Keywords: parallel processing, high performance computing, discrete cracking concrete, probabilistic crack approach, material heterogeneity, size effects, tensile strength domain, Monte Carlo method, non-linear analysis, finite elements.

This work presents a probabilistic crack approach (Rossi et al [1]), based on the Monte Carlo method, that was recently implemented in a 3D fully parallelized finite element code (Paz, [2]). The cracking scheme used is the discrete crack approach introduced by means of 3D interface elements. In this approach the heterogeneity of the material is taken into account by considering the properties (tensile strength, Young Modulus, etc.) to vary spatially following a normal distribution determined by the mean the standard deviation of the considered material properties. samples of a vector of these properties are generated and the corresponding solutions are computed by the FE code. Hence, the average response of the samples corresponding to a Monte Carlo simulation is a function of the mean value and of the standard deviation that define the Gauss density function. If the heterogeneous characteristics of the material are well established and quantified by the statistical moments it is possible that the model displays the size effects related to the material heterogeneity. The problem with this approach is that these statistical moments are not known a priori for the characteristic volume of the finite elements used in the analysis. However, some methods have been proposed to determine these parameters by means of inverse analysis using neural networks [2,3].

The stochastic process is introduced at the material local scale considering that cracks are created within the concrete with different energy dissipation depending on the spatial distribution of constituents and initial defects. The local material behavior in concrete is assumed to be a perfect elastic brittle behavior. This the random distribution of local cracking energies can be replaced by a random distribution of local strengths.

Fracturing is modeled by 3D interface elements that are generated in a previously defined region within the mesh. The interface elements are triangular base prisms connecting adjacent faces of neighboring tetrahedra. These elements simulate crack opening through relative displacements between the triangular faces.

The model is based on the assumption the some particularities of the cracking behavior of concrete, such as strain softening, cracking evolution, and size-effects are derived from the heterogeneous characteristics of the material.

The code achieved a very good level for both parallel performance and vetorization. The most demanding routines, which implement the matrix-vector- multiply computational kernel for the interface and tetrahedral elements, are "fully" parallelized and responsible for over 80 emphasizes the suitability of the implemented code on the parallel-vector machine, CRAY T90 for 2 CPU's, which presented a flop rate of 614 Mflop/s and a parallel speed-up of 3.8 for 4 CPU's.

Extensive use of element-by-element techniques within the computational kernels comprised in the iterative solution drivers provided a natural way for achieving high Flop rates and good parallel speed-up's. Furthermore, element-by-element techniques avoid completely the formation and handling of large sparse matrices. Therefore, the computational strategies presented herein provide a natural way to deal with more complex scenarios, particularly those involving three-dimensional problems.

1

P. Rossi and F.-J. Ulm, "Size effects in the biaxial behavior of concrete: physical mechanisms and modeling", Materials and Structures, 30 (198), 210-216, 1997. doi:10.1007/BF02486178

2

C.N.M. Paz, "Development and Implementation Probabilistic Model for 2D and 3D Discrete Cracking Concrete in Parallel Computing", D.Sc. Thesis, COPPE/UFRJ, Dept, of Civil Engineering, COPPE/UFRJ, Federal University of Rio de Janeiro, Brazil, [in Portuguese] 2000.

3

E.M.R. Fairbairn, N.F.F. Ebecken, C.N.M. Paz, and F-J. Ulm, "Determination of probabilistic parameters of concrete: solving the inverse problem by using artificial neural networks", Computers and Structures, 78, 497-503, 2000. doi:10.1016/S0045-7949(00)00073-0

4

A.L.G.A. Coutinho, M.A.D. Martins, J.L.D. Alves, L.Landau, and A. Moraes, "Edge-based finite element techniques for nonlinear solid mechanics problems", Int. J. for Numerical Methods in Engineering, 50 (9), 2050-2068, 2001. doi:10.1002/nme.107

5

T.J.R. Hughes, R.M. Ferenez, J.O. Hallquist, "Large-Scale Vectorized Implicit Calculation in Solid Mechanics on a CRAY X-MP/48 Utilizing EBE Preconditionated Conjugate Gradients", Computer Methods in applied Mechanics and Engineering, 61, 2115-248, 1987. doi:10.1016/0045-7825(87)90005-3

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