1. Computation of the Navier equation in 2D
H. Notsu and H. Kanayama
CM2
December 28, 2009

2. Finite element approximation How to make an element stiffness matrix
Contents etc.
The Navier equation
Weak form
Finite element approximation
How to make an element stiffness matrix
Numerical results in 2D
December 28, 2009
CM2

3. The Navier equation
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4. Assumption We assume the isotropic elasticity, i.e., December 28, 2009
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5. Weak form
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6. Finite element approximation
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7. How to make an element stiffness matrix d(d+1) x d(d+1)=6x6
December 28, 2009
CM2

8. How to make element stiffness matrix d(d+1) x d(d+1)=6x6
where
for ( i=0; i<3; i++ ) kl[i] = elnp[l][i];
for ( i=0; i<3; i++){
c1[i] = (npxy[kl[(i+1)%3]][1]-npxy[kl[(i+2)%3]][1])/(2.0*sl);
c2[i] = (npxy[kl[(i+2)%3]][0]-npxy[kl[(i+1)%3]][0])/(2.0*sl); }
for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) ae11[i][j] = ((lame1+2.0*lame2)*c1[j]*c1[i]+lame2*c2[j]*c2[i])*sl;
for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) ae12[i][j] = (lame1*c2[j]*c1[i]+lame2*(c1[j]*c2[i]))*sl;
for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) ae21[i][j] = ae12[j][i];
for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) ae22[i][j] = ((lame1+2.0*lame2)*c2[j]*c2[i]+lame2*c1[j]*c1[i])*sl;
for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) ecm[i][j] += ae11[i][j];
for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) ecm[i][3+j] += ae12[i][j];
for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) ecm[3+i][j] += ae21[i][j];
for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) ecm[3+i][3+j] += ae22[i][j];
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9. Test problem 1
Solver: CG
A sample mesh
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10. Test problem 1
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11. Test problem 2
Solver: CG
A sample mesh
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12. Test problem 2
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13. Examples
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14. Examples
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15. Appendix
In order to make the matrix, we consider the following equation;
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16. Appendix
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17. Appendix
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