Gradient- and non-gradient-based optimization

Optimization is understood as the systematic procedure used to find the optimum to a given problem. In my research I have used optimization to design materials and structures. Optimization methods can be broadly classified into gradient- and non-gradient-based. While the former uses the objective function's gradient to improve a given design, the latter just needs the evaluation of the objective function to discern better candidates.

Regarding non-gradient-based optimization, I used genetic algorithms (GAs) to design biomimetic materials for active cooling. GAs are the most common methodology within a broader category of evolutionary algorithms. GAs are population-based algorithms that improve designs by means of selection, crossover, and mutation—genetic operators borrowed from evolutionary theory. GAs can also be combined with a Pareto-selection mechanism to solve Multi-Objective Optimization Problems (MOOPs), where the final Pareto set contains many optimal solutions (showing the trade-offs between objectives). GAs can use discrete variables, and search a wider decision space than gradient-based algorithms, so optima found are likely to be global. On the downside, GAs require a vast number of function evaluations.

As the name suggests, gradient-based optimization algorithms use the derivatives of a function to improve the design. Because of this, one of the main issues in gradient-based optimization is that the algorithm may get stuck in a local optimum. Therefore, non-convex optimization problems require multiple runs from different starting points in the design space when looking for a global optimum (and this is never guaranteed unless the entire design space is tested). In addition, computing gradient information, usually known as sensitivity analysis, may be very complicated depending on the objective function.

Gradient-based optimization is the methodology of choice for topology optimization (TO), a technique widely used in structural optimization. Starting with a computational domain, TO seeks the most appropriate material layout to optimize a given objective function. In standard density-based TO, density values between 0 (void) and 1 (material) are assigned to all finite elements in the computational domain. This approach leads to pixelized or staircased designs that contain "gray values"—elements with density values that do not correspond to material nor void! My research on gradient-based optimization has focused in the use of enriched FEA coupled to a level set function that is used to describe the topology. This approach, although harder to implement than standard TO, obtains crisps "black and white" designs that are smoother than those obtained by standard TO, since the level set can cut elements in any way.

Machine learning

I recently started investigating the use of deep learning in topology optimization. This work does not have yet critical mass, but it will become one of my main areas of research in the near future.

Projects

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Enriched optimization

An enriched FEM for level set-baed topology optimization

  TU Delft
2017 – present
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Enriched optimization

Bandgap maximization in phononic crystals

  TU Delft
2017 – present
1
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Machine learning

Using deep learning for obtaining optimal structural designs

  TU Delft
2019 – present
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