valid/validStiffElas2DP1.m

Go to the documentation of this file.

function validStiffElas2DP1(Num)
% function validStiffElas2DP1(Num)
%  Validation function for the Assembly of the Stiffness Elasticity 
%  Matrix for `P_1`-Lagrange finite elements.
%
%   The Stiffness Elasticity Matrix is given by 
%   ``\StiffElas_{m,l}=\int_{\DOMH} \Odv^t(\BasisFuncTwoD_m) \Ocv(\BasisFuncTwoD_l)dT, \ \forall (m,l)\in\ENS{1}{2\,\nq}^2,``
%   where `\BasisFuncTwoD_m` are `P_1`-Lagrange vector basis functions.
%   Here `\Ocv=(\Occ_{xx},\Occ_{yy},\Occ_{xy})^t` and `\Odv=(\Odc_{xx},\Odc_{yy},2\Odc_{xy})^t`
%   are the elastic stress and strain tensors respectively. 
%
%   This Matrix is computed by functions StiffElasAssembling2DP1{Version} where {Version} is one of
%     'base', 'OptV0', 'OptV1' and 'OptV2'.
%     - Test 1: computation of the Elastic Stiffness Matrix using previous functions giving errors and computation times
%     - Test 2: compute ``\int_\DOM \Odv^t(\vecb{u}) \Ocv(\vecb{v})d\q \approx \DOT{\StiffElas \vecb{U}}{\vecb{V}}``
%       with `\foncdefsmall{\vecb{u}=(u_1,u_2)}{\DOM}{\R^2}` and `\foncdefsmall{\vecb{v}=(v_1,v_2)}{\DOM}{\R^2}`.
%       We have `\vecb{U}_{2i-1}=u_1(\q^i),` `\vecb{U}_{2i}=u_2(\q^i),` `\vecb{V}_{2i-1}=v_1(\q^i),` `\vecb{V}_{2i}=v_2(\q^i)`
%       if Num in {0,2} and `\vecb{U}_{i}=u_1(\q^i),` `\vecb{U}_{i+\nq}=u_2(\q^i),` `\vecb{V}_{i}=v_1(\q^i),` `\vecb{V}_{i+\nq}=v_2(\q^i)`
%       if Num in {1,3}. Use functions `\vecb{u}` and `\vecb{u}` defined in #valid_StiffElas.
%     - Test 3: retrieve order 2 of `P_1`-Lagrange integration 
%       ``|\int_\DOM \Odv^t(\vecb{u}) \Ocv(\vecb{v})d\q - \DOT{\StiffElas \vecb{U}}{\vecb{V}}| \leq C h^2``
%
%   Num :
%    - 0 global alternate numbering with local alternate numbering (classical method), 
%    - 1 global block numbering with local alternate numbering,
%    - 2 global alternate numbering with local block numbering,
%    - 3 global block numbering with local block numbering.
% See also:
%   #StiffElasAssembling2DP1base, #StiffElasAssembling2DP1OptV0,
%   #StiffElasAssembling2DP1OptV1, #StiffElasAssembling2DP1OptV2, #SquareMesh
% 
% @author François Cuvelier @date 2012-11-26
% Copyright:
%   See \ref license

  disp('********************************************')
  disp('*   Stiff Elas Assembling P1 validations   *')
  disp('********************************************')
  switch Num
  case 0
    s=sprintf('Global alternate numbering / local alternate numbering');
  case 1
    s=sprintf('Global block numbering / local alternate numbering');
  case 2
    s=sprintf('Global alternate numbering / local block numbering');
  case 3
    s=sprintf('Global block numbering / local block numbering');
  otherwise
    error('invalid Num value')
  end
  fprintf('  Numbering Choice : %s\n',s);
  
  
  
  Th=SquareMesh(20);

% TEST 1
  disp('-----------------------------------------')
  disp('  Test 1: Matrices errors and CPU times  ')
  disp('-----------------------------------------')
  tic();
  Mbase=StiffElasAssembling2DP1base(Th.nq,Th.nme,Th.q,Th.me,Th.areas,1,1,Num);
  T(1)=toc();
  tic();
  MOptV0=StiffElasAssembling2DP1OptV0(Th.nq,Th.nme,Th.q,Th.me,Th.areas,1,1,Num);
  T(2)=toc();
  Test1.error(1)=norm(Mbase-MOptV0,Inf);
  Test1.name{1}='StiffElasAssembling2DP1OptV0';
  fprintf('    Error P1base vs OptV0 : %e\n',Test1.error(1))
  tic();
  MOptV1=StiffElasAssembling2DP1OptV1(Th.nq,Th.nme,Th.q,Th.me,Th.areas,1,1,Num);
  T(3)=toc();
  Test1.error(2)=norm(Mbase-MOptV1,Inf);
  Test1.name{2}='StiffElasAssembling2DP1OptV1';
  fprintf('    Error P1base vs OptV1 : %e\n',Test1.error(2))
  tic();
  MOptV2=StiffElasAssembling2DP1OptV2(Th.nq,Th.nme,Th.q,Th.me,Th.areas,1,1,Num);
  T(4)=toc();
  Test1.error(3)=norm(Mbase-MOptV2,Inf);
  Test1.name{3}='StiffElasAssembling2DP1OptV2';
  fprintf('    Error P1base vs OptV2 : %e\n',Test1.error(3))

  fprintf('    CPU times base (ref) : %3.4f (s)\n',T(1))
  fprintf('    CPU times OptV0       : %3.4f (s) - Speed Up X%3.3f\n',T(2),T(1)/T(2))
  fprintf('    CPU times OptV1       : %3.4f (s) - Speed Up X%3.3f\n',T(3),T(1)/T(3))
  fprintf('    CPU times OptV2       : %3.4f (s) - Speed Up X%3.3f\n',T(4),T(1)/T(4))
  checkTest1(Test1)

  


% TEST 2
  disp('-----------------------------------------------------')
  disp('  Test 2: Validations by integration on [0,1]x[0,1]  ')
  disp('-----------------------------------------------------')
  Test=valid_StiffElas();
  qf=Th.q;
  for kk=1:length(Test)
    KK=StiffElasAssembling2DP1OptV2(Th.nq,Th.nme,Th.q,Th.me,Th.areas,Test(kk).lambda,Test(kk).mu,Num);
    u=Test(kk).u;
    v=Test(kk).v;
    switch Num
    case {0,2}
      U=[u{1}(qf(1,:),qf(2,:));u{2}(qf(1,:),qf(2,:))];
      U=U(:);
      V=[v{1}(qf(1,:),qf(2,:));v{2}(qf(1,:),qf(2,:))];
      V=V(:);
    case {1,3}
      U=[u{1}(qf(1,:),qf(2,:)) u{2}(qf(1,:),qf(2,:))]';
      V=[v{1}(qf(1,:),qf(2,:)) v{2}(qf(1,:),qf(2,:))]';
    end
    %whos
    Test(kk).error=abs(Test(kk).Stiff-U'*KK*V);
    fprintf('    functions %d : u(x,y)=(%s,%s), v(x,y)=(%s,%s),\n           -> Stiff error=%e\n', ...
            kk,Test(kk).cu{1},Test(kk).cu{2},Test(kk).cv{1},Test(kk).cv{2},Test(kk).error);
  end
  checkTest2(Test)
  
% TEST 3
  disp('--------------------------------')
  disp('  Test 3: Validations by order  ')
  disp('--------------------------------')
  n=length(Test);
  u=Test(n).u;
  v=Test(n).v;
  lambda=Test(n).lambda;
  mu=Test(n).mu;
  ExSol=Test(n).Stiff;

  for k=1:10  
    Th=SquareMesh(20*k+50);
    qf=Th.q;
    fprintf('    Matrix size : %d\n',Th.nq);
    h(k)=GetMaxLengthEdges(Th.q,Th.me);
    tic();
    M=StiffElasAssembling2DP1OptV2(Th.nq,Th.nme,Th.q,Th.me,Th.areas,lambda,mu,Num);
    TT(k)=toc();
    switch Num
    case {0,2}
      U=[u{1}(qf(1,:),qf(2,:));u{2}(qf(1,:),qf(2,:))];
      U=U(:);
      V=[v{1}(qf(1,:),qf(2,:));v{2}(qf(1,:),qf(2,:))];
      V=V(:);
    case {1,3}
      U=[u{1}(qf(1,:),qf(2,:)) u{2}(qf(1,:),qf(2,:))]';
      V=[v{1}(qf(1,:),qf(2,:)) v{2}(qf(1,:),qf(2,:))]';
    end
    Error(k)=abs(ExSol-U'*M*V);
    fprintf('      StiffElasAssembling2DP1OptV2 CPU times : %3.3f(s)\n',TT(k));
    fprintf('      Error                            : %e\n',Error(k));
  end

  loglog(h,Error,'+-r',h,h*1.1*Error(1)/h(1),'-sm',h,1.1*Error(1)*(h/h(1)).^2,'-db')
  legend('Error','O(h)','O(h^2)')
  xlabel('h')
  title(sprintf('Test 3 : Stiff Elas. Matrix (Num=%d)',Num))
  checkTest3(h,Error)
  
  %figure(3)
  %spy(M)
  %title(sprintf('Matrix sparsity for %s numbering',s))
end

function checkTest1(Test)
  I=find(Test.error>1e-14);
  if isempty(I)
    disp('------------------------')
    disp('  Test 1 (results): OK')
    disp('------------------------')
  else
    disp('----------------------------')
    disp('  Test 1 (results): FAILED')
    disp('----------------------------')
  end
end

function checkTest2(Test)
  N=length(Test);
  cntFalse=0;
  for k=1:N
    if (ismember(Test(k).degree,[0 1]))
      if (Test(k).error>1e-12)
        cntFalse=cntFalse+1;
      end
    end
  end
  if (cntFalse==0)
    disp('------------------------')
    disp('  Test 2 (results): OK')
    disp('------------------------')
  else
    disp('----------------------------')
    disp('  Test 2 (results): FAILED')
    disp('----------------------------')
  end
end

function checkTest3(h,error)
  % order 2
  P=polyfit(log(h),log(error),1);
  if abs(P(1)-2)<5e-2
    disp('------------------------')
    disp('  Test 3 (results): OK')
    fprintf('    -> found numerical order %f. Must be 2\n',P(1))
    disp('------------------------')
  else
    disp('----------------------------')
    disp('  Test 3 (results): FAILED')
    fprintf('    -> found numerical order %f. Must be 2\n',P(1))
    disp('----------------------------')
  end
end