Shape Optimization of Free-form Shells Considering Strain Energy and Algebraic Invariants of Parametric Surface

p. 525-535A new approach is proposed for shape optimization of shell surfaces, where requirements on the aesthetic aspect and the constructability as well as the structural rationality are simultaneously considered in the problem formulation. The surface shape is modeled using B'ezier surface to reduce the number of variables, while the ability to generate moderately complex shape is maintained. To apply the new approach to shell structures that have various plan shapes, the surface shape which has a rectangle plan is modeled using a tensor product B'ezier surface, and the surface shape with an irregular plan is modeled using a triangular patch B'ezier surface. The strain energy is used to represent the mechanical performance, and the aesthetic aspects and smoothness of the surface are quantified by algebraic invariants of the surface. The developable surface that has high constructability is created by imposing appropriate algebraic invariants constraints. The e ectiveness of the present approach is confirmed through several numerical examples and the characteristics of the results are discussed.Fujita, S.; Ohsaki, M. (2009). Shape Optimization of Free-form Shells Considering Strain Energy and Algebraic Invariants of Parametric Surface. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/654

Multiobjective robust shape and topology optimization of plane frames using order statistics

This paper presents a worst-case approach to robust optimization of plane frame structures under variation in uncertain parameters. The optimization procedure is first implemented without considering uncertainty, resulting in an optimal structure that may be unstable without bending stiffness. Based on such optimal solution, we then take variation in uncertain parameters into consideration and estimate the quantile response or trimmed mean of order statistics, where the quantile response is used as a relaxation of worst value of structural response. In order to obtain robust optimal solutions at various robustness levels, a multiobjective optimization problem is formulated and solved to simultaneously minimize the several order statistics or trimmed means with different orders. It is demonstrated in the numerical examples that the optimal distribution of cross-sectional areas of elements vary with the change of robustness level, and the convergence by using trimmed mean as estimation of quantile response is better than that of the simple order statistics

Non-Parametric Shape Design of Free-Form Shells Using Fairness Measures and Discrete Differential Geometry

A non-parametric approach is proposed for shape design of free-form shells discretized into triangular mesh. The discretized forms of curvatures are used for computing the fairness measures of the surface. The measures are defined as the area of the offset surface and the generalized form of the Gauss map. Gaussian curvature and mean curvature are computed using the angle defect and the cotangent formula, respectively, defined in the field of discrete differential geometry. Optimization problems are formulated for minimizing various fairness measures for shells with specified boundary conditions. A piecewise developable surface can be obtained without a priori assignment of the internal boundary. Effectiveness of the proposed method for generating various surface shapes is demonstrated in the numerical examples

FDMopt: Force density method for optimal geometry and topology of trusses

This paper presents a new efficient tool for simultaneous optimization of topology and geometry of truss structures. Force density method is applied to formulate optimization problem to minimize compliance under constraint on total structural volume, and objective and constraint functions are expressed as explicit functions of force density only. This method does not need constraints on nodal locations to avoid coalescent nodes, and enables to generate optimal solutions with a variety in topology and geometry. Furthermore, for the purpose of controlling optimal shapes, tensor product Bézier surface is introduced as a design surface. The optimization problem is solved using sensitivity coefficients and the optimizer is compiled as a component compatible with Grasshopper, an algorithmic modeling plug-in for Rhinoceros, which is a popular 3D modeling software. Efficiency and accuracy of the proposed method are demonstrated through two numerical examples of semi-cylindrical and semi-spherical models

Reinforcement Learning and Graph Embedding for Binary Truss Topology Optimization Under Stress and Displacement Constraints

This paper addresses a combined method of reinforcement learning and graph embedding for binary topology optimization of trusses to minimize total structural volume under stress and displacement constraints. Although conventional deep learning methods owe their success to a convolutional neural network that is capable of capturing higher level latent information from pixels, the convolution is difficult to apply to discrete structures due to their irregular connectivity. Instead, a method based on graph embedding is proposed here to extract the features of bar members. This way, all the members have a feature vector with the same size representing their neighbor information such as connectivity and force flows from the loaded nodes to the supports. The features are used to implement reinforcement learning where an action taker called agent is trained to sequentially eliminate unnecessary members from Level-1 ground structure, where all neighboring nodes are connected by members. The trained agent is capable of finding sub-optimal solutions at a low computational cost, and it is reusable to other trusses with different geometry, topology, and boundary conditions

Graph-based reinforcement learning for discrete cross-section optimization of planar steel frames

A combined method of graph embedding (GE) and reinforcement learning (RL) is developed for discrete cross-section optimization of planar steel frames, in which the section size of each member is selected from a prescribed list of standard sections. The RL agent aims to minimize the total structural volume under various practical constraints. GE is a method for extracting features from data with irregular connectivity. While most of the existing GE methods aim at extracting node features, an improved GE formulation is developed for extracting features of edges associated with members in this study. Owing to the proposed GE operations, the agent is capable of grasping the structural property of columns and beams considering their connectivity in a frame with an arbitrary size as feature vectors of the same size. Using the feature vectors, the agent is trained to estimate the accurate return associated with each action and to take proper actions on which members to reduce or increase their size using an RL algorithm. The applicability of the proposed method is versatile because various frames different in the numbers of nodes and members can be used for both training and application phases. In the numerical examples, the trained agents outperform a particle swarm optimization method as a benchmark in terms of both computational cost and design quality for cross-sectional design changes; the agents successfully assign reasonable cross-sections considering the geometry, connectivity, and support and load conditions of the frames

Reinforcement learning for optimum design of a plane frame under static loads

A new method is presented for optimum cross-sectional design of planar frame structures combining reinforcement learning (RL) and metaheuristics. The method starts from RL jointly using artificial neural network so that the action taker, or the agent, can choose a proper action on which members to be increased, reduced or kept their size. The size of the neural network is compressed into small numbers of inputs and outputs utilizing story-wise decomposition of the frame. The trained agent is used in the process of generating a neighborhood solution during optimization with simulated annealing (SA) and particle swarm optimization (PSO). Because the proposed method is able to explore the solution space efficiently, better optimal solutions can be found with less computational cost compared with those obtained solely by metaheuristics. Utilization of RL agent also leads to high-quality optimal solutions regardless of variation of parameters of SA and PSO or initial solution. Furthermore, once the agent is trained, it can be applied to optimization of other frames with different numbers of stories and spans

Gaussian mixture model for robust design optimization of planar steel frames

A new method is presented for an application of the Gaussian mixture model (GMM) to a multi-objective robust design optimization (RDO) of planar steel frame structures under aleatory (stochastic) uncertainty in material properties, external loads, and discrete design variables. Uncertainty in the discrete design variables is modeled in the wide range between the smallest and largest values in the catalog of the cross-sectional areas. A weighted sum of Gaussians is statistically trained based on the sampled training data to capture an underlying joint probability distribution function (PDF) of random input variables and the corresponding structural response. A simple regression function for predicting the structural response can be found by extracting the information from a conditional PDF, which is directly derived from the captured joint PDF. A multi-objective RDO problem is formulated with three objective functions, namely, the total mass of the structure, and the mean and variance values of the maximum inter-story drift under some constraints on design strength and serviceability requirements. The optimization problem is solved using a multi-objective genetic algorithm utilizing the trained GMM for calculating the statistical values of objective and constraint functions to obtain Pareto-optimal solutions. Since the three objective functions are highly conflicting, the best trade-off solution is desired and found from the obtained Pareto-optimal solutions by performing fuzzy-based compromise programming. The robustness and feasibility of the proposed method for finding the RDO of planar steel frame structures with discrete variables are demonstrated through two design examples

Proximal-exploration multi-objective Bayesian optimization for inverse identification of cyclic constitutive law of structural steels

Despite its importance in seismic response analysis, solving an inverse problem to identify the cyclic elastoplastic parameters for structural steels using conventional optimization algorithms still demands a substantial computational cost of repeatedly carrying out many nonlinear analyses. The parameters are commonly identified based on experimental measures from a single loading history, leading them to be biased and giving inaccurate predictions of structural behavior under other loading histories. To address these issues, we formulate a multi-objective inverse problem that simultaneously minimizes the error functions representing the differences between simulated responses and those measured experimentally from various cyclic tests of a steel specimen or a structural component. We then propose proximal-exploration multi-objective Bayesian optimization (MOBO) for solving the formulated inverse problem, resulting in an approximate Pareto front of parameters while limiting the number of costly simulations. MOBO sorts an initial Pareto front and constructs Gaussian process (GP) models for the error functions from a training dataset. It then relies on the hypervolume of the current solutions, the GP models, and a proximal exploration surrounding the current best compromise parameters to formulate an acquisition function that guides the improvement of the current solutions intelligently. Two identification examples show that the parameters obtained from the multi-objective inverse problem can reduce the bias induced using a single loading history for identification. The robustness of MOBO as well as a good prediction performance of the best compromise solution of identified parameters are demonstrated