Keywords: continuum damage mechanics, polynomial wavelets, hybrid-mixed stress models, finite elements.

This paper presents and discusses a hybrid-mixed stress finite element model based on the use of polynomial wavelets for the physically non-linear analysis of concrete structures. The effective stress and the displacement fields in the domain of each element and the displacements on the static boundary are independently approximated. None of the fundamental relations is enforced a priori and all field equations are enforced in a weighted residual form, ensuring that the discrete numerical model embodies all the relevant properties of the continuum it represents. The Mazars' isotropic model is adopted and a non-local integral formulation where the equivalent strain is taken as the non-local variable is considered [1,2].

As none of the fundamental equations is locally enforced a priori, the hybrid-mixed stress formulation enables the use of a wide range of functions. In the numerical model reported in this communication, all approximations are defined using complete sets of polynomial wavelets [3].

Wavelet systems present several properties that make their application very attractive in the context of hybrid-mixed stress models. As a result of its hierarchical nature, they are well suited for the development of highly efficient adaptive procedures, since scaling functions capture the average information and wavelets are responsible for the representation of the details. As a result of this fact, it is possible to enrich the approximation in the neighbourhood of singularities without remeshing, simply by adding to the basis wavelets at higher levels of resolution. The polynomial wavelet system has been selected due to some specific properties it presents. On the one hand, the functions are orthogonal, which is an important issue when implementing hybrid-mixed stress elements as it ensures high levels of sparsity in the global governing system. On the other hand, the polynomial wavelet basis is defined through linear combinations of Legendre polynomials. This fact enables the use of closed-form solutions for the computation of the integrations involved in the definition of all linear structural operators. Numerical integration schemes can be avoided, with clear advantages both in terms of accuracy and numerical performance.

Because the hybrid-mixed stress model used here is built on a naturally hierarchical basis, it can be implemented using coarse meshes of macro-elements, where the refinement is achieved by increasing the degree of the approximation. The numerical models are both incremental and iterative and are solved using a modified Newton-Raphson method that uses the secant matrix. A set of classical benchmark tests is chosen to illustrate the use of such models and to assess and compare their numerical performance.

1

C.M. Silva, L.M.S.S. Castro, "Hybrid-Mixed Stress Model for the Non-Linear Analysis of Concrete Structures", Computers & Structures, 83, 2381-2394, 2005. doi:10.1016/j.compstruc.2005.03.031

2

C.M. Silva, L.M.S.S. Castro, "Hybrid-mixed stress formulation using continuum damage models", Communications in Numerical Methods in Engineering, 22, 605-617, 2006. doi:10.1002/cnm.837

3

L.M.S.S. Castro, "Polynomial wavelets in hybrid-mixed stress finite element models", Communications in Numerical Methods in Engineering,26(10), 1293-1312, 2009. doi:10.1002/cnm.1215

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