dohrmann Episode 1 Part 8 pot

Conclusions A transition element for meshes containing uniform strain hexahedral and tetrahedral elements is presented.. Meshes containing the transition element satisfy first-order patc

to those obtained using existing methods based on master-slave concepts for connecting two meshes

5 Conclusions

A transition element for meshes containing uniform strain hexahedral and tetrahedral elements is presented Meshes containing the transition element satisfy first-order patch tests and converge for second-order patch tests under mesh refinement Comparisons with all-hexahedral meshes show that mixed-element meshes do not cause any significant degradation

in accuracy convergence rates or locking behavior for a variety of problems Guidelines are established for cxt ending the present approach to higher-order elements and for connecting dissimilar finite element meshes at a shared boundary

References

1 D P Flanagan and T Belytschko, ‘A Uniform Strain Hexahedron and Quadrilateral

wit h Orthogonal Hourglass Control’, International Journal for Numerical Methods in Engmecmng 17, 679-706 (1981).

2 C R Dohrmann, S W Key, M W Heinstein and J Jung, ‘A Least Squares Approach

for Uniform Strain Triangular and Tetrahedral Finite Elements’, International Journal for :Yumcrzcal Methods in Engineering, 42, 1181-1197 (1998).

3 S \\’ Key \l W Heinstein, C M Stone, F J Mello, M L Blanford and K G Budge,

A Suitable Low-Order, 8-Node Tetrahedral Finite Element for Solids’, to appear in

Intemlat~onal Journal for Numerical Methods in Engineering.

4 S J Owen S A Canann and S Saigal, ‘Pyramid Elements for Maintaining Tetrahe dra to Hexahedra Conformability’, Trends in Unstructured Mesh Generation,

AMD-Vol 220 ASME; 123-129 (1997)

5 K \f Heal hf L Hansen and K M Rickard, Maple V Learning Guide, Springer, New York Sew York, 1996

6 0 C Zienkiewicz and R L Taylor, The Finite Element Method, Vol 1, 4th Ed.,

McGraw Hill, New York, New York, 1989

7 C R Dohrmann, S W Key and M W Heinstein, ‘A Method for Connecting Dissimilar

Finite Element Meshes in Two Dimensions’, submitted to International Journal for Numerical methods in Engineering.

Table 1: Example 1 strain energies for meshes 3m, 6m, 9m and 9h.

v

0.0

0.1

0.2

0.3

0.4

0.499

0.4999

0.49999

9h

Edev 278 306 333 361 389 416 417 417

E.Ol

139 111 83.3 55.6 27.8 0.27 2.78e-2 2.78e-3

Edev

302

332

362

393

423

454

454

454

E.ol

150 121 90.9 60.9 30.6 0.308 3.07e-2 3.07e-3

Edev 283 312 340 369 397 425 426 426

EVOl

142 113 85.2 56.9 28.5 0.285 2.85e2 2.85e-3

Edev 280 308 336 364 393 420 421 421

E.ol

140 112 84,2 56.1 28.1 0.281 2.81e-2 2.81e-3

Edev 280 308 336 364 393 420 421 421

E.Ol

140 112 84.2 56.1 28.1 0.281 2.81e-2 2.81e-3

Table 2: Example 2 strain energies for meshes 3m, 6m, 9m and 9h.

m

ma=

0.1 1029

0.2 944

0.3 871

0.4 809

0.499 755

0.4999 755

0.49999 755

0.09 0.09 0.11 0.17 14 1.4e2 1.4e3

1043 956 882 819 765 765

765 u 0.02 1045 0.01

0.02 958 0.01 0.02 884 0.01 0.01 821 0.003 0.04 767 0.02 0.18 767 0.06 0.10 767 0.03

1045 958 884 821 767 767 767

0 0 0 0 0 0 0

1047 960 886 823 769 768 768

Table 3: Example 3 strain energies for meshes 3m and 3h.

3h

E.Ol

0.0835 0.0908 0.105 0.136 0.232 4.62 1.83 2.13 16.2

EVO1

0.0174 0.0208 0.0262 0.0359 0.0621 3.28 21.4 117 854

.&ev 1132 1029 944 871 809 760 769 772 773

Edev 1132 1029 943 871 809 756 759 794 863

0.0 0.1 0.2 0.3 0.4 0.499 0.4999 0.49999 0.499999

13

5

3

1

2

Figure 2: Connecting face of transitionelement

15

Figure 3: Element geometry of uniform strain tetrahedron.

Figure 4: Mesh 6m with 12 transition elements and 92 hexahedral elements removed.

1.8

1.6

1.4

1.2

1

0.8’

-2.2

x x mixed–element meshes

❑ H all-hexahedral meshes

log(lln)

–1 2 -1 -0.8

I.8

1.6

1.4

1.2

1

O.E

-:

x x mixed–element meshes

Q ❑ all–hexahedral meshes

log(lln)

Figure 6: Energy norm of the error for Example 1(v= 0.4999).

1.4

1.2

1

0.8

0.6

0.4

o.~

log(lhz)

Figure 7: Energy norm of the error for Example 2 (v = 0.3).

-??.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8

Iog(lln) Figure 8: Energy nom of the error for Example 2 (v = 0.4999).

21