Modeling of nowadays structures is based mainly on finite element method and sophisticated material models. Although, the progress in numerical methods and material modeling is impressive we are still facing numerous problems concerning the methods themselves and also computational demands. Typically, model of a real structure or a specimen contains fine and large finite element mesh. For modeling of concrete it is necessary to capture several important phenomena such as development of anisotropy, damage and ductility (possibly softening). For these purposes different approaches based on e.g. multi-surface plasticity or damage mechanics have been developed.

Another example of such a material model is the microplane model for concrete [3], [4]. This model uses a non-tensorial formulation of constitutive relations that are defined on the microlevel between strain and stress components belonging to the particular direction. This definition allows a straightforward modeling of the phenomena that are in connection with certain direction or a plane. Projection of the macroscopic strain tensor has to be performed to obtain particular strain components. Then constitutive relations are applied and finally the principle of virtual work is used to obtain macroscopic stress tensor. This procedure is computationally very demanding and involves numerous operations within the material (Gauss) point. Computational algorithm of the model puts also high demands on computer memory because a large number of data have to be stored during the solution step.

There are several limitations for using the microplane model. The first one is that the model is not suitable for implicit numerical schemes. It is due to the formulation of constitutive laws that give no direct formula for tangential stiffness matrix [3]. This lack of tangential stiffness can be overcome by either using the initial stiffness matrix throughout the whole computation (which is very ineffective, of course) or preferably by using explicit methods. One also have to keep in mind that using only local constitutive law leads to non-realistic response for energy dissipative material. The possible solution can be using of non-local approach. Otherwise, the material parameters would have to be fitted for a particular size of an element separately. In case of the lack of non-local model and in order to overcome difficulties with the need of using different parameters for each element it is possible to use a structured mesh (i.e. the size of elements is constant) to keep the dissipation the same (e.g. [7]).

In order to illustrate the above mentioned statements we have taken an example of reinforced concrete column. This is a typical representative of a common structural element where we can encounter e.g. compression damage and softening, confinement and steel-concrete interaction. For this purpose we constructed a structured mesh containing brick elements with microplane model for concrete and 3D geometrically non-linear beam element with elastoplasticity for steel. Outward dimensions of the column were approximately 1000x150x150mm. Reinforcement consisted of four longitudinal bars 12mm and closed stirrups 6mm. Final mesh contained 33159 degrees of freedom, 8928 brick elements (approx. one half contained microplane model, the rest was modeled by linear elasticity). Columns were loaded in compression with small eccentricity. Explicit integration was engaged to minimize computational effort as reported above. We obtained very satisfactory results but the computational times were unreasonable (about 3 days).

In order to decrease also computational times we adopted parallel approach based on a domain decomposition technique. Original domain containing the whole body is decomposed into several partitions (subdomains) using so called node-cut concept. Parallelized version of a central difference integration scheme [8], [11] was employed. The standard sequential algorithm can be used for solving the displacements in each subdomain. But the algorithm must be enlarged by an additional step which is the exchange of the real nodal forces contributions for shared nodes. Shared nodes are located only on the boundary of each subdomain. Similarly, at the beginning of the solution, mass matrix must be assembled with respect to contributions from share nodes in each subdomain.

Only static partitioning (no changes in the size of subdomains during computation) was considered because the microplane model takes in fact the same computational effort in linear and non-linear regimes. Parallel version of OOFEM (finite element environment developed at the Department of Structural Mechanics in Prague) was used [9]. The computational system is running on a Linux cluster (PII Xeon 400-450 MHz, 512 MB RAM, fast Ethernet 100 Mb). MPIPro message passing library [6] was adopted to handle interprocessor communication.

Significant reduction of a computational time and almost linear speedup was achieved (see Fig. 6). This particular finding shows us large capabilities of a PC cluster. Complex and computationally demanding problems can be solved effectively using standard computer network. In many cases, parallel approach can be the only way how to solve the problem at all.

*Daniel Rypl
2005-12-03*