PSI - Issue 52

Pascal Alexander Happ et al. / Procedia Structural Integrity 52 (2024) 401–409 Author name / Structural Integrity Procedia 00 (2019) 000–000

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5

Algorithm 1 . Surrogate Creation

Algorithm 2 . Voigt Approximation 1:

Calculate the volume ratio of the matrix material in the shell ℎ � = � ����� �� ����� � ����� Calculate the volume ratio of the layer material in the shell ℎ � =1−ℎ � Calculate the Young’s modulus of the shell ����� =ℎ � ������ +ℎ � �������� Calculate the Poisson’s ratio of the shell ����� =ℎ � ������ +ℎ � �������� Calculate the volume ratio of the matrix material in the shell ℎ � = � ����� �� ����� � ����� Calculate the volume ratio of the layer material in the shell ℎ � =1−ℎ � Calculate the Young’s modulus of the shell ����� =� � � � ������ + � � � �������� � �� Calculate the Poisson’s ratio of the shell ����� =� � � � ������ + � � � �������� � ��

1: 2:

Assign number of layers

Evaluate the inner most point of the particle surface Evaluate the outer most point of the particle surface Calculate inner shell radius ����� Calculate outer shell radius ����� Cut the particle according to the ����� and ����� Create shell with inner and outer shall radius Evaluate volume of shell ����� Evaluate volume of particle layer ����� Calculate Young’ modulus ����� and Poisson’s ratio ����� using Algorithm 2 or Algorithm 3 Assign ����� and ����� to each shell end for for each layer do

2:

3:

3:

4: 5: 6: 7:

4:

Algorithm 3 . Reuss Approximation

8:

1:

9:

10:

2:

3:

11:

4:

12: 13:

Fig. 4: Algorithms for surrogate creation: Algorithm 1 describes the main program of the surrogate creations and Algorithm 2 and 3 describe calculations using Voigt and Reuss bounds.

3.2 Optimization of the elastic properties of the layers To meliorate the results of the Voigt and Reuss bounds of elastic properties for each surrogate layer (with exception of the starting layer consisting of the particle material), the heuristic optimization theory of Particle Swarm Optimization (PSO) as proposed by Eberhart and Kennedy (1995) was chosen. So called agents were spawned in the solution space using the results provided by Reuss as initial positions for the optimization procedure. The agents then swarm the solution space in search of the optimal solution. The equations proposed by Chopard and Tomassini (2018) were used to describe the agents swarm behavior: v � ( + 1) = v � ( ) + � � ( + 1)� � ���� ( ) − � ( )� + � � ( + 1)[ ( ) − � ( )] (1) � ( + 1) = � ( )+v � ( +1) (2) with describing the inertia of the agent, � , � denoting the cognitive and the social coefficient respectively, representing the iteration time, ( ) is the current global best position, r � and � are random numbers from an evenly distributed interval, v � and � are the velocity and the position of the agents at the i th iteration. The PSO algorithm was implemented in Python. The optimization process for the layers was started at the initial position � and elastic properties (Young’s modulus and Poisson’s ratio ) calculated using the Reuss method; this was done to reduce optimization time. A random initial velocity was chosen for each agent. Its position and velocity changes according to the equations (1) and (2) with every iteration. The elastic properties are calculated for every agent’s position and compared with each other and with the elastic properties of the current global best position. ( ) is updated, if one of the new positions of the agent is achieving an effective value that is closer to the values obtained from the calculations for accurate particles. These agents move in the ℝ � , where = 2 ∗ , with representing the number of layers used for the surrogate model and for each layer two elastic material properties and are used.