| OptFEM2D Toolbox for Matlab V1.2b1 Matlab/Octave Optimized P1-Lagrange Finite Element Method in 2D

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validMassP1.m
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1 function validMassP1()
2 % function validMassP1()
3 % Validation function for the assembly of the mass matrix for
4 % `P_1`-Lagrange finite element method.
5 %
6 % The Mass Matrix `\Masse` is given by
7 % ``\Masse_{i,j}=\int_\DOMH \FoncBase_i(\q) \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 MassAssemblingP1{Version} where {Version} is one of
10 % 'base', 'OptV0', 'OptV1' and 'OptV2'.
11 % - Test 1: Computation of the Mass Matrix using all the versions giving errors and cputimes
12 % - Test 2: Computation of the integral ``\int_\DOM u(x,y) v(x,y) dxdy \approx \DOT{\Masse \vecb{U}}{\vecb{V}}``
13 % where `\vecb{U}_i=u(\q^i)` and
14 % `\vecb{V}_i=v(\q^i)`. Functions `u` and `v` are those defined in #valid_FEMmatrices.
15 % - Test 3: Ones retrieves the order 2 of `P_1`-Lagrange integration ``|\int_\DOM u\,v -\Pi_h(u)\,\Pi_h(v)d\DOM| \leq C h^2``
16 %
17 % See also:
18 % #MassAssemblingP1base, #MassAssemblingP1OptV0,
19 % #MassAssemblingP1OptV1, #MassAssemblingP1OptV2,
20 % #valid_FEMmatrices, #SquareMesh, #GetMaxLengthEdges
21 %
22 % Results:
23 % \image html images/validMassP1.png "figure : validMassP1 Test 3 result"
24 %
25 % @author François Cuvelier @date 2012-11-26
26 %
27 % OptFEM2DP1 [V1.2b1] - Copyright (C) 2013 CJS (LAGA)
28 %
29 % This file is part of OptFEM2DP1.
30 % OptFEM2DP1 is free software: you can redistribute it and/or modify
31 % it under the terms of the GNU General Public License as published by
32 % the Free Software Foundation, either version 3 of the License, or
33 % (at your option) any later version.
34 %
35 % OptFEM2DP1 is distributed in the hope that it will be useful,
36 % but WITHOUT ANY WARRANTY; without even the implied warranty of
37 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
38 % GNU General Public License for more details.
39 %
40 % You should have received a copy of the GNU General Public License
41 % along with this program. If not, see <http://www.gnu.org/licenses/>.
42
43 disp('******************************************')
44 disp('* Mass Assembling P1 validations *')
45 disp('******************************************')
46
47 Th=SquareMesh(50);
48
49 % TEST 1
50 disp('-----------------------------------------')
51 disp(' Test 1: Matrices errors and CPU times ')
52 disp('-----------------------------------------')
53 tic();
54 Mbase=MassAssemblingP1base(Th.nq,Th.nme,Th.me,Th.areas);
55 T(1)=toc();
56 tic();
57 MOptV0=MassAssemblingP1OptV0(Th.nq,Th.nme,Th.me,Th.areas);
58 T(2)=toc();
59 Test1.error(1)=norm(Mbase-MOptV0,Inf);
60 Test1.name{1}='MassAssemblingP1OptV0';
61 fprintf(' Error P1base vs OptV0 : %e\n',Test1.error(1))
62 tic();
63 MOptV1=MassAssemblingP1OptV1(Th.nq,Th.nme,Th.me,Th.areas);
64 T(3)=toc();
65 Test1.error(2)=norm(Mbase-MOptV1,Inf);
66 Test1.name{2}='MassAssemblingP1OptV1';
67 fprintf(' Error P1base vs OptV1 : %e\n',Test1.error(2))
68 tic();
69 MOptV2=MassAssemblingP1OptV2(Th.nq,Th.nme,Th.me,Th.areas);
70 T(4)=toc();
71 Test1.error(3)=norm(Mbase-MOptV2,Inf);
72 Test1.name{3}='MassAssemblingP1OptV2';
73 fprintf(' Error P1base vs OptV2 : %e\n',Test1.error(3))
74
75 fprintf(' CPU times base (ref) : %3.4f (s)\n',T(1))
76 fprintf(' CPU times OptV0 : %3.4f (s) - Speed Up X%3.3f\n',T(2),T(1)/T(2))
77 fprintf(' CPU times OptV1 : %3.4f (s) - Speed Up X%3.3f\n',T(3),T(1)/T(3))
78 fprintf(' CPU times OptV2 : %3.4f (s) - Speed Up X%3.3f\n',T(4),T(1)/T(4))
79 checkTest1(Test1)
80
81 M=Mbase;
82
83 % TEST 2
84 disp('-----------------------------------------------------')
85 disp(' Test 2: Validations by integration on [0,1]x[0,1] ')
86 disp('-----------------------------------------------------')
87 Test=valid_FEMmatrices();
88 for kk=1:length(Test)
89 U=Test(kk).u(Th.q(1,:),Th.q(2,:));
90 V=Test(kk).v(Th.q(1,:),Th.q(2,:));
91 Test(kk).error=abs(Test(kk).Mass-U*M*V');
92 fprintf(' function %d : u(x,y)=%s, v(x,y)=%s,\n -> Mass error=%e\n',kk,Test(kk).cu,Test(kk).cv,abs(Test(kk).Mass-U*M*V'));
93 end
94 checkTest2(Test)
95
96 % TEST 3
97 disp('--------------------------------')
98 disp(' Test 3: Validations by order ')
99 disp('--------------------------------')
100 n=length(Test);
101 u=Test(n).u;
102 v=Test(n).v;
103 ExSol=Test(n).Mass;
104
105 for k=1:10
106 Th=SquareMesh(50*k+50);
107 fprintf(' Matrix size : %d\n',Th.nq);
108 h(k)=GetMaxLengthEdges(Th.q,Th.me);
109 tic();
110 M=MassAssemblingP1OptV2(Th.nq,Th.nme,Th.me,Th.areas);
111 TT(k)=toc();
112 U=u(Th.q(1,:),Th.q(2,:));
113 V=v(Th.q(1,:),Th.q(2,:));
114 Error(k)=abs(ExSol-U*M*V');
115 fprintf(' MassAssemblingP1OptV2 CPU times : %3.3f(s)\n',TT(k));
116 fprintf(' Error : %e\n',Error(k));
117 end
118
119 loglog(h,Error,'+-r',h,h*1.1*Error(1)/h(1),'-sm',h,1.1*Error(1)*(h/h(1)).^2,'-db')
120 legend('Error','O(h)','O(h^2)')
121 xlabel('h')
122 title('Test 3 : Mass Matrix')
123 checkTest3(h,Error)
124 end
125
126 function checkTest1(Test)
127 I=find(Test.error>1e-14);
128 if isempty(I)
129 disp('------------------------')
130 disp(' Test 1 (results): OK')
131 disp('------------------------')
132 else
133 disp('----------------------------')
134 disp(' Test 1 (results): FAILED')
135 disp('----------------------------')
136 end
137 end
138
139 function checkTest2(Test)
140 N=length(Test);
141 cntFalse=0;
142 for k=1:N
143 if (Test(k).degree<=1)
144 if (Test(k).error>1e-14)
145 cntFalse=cntFalse+1;
146 end
147 end
148 end
149 if (cntFalse==0)
150 disp('------------------------')
151 disp(' Test 2 (results): OK')
152 disp('------------------------')
153 else
154 disp('----------------------------')
155 disp(' Test 2 (results): FAILED')
156 disp('----------------------------')
157 end
158 end
159
160 function checkTest3(h,error)
161 % order 2
162 P=polyfit(log(h),log(error),1);
163 if abs(P(1)-2)<1e-2
164 disp('------------------------')
165 disp(' Test 3 (results): OK')
166 fprintf(' -> found numerical order %f. Must be 2\n',P(1))
167 disp('------------------------')
168 else
169 disp('----------------------------')
170 disp(' Test 3 (results): FAILED')
171 fprintf(' -> found numerical order %f. Must be 2\n',P(1))
172 disp('----------------------------')
173 end
174 end