Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element
Abstract
:1. Introduction
2. The Pure Displacement Problem
The Lower Bounds for the Eigenvalues of the Pure Displacement Problem
- A1
- V is a Hilbert space of real function on with the inner product and the corresponding norm .
- A2
- is another inner product of V. The corresponding norm is compact for V with respect to , i.e., every sequence in V which is bounded in has a subsequence, which is Cauchy in .
- A3
- is a finite dimensional space of real function over , Dim() = n. Define .
- A4
- Bilinear forms and on are extension of and to , such that
- (1)
- , .
- (2)
- and are symmetric and positive definite on .
3. The Pure Traction Problem
The Lower Bounds for Eigenvalues of the Pure Traction Problem
4. Numerical Experiments
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. The MATLAB Codes for the Planar Linear Elastic Eigenvalue Problem
Listing A1. MATLAB® codes for the pure displacement problem. |
function [eigv,eigvlower]=Elasticuniformintereig(nodeH,elemH) |
nodeH=[0 0;1 0;1 1;0 1]; |
elemH=[2 3 1;4 1 3]; |
mu=1;Lbd=10^8;rou=1; |
for i=1:8 |
[nodeH,elemH]=uniformrefine(nodeH,elemH); |
end |
node=nodeH;elem=elemH; |
figure(1);showmesh(node,elem) |
T=auxstructure(elem); |
elem2edge=T.elem2edge;edge=T.edge; |
N=size(node,1);NE=size(edge,1);NT=size(elem,1); |
Ndof=2*NE;elem2dof=[elem2edge elem2edge+NE]; |
ve1 = node(elem(:,3),:)-node(elem(:,2),:); |
ve2 = node(elem(:,1),:)-node(elem(:,3),:); |
ve3 = node(elem(:,2),:)-node(elem(:,1),:); |
area = 0.5*abs(-ve3(:,1).*ve2(:,2) + ve3(:,2).*ve2(:,1)); |
Dlambda(1:NT,:,1) = [-ve1(:,2)./(2*area), ve1(:,1)./(2*area)]; |
Dlambda(1:NT,:,2) = [-ve2(:,2)./(2*area), ve2(:,1)./(2*area)]; |
Dlambda(1:NT,:,3) = [-ve3(:,2)./(2*area), ve3(:,1)./(2*area)]; |
[lambda,weight] = quadpts(2); nQuad=length(weight); |
phi(:,1)=1-2*lambda(:,1); |
phi(:,2)=1-2*lambda(:,2); |
phi(:,3)=1-2*lambda(:,3); |
Dphi(:,:,1)=(-2)*Dlambda(:,:,1); |
Dphi(:,:,2)=(-2)*Dlambda(:,:,2); |
Dphi(:,:,3)=(-2)*Dlambda(:,:,3); |
A=sparse(Ndof,Ndof);M=sparse(Ndof,Ndof); |
uphi1=zeros(nQuad,6);uphi2=uphi1; |
Dphi1x=zeros(NT,2,6);Dphi1y=Dphi1x;Dphi2x=Dphi1x;Dphi2y=Dphi1x; |
for i=1:6 |
if i<=3; |
uphi1(:,i)=phi(:,i);uphi2(:,i)=0; |
Dphi1x(:,1,i)=Dphi(:,1,i);Dphi1y(:,2,i)=Dphi(:,2,i); |
Dphi2x(:,:,i)=0;Dphi2y(:,:,i)=0; |
else |
uphi1(:,i)=0;uphi2(:,i)=phi(:,i-3); |
Dphi1x(:,1,i)=0;Dphi1y(:,2,i)=0; |
Dphi2x(:,1,i)=Dphi(:,1,i-3);Dphi2y(:,2,i)=Dphi(:,2,i-3); |
end |
end |
for i=1:6 |
for j=i:6 |
Aij=mu*(Dphi1x(:,1,i).*Dphi1x(:,1,j)+Dphi1y(:,2,i).*Dphi1y(:,2,j)+... |
Dphi2x(:,1,i).*Dphi2x(:,1,j)+Dphi2y(:,2,i).*Dphi2y(:,2,j)).*area; |
Bij=(mu+Lbd)*((Dphi1x(:,1,i)+… |
Dphi2y(:,2,i)).*(Dphi1x(:,1,j)+Dphi2y(:,2,j))).*area; |
Kij=Aij+Bij; |
if (i==j) |
A=A+sparse(double(elem2dof(:,i)),double(elem2dof(:,j)),Kij,Ndof,Ndof); |
else |
A=A+sparse([double(elem2dof(:,i));… |
double(elem2dof(:,j))],[double(elem2dof(:,j));… |
double(elem2dof(:,i))],[Kij,Kij],Ndof,Ndof); |
end |
end |
end |
for i=1:6 |
for j=i:6 |
Mij=0; |
for p=1:nQuad |
Mij=Mij+weight(p)*(dot(uphi1(p,i),uphi1(p,j),2)+… |
dot(uphi2(p,i),uphi2(p,j),2)); |
end |
Mij=rou*Mij.*area; |
if (i==j) |
M=M+sparse(double(elem2dof(:,i)),double(elem2dof(:,j)),Mij,Ndof,Ndof); |
else |
M=M+sparse([double(elem2dof(:,i));… |
double(elem2dof(:,j))],[double(elem2dof(:,j));… |
double(elem2dof(:,i))],[Mij,Mij],Ndof,Ndof); |
end |
end |
end |
bdEage=setboundary(node,elem,’Dirichlet’); |
bdedgejudge=false(NE,1); |
bdedgejudge(elem2edge(bdEage==1))=1; |
inedge=find(bdedgejudge~=1); |
bdedge=find(bdedgejudge==1); |
bdedge=[bdedge;bdedge+NE]; |
A(bdedge,:)=[];A(:,bdedge)=[]; |
M(bdedge,:)=[];M(:,bdedge)=[]; |
[eigf,eigv]=eigs(A,M,8,’sm’); |
eigv=sort(diag(eigv)) |
i=6; |
eigvi=eigv(i) |
hc=sqrt(ve1(:,1).^2+ve1(:,2).^2); |
ha=sqrt(ve2(:,1).^2+ve2(:,2).^2); |
hb=sqrt(ve3(:,1).^2+ve3(:,2).^2); |
hh=max([hc,ha,hb]); h=max(hh); |
Ch=0.1893*h; |
eigvlower=eigvi/(1+eigvi*(1/mu)*(Ch^2)); |
Listing A2. MATLAB® codes for the pure traction problem. |
function [eigv1,eigvlower]=Elastictractionuniformeig(nodeH,elemH) |
nodeH=[0 0;1 0;1 1;0 1]; |
elemH=[2 3 1;4 1 3]; |
mu=1;Lbd=10^8;rou=1; |
node=nodeH;elem=elemH; |
for i=1:9 |
[node,elem]=uniformrefine(node,elem); |
end |
figure(1);showmesh(node,elem) |
T=auxstructure(elem); |
elem2edge=T.elem2edge; |
edge=T.edge; |
N=size(node,1); |
NE=size(edge,1);NT=size(elem,1); |
Ndof=2*NE; |
elem2dof=[elem2edge elem2edge+NE]; |
ve1 = node(elem(:,3),:)-node(elem(:,2),:); |
ve2 = node(elem(:,1),:)-node(elem(:,3),:); |
ve3 = node(elem(:,2),:)-node(elem(:,1),:); |
area = 0.5*abs(-ve3(:,1).*ve2(:,2) + ve3(:,2).*ve2(:,1)); |
Dlambda(1:NT,:,1) = [-ve1(:,2)./(2*area), ve1(:,1)./(2*area)]; |
Dlambda(1:NT,:,2) = [-ve2(:,2)./(2*area), ve2(:,1)./(2*area)]; |
Dlambda(1:NT,:,3) = [-ve3(:,2)./(2*area), ve3(:,1)./(2*area)]; |
[lambda,weight] = quadpts(2); nQuad=length(weight); |
phi(:,1)=1-2*lambda(:,1); |
phi(:,2)=1-2*lambda(:,2); |
phi(:,3)=1-2*lambda(:,3); |
Dphi(:,:,1)=(-2)*Dlambda(:,:,1); |
Dphi(:,:,2)=(-2)*Dlambda(:,:,2); |
Dphi(:,:,3)=(-2)*Dlambda(:,:,3); |
A=sparse(Ndof,Ndof);M=sparse(Ndof,Ndof); |
uphi1=zeros(nQuad,6);uphi2=uphi1; |
Dphi1x=zeros(NT,2,6);Dphi1y=Dphi1x;Dphi2x=Dphi1x;Dphi2y=Dphi1x; |
for i=1:6 |
if i<=3; |
uphi1(:,i)=phi(:,i);uphi2(:,i)=0; |
Dphi1x(:,1,i)=Dphi(:,1,i);Dphi1y(:,2,i)=Dphi(:,2,i); |
Dphi2x(:,:,i)=0;Dphi2y(:,:,i)=0; |
else |
uphi1(:,i)=0;uphi2(:,i)=phi(:,i-3); |
Dphi1x(:,1,i)=0;Dphi1y(:,2,i)=0; |
Dphi2x(:,1,i)=Dphi(:,1,i-3);Dphi2y(:,2,i)=Dphi(:,2,i-3); |
end |
end |
for i=1:6 |
for j=i:6 |
Aij=mu*(Dphi1x(:,1,i).*Dphi1x(:,1,j)+Dphi1y(:,2,i).*Dphi1y(:,2,j)+… |
Dphi2x(:,1,i).*Dphi2x(:,1,j)+Dphi2y(:,2,i).*Dphi2y(:,2,j)).*area; |
Bij=(mu+Lbd)*((Dphi1x(:,1,i)+… |
Dphi2y(:,2,i)).*(Dphi1x(:,1,j)+Dphi2y(:,2,j))).*area; |
Kij=Aij+Bij; |
if (i==j) |
A=A+sparse(double(elem2dof(:,i)),double(elem2dof(:,j)),Kij,Ndof,Ndof); |
else |
A=A+sparse([double(elem2dof(:,i));… |
double(elem2dof(:,j))],[double(elem2dof(:,j));… |
double(elem2dof(:,i))],[Kij,Kij],Ndof,Ndof); |
end |
end |
end |
for i=1:6 |
for j=i:6 |
Mij=0; |
for p=1:nQuad |
Mij=Mij+weight(p)*(dot(uphi1(p,i),uphi1(p,j),2)+… |
dot(uphi2(p,i),uphi2(p,j),2)); |
end |
Mij=rou*Mij.*area; |
if (i==j) |
M=M+sparse(double(elem2dof(:,i)),double(elem2dof(:,j)),Mij,Ndof,Ndof); |
else |
M=M+sparse([double(elem2dof(:,i));… |
double(elem2dof(:,j))],[double(elem2dof(:,j));… |
double(elem2dof(:,i))],[Mij,Mij],Ndof,Ndof); |
end |
end |
end |
A=A+M; |
cA=condest(A,2) |
cB=condest(M,2); |
[eigf,eigv]=eigs(A,M,12,’sm’); |
eigv=sort(diag(eigv)) |
i=12; |
eigvi=eigv(i) |
eigv3=eigvi-1; eigv1=eigv(1) |
eigf=eigf(:,3); |
hc=sqrt(ve1(:,1).^2+ve1(:,2).^2); |
ha=sqrt(ve2(:,1).^2+ve2(:,2).^2); |
hb=sqrt(ve3(:,1).^2+ve3(:,2).^2); |
hh=max([hc,ha,hb]); h=max(hh); |
Ch=((1/sqrt(eigv1))+1)*0.1893*h*(1/sqrt(mu)); |
eigvlower=eigvi/(1+eigvi*(1/mu)*(Ch^2)); |
eigvlower=eigvlower-1; |
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h | ||||||||
---|---|---|---|---|---|---|---|---|
17.886861 | 26.322914 | 19.651300 | 30.330628 | 19.692868 | 30.429766 | 19.693293 | 30.430780 | |
29.113233 | 33.479158 | 38.248713 | 46.156631 | 38.253602 | 46.163752 | 38.253652 | 46.163825 | |
34.755856 | 36.163354 | 47.885721 | 50.599032 | 47.903694 | 50.619099 | 47.903877 | 50.619303 | |
36.589358 | 36.968038 | 51.107816 | 51.849680 | 51.134479 | 51.877123 | 51.134741 | 51.877393 | |
37.092030 | 37.188573 | 52.003773 | 52.193743 | 52.033718 | 52.223907 | 52.033972 | 52.224163 | |
37.222052 | 37.246310 | 52.235248 | 52.283033 | 52.266137 | 52.313979 | 52.266233 | 52.314075 | |
37.255010 | 37.261082 | 52.293639 | 52.305604 | 52.324774 | 52.336753 | 52.324211 | 52.336190 | |
37.263300 | 37.264818 | 52.308271 | 52.311263 | 52.339468 | 52.342464 | 52.336206 | 52.339202 | |
37.265378 | 37.265758 | 52.311931 | 52.312679 | 52.343142 | 52.343891 | 52.329225 | 52.329973 | |
37.265378 | 52.311931 | 52.343142 | 52.329225 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
46.751983 | 288.000000 | 53.879884 | 1556.281308 | 55.791490 | 150701.95 | 55.812150 | 1506524739 | |
74.701495 | 112.267444 | 75.756482 | 114.667336 | 75.851660 | 114.885537 | 75.852618 | 114.887735 | |
134.735582 | 158.676849 | 158.167030 | 192.211501 | 160.002392 | 194.928781 | 160.018233 | 194.952294 | |
162.768463 | 170.539641 | 217.346150 | 231.427979 | 219.753392 | 234.159216 | 219.773060 | 234.181547 | |
171.369078 | 173.449429 | 236.789056 | 240.779418 | 239.244782 | 243.319052 | 239.264608 | 243.339559 | |
173.649500 | 174.178724 | 242.042378 | 243.071809 | 244.508667 | 245.559229 | 244.528375 | 245.579106 | |
174.228772 | 174.361658 | 243.383107 | 243.642498 | 245.852024 | 246.116707 | 245.871086 | 246.135810 | |
174.374229 | 174.407488 | 243.720067 | 243.785043 | 246.189635 | 246.255934 | 246.206029 | 246.272337 | |
174.410641 | 174.418958 | 243.804421 | 243.820673 | 246.274148 | 246.290731 | 246.280228 | 246.296812 | |
Trend | ↗ | ↗ | ↗ | ↗ | ||||
174.410641 | 243.804421 | 246.274148 | 246.280228 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
20.494339 | 32.386862 | 21.134242 | 34.014378 | 21.148868 | 34.052280 | 21.149018 | 34.052668 | |
38.771461 | 46.920038 | 55.671045 | 74.165556 | 55.710938 | 74.236376 | 55.711332 | 74.237075 | |
48.852331 | 51.679521 | 94.220010 | 105.333799 | 94.429370 | 105.595531 | 94.431453 | 105.598135 | |
52.534607 | 53.318789 | 115.066100 | 118.896168 | 115.432059 | 119.286937 | 115.435625 | 119.290745 | |
53.737136 | 53.940006 | 123.207359 | 124.279040 | 123.656871 | 124.736421 | 123.661198 | 124.740824 | |
54.137166 | 54.188497 | 126.144324 | 126.423364 | 126.635772 | 126.916993 | 126.640338 | 126.921579 | |
54.278441 | 54.291332 | 127.234951 | 127.305805 | 127.747230 | 127.818656 | 127.751345 | 127.822776 | |
54.331611 | 54.334839 | 127.666037 | 127.683864 | 128.188741 | 128.206713 | 128.190307 | 128.208281 | |
54.352676 | 54.353484 | 127.846819 | 127.851288 | 128.374768 | 128.379273 | 128.365844 | 128.370349 | |
54.352676 | 127.846819 | 128.374768 | 128.365844 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
46.751983 | 288.000000 | 52.535946 | 894.981566 | 55.775556 | 85063.9081 | 55.812148 | 850196897 | |
76.421607 | 116.198092 | 77.875935 | 119.593964 | 78.440738 | 205.401788 | 78.445830 | 120.943275 | |
129.040140 | 150.836415 | 163.305953 | 199.854199 | 166.991337 | 257.063840 | 167.027271 | 205.456157 | |
154.609650 | 161.604549 | 231.201955 | 247.202534 | 239.805783 | 257.063840 | 239.855617 | 210.516825 | |
162.809875 | 164.686468 | 257.145280 | 261.858045 | 266.073733 | 271.122660 | 266.121735 | 271.172501 | |
165.032625 | 165.510557 | 265.071803 | 266.306946 | 274.236848 | 275.559093 | 274.284810 | 275.607518 | |
165.605303 | 165.725356 | 267.434666 | 267.747890 | 276.732368 | 277.067763 | 276.779836 | 277.115346 | |
165.751008 | 165.781058 | 268.170017 | 268.248685 | 277.534251 | 277.618510 | 277.579130 | 277.663417 | |
165.788047 | 165.795562 | 268.417541 | 268.43724 | 277.813925 | 277.835028 | 277.848258 | 277.869366 | |
Trend | ↗ | ↗ | ↗ | ↗ | ||||
165.788047 | 268.417541 | 277.813925 | 277.848258 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
0.6115 | 14.323135 | 39.913359 | 15.160216 | 47.171428 | 15.173315 | 47.298479 | 15.173447 | 47.299765 |
0.3057 | 32.193412 | 50.323217 | 45.995245 | 94.780750 | 46.052329 | 95.023469 | 46.052905 | 95.025918 |
0.1529 | 46.126594 | 52.961082 | 87.795384 | 116.381345 | 87.987716 | 116.719553 | 87.989637 | 116.722933 |
0.0764 | 51.903333 | 53.858515 | 113.841249 | 123.689774 | 114.214590 | 124.130630 | 114.218245 | 124.134948 |
0.0382 | 53.667221 | 54.175603 | 123.594495 | 126.324507 | 124.063842 | 126.814858 | 124.068402 | 126.819623 |
0.0191 | 54.163439 | 54.291985 | 126.603166 | 127.307725 | 127.110189 | 127.820418 | 127.115465 | 127.825753 |
0.0096 | 54.304519 | 54.336766 | 127.519231 | 127.697188 | 128.040936 | 128.220353 | 128.047000 | 128.226434 |
0.0048 | 54.346670 | 54.354741 | 127.816022 | 127.860672 | 128.343900 | 128.388919 | 128.349842 | 128.394866 |
54.346670 | 127.816022 | 128.343900 | 128.349842 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
0.6115 | 21.133209 | 391.229908 | 21.454746 | 541.452504 | 15.173315 | 542.726367 | 15.173447 | 542.736559 |
0.3057 | 61.835112 | 200.749622 | 61.983149 | 202.318364 | 46.052329 | 204.054970 | 46.052905 | 204.066711 |
0.1529 | 169.356563 | 321.851420 | 179.023153 | 358.655407 | 87.987716 | 359.549948 | 87.989636 | 359.558633 |
0.0764 | 287.588211 | 360.000427 | 324.087253 | 419.081749 | 114.214590 | 420.415655 | 114.218245 | 420.426716 |
0.0382 | 346.264391 | 368.580471 | 404.968837 | 435.830320 | 124.063842 | 437.248819 | 124.068401 | 437.260318 |
0.0191 | 364.808130 | 370.720047 | 431.948559 | 440.261613 | 127.110189 | 441.687545 | 127.115479 | 441.699321 |
0.0096 | 369.758806 | 371.259020 | 439.288760 | 441.407842 | 128.040936 | 442.831032 | 128.046694 | 442.842850 |
0.0048 | 371.019253 | 371.395723 | 441.171930 | 441.704329 | 128.343900 | 443.124811 | 128.349798 | 443.138871 |
Trend | ↗ | ↗ | ↗ | ↗ | ||||
371.019253 | 441.171930 | 128.343900 | 128.349798 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
4.679529 | 8.578365 | 4.684617 | 8.592845 | 4.684738 | 8.593190 | 4.684740 | 8.593194 | |
8.054527 | 9.807919 | 8.102456 | 9.876278 | 8.103670 | 9.878011 | 8.103683 | 9.878029 | |
9.625991 | 10.157034 | 9.722357 | 10.263320 | 9.724900 | 10.266126 | 9.724919 | 10.266149 | |
10.107923 | 10.247834 | 10.223430 | 10.366285 | 10.226510 | 10.369444 | 10.226525 | 10.369460 | |
10.235374 | 10.270826 | 10.356354 | 10.392575 | 10.359589 | 10.395830 | 10.359569 | 10.395811 | |
10.267705 | 10.276598 | 10.390111 | 10.399198 | 10.393386 | 10.402479 | 10.393453 | 10.402546 | |
10.275819 | 10.278044 | 10.400859 | 10.400859 | 10.401871 | 10.404146 | 10.400585 | 10.402862 | |
10.277849 | 10.278405 | 10.400706 | 10.401274 | 10.403995 | 10.404564 | 10.396304 | 10.396875 | |
10.278357 | 10.278496 | 10.401236 | 10.401378 | 10.404525 | 10.404667 | 10.428847 | 10.428988 | |
10.278357 | 10.401236 | 10.404525 | 10.428847 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
11.570888 | 125.905191 | 11.570972 | 125.913754 | 11.570973 | 125.913931 | 11.570972 | 125.913933 | |
32.080588 | 80.221812 | 33.033058 | 86.214751 | 33.044201 | 86.287965 | 33.044322 | 86.288716 | |
66.683469 | 96.131275 | 69.383473 | 101.790033 | 69.424780 | 101.878160 | 69.425136 | 101.879058 | |
89.859480 | 100.151320 | 94.542505 | 105.989428 | 94.605273 | 106.068143 | 94.605816 | 106.068933 | |
98.316148 | 101.156533 | 103.875452 | 107.047802 | 103.947626 | 107.124410 | 103.948167 | 107.125126 | |
100.679017 | 101.407796 | 106.498587 | 107.313510 | 106.573562 | 107.389626 | 106.574174 | 107.390327 | |
101.287213 | 101.470609 | 107.174904 | 107.380042 | 107.250613 | 107.456039 | 107.249767 | 107.455353 | |
101.440388 | 101.486313 | 107.345312 | 107.396686 | 107.421208 | 107.472653 | 107.412503 | 107.464208 | |
101.478753 | 101.490238 | 107.387999 | 107.400847 | 107.463940 | 107.476806 | 107.485870 | 107.498623 | |
Trend | ↗ | ↗ | ↗ | ↗ | ||||
101.478753 | 107.387999 | 107.463940 | 107.485870 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
4.652581 | 8.501969 | 4.652581 | 8.501969 | 4.652581 | 8.501969 | 4.652581 | 8.501968 | |
7.891788 | 9.576853 | 7.900475 | 9.589148 | 7.900696 | 9.589461 | 7.900699 | 9.589464 | |
9.302245 | 9.800662 | 9.305798 | 9.804567 | 9.305887 | 9.804666 | 9.305880 | 9.804659 | |
9.722305 | 9.852614 | 9.723299 | 9.853632 | 9.723324 | 9.853658 | 9.723309 | 9.853644 | |
9.832421 | 9.865372 | 9.832677 | 9.865629 | 9.832683 | 9.865635 | 9.832615 | 9.865568 | |
9.860286 | 9.868547 | 9.860350 | 9.868612 | 9.860352 | 9.868613 | 9.860272 | 9.868534 | |
9.867273 | 9.869340 | 9.867289 | 9.869356 | 9.867290 | 9.869357 | 9.865821 | 9.867889 | |
9.869022 | 9.869538 | 9.869026 | 9.869542 | 9.869026 | 9.869543 | 9.859892 | 9.860410 | |
9.869459 | 9.869588 | 9.869460 | 9.869589 | 9.869459 | 9.869588 | 9.890587 | 9.890715 | |
9.869459 | 9.869460 | 9.869459 | 9.890587 |
h | ||||||||
---|---|---|---|---|---|---|---|---|
9.860459 | 48.000000 | 9.860459 | 48.000000 | 9.860459 | 48.000000 | 9.860457 | 48.000000 | |
20.827963 | 34.848012 | 20.925542 | 35.111955 | 20.927973 | 35.118549 | 20.928000 | 35.118617 | |
32.475281 | 38.380191 | 32.522585 | 38.445672 | 32.523776 | 38.447321 | 32.523768 | 38.447332 | |
37.474579 | 39.206889 | 37.489648 | 39.223346 | 37.490026 | 39.223758 | 37.490005 | 39.223749 | |
38.958651 | 39.410714 | 38.962681 | 39.414835 | 38.962782 | 39.414938 | 38.962698 | 39.414873 | |
39.347245 | 39.461503 | 39.348269 | 39.462533 | 39.414938 | 39.462559 | 39.348215 | 39.462490 | |
39.445546 | 39.474190 | 39.445804 | 39.474447 | 39.445810 | 39.474454 | 39.444329 | 39.472993 | |
39.470195 | 39.477361 | 39.470259 | 39.477425 | 39.470261 | 39.477427 | 39.461186 | 39.468386 | |
39.476362 | 39.478153 | 39.476378 | 39.478169 | 39.476377 | 39.478169 | 39.498316 | 39.500093 | |
Trend | ↗ | ↗ | ↗ | ↗ | ||||
39.476362 | 39.476378 | 39.476377 | 39.498316 |
1 | |||||
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cond(CR) | |||||
cond(P1) |
1 | |||||
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cond(CR) | |||||
cond(P1) |
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Zhang, X.; Zhang, Y.; Yang, Y. Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element. Mathematics 2020, 8, 1252. https://doi.org/10.3390/math8081252
Zhang X, Zhang Y, Yang Y. Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element. Mathematics. 2020; 8(8):1252. https://doi.org/10.3390/math8081252
Chicago/Turabian StyleZhang, Xuqing, Yu Zhang, and Yidu Yang. 2020. "Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element" Mathematics 8, no. 8: 1252. https://doi.org/10.3390/math8081252