Research · 04
Multiphysics & Topology Optimization
How do you design a structure that has to be electromagnetically excellent and mechanically strong at the same time — when improving one usually costs you the other? This research studies the loop that answers that question: adjoint sensitivity for the fields, SIMP for the material.
The idea
Let the algorithm decide where the material goes.
Conventional design starts from a shape and tunes its parameters. Topology optimization starts from a block and asks a different question: given a fixed budget of material, where should it be? The design domain is divided into voxels, each with a density between 0 (void) and 1 (solid), and the optimizer drives those densities toward an answer.
Under the SIMP model — Solid Isotropic Material with Penalization — a penalty factor pushes intermediate densities toward the extremes, so the result is a structure you can actually build rather than a cloud of half-material.
Compliance, plainly
Mechanically, the objective is compliance: how much the structure bends under load. c = ½·UᵀKU, with displacements U solved from KU = F. Lower compliance means a stiffer structure. Raise a voxel's density and its elasticity rises, which raises stiffness, which lowers displacement, which lowers compliance — the chain the optimizer is pulling on.
The trick
Two simulations instead of thousands.
The naive way to find out how each voxel affects antenna gain is to change one, re-run the electromagnetic simulation, and repeat — thousands of simulations for one design.
Adjoint optimization gets the same information from two. A forward simulation energises the antenna and measures the radiated field. An adjoint simulation runs it backwards, injecting a wave from the direction where gain should be maximised. Multiply the forward field by the adjoint field and you have the gradient at every voxel at once: positive means add permittivity there, negative means remove it, near zero means leave it alone.
Permittivity itself is controlled physically — by printing voids of different sizes into each voxel, so more air means lower permittivity. The design becomes manufacturable by construction.
The loop
One iteration of a coupled optimization.
Electromagnetic and mechanical gradients are computed separately, then combined before the voxel densities are updated. The loop runs until gain stops rising and compliance stops falling.
Applications studied
Two structures, one method.
GRIN lenses
A gradient-index lens is made from a single material whose internal structure varies, bending waves to focus an antenna's beam. Designed by adjoint optimization rather than transformation optics, the resulting lenses avoid the anisotropic or extreme-permittivity materials that make a design impossible to fabricate.
Reference: E. B. Whiting et al., "Adjoint Sensitivity Optimization of Three-Dimensional Directivity-Enhancing, Size-Reducing GRIN Lenses," IEEE Antennas and Wireless Propagation Letters, vol. 21, no. 11, pp. 2166–2170, Nov. 2022.
Meta-radomes
A radome protects an antenna from the weather — but it sits directly in the beam. The design problem is genuinely multiphysics: improve gain, remain manufacturable, and survive a 4400 N mechanical load. Electromagnetic and structural objectives are optimized in the same loop rather than one after the other.
Reference: T. Burns, A. Huynh, G. Mackertich-Sengerdy, S. Campbell, D. Werner, "Multiphysics Inverse-Design of Multifunctional Freeform Meta-Radomes," Small Structures, vol. 6, 2025.
Forward and adjoint, instead of one per voxel.
Reported runtime for the coupled multiphysics loop.
Sigmund's classic topology code, and its vectorised successor — 10× to 100× faster.
The mechanical constraint the meta-radome had to survive.
Documentation
The full deck, readable here.
Twenty-five slides covering optimization fundamentals, the adjoint method, SIMP, and a walkthrough of the 99-line MATLAB topology optimization code. Arrow keys navigate.
Also built
An interactive tool for reading the results
Alongside the study I built a finite-element-based topology optimization tool using Sigmund's SIMP method, generating the stiffest possible structure under specified loading conditions, plus an interactive way to visualise the results — so the output of the loop connects to an engineering decision instead of stopping at a field plot. That tool is what became PlateOptimize, carried from research code through to a published application. The work also extended these methods toward thermal systems, minimising thermal compliance to improve heat transfer, and combined thermal, mechanical, fluid and electromagnetic objectives in one multiphysics framework.