General concepts » Shape and topology optimization

Introduction

Shape optimization seeks a domain minimizing an objective

where solves a PDE posed on (the state equation) and is a Lagrange multiplier penalizing volume. The canonical benchmark is compliance minimization: the state is linear elasticity and is the work of the loads, . Rodin was designed around this problem class — this guide explains the moving parts and how the examples assemble them.

Working examples, in increasing sophistication:

The shape derivative

Perturbing the domain along a vector field , , the shape derivative is

The Hadamard structure theorem states that, for smooth data, the derivative concentrates on the boundary and depends only on the normal component:

with a scalar density computed from the state (and, for non-self-adjoint objectives, an adjoint state). For compliance with a traction-free optimizable boundary, .

Two equivalent discrete realizations exist, and the choice matters:

• Boundary (Hadamard) form. Assemble on the interface facets. Exact in the continuous setting, but discretely concentrated on the interface: the density is quadratic in the strain and inherits mesh noise.
• Distributed (volume) form. Assemble with an Eshelby-type tensor ; integrating by parts with the state equation recovers the boundary form analytically. Discretely this yields a smoother, more robust gradient and is generally preferred.

From derivative to velocity: Hilbertian regularization

is a linear functional, not a vector field. To descend, compute its Riesz representative in an -type inner product — this simultaneously extends the boundary information into the bulk and smooths it. This is the Hilbert::H1a operator, which solves

and returns :

Hilbert::H1a extension(vectorFes);
extension.setAlpha(alpha);                  // regularization length scale
auto theta = extension(dJ);                 // dJ: the assembled shape differential

The parameter has units of length and is the smoothing radius* of the descent direction:

• larger → smoother, more global velocities; fewer mesh-frequency oscillations; slower resolution of fine features;
• too small → the velocity inherits interface noise and the descent stagnates or oscillates.

Level-set representation

Representing the shape implicitly, with interface , allows topology changes (holes merging and splitting) without any surgery. The shape is advanced by transporting with the descent velocity — the Hamilton–Jacobi/advection step (Advection module):

Two maintenance obligations follow from the mathematics, not from implementation taste:

1. Redistancing. Transport degrades the signed-distance property ( ); normals, band widths, and step-size control all silently assume it. Restore it periodically with the fast marching method (Eikonal::FMM) or the PDE-based models in Distance (Eikonal, Poisson, SignedPoisson, or the Rvachev / SpaldingTucker normalizations).
2. Step-size (CFL) control. The interface should move at most a bounded number of cells per iteration; combine a time-step cap with a line search on the true objective. A descent step accepted without an objective decrease check can jump out of the basin entirely (a classic failure mode: the "optimized" material floods the whole bounding box).

Getting a computable domain from the level set

The state equation needs a domain to integrate over. Two strategies:

• Body-fitted discretization. Cut the zero set into the mesh so that becomes a set of interface facets: MMG's LevelSetDiscretizer produces the body-fitted mesh, MMG::Adapt and MMG::MeshOptimizer maintain metric conformity and element quality across iterations. Accurate domain and interface integrals; the cost is remeshing every iteration.
• Interface registration (WNGIR). Instead of remeshing, displace the existing mesh so its interface tracks : the Adaptation module's WNGIR solver. Keeps the mesh topology and connectivity stable between iterations.
• Ersatz material / density. Keep a fixed mesh and interpolate material properties (void as weak material) — the examples/DensityOptimization family. No remeshing, at the price of a modeling error controlled by the material contrast.

The optimization loop

Putting it together, one iteration of a level-set shape optimization reads:

// 1. Discretize the current domain from phi (remesh / register).
// 2. Solve the state equation on the current domain.
// 3. Evaluate the objective; assemble the shape differential dJ.
// 4. Extend/regularize: theta = H1a(vectorFes).setAlpha(alpha)(dJ).
// 5. Line search on J along theta (with a CFL cap on the step).
// 6. Advect phi by the accepted step.
// 7. Redistance phi (every few iterations).

Judge a run by measured quantities: a monotone objective history (the line search guarantees it), the volume tracking its multiplier target, interface quality (no inverted elements), and — after changes to any derivative — the finite-difference check of the shape derivative.

See also

• Variational formulations in Rodin — assembling the state problem
• Solvers — solving the state and extension systems
• Hilbert — extension-regularization
• Distance, Eikonal — redistancing
• MMG — remeshing and implicit-domain discretization