## Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods.(English)Zbl 0737.73008

The paper deals with composite materials. Their high degree of heterogeneity needs to find some kind of equivalent medium with the same average mechanical behavior. The homogenization method is a rigorous mathematical theory to compute the equivalent material properties. It is usually assumed that the composite is locally formed by the spatial repetition of small microstructures, the microscopic cells, when compared with the macroscopic dimensions of the structure of interest. This enables the computation of equivalent material properties by a limiting process when the microscopic cell size, $$\varepsilon$$, tends to zero.
In this paper the method is applied to a linear elastic material with a periodic distribution of holes. The paper gives at first a clear review of the homogenization theory. The elastic connected domain is subjected to body forces, to traction and prescribed displacement on its boundary, together with traction inside the holes. Homogenization leads to the solution of three distinct problems: two in the microscopic cell, which give the homogenized elastic coefficients $$D_{ijkl}$$ and the residual stress $$\tau_{ij}$$ (due to the traction inside the holes), and the other on the macroscopic level, to obtain the macroscopic displacement field $$\vec u^ 0$$ of the homogenized problem.
The finite element solution procedure to obtain the homogenized elastic coefficients $$D_{ijkl}$$, the residual stress $$\tau_{ij}$$ and the macroscopic displacement field $$\vec u^ 0$$ is presented. Then n the homogenization method is used to introduced the idea of a material preprocessor (PREMAT) for the computation of the homogenized properties and a material postprocessor (POSTMAT) for the computation of stress and strain distribution within the composite microstructure after a finite element procedure has been used to solve the general structure. Several examples are presented: fibers in elasticity, honeycomb structures, woven fiber reinforced composites. Finally, an adaptive finite element method is introduced in order to improve the accuracy of the numerical results.
Reviewer: Th.Lévy (Paris)

### MSC:

 74E05 Inhomogeneity in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 74E30 Composite and mixture properties
Full Text:

### References:

 [1] Hashin, Z., Theory of composite materials, (Wend, F. W.; Liebowitz, H.; Perrone, N., Mechanical of Composite Materials (1970), Pergamon: Pergamon Oxford) · Zbl 0542.73092 [2] Lions, J. L., Some Methods in the Mathematical Analyses of Systems and their Control (1981), Science Press: Science Press Beijing, China, and (Gordon and Breach, New York) · Zbl 0542.93034 [3] Benssousan, A.; Lions, J. L.; Papanicoulau, G., Asymptotic Analysis for Periodic Structures (1978), North-Holland: North-Holland Amsterdam [4] Sanchez-Palencia, E., Non Homogeneous Media and Vibration Theory, (Lecture Notes in Physics (1980), Springer: Springer Berlin), No. 127 · Zbl 0432.70002 [5] Duvaut, G., Analyse functionelle et mécanique des milieux continues. Applications à l’etude de matérieux composites elastiques à structure périodiques Homogénéisation, (Koiter, W. T., Theoretical and Applied Mechanics (1976), North-Holland: North-Holland Amsterdam), 119-132 [6] Duvaut, G., Homogeneization et materiaux composite, (Ciarlet, P. G.; Rouseau, M., Trends and Applications of Pure Mathematics to Mechanics. Trends and Applications of Pure Mathematics to Mechanics, Lecture Notes in Physics (1984), Springer: Springer Berlin), No. 195 · Zbl 0541.73023 [7] Lene, F., (Thése de Doctorat d’Etat (1984), Université Pierre et Marie Curie: Université Pierre et Marie Curie Paris VI) [8] Oleinik, O. A., On homogenization problems, (Ciarlet, P. G.; Rouseau, M., Trends and Applications of Pure Mathematics to Mechanics (1984), Springer: Springer Berlin) · Zbl 0870.35012 [9] Murat, F.; Tartar, L., Calculs des variations et homogénéization, (Les Methodes de l’Homogenéization: Theory et Applications en Physique (1985), Coll. de la Dir. de Etudes et Recherches d’Electricité de France: Coll. de la Dir. de Etudes et Recherches d’Electricité de France Eyrolles), 319 [11] Willis, J. R., Bounds and selfconsistent estimates for overall properties of anisotropic composites, J. Mech. Phys. Solids, 25, 3, 389-393 (1977) [12] Kroner, E., Effective moduli of random elastic media—Unified calculation of bounds and self consistent values, Mech. Res. Comm., 4, 6, 389-393 (1977) [14] Necas, J.; Hlavacek, I., Mathematical Theory of Elastic and Elastico-Plastic Bodies: An Introduction (1981), Elsevier: Elsevier Amsterdam · Zbl 0448.73009 [15] Cioranescu, D., Thesis (1978), Paris [16] Cioranescu, D.; Paulin, J. S.J., Homogenization in open sets with holes, J. Math. Anal. Appl., 71, 590-607 (1979) · Zbl 0427.35073 [17] Oden, J. T.; Reddy, J. N., An Introduction to the Mathematical Theory of Finite Elements (1976), Wiley: Wiley New York · Zbl 0336.35001 [18] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 [19] Babuška, I.; Aziz, A. K., Survey lectures on the mathematical foundations of the finite element method, (Aziz, A. K., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (1972), Academic Press: Academic Press New York) · Zbl 0268.65052 [20] Koh, B. C.; Kikuchi, N., New improved hourglass control for bilinear and trilinear elements in anisotropic linear elasticity, Comput. Methods Appl. Mech. Engrg., 65, 1, 1-46 (1987) · Zbl 0621.73104 [21] (Lynch, C. T., CRC—Handbook of Materials Science, Vol. II (1975), CRC Press: CRC Press Cleveland), 392 [22] Arantes, E. R.; Oliveira, E., Optimization of the finite element solution, (Proc. Third Conference of Matrix Methods in Structural Mechanics (1971), Wright-Patterson Air Force Base: Wright-Patterson Air Force Base Dayton, OH) · Zbl 0174.41601 [23] Babuška, I., The adaptive methods and error estimation for elliptic problems of structural mechanics, (Proc. ARO Workshop on Adaptive Methods for Partial Differential Equations (14-16 February 1983), Univ. of Maryland: Univ. of Maryland College Park) [24] Babuška, I.; Rheinboldt, W. C., Reliable error estimation and mesh adaptation for the finite element method, (Oden, J. T., Computational Methods in Nonlinear Mechanics (1980), North-Holland: North-Holland Amsterdam), 67-109 [25] Babuška, I.; Rheinboldt, W. C., Error estimates for adaptive finite element computations, Internat. J. Numer. Methods Engrg., 15, 736-754 (1980) · Zbl 0398.65069 [26] Babuška, I.; Dorr, M. R., Error estimates for the combined $$h$$ and $$p$$ version of the finite element method, Numer. Math., 25, 257-277 (1981) · Zbl 0487.65058 [27] Diaz, A. R.; Kikuchi, N.; Taylor, J. E., Optimal design formulations for finite element grid adaptation, (Komkov, V., Sensitivity of Functionals with Applications to Engineering Science. Sensitivity of Functionals with Applications to Engineering Science, Lecture Notes in Mathematics, 1086 (1984), Springer: Springer Berlin), 56-76 [28] Diaz, A. R.; Kikuchi, N.; Taylor, J. E., A method of grid optimization for finite element methods, Comput. Methods Appl. Mech. Engrg., 41, 1, 29-45 (1983) · Zbl 0509.73071 [29] Kikuchi, N., Adaptive grid design methods for finite element analysis, Comput. Methods Appl. Mech. Engrg., 55, 129-160 (1986) · Zbl 0572.65098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.