Keywords: microplane model, inverse analysis, approximations, neural networks, evolutionary algorithms, stochastic sensitivity analysis.

The problem of an inverse analysis appears in many engineering tasks. Generally speaking, the aim of an inverse analysis is to rediscover unknown inputs from the known outputs. In common engineering applications, a goal is to determine original conditions and properties from physical experiments or, equivalently, to find a set of parameters for a numerical model describing the experiment. Therefore, the existence of such a numerical model is assumed in this work and the task is to find parameters of the model to match the outputs to the experimental data.

In overall, there are two main philosophies to the solution of this problem. A forward (classical) mode-direction is based on the definition of an error function of the difference between the outputs of the model and the experimental measurements. A solution comes with the minimum of this function. The main advantage of this approach is that the forward mode is general in all possible aspects and is able to find an appropriate solution if one exists. This statement is confirmed with special cases like:

(a)

A problem with the same values of outputs for different inputs, i.e. the existence of several global optima. This case leads to a multi-modal optimization [1] but is solvable by an appropriate modification of an optimization algorithm.

(b)

There are different outputs for one input. This is the case of stochastic and probability calculations as well as experiments burdened with noise or errors. This obstacle can be tackled for example by the introduction of stochastic parameters for the outputs.

(c)

There is more than one experiment for the model. This task can be handled as a multi-objective optimization problem [2].

The biggest disadvantage of the forward mode is the need for a great number of error function evaluations, especially when employing derivative-free optimization algorithms for error minimization. This problem can be managed by two approaches: the first one is based on parallel decomposition and parallel implementation, the second one employs a computationally inexpensive approximation or interpolation methods.

The second philosophy, an inverse mode, assumes existence of an inverse relationship between outputs and inputs. If such relationship is constructed, then the retrieval of desired inputs is a matter of seconds. This is of a great value especially for repeated identification of one model.

On the contrary, the main disadvantage is an extremely demanding search for the inverse relationship. Further obstacles are the existence problems for the whole search domain and inability to solve the first case (a) mentioned above. The second case (b) can be handled by introducing stochastic parameters, case (c) can be solved by sequential, hierarchical or iterative processes. Nowadays, artificial neural networks [3, 4] are commonly used due to their ability to approximate complex non-linear functions and their straightforward implementation and utilization.

Both philosophies are introduced in the paper and thoroughly discussed. As an example, the identification of parameters for the microplane material model for concrete is presented. The forward mode is shown to lead to a multi-modal parallel optimization problem. The larger part of the work is devoted to the inverse mode employing a multi-layered neural network along with stochastic sensitivity analysis. We examine three specific experimental tests in detail: uniaxial compression, a hydrostatic test and a triaxial test.

1

S.W. Mahfoud, "Niching methods for genetic algorithms", PhD thesis, University of Illinois at Urbana-Champaign, Urbana, IL, USA, 1995.

2

C.A.C. Coello, "List of references on evolutionary multi objective optimization", URL

3

S. Haykin, "Neural Networks: A Comprehensive Foundation", Prentice Hall, 2nd edition, 1998.

4

Ch. Bishop, "Neural Networks for Pattern Recognition", Clarendon Press, Oxford, 1995.

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