Input: A query function Q(q, g) that given a query graph q and a data graph g, return a Bool indicating if q and g is relevant, say the set of all relevant graphs L(q) . i.e. Q: if GED of q and g is less than 5 then q and g is relevant. A graph database. A constant k: k graphs in the answer set Neighborhood of a graph g (N(g)): The number of relevant graphs to q, say g’, such that distance(g, g’) is less or equal to theta. i.e. GED of g and g’ is less than theta. Representativeness of a set of graphs S: |The union of all the neighborhoods of all graph in S|/|L(q)|, say π(S). π(g) means the special case that S only has 1 graph g. Output: k graphs in all the relevant graphs such that the k graph’s representativeness is maximized.

The Problem of Formulation

In this review we identify a new category of methods for implementing and solving structural optimization problems that has emerged over the last 20 years, which we propose to call feature-mapping methods. The two defining aspects of these methods are that the design is parameterized by a high-level geometric description and that features are mapped onto a non-body-fitted mesh for analysis. One motivation for using these methods is to gain better control over the geometry to, for example, facilitate imposing direct constraints on geometric features, while avoiding issues with re-meshing. The review starts by providing some key definitions and then examines the ingredients that these methods use to map geometric features onto a fixed mesh. One of these ingredients corresponds to the mechanism for mapping the geometry of a single feature onto a fixed analysis grid, from which an ersatz material or an immersed-boundary approach is used for the analysis. For the former case, which we refer to as the pseudo-density approach, a test problem is formulated to investigate aspects of the material interpolation, boundary smoothing, and numerical integration. We also review other ingredients of feature-mapping techniques, including approaches for combining features (which are required to perform topology optimization) and methods for imposing a minimum separation distance among features. A literature review of feature-mapping methods is provided for shape optimization, combined feature/free-form optimization, and topology optimization. Finally, we discuss potential future research directions for feature-mapping methods.

A review on feature-mapping methods for structural optimization

NUMERISCHE OPTIMIERUNG VON COMPUTER-MODELLEN MITTELS DER EVOLUTIONSSTRATEGIE Hans-Paul Schwefel Birkhäuser, Basel and Stuttgart, 1977 370 pages Hardback SF/48 ISBN 3-7643-0876-1

Topology optimization using material-field series expansion and Kriging-based algorithm: An effective non-gradient method

Kriging-assisted topology optimization of crash structures

Structural topology optimisation (TO) methods are well established in many engineering disciplines However, for highly non-linear problems, including crashworthiness, methods are less developed and still subject to active research In particular, due to the simplifications made in the state-of-the-art methods as well as heuristic character of most of them, the optimality of the obtained structures is arguable In this paper, a TO approach based on the level set method and evolutionary algorithms is presented The method is free from the heuristic assumptions and therefore, allows for a direct optimisation of the underlying crash problem As a result, a reliable information about the optima of crash problems can be obtained In order to assure the consistency of the results, a comprehensive study of optimisation settings is performed and discussed Optimisation results for 2D crash test cases are presented and general conclusions concerning the behaviour of the optimal crash structures are derived

Identification of optimal topologies for crashworthiness with the evolutionary level set method

Vehicle crashworthiness design belongs to one of the most complex problems considered in the design optimization. Physical phenomena that are taken into account in crash simulations range from complex contact modeling to mechanical failure of materials. This results in high nonlinearity of the optimization problem and involves remarkable amount of numerical noise and discontinuities of the objective functions that are optimized. Consequently, the sensitivity information, which is necessary for the majority of Topology Optimization approaches, can be obtained analytically only for considerably simplified problems, which, in most cases, excludes the use of the gradient-based optimization methods. As a result, in the state-of-the-art methods for crashworthiness Topology Optimization, strong and thus arguable assumptions about the properties of the optimization problem are made and heuristic approaches are used. This problem can be solved with use of Evolutionary Algorithms, where no assumptions about the optimization problem have to be made and which perform well even for highly nonlinear and discontinuous problems. We propose a novel approach using evolutionary optimization
techniques together with a geometric Level-Set Method in crashworthiness Topology Optimization. Both standard Evolution Strategies and the state-of-the-art Covariance Matrix Adaptation Evolution Strategy are used. In order to evaluate the proposed method, an energy maximization problem for a rectangular beam, fixed at both ends and impacted in the middle by a cylindrical pole, is considered. The results show that the evolutionary optimization methods can be efficiently
used for an optimization of crash-loaded structures, while defining the objective function explicitly.

Evolutionary Level Set Method for Crashworthiness Topology Optimization

A basis for most of the state-of-the-art methods for crashworthiness topology optimization form voxel elements, being three-dimensional, regular brick finite elements.
Such voxel-based optimization techniques lead to creation of so-called zigzag structures that are used as a reference for positioning of the structural beams. On the other hand,
important vehicle body components are made of thin-walled sheet metal structures and the use of an optimized design obtained from any voxel-based optimization method as an
inspiration for the final thin-walled structure is questionable and requires considerable manual post-processing. In this paper we propose a novel approach using evolutionary algorithms for optimization of thin-walled structures. For evaluation of the method, a 2D transverse bending of a rib-reinforced thin-walled structure is considered. A state-of-the-art Covariance Matrix Adaptation Evolution Strategy (CMA-ES), combined with
a suitable representation, is used for optimization of the layout of the reinforcing ribs. The results show that evolutionary optimization algorithms can be efficiently used for topology optimization of crash-loaded thin-walled structures.

Evolutionary Crashworthiness Topology Optimization of Thin-Walled Structures

Bayesian Optimization (BO) is a surrogate-assisted global optimization technique that has been successfully applied in various fields, e.g., automated machine learning and design optimization. Built upon a so-called infill-criterion and Gaussian Process regression (GPR), the BO technique suffers from a substantial computational complexity and hampered convergence rate as the dimension of the search spaces increases. Scaling up BO for high-dimensional optimization problems remains a challenging task.

Mariusz Bujny

Papers

Kriging-assisted topology optimization of crash structures

Identification of optimal topologies for crashworthiness with the evolutionary level set method

Evolutionary Level Set Method for Crashworthiness Topology Optimization

Evolutionary Crashworthiness Topology Optimization of Thin-Walled Structures

High Dimensional Bayesian Optimization Assisted by Principal Component Analysis