Differentiable FFT Homogenization for CTE-Driven Design of Thermoelastic Metamaterials with Double-Negative Response

Abstract

We present a differentiable topology-optimization framework for the inverse design of three-phase (void plus two solids) periodic thermoelastic metamaterials, with the primary objective of achieving a negative mean effective coefficient of thermal expansion (CTE). The framework combines a matrix-free FFT-based conjugate-gradient (FFT-CG) homogenization solver with JAX-native automatic differentiation: the equilibrium solve is wrapped in an implicit linear-solve boundary, so reverse-mode gradients follow from adjoint solves without hand-coding the full sensitivity, and a staged continuation of the SIMP exponent and projection sharpness guides multiphase designs from a gray field to a near-binary layout. The solver is verified against homogeneous limits, symmetry and positive-definiteness checks,finite-difference gradient comparisons, and an independent finite-element reference. For a $100 \times 100$  plane-stress cell, CTE-driven optimization produces a design with a negative mean in-plane homogenized CTE at feasible modulus floors. Postprocessing of the same homogenized stiffness shows that the CTE-optimized design also has a negative effective Poisson ratio; this auxetic response is an observed outcome rather than an optimization target. Five of the six screened initializations yield designs with negative mean in-plane CTE, and a $40 \times 40 \times 40$hree-dimensional run reaches a negative mean CTE as budget-limited feasibility evidence.