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Efficient FGM optimization with a novel design space and DeepONet

Piyush Agrawal, Ihina Mahajan, Shivam Choubey, Manish Agrawal

TL;DR

The paper tackles thermoelastic optimization of FGMs by introducing a novel smooth-profile design-space generator, a DL-based surrogate for maximum effective stress, and a DeepONet surrogate for point-wise temperature fields, all integrated within a genetic algorithm. The design space extends beyond conventional power-law distributions, enabling smooth gradations and the inclusion of thermal constraints through a hybrid fitness that relies on DL predictions and FEM for limiting cases. DeepONet provides accurate temperature-field predictions, allowing efficient enforcement of thermal limits, while the overall DL-FEM-GA pipeline delivers accurate, computationally efficient optimization across multiple numerical examples. Results show substantial reductions in peak stress or ceramic content under thermoelastic constraints, highlighting the framework’s potential for practical FGM design in aerospace and energy applications.

Abstract

This manuscript proposes an optimization framework to find the tailor-made functionally graded material (FGM) profiles for thermoelastic applications. This optimization framework consists of (1) a random profile generation scheme, (2) deep learning (DL) based surrogate models for the prediction of thermal and structural quantities, and (3) a genetic algorithm (GA). From the proposed random profile generation scheme, we strive for a generic design space that does not contain impractical designs, i.e., profiles with sharp gradations. We also show that the power law is a strict subset of the proposed design space. We use a dense neural network-based surrogate model for the prediction of maximum stress, while the deep neural operator DeepONet is used for the prediction of the thermal field. The point-wise effective prediction of the thermal field enables us to implement the constraint that the metallic content of the FGM remains within a specified limit. The integration of the profile generation scheme and DL-based surrogate models with GA provides us with an efficient optimization scheme. The efficacy of the proposed framework is demonstrated through various numerical examples.

Efficient FGM optimization with a novel design space and DeepONet

TL;DR

The paper tackles thermoelastic optimization of FGMs by introducing a novel smooth-profile design-space generator, a DL-based surrogate for maximum effective stress, and a DeepONet surrogate for point-wise temperature fields, all integrated within a genetic algorithm. The design space extends beyond conventional power-law distributions, enabling smooth gradations and the inclusion of thermal constraints through a hybrid fitness that relies on DL predictions and FEM for limiting cases. DeepONet provides accurate temperature-field predictions, allowing efficient enforcement of thermal limits, while the overall DL-FEM-GA pipeline delivers accurate, computationally efficient optimization across multiple numerical examples. Results show substantial reductions in peak stress or ceramic content under thermoelastic constraints, highlighting the framework’s potential for practical FGM design in aerospace and energy applications.

Abstract

This manuscript proposes an optimization framework to find the tailor-made functionally graded material (FGM) profiles for thermoelastic applications. This optimization framework consists of (1) a random profile generation scheme, (2) deep learning (DL) based surrogate models for the prediction of thermal and structural quantities, and (3) a genetic algorithm (GA). From the proposed random profile generation scheme, we strive for a generic design space that does not contain impractical designs, i.e., profiles with sharp gradations. We also show that the power law is a strict subset of the proposed design space. We use a dense neural network-based surrogate model for the prediction of maximum stress, while the deep neural operator DeepONet is used for the prediction of the thermal field. The point-wise effective prediction of the thermal field enables us to implement the constraint that the metallic content of the FGM remains within a specified limit. The integration of the profile generation scheme and DL-based surrogate models with GA provides us with an efficient optimization scheme. The efficacy of the proposed framework is demonstrated through various numerical examples.
Paper Structure (19 sections, 34 equations, 28 figures, 4 tables)

This paper contains 19 sections, 34 equations, 28 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Generation of the FGM design space
  3. Generation of 1D profile
  4. Generation of 2D profile
  5. Finite element analysis of thermoelasticity problems
  6. Governing equation
  7. Finite element formulation
  8. Deep learning based surrogate models
  9. Deep neural network implementation
  10. DeepONet implementation
  11. Genetic algorithm
  12. Numerical examples
  13. Problem 1: $Ni/Al_2O_3$ FGM plate uniformly cooled from a high temperature
  14. Problem 2: $Al/ZrO_2$ FGM plate subjected to sinusoidal temperature loading
  15. Case 1
  16. ...and 4 more sections

Figures (28)

  • Figure 1: FGM profile with piece-wise linear interpolation.
  • Figure 2: Flowchart of proposed FGM profile generation scheme.
  • Figure 3: Comparison of 1D FGM profiles from (a) proposed scheme, (b) power law.
  • Figure 4: Sample profiles from the proposed profile generation scheme, that can not be represented by power law.
  • Figure 5: Sample two-dimensional FGM profiles obtained using profile generation scheme.
  • ...and 23 more figures