In optimization tasks, it is interesting to achieve a set of efficient solutions instead of one single output, in the case the best solution is not suitable. Many niching methods offer a diversified response, yet some important problems are common: (1) The most interesting solutions of each local optimum are not identified. Thus, the output is the overall population of solutions, which increases the work of the designer in verifying which solution is the most interesting. (2) Existing niching algorithms tend to distribute the solutions on the most promising regions, over-populating some local optima and sub-populating others, which leads to poor optimization.To solve these challenges, a novel niching method is presented, named local optimum ranking 2 (LOR2). This sorting methodology favors the exploration of a defined number of local optima and ranks each local population by objective value within each local optimum. Thus, is performed a multi-focus exploration, with an equalized number of solutions on each local optimum, while identifying which solutions are the local apices. To exemplify its application, the LOR2 algorithm is applied in the design optimization of a metallic cantilever beam. It achieves a set of efficient and diverse design configurations, offering both performance and diversity for structural design challenges.In addition, a second experiment describes how the algorithm can be applied to segment the domain of any function, into a mesh of similar sized or custom-sized elements. Thus, it can significantly simplify metamodels and reduce their computation time.

Artificial neural networks (ANNs) gained much attention due to its capacity to mimic neural cells of living organisms which have intelligence. ANNs and some of its variations, have been applied in text and image recognition, among other applications, and are the basic building blocks of artificial intelligence (AI). A known problem, however, is the impossibility to extract the knowledge achieved by training. To solve this problem, a new neural network is hereby presented, called functional network (FN). This functional network differs from the ANNs basically by the fact that the sigmoid function can be substituted by any mathematical expression, that each functional neuron has multiple inputs and multiple function outputs, and that the trained information can be extracted in the form of a mathematical expression, denominated generative mathematical expression (GME).Although a moderate size FN may give a very large number of possible combinations, requiring very expensive computation efforts on training, recent developments in computation such as quantum computers indicate it increasingly becomes a viable alternative for large patterns and complex models. Thus, the GME of pattern representations can be extracted from a FN in order to simplify AI and numerical modeling tasks in general, such as finite element models (FEM) of structures, or computer fluid dynamic (CFD) models, decreasing overall computation costs.

Artificial intelligence in general and optimization tasks applied to the design of very efficient structures rely on response surfaces to forecast the output of functions, and are vital part of these methodologies. Yet they have important limitations, since greater precisions require greater data sets, thus, training or updating larger response surfaces become computationally expensive or unfeasible. This has been an important bottle neck limitation to achieve more promising results, rendering many optimization and AI tasks with a low performance.To solve this challenge, a new methodology created to segment response surfaces is hereby presented. Differently than other similar methodologies, this algorithm named outer input method has a very simple and robust operation, generating a mesh of near isopopulated partitions of inputs which share similitude. The great advantage it offers is that it can be applied to any data set with any type of distribution, such as random, Cartesian, or clustered, for domains with any number of coordinates, significantly simplifying any metamodel with a mesh ensemble.This study demonstrates how one of the most known and precise metamodel denominated Kriging, yet with expensive computation costs, can be significantly simplified with a response surface mesh, increasing training speed up to 567 times, while using a quad-core parallel processing. Since individual mesh elements can be parallelized or updated individually, its faster operational speed has its speed increased.

Artificial intelligence in general and optimization tasks applied to the design of aerospace, space,and automotive structures, rely on response surfaces to forecast the output of functions, and are vital part of these methodologies. Yet they have important limitations, since greater precisions require greater data sets, thus, training or updating larger response surfaces become computationally expensive, sometimes unfeasible. This has been a bottle neck limitation to achieve more promising results, rendering many AI related task with a low efficiency.To solve this challenge, a new methodology created to segment response surfaces is hereby presented. Differently than other similar methodologies, the novel algorithm here presented named outer input method, has a very simple and robust operation. With only one operational parameter, maximum element size, it efficiently generates a near isopopulated mesh for any data set with any type of distribution, such as random, Cartesian, or clustered, for domains with any number of coordinates.Thus, it is possible to simplify the response surfaces by generating an ensemble of response surfaces, here denominated response surface mesh. This study demonstrates how a metamodel denominated Kriging, trained with a large data set, can be simplified with a response surface mesh, significantly reducing its often expensive computation costs> experiments here presented achieved an speed increase up to 180 times, while using a dual core parallel processingcomputer. This methodology can be applied to any metamodel, and metamodel elements can be easily parallelized and updated individually. Thus, its already faster training operation has its speed increased.

In optimization tasks, it is interesting to achieve a set of efficient solutions instead of one single output, in the case the best solution is not suitable. Many niching methods offer a diversified response, yet some important problems are common: (1) The most interesting solutions of each local optimum are not identified. Thus, the output is the overall population of solutions, which increases the work of the designer in verifying which solution is the most interesting. (2) Existing niching algorithms tend to distribute the solutions on the most promising regions, over-populating some local optima and sub-populating others, which leads to poor optimization.To solve these challenges, a novel niching method is presented, named local optimum ranking 2 (LOR2). This sorting methodology favors the exploration of a defined number of local optima and ranks each local population by objective value within each local optimum. Thus, is performed a multi-focus exploration, with an equalized number of solutions on each local optimum, while identifying which solutions are the local apices. To exemplify its application, the LOR2 algorithm is applied in the design optimization of a metallic cantilever beam. It achieves a set of efficient and diverse design configurations, offering both performance and diversity for structural design challenges.In addition, a second experiment describes how the algorithm can be applied to segment the domain of any function, into a mesh of similar sized or custom-sized elements. Thus, it can significantly simplify metamodels and reduce their computation time.