TopOpt.jl Tutorials
Interactive, executable examples that walk through complete topology optimization workflows — from problem setup to visualization.
Density-based methods
BESO: Bi-directional Evolutionary Structural Optimization
2D compliance minimization on the HalfMBB beam with problem setup, FEA solver, sensitivity filtering, and BESO algorithm loop.
SIMP: Solid Isotropic Material with Penalization
3D compliance minimization on a cantilever with density filtering, Nonconvex.jl + MMA87, and Makie visualization.
GESO: Genetic Evolutionary Structural Optimization
2D compliance minimization with genetic algorithm — binary encoding, crossover, and mutation for global search.
Penalty ramp (1.0 → 5.0) for improved convergence on cantilever, Half MBB, L-beam, and tie-beam benchmarks.
TOBS: Topological Optimization of Binary Structures
Binary (0/1) topology optimization with sequential linearization and Cbc.jl branch-and-cut for 16,000+ variables.
Power, rational, hyperbolic-sine, and projected penalties: their curves, their effect on gray elements, and parameter sensitivity.
Filter + Heaviside/Sigmoid projection for near-binary designs, with β-continuation and comparison to the plain density filter.
Minimizing Volume Subject to a Compliance Constraint
The dual of compliance minimization: least material for a required stiffness, on continuum cantilevers and 2D/3D trusses.
Level-set method
Level-Set Topology Optimization (OpenLSTO)
Boundary-based optimization with a signed-distance level set: compliance minimization on a cantilever and p-norm stress minimization on an L-beam, using the OpenLSTO port.
Heat conduction
Heat Conduction: Conductivity Tree
Thermal compliance minimization with the classic branching tree benchmark — heat flux on top, fixed temperature at bottom.
Heat Sink with Temperature BCs
Asymmetric temperature BCs (T=100 left, T=0 right), distributed flux from top, and Zygote gradient verification.
Stress and buckling
Stress-constrained L-bracket with relaxed stress and p-norm/KS/ε-relaxation aggregation. Minimizes volume subject to a global stress constraint, reproducing the classic rounded-corner design.
Element-wise stress limits using Percival.jl for large-scale constrained optimization with continuation SIMP.
Buckling-Constrained Truss Optimization
Semidefinite programming (SDP) constraints for stability via NonconvexSemidefinite.jl. Enforces K + c·Kσ ≽ 0.
Stochastic and uncertainty
Stochastic Compliance Minimization
Mean and standard-deviation compliance objectives over random load scenarios, exact and randomized (Hutchinson/Hadamard/SVD) trace estimators, and a maximum-compliance-constrained problem solved with the augmented Lagrangian method.
Advanced parametrization
Softmax parametrization for distributing 3+ candidate materials with mass constraints and MaterialInterpolation.
Neural Network Parametrization (IPOPT)
4-layer MLP parametrization with feasibility restoration and augmented-Lagrangian refinement using IPOPT.
Neural Network Parametrization (Adam)
6-layer MLP with Adam optimizer and continuation on penalty and constraint aggregation weight.
Reserve solid/void regions with FixedElementProjectorFun — support and load regions, keep-out zones — and optimize only the free elements.
Simultaneous Analysis and Design (SAND)
Treat displacements as design variables and enforce K u = f as a constraint with the augmented Lagrangian method, using the differentiable assembly blocks.
Truss optimization
Compliance minimization on bar structures with JSON-defined geometry, power-law penalization, and MMA87 optimizer.
Mixed-Integer Truss Optimization
Binary (0/1) truss design via Juniper.jl branch-and-bound with IPOPT relaxation — crisp layouts with no intermediate densities.
3D Truss Topology Optimization
Compliance minimization on a 3D space truss with the interactive static browser viewer and camera controls.
Solvers, IO and post-processing
Iterative and Matrix-Free Solvers
DirectSolver vs CGAssemblySolver vs CGMatrixFreeSolver on a 3D problem, with preconditioning and convergence criteria.
Custom Linear Solver and Preconditioner
Define your own linear solver (subtyping AbstractLinearSolver) and a from-scratch Jacobi preconditioner that plugs into the FEASolver interface.
Import a CAD/FEA mesh from an Abaqus .inp file with InpStiffness and optimize it directly.
Write optimized designs (and heat-conduction temperature fields) as .vtu files for ParaView and other post-processors.
Problem types
Standard benchmarks: point load cantilever (2D/3D), Half MBB, L-beam, tie-beam, INP import, and defining a custom problem type.
Ground structure via JSON files, programmatic truss cantilever construction, and 2D vs 3D truss differences.