Multi-Material Topology Optimization

with General Convex Polytopal Constraints

Dohyun Kim

joint work with P. Gangl, B. Keith, B. S. Lazarov, T. M. Surowiec

Division of Applied Mathematics, Brown University

Engineering Conference · 2026

Topology optimization — one material, then many

Where should material go?

$$ \min_{\rho}\ J(\rho)=\int_\Omega f\cdot u(\rho)\,\dd x \quad \text{s.t.}\ \ \rho(x)\in[0,1],\ \ \int_\Omega \rho\,\dd x = V|\Omega| $$

pointwise bound + one volume budget (linear)
gradients: one adjoint solve

…and which material?

$n$ phases $\{\mathsf{C}_i\}$ — stiff / light / cheap, or fiber angles
one-of-$n$ choice at every point,
per-phase budgets (still linear)
relax the discrete choice

SiMPL in one slide — mirror descent

Projected gradient → mirror descent (two edits)

              given $\rho^0$;  for $k=0,1,2,\dots$
              $\rho^k \leftarrow \mathrm{sigmoid}(\psi^k)$  // latent → primal
              state + adjoint solve → $g^k=\nabla J(\rho^k)$
              $\rho^{k+1} \leftarrow \mathrm{proj}_{[0,1]}\!\big(\rho^k - \alpha\, g^k\big)$
              $\psi^{k+1} \leftarrow \psi^k - \alpha_k\, g^k$
            

Each step solves a subproblem

$$ \rho^{k+1} \;=\; \argmin_{\rho}\ \langle g^k,\rho\rangle \;+\; \tfrac{1}{\alpha_k}\,D_R(\rho,\rho^k) $$

first-order optimality:

$\nabla R(\rho^{k+1}) \;=\; \nabla R(\rho^k) \;-\; \alpha_k\, g^k$

latent variable $\psi \coloneqq \RED{\nabla R}(\rho)$:

$\BLUE{\psi^{k+1}} \;=\; \BLUE{\psi^{k}} \;-\; \alpha_k\, g^k$

inverse relation:

$\rho \;=\; \BLUE{\nabla R^*}(\psi) \;=\; \mathrm{sigmoid}(\psi)$

no projection
$\psi\to\pm\infty$ ⇒ $\rho\to\{0,1\}$ exponentially
$\alpha_k \approx \kappa_{\text{geom}} / \kappa_{\text{obj}}$
secant guess, then backtrack
mesh-independent iteration counts

Keith, Kim, Lazarov, Surowiec — SIAM J. Optim. 2025; Struct. Multidisc. Optim. 2025 (SiMPL).

What convergence looks like

step size $\alpha_k$ grows — $|\psi|\to\infty$, the design binarizes

Curves for $h = 1/64,\ 1/128,\ 1/256$ lie on top of each other — mesh-independent iteration counts, measured.

Multi-material: is the simplex the natural choice?

Conventional: lift to fractions

weights $\lambda\in\Delta^{n-1}$, design $\rho=\sum_i\lambda_i v_i$
$n$ discrete angles ⇒ $n$ fields for a 2-D quantity

materials = vertices of a polytope $P=\mathrm{conv}\{v_i\}$

Orthotropic: the angles already live in 2-D

double-angle plane: vertex $k$ sits at $2\theta_k$

more angles = finer polygon, still dim 2

Entropy handles Geometry (Hidden in $D_\RED{R}$)

A Legendre entropy $R:C\to \mathbb{R}^n$ gives a bijection $\nabla R^*:\mathbb{R}^d\to \mathrm{int}\,C$

box — Fermi–Dirac ⇒ sigmoid

ball — Hellinger

simplex — Gibbs ⇒ softmax

polytope — ? (this talk)

Key idea

A Legendre entropy for a general polytope $P=\mathrm{conv}\{v_1,\dots,v_n\}\subset \mathbb{R}^n$, with closed-form $\nabla R^*$.

The polytopal entropy

Collect the vertices in $V=[v_1\ \cdots\ v_n]$ and push the Gibbs entropy through $V$:

$$ R^*(\psi) \;=\; \log\!\sum_{k} \exp\!\big(v_k^{\!\top}\psi\big) \qquad\Longrightarrow\qquad \BLUE{\rho \;=\; \nabla R^*(\psi) \;=\; V\,\mathrm{softmax}(V^{\!\top}\psi)} $$

$R$ = Gibbs entropy inf-projected onto $P$
closed-form $\nabla R^*$ and Hessian — cheap at every quadrature point
recovers the classics: box → sigmoid, simplex → softmax

Octagon-base polytope in $\mathbb{R}^3$
void + orthotropic + isotropic

Intuition

$|\psi|\to+\infty \;\iff \lambda=\text{softmax}(V^T\psi)\to e_i\;\iff\; \rho\to\text{vertex}$

The algorithm — one line changes

SiMPL, multi-material edition

          given $\psi^0$;  for $k=0,1,2,\dots$
          $\rho^k \leftarrow$ $\mathrm{sigmoid}(\psi^k)$$\BLUE{V\,\mathrm{softmax}(V^{\!\top}\psi^k)}$  // the only change
          state + adjoint solve → $g^k=\nabla J(\rho^k)$
          $\alpha_k\leftarrow$  secant curvature guess
          $\psi^{k+1} \leftarrow \psi^k - \alpha_k\, g^k - \mu_k$, backtrack $\alpha_k$ until $J$ decreases
          // $\mu_k$: a small dual solve for the global linear constraints
        

Everything else is inherited

Adaptive step size, monotone backtracking, KKT gap
— from single-material SiMPL.

first-order, gradients only
feasible at every iterate
$\psi(x)\in \mathbb{R}^d$, not $\mathbb{R}^n$

Results — 2D multi-material

three materials, per-phase budgets 32 it

Results — 3D and orthotropic compliance

orthotropic cantilever, 8 discrete fiber angles — color = angle 38 it

Continuous angles — the limit of the polygon

cone = latent ball mapped by $\nabla R^*$:
rim = solid at any $\theta$, apex = void

converged design — continuous fiber angle (color), volume fraction $0.5$, $h=1/256$

mesh size $h$	control unknowns	iterations	compliance	KKT measure
1/64	36,864	28	$2.76\times10^{-3}$	$9.1\times10^{-5}$
1/128	147,456	30	$2.77\times10^{-3}$	$9.9\times10^{-5}$
1/256	589,824	31	$2.77\times10^{-3}$	$9.7\times10^{-5}$

Permanent Magnet Placement

Maximize magnetic flux through the airgap:

$$ \max\ \int_\Gamma \operatorname{curl} u\cdot \mathbf{n}\,\dd s $$

Red = permanent magnet direction, Black = iron

optimized rotor: material + orientation per point 38 it

Conclusion

SiMPL for physics-encoded polytopal constraints
— same algorithm, different pointwise constraint set.

the loop never changes — only $\nabla R^*$ is swapped
box, simplex, polytope, orientation polygon — one framework, feasible by construction
mesh-independent iterations, first-order cost

Ongoing work

general global constraints (min volume subject to critical compliance)(WIP)
robust dual solver (WIP)

Gangl, Keith, Kim, Lazarov, Surowiec — Multi-material SiMPL (2026) · builds on Keith, Kim, Lazarov, Surowiec — SIAM J. Optim. 2025 & Struct. Multidisc. Optim. 2025.

Thank you

Dohyun Kim · dohyun_kim@brown.edu

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