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Keywords: mixed finite elements, method of incompatible modes, method of enhanced strains, area bubble functions.

Summary

It is well known, that standard bi-linear finite element formulations exhibit rather poor performance when extra (physical) constraints occur. Typical examples are the incompressibility constraint, leading to volume locking and the shear constraint for bending dominated problems, which induces shear locking. Although these problems can be circumvated by high order elements, low order elements remain quite popular. In this way, the combination of incompatible modes with bubble functions for formulation of incompatible displacements at small and finite elastic strains for tetrahedral elements is presented by Taylor [1]. The assumed enhanced strain method is analysed extensively in [2].

In our work we address the solution of geometrically linear elastic problems for three dimensional solids using tetrahedral finite elements. Both the method of incompatible modes and the assumed enhanced strain method are used for stabilization. As a key idea area bubble functions are used for both mixed finite element formulations in order to enriche the displacement field and the enhanced strain field, respectively. As for volume bubble functions they serve to enrich the standard Galerkin space in an element-by-element technique. They satisfy a residual equation at each element and can statically be condensed from the global formulation.

In the numerical example we compare results for the stress distributions along the clamped edge for Cook's membrane problem between our formulations and those from the literature. The results show, that our elements reduce the stress oscillations more significantly than volume bubble elements. We also observe, that the enhanced strain version provides slightly better results than the incompatible mode version.

References

1: R.L. Taylor, " A mixed-enhanced formulation for tetrahedral finite elements," Int. J. Num. Meths. Eng., 47:205-225, 2000. doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<205::AID-NME768>3.0.CO;2-J
2: J.C. Simo, M.S. Rifai, "A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes, Int. J. Num. Meths. Eng., 29:1595-1638, 1990. doi:10.1002/nme.1620290802

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 85 PROCEEDINGS OF THE FIFTEENTH UK CONFERENCE OF THE ASSOCIATION OF COMPUTATIONAL MECHANICS IN ENGINEERING Edited by: B.H.V. Topping Paper 39 Area Bubble Functions for Stabilization of Mixed Finite Tetrahedral Elements R. Mahnken¹, I. Caylak¹ and G. Laschet² ¹Chair of Engineering Mechanics, University of Paderborn, Germany ²ACCESS e.V., Aachen University, Germany doi:10.4203/ccp.85.39 purchase the full-text of this paper Full Bibliographic Reference for this paper R. Mahnken, I. Caylak, G. Laschet, "Area Bubble Functions for Stabilization of Mixed Finite Tetrahedral Elements", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 39, 2007. doi:10.4203/ccp.85.39 Keywords: mixed finite elements, method of incompatible modes, method of enhanced strains, area bubble functions. Summary It is well known, that standard bi-linear finite element formulations exhibit rather poor performance when extra (physical) constraints occur. Typical examples are the incompressibility constraint, leading to volume locking and the shear constraint for bending dominated problems, which induces shear locking. Although these problems can be circumvated by high order elements, low order elements remain quite popular. In this way, the combination of incompatible modes with bubble functions for formulation of incompatible displacements at small and finite elastic strains for tetrahedral elements is presented by Taylor [1]. The assumed enhanced strain method is analysed extensively in [2]. In our work we address the solution of geometrically linear elastic problems for three dimensional solids using tetrahedral finite elements. Both the method of incompatible modes and the assumed enhanced strain method are used for stabilization. As a key idea area bubble functions are used for both mixed finite element formulations in order to enriche the displacement field and the enhanced strain field, respectively. As for volume bubble functions they serve to enrich the standard Galerkin space in an element-by-element technique. They satisfy a residual equation at each element and can statically be condensed from the global formulation. In the numerical example we compare results for the stress distributions along the clamped edge for Cook's membrane problem between our formulations and those from the literature. The results show, that our elements reduce the stress oscillations more significantly than volume bubble elements. We also observe, that the enhanced strain version provides slightly better results than the incompatible mode version. References 1 R.L. Taylor, " A mixed-enhanced formulation for tetrahedral finite elements," Int. J. Num. Meths. Eng., 47:205-225, 2000. doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<205::AID-NME768>3.0.CO;2-J 2 J.C. Simo, M.S. Rifai, "A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes, Int. J. Num. Meths. Eng., 29:1595-1638, 1990. doi:10.1002/nme.1620290802 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £75 +P&P)
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