PSI - Issue 33

Boris Fedulov et al. / Procedia Structural Integrity 33 (2021) 843–849 Fedulov B.N., Fedorenko A.N. / Structural Integrity Procedia 00 (2019) 000–000

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material. There are some works describing algorithms for this type of optimization [8-9], but subject is relatively new, thus there are no a lot of examples of optimization described in the literature. The most popular type of optimization problem is the minimization of compliance of optimization solid with restriction to maximum solid mass [1-5]. 2. Problem statement To formalize orientation of the material it is possible to introduce two parameters, which are angles of coordinates transformation using rotation around axis ( , ). Initial coordinate system, where =0 and � 0 , coincides with principle axes of anisotropy. Thus, after subdivision of optimization volume ٠into small subvolumes ٠� , which usually associated with a finite element volumes, each ٠� or element has three parameters for optimization � � , � , � � . Therefore, constitutive relations of the material can be formulated as following: �� � ���� � � �� , ���� � � � � � � ��� � �, � � ��� � � � � �� � � � �� � � � � � � � � � � � � �� � � � � �� � � � � � � � � � � � � ,

� ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 1/ � �� �� / � �� �� / � 0 0 0 �� �� / � 1/ � �� �� / � 0 0 0 �� �� / � �� �� / � 1/ � 0 0 0 0 0 0 1/ �� 0 0 0 0 0 0 1/ �� 0 0 0 0 0 0 1/ �� ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ �� , ⎝ ⎜ ⎜⎛ cos � � sin � � 0 � s 0 0 0 0 1 0 0 0 � s � cos � s � cos � 0 cos � � s � 0 0 0 0 cos 0 0 0 0 � s cos ⎟ ⎠ ⎟⎞ , � � � ⎜ ⎝ ⎜⎛ cos � � 0 1 0 0 0 0 sin � � 0 � s 0 0 0 0 cos � s cos 0 s cos 0 s � 0 0 0 � s 0 cos ⎟ ⎠ ⎟⎞ .

where

(1)

� � �

Constant p is the penalization factor which usually taken as 3.0. Matrix � � and � � are derived from sequential coordinate system transformations using rotation around Z axis with angle , and then around Y’ axis with angle . Where Y’ is the modified axis Y after first transformation. Eventually the problem of compliance minimization has the form: m � � ,� � ,� � � � �� �� under conditions of: