Keywords: Hu-Washizu, well-posedness, approximation, convergence.

A three-field approximate method for the analysis of linear elastostatic problems is analysed on the basis of the Hu-Washizu principle. A discussion on existence and uniqueness of the solution of the discrete problem is developed with reference to the mixed formulation following the approach presented in [1].

The analysis performed in this paper shows that the three-field mixed problem can be split in a reduced problem, formulated in terms of strains and stresses, and in a displacement recovery problem depending on the solution of the previous one.

A basic issue which has been only partially treated in the literature is the one concerning the well-posedness of discrete mixed problems. Here well-posedness means existence and uniqueness of the discrete solution for any data. A full discussion of well-posedness is provided in this paper and necessary and sufficient conditions are contributed. Further, by a suitable specialization of known results [1], error bound estimates in the energy norm are provided.

A peculiar feature of the finite element method is that well-posedness and convergence properties must be assessed in terms of the shape functions defined in the reference element since the assembly operation is a priori unknown. This issue is discussed in the last part of the paper and sufficient applicable criteria are provided.

Previous treatments of three-field methods in the finite element literature are quoted in the textbook of Zienkiewicz and Taylor [2]. A rough necessary nonsingularity condition of the structural matrix is provided in terms of the dimension of the interpolating subspaces. This condition is usually referred to as a rank condition.

More recently, a three-field method, labelled Mixed Enhanced Strain (MES) method, has been proposed by Kasper and Taylor in [3] and applied to some numerical benckmarks. Anyway no well-posedness and convergence analysis has been carried out in [3].

The treatment of the MES method discussed in [3] appears to be strongly influenced by the presentation of the enhanced assumed strain (EAS) method proposed by Simo and Rifai in [4]. A critical analysis of the EAS method and a new mixed formulation has been recently contributed by the authors in [5,6,7].

Finally, finite element approximations of two-dimensional elastostatic problems are discussed and a sufficient convergence criteria for Q₁ elements is assessed. Two-dimensional elastostatic problems, commonly adopted in the literature as significant benchmarks, are investigated to get numerical evidence about convergence of the method.

1

Brezzi, F., Fortin, M., "Mixed and Hybrid Finite Element Methods", Springer-Verlag, New York, 1991.

2

Zienkiewicz, O.C., Taylor, R.L., "The Finite Element Method", vol I, IV, ed., Mc Graw-Hill, London, 1989.

3

Kasper, E.P., Taylor, R.L., "A mixed-enhanced strain method problems", Univ. of California at Berkeley, Re. UCB/SEMM-97/02, 1997.

4

Simo, J.C., Rifai, M.S., "A class of mixed assumed strain methods and the method of incompatible modes", Int. J. Num. Methods Engrg., 29, 1595-1638, 1990. doi:10.1002/nme.1620290802

5

Romano, G., Marotti de Sciarra, F., Diaco, M., "An analysis of enhanced strain methods",in: CD-Rom Proceedings, Numerical Methods in Continuum Mechanics 2000, Slovak Republic, 2000.

6

Romano, G., Marotti de Sciarra, F., Diaco, M., "Well-posedness and numerical performances of the strain gap method", Int. J. Num. Methods Engrg., 51, 103-126, 2001. doi:10.1002/nme.173¬

7

Romano, G., Marotti de Sciarra, F., Diaco, M., "Well-posedness and convergence of the strain gap method", submitted to SIAM. Journal on Numerical Analysis, 2000.

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 £123 +P&P)