In node-based shape optimization, there are a vast amount of design parameters, and the objectives, as well as the physical constraints, are non-linear in state and design. Robust optimization algorithms are required. The methods of feasible directions are widely used in practical optimization problems and know to be quite robust. A subclass of these methods is the gradient projection method. It is an active-set method, it can be used with equality and non-equality constraints, and it has gained significant popularity for its intuitive implementation. One significant issue around efficiency is that the algorithm may suffer from zigzagging behavior while it follows non-linear design boundaries. In this work, we propose a modification to Rosen’s gradient projection algorithm. It includes the efficient techniques to damp the zigzagging behavior of the original algorithm while following the non-linear design boundaries, thus improving the performance of the method.
The latest updates on the Vertex Morphing technique for large optimization problems are shown in this work. Discussions about the challenges of node-based shape optimization in academic and industrial applications are included. The adaptive Vertex Morphing technique is demonstrated, which is easy to use in practice and allows the full exploitation of the potential of node-based shape optimization to find new designs in large-scale applications. We also show an efficient optimization method to handle different physical responses with many geometrical constraints. A state-of-the-art example of industrial importance supports the work.
Node-based shape optimization methods like Vertex Morphing use every node of the surface mesh as design variables. The method's key advantages are the minimal effort for problem setup and the resulting largest possible design space that allows finding new and innovative shapes. Due to a large number of design variables, the application of gradient-based optimization methods becomes mandatory, and adjoint sensitivity analysis is the preferred way for efficient computation of the shape gradients.For some industrial applications, it is necessary to restrict the large design freedom in some sense, for example by limiting the geometric design space. In recent work, nodal nonpenetration constraints have been successfully applied in the context of Vertex Morphing, resulting in a potentially large number of active constraints. Aggregation methods are commonly used to reduce the number of constraints in the optimization problem and are investigated here to combine nodal geometric constraints in a single global constraint function.
In the original publication of the article, the author would like to update the following changes in the equations 1. After Eq. 18, the changes in bold ("where y (i) = ∇f (x (i−1) ) − ∇f (x (i) ) ") to new ("where y (i) = ∇f (x (i) ) − ∇f (x (i−1) )"); 2. Adding the missing equation between Eqs. 20-22. The addition of equation in bold:
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.