Keywords: multiscale, finite element method, object oriented design.

The paper deals with the hierarchical and concurrent multiscale methods and their object-oriented implementation. Multiscale approaches have attracted increased attention in the modeling engineering materials and structures. The fundamental reason is that the multiscale approach allows the prediction of the behavior and properties of complex heterogeneous materials from intrinsic properties and physical mechanisms. A multiscale analysis exploits the fact that virtually all materials are heterogeneous at certain scales. The analysis starts from a definition of a unit cell, which sufficiently describes the underlying microstructure and mechanisms. Depending on the solution strategy, homogenization methods are classified in two broad groups. The hierarchical methods belong to a first category and are normally based on the assumptions of scale separation, a uniformity of the macroscopic field in the vicinity of a material point, and a local periodicity. The solution proceeds from a value of the macroscopic field, which is passed to the lower scale where it forms the boundary conditions of the microproblem. Homogenization of the microproblem yields an effective property, which is passed back to the material point of the higher scale. The material point often corresponds to the integration point of the finite element (FE). At the end, a sequential processing of many sub-problems leads to the solution of the macroproblem. The scheme is often enhanced with iterations between scales so equilibrium is reached on both scales. Material data for a multi-scale model are typically obtained either through micro level tests (nanoindentation), or at meso level, such as uniaxial tests. Micromechanical tests are faster, cheaper, and, in combination with a computer model, relatively easy to generalize, as they rely on the properties of the composite components only.

The implementation of multiscale support, particularly its seamless integration into an existing code, is a complex and challenging task. In this paper, a design of an object-oriented multiscale toolkit is presented and its integration into an existing object-oriented FE environment [1,2] is discussed. The overall structure of the proposed multiscale toolkit consists of several new classes, that represents the macro-level material model or macro-level element. The primary role of these classes is to hide all details of lower level representation while preserving a general interface, provided by all material models or elements. Additional classes representing a general data stream were introduced to allow transparent management of history variables for the unit cell. This allows the unit cell model to be shared among all integration points at the macro level. The application of the object-oriented tool developed is demonstrated using a simple three-dimensional example of the analysis of a cantilever beam subjected to tension, where unit cells are represented by a solid rectangle with a circular hole, made of plastic material.

1

B. Patzák, Z. Bittnar, "Design of object oriented finite element code", Advances in Engineering Software, 32(10-11), 759-767, 2001. doi:10.1016/S0965-9978(01)00027-8

2

B. Patzák, "OOFEM home page", http://www.oofem.org, 2008.

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