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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4. Numerical Results A sensitivity analysis was conducted for understanding the influence of the number of layers used for the surrogate model onto the engineering constants. Representative volume elements (RVEs) including one particle embedded in the matrix material with ideal contact between particle and the matrix, were created. The size of the particle in RVE was chosen in such matter, that the particle content in the RVE corresponds to the volume fraction of the particles in the composite and was set at ℎ � = 7% in the presented studies. The RVE for the accurate particles with shape undulations � � , � � � (see Fig. 2) as well as for the corresponding surrogate models were created. The RVE was meshed using tetrahedron elements of linear order and periodic boundary conditions were applied as described in detail in Happ and Piat (2022). The following FEM calculation was conducted with the ABAQUS CAE 2021 software package (Dassault Systèmes (2023)). The results obtained from the method proposed by Reuss for the individual layer regarding the and were used as starting point for the subsequent PSO optimization procedure. The parameters of the PSO are presented in Table 1. Social and cognitive coefficients were set to the same value to not bias a specific behavior of the agents. The inertia constant was selected so that the results converge after exploration of the solution space. The PSO algorithm was used to determine the values of the engineering constants for each individual layer. The numerical value for each parameter of the corresponding surrogate layer was optimized until the effective material behavior of the composite material was close to the material properties of the corresponding accurate particle. The calculations are provided for the surrogate and accurate particles with the undulations � � , � � � . Table 1. Parameters used for the PSO algorithm. Coefficients Variable Value Agent Swarm Size 10 Iterations 10 Cognitive Coefficient � 1 Social Coefficient � 1 Inertia Constant 0.75 An agent swarm size of 10 agents was chosen, which is small in comparison to the traditionally recommended PSO size of 20-50 (see Piotrowski et al. (2020)). The reasoning for an agent swarm size of 10 is that for every agent of the swarm a FEM simulation is spawned, which makes higher agent counts computationally demanding. Cognitive and social coefficients were set to 1 to not bias the agent towards a “Social-only” or “Cognitive-only” (Ozcan and Mohan (1999)) behavior. The inertia constant was chosen as 0.75, which resulted from previous parameter sensitivity studies to achieve a convergence of the agent swarm while also allowing enough exploration of the search space throughout the optimization process. The FEM analysis was conducted for RVEs containing particles obtained with the accurate particles and the corresponding surrogate models and elastic properties were calculated. The obtained ��������� modulus of the surrogates were then normalized with the � of the corresponding accurate particle. This procedure was repeated for the surrogate models considering different numbers of layers. The results are displayed in the Fig. 5 and Fig. 6. Fig. 5 (left) displayed the results of the study conducted for the particle shape � � which is then applied to different spheres of base radius = 0.6, 1.1, 1.6 . The results in Fig. 5 show that the normalized Young’s modulus obtained by using the Reuss or Voigt scheme never achieves the same numeric values compared to the Young’s modulus obtained for the accurate particles. The study shows that the number of layers used to create the surrogate models has a small influence on the obtained effective modulus. An increase in accuracy is achieved by increasing the number of layers up to 5 layers, but a further increase in numbers of layers has no effect onto the obtained effective modulus. The same trend can also be seen in Fig. 6, where the results for the particle with the surface shape � � � are displayed. The Reuss and Voigt bounds lack in accuracy in comparison to the heuristic PSO method. The deviation between the modulus obtained with the use of Voigt and Reuss is the greatest for undulations applied to a sphere of radius 0.6. The deviation shrinks with the