| OptFEM2DP1 Toolbox V1.2b2 Matlab/Octave Optimized P1-Lagrange Finite Element Method in 2D

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validStiffP1.m
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1 function validStiffP1()
2 % function validStiffP1()
3 % Validation function for the assembly of the stiffness matrix for
4 % `P_1`-Lagrange finite element method
5 %
6 % The Stiffness Matrix `\Stiff` is given by
7 % ``\Stiff_{i,j}=\int_\DOMH \DOT{\GRAD\FoncBase_i(\q)}{\GRAD\FoncBase_j(\q)}d\q,\ \forall (i,j)\in\ENS{1}{\nq}^2``
8 % where `\FoncBase_i` are `P_1`-Lagrange basis functions.
9 % This Matrix is computed by functions StiffAssemblingP1{Version} where {Version} is one of
10 % 'base', 'OptV0', 'OptV1' and 'OptV2'.
11 % - Test 1: Computation of the Stiffness Matrix using all the versions giving errors and computation times
12 % - Test 2: Computation of the integral ``\int_\DOM \DOT{\GRAD u(\q)}{\GRAD v(\q)}d\q \approx \DOT{\Stiff \vecb{U}}{\vecb{V}}``
13 % where `\vecb{U}_i=u(\q^i)` and `\vecb{V}_i=v(\q^i)`.
14 % Functions `u` and `v` are those defined in #valid_FEMmatrices.
15 % - Test 3: One retrieves the order 2 of `P_1`-Lagrange integration
16 % ``|\int_\DOM \DOT{\GRAD u}{\GRAD v} -\DOT{\GRAD \Pi_h(u)}{\GRAD \Pi_h(v)}d\DOM| \leq C h^2``
17 %
18 % See also:
19 % #StiffAssemblingP1base, #StiffAssemblingP1OptV0,
20 % #StiffAssemblingP1OptV1, #StiffAssemblingP1OptV2,
21 % #valid_FEMmatrices, #SquareMesh, #GetMaxLengthEdges
22 %
23 % Results:
24 % \image html images/validStiffP1.png "figure : validStiffP1 Test 3 result"
25 %
26 % @author François Cuvelier @date 2012-11-26
27 % Copyright:
28 % See \ref license
29
30 disp('*******************************************')
31 disp('* Stiff Assembling P1 validations *')
32 disp('*******************************************')
33
34 Th=SquareMesh(50);
35
36 % TEST 1
37 disp('-----------------------------------------')
38 disp(' Test 1: Matrices errors and CPU times ')
39 disp('-----------------------------------------')
40 tic();
41 Mbase=StiffAssemblingP1base(Th.nq,Th.nme,Th.q,Th.me,Th.areas);
42 T(1)=toc();
43 tic();
44 MOptV0=StiffAssemblingP1OptV0(Th.nq,Th.nme,Th.q,Th.me,Th.areas);
45 T(2)=toc();
46 Test1.error(1)=norm(Mbase-MOptV0,Inf);
47 Test1.name{1}='StiffAssemblingP1OptV0';
48 fprintf(' Error P1base vs OptV0 : %e\n',Test1.error(1))
49 tic();
50 MOptV1=StiffAssemblingP1OptV1(Th.nq,Th.nme,Th.q,Th.me,Th.areas);
51 T(3)=toc();
52 Test1.error(2)=norm(Mbase-MOptV1,Inf);
53 Test1.name{2}='StiffAssemblingP1OptV1';
54 fprintf(' Error P1base vs OptV1 : %e\n',Test1.error(2))
55 tic();
56 MOptV2=StiffAssemblingP1OptV2(Th.nq,Th.nme,Th.q,Th.me,Th.areas);
57 T(4)=toc();
58 Test1.error(3)=norm(Mbase-MOptV2,Inf);
59 Test1.name{3}='StiffAssemblingP1OptV2';
60 fprintf(' Error P1base vs OptV2 : %e\n',Test1.error(3))
61
62 fprintf(' CPU times base (ref) : %3.4f (s)\n',T(1))
63 fprintf(' CPU times OptV0 : %3.4f (s) - Speed Up X%3.3f\n',T(2),T(1)/T(2))
64 fprintf(' CPU times OptV1 : %3.4f (s) - Speed Up X%3.3f\n',T(3),T(1)/T(3))
65 fprintf(' CPU times OptV2 : %3.4f (s) - Speed Up X%3.3f\n',T(4),T(1)/T(4))
66 checkTest1(Test1)
67
68 M=Mbase;
69
70 % TEST 2
71 disp('-----------------------------------------------------')
72 disp(' Test 2: Validations by integration on [0,1]x[0,1] ')
73 disp('-----------------------------------------------------')
74 Test=valid_FEMmatrices();
75 for kk=1:length(Test)
76 U=Test(kk).u(Th.q(1,:),Th.q(2,:));
77 V=Test(kk).v(Th.q(1,:),Th.q(2,:));
78 Test(kk).error=abs(Test(kk).Stiff-U*M*V');
79 fprintf(' functions %d : u(x,y)=%s, v(x,y)=%s,\n -> Stiff error=%e\n',kk,Test(kk).cu,Test(kk).cv,Test(kk).error);
80 end
81 checkTest2(Test)
82
83 % TEST 3
84 disp('--------------------------------')
85 disp(' Test 3: Validations by order ')
86 disp('--------------------------------')
87 n=length(Test);
88 u=Test(n).u;
89 v=Test(n).v;
90 ExSol=Test(n).Stiff;
91
92 for k=1:10
93 Th=SquareMesh(50*k+50);
94 fprintf(' Matrix size : %d\n',Th.nq);
95 h(k)=GetMaxLengthEdges(Th.q,Th.me);
96 tic();
97 M=StiffAssemblingP1OptV2(Th.nq,Th.nme,Th.q,Th.me,Th.areas);
98 TT(k)=toc();
99 U=u(Th.q(1,:),Th.q(2,:));
100 V=v(Th.q(1,:),Th.q(2,:));
101 Error(k)=abs(ExSol-U*M*V');
102 fprintf(' StiffAssemblingP1OptV2 CPU times : %3.3f(s)\n',TT(k));
103 fprintf(' Error : %e\n',Error(k));
104 end
105
106 loglog(h,Error,'+-r',h,h*1.1*Error(1)/h(1),'-sm',h,1.1*Error(1)*(h/h(1)).^2,'-db')
107 legend('Error','O(h)','O(h^2)')
108 xlabel('h')
109 title('Test 3 : Stiffness Matrix')
110 checkTest3(h,Error)
111 end
112
113 function checkTest1(Test)
114 I=find(Test.error>1e-14);
115 if isempty(I)
116 disp('------------------------')
117 disp(' Test 1 (results): OK')
118 disp('------------------------')
119 else
120 disp('----------------------------')
121 disp(' Test 1 (results): FAILED')
122 disp('----------------------------')
123 end
124 end
125
126 function checkTest2(Test)
127 N=length(Test);
128 cntFalse=0;
129 for k=1:N
130 if ( ismember(Test(k).degree,[0 1]) )
131 if (Test(k).error>1e-12)
132 cntFalse=cntFalse+1;
133 end
134 end
135 end
136 if (cntFalse==0)
137 disp('------------------------')
138 disp(' Test 2 (results): OK')
139 disp('------------------------')
140 else
141 disp('----------------------------')
142 disp(' Test 2 (results): FAILED')
143 disp('----------------------------')
144 end
145 end
146
147 function checkTest3(h,error)
148 % order 2
149 P=polyfit(log(h),log(error),1);
150 if abs(P(1)-2)<1e-2
151 disp('------------------------')
152 disp(' Test 3 (results): OK')
153 fprintf(' -> found numerical order %f. Must be 2\n',P(1))
154 disp('------------------------')
155 else
156 disp('----------------------------')
157 disp(' Test 3 (results): FAILED')
158 fprintf(' -> found numerical order %f. Must be 2\n',P(1))
159 disp('----------------------------')
160 end
161 end