FGM optimization in complex domains using Gaussian process regression based profile generation algorithm

TL;DR

The paper tackles the design of functionally graded materials over complex geometries by introducing a Gaussian Process Regression–based profile generation scheme that yields smooth, boundary-constrained volume-fraction profiles. This design space is coupled with a genetic algorithm, where a projection operator preserves profile smoothness during crossover and a Gaussian mutation maintains consistency with the GPR space. The framework is demonstrated on thermoelastic FGMs of Al/ZrO2 in several geometries, showing significant stress reduction and effective adherence to boundary constraints, including constrained optimization scenarios. The approach enables efficient exploration of irregular domains and offers a flexible, generalizable tool for FGM design beyond standard geometries, with potential application to broader FGM problems.

Abstract

This manuscript addresses the challenge of designing functionally graded materials (FGMs) for arbitrary-shaped domains. Towards this goal, the present work proposes a generic volume fraction profile generation algorithm based on Gaussian Process Regression (GPR). The proposed algorithm can handle complex-shaped domains and generate smooth FGM profiles while adhering to the specified volume fraction values at boundaries/part of boundaries. The resulting design space from GPR comprises diverse profiles, enhancing the potential for discovering optimal configurations. Further, the algorithm allows the user to control the smoothness of the underlying profiles and the size of the design space through a length scale parameter. Further, the proposed profile generation scheme is coupled with the genetic algorithm to find the optimum FGM profiles for a given application. To make the genetic algorithm consistent with the GPR profile generation scheme, the standard simulated binary crossover operator in the genetic algorithm has been modified with a projection operator. We present numerous thermoelastic optimization examples to demonstrate the efficacy of the proposed profile generation algorithm and optimization framework.

Figures (19)

• Figure 1: Sample domain discretized into the elements and nodes. The volume fraction at nodes are generated from the multivariate Gaussian distribution.
• Figure 2: Posterior samples of FGM profiles obtained using GPR, satisfying pure metal $(V_{c} = 0)$ and pure ceramic $(V_{c} = 1)$ boundary conditions: (a) square plate with circular hole, having pure metal at the bottom and top edge, and pure ceramic at the periphery of hole, (b) ellipse geometry with two circular holes, having pure metal at the periphery of ellipse and pure ceramic at the periphery of holes, and (c) half ellipse geometry with two circular holes, having pure metal at the outer curved surface and pure ceramic at the periphery of holes.
• Figure 3: Comparison of FGM profiles before ($P_1$, $P_2$) and after crossover ($C_1$, $C_2$) operation, highlighting the reduction in the smoothness of the FGM design.
• Figure 4: Projection of the FGM profiles obtained after the crossover operation into a smoother design space.
• Figure 5: Eigenvectors corresponding to the (a) $5^{th}$, (b) $50^{th}$, (c) $100^{th}$, and (d) $200^{th}$ largest eigenvalue of the covariance matrix. This illustrates that the smoother eigenvector profiles are obtained for the higher eigenvalues.