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. 12 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 3m 6m 9m exact 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. v 3m 6m 9m 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 v 3m 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 8 5 3 1 2 Figure 1: Sketch of polyhedron showing 12 of the 24 triangularfaces. 3 Figure 2: Connecting face of transitionelement. 15 Figure 3: Element geometry of uniform strain tetrahedron. 16 Figure 4: Mesh 6m with 12 transition elements and 92 hexahedral elements removed. 2 1.8 1.6 1.4 1.2 1 0.8’ -2.2 I I I I 1 1 x x mixed–element meshes ❑ H all-hexahedral meshes -2 -1.8 -1.6 -1.4 log(lln) –1 .2 -1 -0.8 Figure 5: Energy norm of the error for Example 1 (v= 0.3). 2 I.8 1.6 1.4 1.2 1 O.E -: I I I [ I # x x mixed–element meshes Q ❑ all–hexahedral meshes ) -2 -1.8 -1.6 –1.4 -1.2 -1 -0.8 log(lln) Figure 6: Energy norm of the error for Example 1 (v= 0.4999). m 1.6 1.4 1.2 1 0.8 0.6 0.4 o.~ ~ -2 -1.8 -1.6 –1.4 -1.2 -1 -0.8 log(lhz) Figure 7: Energy norm of the error for Example 2 (v = 0.3). 20 -??.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 . 9m exact Edev 2 78 306 333 3 61 389 416 417 417 E.Ol 13 9 11 1 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 15 0 12 1 90.9 60.9 30.6 0.3 08 3.07e-2 3.07e-3 Edev 283 312 340 369 397 425 426 426 EVOl 14 2 11 3 85 .2 56.9 28. 5 0. 285 2 .85 e2 2 .85 e-3 Edev 280 3 08 336 364 393 420 4 21 4 21 E.ol 14 0 11 2 84 ,2 56 .1 28 .1 0.2 81 2 .81 e-2 2 .81 e-3 Edev 280 3 08 336 364 393 420 4 21 4 21 E.Ol 14 0 11 2 84 .2 56 .1 28 .1 0.2 81 2 .81 e-2 2 .81 e-3 Table 2:. 3h. 3h v 3m E.Ol 0. 083 5 0.09 08 0 .10 5 0 .13 6 0.232 4.62 1. 83 2 .13 16 .2 EVO1 0. 017 4 0.02 08 0.0262 0.0359 0.06 21 3. 28 21. 4 11 7 85 4 .&ev 11 32 10 29 944 8 71 80 9 760 769 772 773 Edev 11 32 10 29 943 8 71 80 9 756 759 794 86 3 0.0 0 .1 0.2 0.3 0.4 0.499 0.4999 0.49999 0.499999 13 8 5 3 1 2 Figure. 9m m ma= 0 .1 1029 0.2 944 0.3 8 71 0.4 80 9 0.499 755 0.4999 755 0.49999 755 0.09 0.09 0 .11 0 .17 14 1. 4e2 1. 4e3 10 43 956 88 2 81 9 765 765 765 u 0.02 10 45 0. 01 0.02 9 58 0. 01 0.02 88 4 0. 01 0. 01 8 21 0.003 0.04