Robust Design Optimization with Limited Data for Char Combustion

TL;DR

This paper tackles robust design optimization of a char combustion process under limited data, where high-fidelity simulations are expensive. It combines transformed input variables with a polynomial dimensional decomposition (PDD) surrogate trained via sparsity-promoting sD-MORPH regression and a single-pass training strategy to avoid repeated data generation during optimization. The approach enables analytical computation of mean and variance of the quantity of interest, the total thermal energy $Q$, and employs a derivative-free Nelder-Mead solver to locate Pareto-optimal designs that balance energy yield and its fluctuations. Empirical results on a PIC-based char combustion model show that sD-MORPH yields higher accuracy (e.g., $R^2$ up to 0.997) and faster convergence, with Pareto solutions indicating robust guidance on operating parameters such as air inflow and glass bead diameter. The work demonstrates a computationally efficient framework for reliability-focused design under uncertainty in complex combustion systems, with potential extensions to higher dimensions and additional objectives or constraints.

Abstract

This work presents a robust design optimization approach for a char combustion process in a limited-data setting, where simulations of the fluid-solid coupled system are computationally expensive. We integrate a polynomial dimensional decomposition (PDD) surrogate model into the design optimization and induce computational efficiency in three key areas. First, we transform the input random variables to have fixed probability measures, which eliminates the need to recalculate the PDD's basis functions associated with these probability quantities. Second, using the limited data available from a physics-based high-fidelity solver, we estimate the PDD coefficients via sparsity-promoting diffeomorphic modulation under observable response preserving homotopy regression. Third, we propose a single-pass surrogate model training that avoids the need to generate new training data and update the PDD coefficients during the derivative-free optimization. The results provide insights for optimizing process parameters to ensure consistently high energy production from char combustion.

Figures (6)

• Figure 1: Fluidized bed furnace: (a) an industrial-scale model of the fluidized bed furnace (left) is scaled down to a lab-scale version (right), which details the geometry and initial conditions with the diameters of glass beads and char particles, the air inlet, and char inlet; (b) in one simulation using the PIC method, we use 2,520 cells to discretize the fluid (left) and the parcel model uses 8,344 parcels to predict solid behavior (right).
• Figure 2: A flow chart for a single-pass surrogate model training process that repeatedly computes PDD for the updated design $\mathbf{r}'$ using the PDD computed at $\mathbf{r}$ with a sD-MORPH.
• Figure 3: Flowchart for the RDO with limited data.
• Figure 4: Change in objective function value with increasing iterations for different combinations of $w_1$ and $w_2$ in \ref{['opt']}, comparison of LASSO-based and sD-MORPH-based design methods over different sample numbers (150 and 200).
• Figure 5: Pareto solutions in the design space $\mathbb{E}[X_2]=d_1$ and $\mathbb{E}[X_3]=d_2$ in comparison of sD-MORPH-and LASSO-based design methods over different sample numbers (150, 170, 190, 200).