Keywords: nonlinear programming, symbolic-numerical algorithm, hybrid genetic algorithm.

The material parameter identification procedure for advanced yield criteria is developed. The procedure proposed include two basic steps. First the dimension of the posed problem is reduced by the use of theoretical analysis and symbolic calculation. In the case of the plane stress yield criterion BBC2003 [1] the dimensions of the problem are reduced from 6 to 2. This step includes the analysis of the nonlinear algebraic system, selection of variables for elimination, determining the order of elimination, exchange of variables, factorizations, simplifications, recursive substitutions performed by the use of symbolic calculation. As a result the remaining nonlinear equations are higher order algebraic equations in comparison with equations in the initial system. In the case of BBC2003 there remains two nonlinear equations only, which makes three-dimensional visualization of the corresponding optimization problem available (two variables and objective function). The second step of the solution procedure proposed contains numerical solution of the nonlinear algebraic system derived in the first step. The following numerical methods are considered for solving the algebraic system: a Newton solver and an error minimization function based algorithm. In the latter case the nonlinear optimal design problem is formulated, which allows over-constraining to be considered and is preferred in the current study. The error minimization function based algorithm is also used for 'validation' of the new yield criteria. In [1] an error function is defined by means of a Gaussian square of error, the steepest descent and downhill-simplex methods are employed, respectively. In the current study the error minimization problem is formulated and solved by use of a hybrid genetic algorithm. The global and local search for the minimum of the error function is performed by the use of real-coded genetic algorithm and gradient method, respectively [2].

An analysis of the nonlinear system, covering the material parameter identification problem considered, has been utilized for the yield criterion BBC2003. However, the results of the analysis hold good for more general cases (similar yield criteria: BBC2002, BBC2000). Some advantages of the approach proposed are:

• the convergence to global extreme can be expected (although not guaranteed);
• material parameter identification problems, where the number of unknown parameters is not equal to the number of given constraints, are covered (error minimization algorithm);
• higher accuracy and speed in comparison with the standard genetic algorithm.

The algorithm proposed is tested on a 6000 series aluminium alloy sheet AA6181-T4. The results obtained are found to be in good agreement with the results given in [1]. The solutions corresponding to global extreme coincide with the solutions achieved in [1] using a Newton solver.

1

D. Banabic, H. Aretz, D.S. Comsa, L. Paraianu, "An improved analytical description of orthotropy in metallic sheet", Int. J. Plasticity, 21, 493-512, 2005. doi:10.1016/j.ijplas.2004.04.003

2

M. Pohlak, J. Majak, K. Karjust, R. Küttner, "Multicriteria optimization of large composite parts", Composite Structures, 92(9),2146-2152, 2010. doi:10.1016/j.compstruct.2009.09.039

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