Optimization (Optimization) — CAE Glossary

The basic idea is the same: finding design variables that minimize (or maximize) an objective function. However, in CAE, one function evaluation requires FEM or CFD analysis, so computation costs are orders of magnitude higher. For example, if one structural body analysis takes several hours, you can't afford to do 1,000 trial-and-error iterations.

Broadly, there are two approaches. One is gradient-based methods, which use sensitivity (derivative information) to efficiently explore descent directions. The other is metaheuristics, which use probabilistic search without derivatives, like genetic algorithms or particle swarm optimization. The former is faster but prone to local minima; the latter can explore globally but requires more evaluations.

The three differ in "how much freedom you have in redesign."

Exactly right. Topology optimization results often have organic shapes similar to biological bone, which is difficult to manufacture with traditional machining or casting. However, additive manufacturing (3D printers) can directly produce such complex shapes. GE Aviation's jet engine fuel nozzle is a famous success story—they consolidated 20 parts into one, reducing weight by 25%.

By calculating sensitivity (the derivative). Roughly speaking, "if I increase this plate thickness by 0.1 mm, how much will the stress decrease?" There are three approaches:

Metaheuristics excel in three scenarios. First, when the objective function is discontinuous or discrete. For example, integer variables like the number of bolts (2, 4, or 6) cannot be differentiated. Second, when there are many local minima (multimodal problems). Gradient methods fall into the nearest valley, but genetic algorithms can explore broadly across the entire population. Third, when the analysis code is a black box with no access to sensitivity. This happens often with commercial software.

Particle Swarm Optimization (PSO), Simulated Annealing, and Differential Evolution are standard. Recently, Bayesian optimization is gaining popularity. It uses a surrogate model (approximation model) to approximate before searching, so even costly CFD or crash analyses can find good solutions with just dozens of samples.

The key insight is: even with millions of design variables, only a single additional linear system is needed for sensitivity calculation. With ordinary finite difference, if you have $n$ variables, you need $n$ analyses. But with the adjoint method, a single adjoint analysis gives you sensitivities for all variables at once.

The key is Lagrange multipliers. Consider the augmented Lagrangian combining the objective function $J$ and the constraint residual $R(\mathbf{u}, \mathbf{x}) = 0$:

Good question. OpenFOAM's adjointOptimisationFoam supports steady RANS adjoints and is production-ready. Key points: First, the original primal flow field must be well-converged. Large residuals lead to noisy adjoints. Second, adjoint equation convergence is often harder than the primal—use smaller relaxation factors. Finally, you decide whether to freeze turbulence model adjoints or solve them rigorously. With adequate computational resources, rigorous adjoints give better accuracy.

That's multi-objective optimization. For example, in automotive body design, "minimize mass" and "maximize energy absorption in crash" conflict. Making it lighter reduces absorption; increasing absorption makes it heavier. The boundary of this trade-off is called the Pareto front, where all solutions on the curve are "optimal in a sense."

Mathematically they're equivalent, so the final choice is made by engineer judgment. Three common methods:

You need evaluations = population size × generations. For example, 100 individuals × 50 generations = 5,000 evaluations. If one analysis takes 1 hour, that's 5,000 hours by simple math. In practice, you either use a surrogate model to reduce evaluations to 100–200, or run in parallel on HPC clusters. Platforms like modeFRONTIER and HyperStudy automate this.