PSI - Issue 18

4

Boris Fedulov et al. / Procedia Structural Integrity 18 (2019) 399–405 Boris Fedulov and Alexey Fedorenko / Structural Integrity Procedia 00 (2019) 000–000

402

� � � 1 � � � � � � �� � �� �� � � � � 1� � � � .

subjected to constrains:

(5)

Consider gradient of the objective function for finite volume used in topology optimization method: � ���� � �� �� � � � ���� � � � �� �� (6) where � - damage of the material at volume Ω � (undamaged material corresponds to � 1 ), index e is associated with volume and damage of one element in FEM analysis. Each tensor component �� ���� �� � can be calculated directly using matrix of (4). The SIMP optimization algorithm described in [19] has been implemented into Abaqus software using special user subroutines. The iterative procedure for finding of element damage value �� at step K has the form ���� � � ��1 � � �� , ��� � �� � �� � ��1 � � �� , ��� � ��1 � � �� , 1� ��1 � � �� , 1� � �� � �� �� � �� (7) where � � Λ �� � � �� ���� �� � � � �� �� , and Λ � is a multiplier that satisfies the total damage energy constraint and is found by a bi-sectioning algorithm. The variable is a tuning parameter with typical value of 0.5 and is a move limit with reasonable value of 0.2. 4. Comparison with topology optimization algorithm As it was mentioned above, that the presented method is very close to topology optimization algorithm based on material stiffness penalization. Table 1 shows comparison of main points of presented and conventional methods.

Table 1. An example of a table.

Presented method

Topology optimization

Degradation of the material based on damage parameter Strain energy: 1/2 � ���� � �� �� Strain energy: 1/2 � ���� � �� �� ���,�,�� �1/2 � ���� �� �� � ���,�,�� �1/2 � ���� �� �� � Reduction of the stiffness of the material based on density parameter

Material transformation

Objective function

� � � 1 � � � 1 � 1� Ω ���� � � – based on eq.(4) � �� ���� �� ��� Ω � �� �� Ω Ω � � � � 2

� � � 1 ٠٠� � ���� � � � ���� � �� ���� �� ��� ٠� �� �� �

Max/Min

Range constrain “Mass” constrain

Stiffness reduction

Gradient

Ω

It is possible to see that the main difference in material stiffness formulation in dependence on variation parameters and . This cause essential difference in gradient formulation, which are local for both methods but have different final analytical formulation. Nevertheless, keeping gradient formulation in general form as in Table 1, algorithm for searching of maximum of energy is completely the same as (7), which is performed for topology optimization in [19].