a3(:,k))*Gr + Gz'*diag(a4(:,k))*M + Gr'*diag(a5(:,k))*Gz,...
	  Gz'*diag(a1(:,k))*Gz + Gr'*diag(a5(:,k))*Gr];  
  vec1 = area*M'*(w.*f1V(:,k));   vec2 = area*M'*(w.*f2V(:,k));
  dofs1 = Mesh.node2DOF(Mesh.elem(k,:),1);
  dofs2 = Mesh.node2DOF(Mesh.elem(k,:),2);
  for k1 = 1:6
    %%%%%%%%%%% k1 is free for u1  %%%%%%%%%%%%%%%%
    if dofs1(k1)>0 % k1 is free node for u1
      gVec(dofs1(k1))      -= vec1(k1);
      for k2 = 1:6
 	%% gMat(dofs(k1),dofs(k2)) = gMat(dofs(k1),dofs(k2)) + mat(k1,k2);
	if dofs1(k2)>0  % k2 is free node for u1
	  Si(ptrDOF)   = dofs1(k1);   Sj(ptrDOF) = dofs1(k2);
	  Sval(ptrDOF) = mat1(k1,k2);  ptrDOF++;
	else  %% k2 is a Dirichlet node for u1
	  if   ischar(gDFunc{1}) gD1 = feval(gDFunc{1},cor(k2,:));
	  else                   gD1 = gDFunc{1};
	  endif% ischchar
 	  gVec(dofs1(k1)) += mat1(k1,k2)*gD1;
	endif %%dofs1(k2)
	
	if dofs2(k2)>0  % k2 is free node for u2
	  Si(ptrDOF)   = dofs1(k1); Sj(ptrDOF) = nDOF(1)+dofs2(k2);
	  Sval(ptrDOF) = mat1(k1,k2+6);  ptrDOF++;
	else  %% k2 is a Dirichlet node for u2
	  if   ischar(gDFunc{2}) gD2 = feval(gDFunc{2},cor(k2,:));
	  else                   gD2 = gDFunc{2};
	  endif% ischchar
 	  gVec(     dofs1(k1)) += mat1(k1,k2+6)*gD2;
	endif %%dofs1(k2)
      endfor  %% k2 = 1:6
    endif %% dofs1(k1)>0

    %%%%%%%%%%% k1 is free for u2  %%%%%%%%%%%%%%%%
    if dofs2(k1)>0 % k1 is free node for u2
      gVec(nDOF(1)+dofs2(k1))      -= vec2(k1);
      for k2 = 1:6
 	%% gMat(dofs(k1),dofs(k2)) = gMat(dofs(k1),dofs(k2)) + mat(k1,k2);
	if dofs1(k2)>0  % k2 is free node for u1
	  Si(ptrDOF)   = nDOF(1)+dofs2(k1); Sj(ptrDOF) = dofs1(k2);
	  Sval(ptrDOF) = mat2(k1,k2);  ptrDOF++;
	else  %% k2 is a Dirichlet node for u1
	  if   ischar(gDFunc{1}) gD1 = feval(gDFunc{1},cor(k2,:));
	  else                   gD1 = gDFunc{1};
	  endif% ischchar
 	  gVec(nDOF(1)+dofs2(k1)) += mat2(k1,k2)*gD1;
	endif %%dofs1(k2)
	
	if dofs2(k2)>0  % k2 is free node for u2
	  Si(ptrDOF)   = nDOF(1)+dofs2(k1); Sj(ptrDOF) = nDOF(1)+dofs2(k2);
	  Sval(ptrDOF) = mat2(k1,k2+6);  ptrDOF++;
	else  %% k2 is a Dirichlet node for u2
	  if   ischar(gDFunc{2}) gD2 = feval(gDFunc{2},cor(k2,:));
	  else                   gD2 = gDFunc{2};
	  endif% ischchar
 	  gVec(nDOF(1)+dofs2(k1)) += mat2(k1,k2+6)*gD2;
	endif %%dofs1(k2)
      endfor %% k2 = 1:6
    endif %% dofs2(k1)>0
  endfor %% k1 = 1:6

endfor % k (elements)

%% add up to create the sparse matrix
%%Sval(find(abs(Sval)<1e-15)) = 0;  % eliminate very small entries
Si = Si(1:ptrDOF-1); Sj = Sj(1:ptrDOF-1); Sval = Sval(1:ptrDOF-1);
gMat = sparse(Si,Sj,Sval,sum(nDOF),sum(nDOF));

%% insert the edge contributions
alpha = sqrt(0.6)/2; %% the edge interpolation matrix, transposed
Mb = [alpha*(+1+2*alpha) 1-4*alpha^2 alpha*(-1+2*alpha);
      0 1 0;
      alpha*(-1+2*alpha) 1-4*alpha^2 alpha*(+1+2*alpha)];
Mbc = 1/18*Mb'*diag([5 8 5]);

for k = 1:size(Mesh.edges,1)
  EdgeType = Mesh.edgesT(k);
  EdgeType_x = fix(EdgeType/10);  EdgeType_y = mod(EdgeType,-10);
  if ((EdgeType_x==-3)||(EdgeType_y==-3))
    cor = Mesh.nodes(Mesh.edges(k,:),:); % the three nodes on the edge
    p2 =  cor(2,:); % the three Gauss points on the edge
    p3 =  cor(3,:); 
    vec_diff = sqrt(0.6)*(p3-p2);
    p1 = p2 - vec_diff;  %% the Gauss points on the edge
    p3 = p2 + vec_diff;
    L = norm(cor(3,:)-cor(1,:));
    dofs = Mesh.node2DOF(Mesh.edges(k,:),:);
    dofs1 = dofs(:,1);  dofs2 = dofs(:,2);
    if EdgeType_x == -3  % nonzero force in x-direction
      if ischar(gNFunc{1}) g1 = feval(gNFunc{1},[p1;p2;p3]).*[p1(1);p2(1);p3(1)];
      else                 g1 = gNFunc{1}*[p1(1);p2(1);p3(1)];
      endif
      edgeVec1 = L*Mbc*g1;
      if (dofs1(1)>0)&&(dofs1(3)>0)        %% both end points free
	gVec(dofs1)         -= edgeVec1;
      elseif (dofs1(1)>0)&&(dofs1(3)==0)   %% node 1 free, node 3 Dirichlet
	gVec(dofs1([1 2]))  -= edgeVec1([1 2]);
      elseif (dofs1(1)==0)&&(dofs1(3)>0)   %% node 1 Dirichlet, node 3 free
	gVec(dofs1([2,3]))  -= edgeVec1([2,3]);
      endif    
    endif  % EdgeType_x

    if EdgeType_y == -3  % nonzero force in y-direction
      if ischar(gNFunc{2}) g2 =feval(gNFunc{2},[p1;p2;p3]).*[p1(1);p2(1);p3(1)];
      else                 g2 = gNFunc{2}*[p1(1);p2(1);p3(1)];
      endif
      edgeVec2 = L*Mbc*g2;
      if (dofs2(1)>0)&&(dofs2(3)>0)        %% both end points free
	gVec(nDOF(1)+dofs2)         -= edgeVec2;
      elseif (dofs2(1)>0)&&(dofs2(3)==0)   %% node 1 free, node 3 Dirichlet
	gVec(nDOF(1)+dofs2([1 2]))  -= edgeVec2([1 2]);
      elseif (dofs2(1)==0)&&(dofs2(3)>0)   %% node 1 Dirichlet, node 3 free
	gVec(nDOF(1)+dofs2([2,3]))  -= edgeVec2([2,3]);
      endif
    endif  % EdgeType_y
  endif %% EdgeType_x||EdgeType_y
endfor  % k edges
endfunction
