As Example 3, the agent is appli

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As Example 3, the agent is appli

As Example 3, the agent is applied to a larger-scale truss, as shown in Figure 10, without re-training. Solids Struct. 6, 679684. The truss is optimized for two loading conditions. doi: 10.1515/9781400874668, PubMed Abstract | CrossRef Full Text | Google Scholar, Cai, H., Zheng, V. W., and Chang, K. C. (2017). Although stress and displacement bounds have the same value and , respectively, for each member and DOF in this study, it should be noted that each member could have a different stress bound and each DOF could have a different displacement bound for each load case, which provides a versatility to the proposed method. doi: 10.1109/CVPR.2016.90, Khandelwal, M. (2011). In Equation (10), the action value is updated so as to minimize the difference between the sum of observed reward and estimated action value at the next state r(s)+maxaQ(s,a) and estimated action value at the previous state Q(s, a). In this paper, a machine-learning based method combining graph embedding and Q-learning is proposed for binary truss topology optimization to minimize total structural volume under stress and displacement constraints. The upper-bound displacement for each boundary condition is computed by multiplying 100 to the maximum absolute value of displacement among the all DOFs of the initial GS with the same loading and boundary conditions; hence, varies depending on the structure and the loading and boundary conditions. Watkins, C. J. C. H., and Dayan, P. (1992). Comparison between proposed method (RL+GE) and GA in view of total structural volume V[m3] and CPU time for one optimization t[s] using benchmark solutions. Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and Dahl, G. E. (2017). Loading condition L2 of Example 1; (A) initial GS, (B) removal sequence of members. Generative adversarial networks.

Eng. doi: 10.1007/s00158-004-0480-2, Papadrakakis, M., Lagaros, N. D., and Tsompanakis, Y. From these results, the agent is confirmed to behave well for a different loading condition. GA algorithm is run for 10 times with different initial solutions that are generated randomly, and only the best result that yields a solution with the least total structural volume is provided in the GA column in Table 3. In this study, = 0.99 is adopted, because cumulative reward indicating the amount of reduction of structural volume as a result of the action is much more important than the instant reward. If stress and displacement constraints are satisfied, penalty terms become zero and the cost function becomes equivalent to the total structural volume V(A). doi: 10.1007/s10589-007-9152-7, Bellman, R. (1957). Topology optimization of trusses by growing ground structure method. Machine learning prediction errors better than DFT accuracy. Knowl. Figure 2. different removal sequences can be obtained for different order of the same set of load cases in node state data vk; for example, exchanging the values at indices 2 and 4 and those at indices 3 and 5 in vk maintains the original loading condition but may lead to different action to be taken during each member removal process, because the neural network outputs different Q values due to the exchange.

The one just before the terminal state is a sub-optimal truss; however, instability exists at the loaded node. Note that the nodes highlighted in blue are pin-supported, those in yellow are loaded. COURSERA: Neural Netw. Comput. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). This work was kindly supported by Grant-in-Aid for JSPS Research Fellow No.JP18J21456 and JSPS KAKENHI No. Math. 29, 190197. From this result, it is confirmed that the agent is capable of eliminating unnecessary members properly for a different-scale truss. One of the loads applied at node 4 is an irregular case where pin-supports and the loaded node aligns on the same straight line. Int. Figure 12. Figure 8. The final truss of removal process of members presented in Figure 5 is a terminal state, where displacement constraint is violated at the nodes highlighted in red. Example 2: 3 2-grid truss (V = 0.0340 [m3]). According to Table 3, the proposed RL+GE method is much more efficient than GA; the CPU time exponentially increases as the number of variable increases in GA; on the other hand, the CPU time increases almost linearly in RL+GE. doi: 10.1038/nature24270, Sutton, R. S., and Barto, A. G. (1998). KH and MO approved the final version of the manuscript, and agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. Struct. The boundary conditions are given at the beginning of each episode. (1996). Copyright 2020 Hayashi and Ohsaki. 37, 377393. This way, features of each member considering connectivity can be extracted. 6:59. doi: 10.3389/fbuil.2020.00059. Mech. Eng. Because the optimization problem (Equation 3) contains constraint functions, the cost function F used in GA is defined using the penalty term as: where 1 and 2 are penalty coefficients for stress and displacement constraints; both are set to be 1000 in this study. Lecture 6.5RmsProp: Divide the gradient by a running average of its recent magnitude. Jpn. Mastering the game of go without human knowledge. The performance is also tested for other different trusses in sections 4.24.4 without re-training the results in section 4.1. Optim. Optim. It is confirmed from this history that the agent successfully improves its policy to eliminate unnecessary members as the training proceeds. Comput. Indiana Univ. (2017). AIAA J. Genetic algorithm used in this study. Since ^ is also computed using {1, , 6}, the action value Q(^,i) is dependent on = {1, , 9}. Loading condition L2 of Example 3; (A) initial GS, (B) removal sequence of members. Received: 22 November 2019; Accepted: 09 April 2020; Published: 30 April 2020. Appl. Trivial members close to pin-supports or placed upper-right or bottom-right that do not bear stress are first removed, and important members along the load path are removed afterwards. (2015). Figure 4. 30, 16161637. Similarly to loading condition L1 in Figure 5, several symmetric topologies are observed during the removal process, and the sub-optimal topology is a well-converged solution that does not contain unnecessary members. The removal sequence of members is illustrated in Figure 6B. A Markovian decision process. 10, 155162. The other solutions are assumed to be global optima which have not been verified through enumeration due to extremely high computational cost. The agent is trained using a 72-member truss with 4 4 grids. Topology optimization of trusses with stress and local constraints on nodal stability and member intersection. (1986). It is notable that the agent was able to optimize the structure with the unforeseen boundary conditions which the agent has never experienced during the training. (Princeton, NJ: Princeton University Press). If the cumulative reward is larger than the previous best score, at that step is saved. Struct. Figure 5. Since removal of any remaining member will cause violation of the displacement constraint, there is no unnecessary member in the sub-optimal topology. The datasets generated for this study are available on request to the corresponding author. Table 3. Deepwalk: online learning of social representations. 4, 2631. doi: 10.1007/s00366-019-00753-w. [Epub ahead of print]. The left two corners 1 and 7 are pin-supported and rightward and downward unit loads are separately applied at the bottom-right corner 43, as shown in Figure 11A in the loading condition L1. Another approach may be to incorporate a rule-based method to create a hybrid optimization agent. The same material property and constraints as the examples of RL are applied to the following problems. (2015). As shown in Figure 8B, the agent utilizes an reasonable policy to eliminate obviously unnecessary members connecting to supports at first, non-load-bearing members around the supports next, and members in the load path at last. Mach. Optimal topologies of truss structures. doi: 10.1007/s00158-017-1710-8, Ohsaki, M., and Katoh, N. (2005). Algorithm 1. Figure 9. Topologies at steps 37, 60, 84, 100, 144, and 145 in the removal sequence are illustrated in Figure 11B. 72, 1528. JP18K18898. Enhanced artificial intelligence for ensemble approach to predicting high performance concrete compressive strength. Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. The upper-bound stress is 200 N/mm2 for both tension and compression for all examples. doi: 10.1080/03052158608902532, Rumelhart, D. E., Hinton, G. E., and Williams, R. J. Yu, Y., Hur, T., and Jung, J. (2017).

After the 18th step, the topology became asymmetric despite the symmetry of problem definition; however, the asymmetric solutions should be accepted, because Guo et al. A simple GA algorithm used in this section is explained in Algorithm 1. Optim. In GA, a set of solutions are repeatedly modified using the operations such as selection, where superior solutions at current generation are selected for new generation, crossover, where the selected solutions are combined to breed child solutions sharing the same characteristics as the parents, and mutation, where the selected solutions randomly change their values with low probability. Each grid is a square whose side length is 1 m. The intersection of bracing members is not connected. Furthermore, the trained agent is applicable to a truss with different topology, geometry and loading and boundary conditions after it is trained for a specific truss with various loading and boundary conditions. https://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf (accessed April 23, 2020). Optim. doi: 10.1109/5.726791, Lee, S., Ha, J., Zokhirova, M., Moon, H., and Lee, J. J. Optim. Figure 3. Architect. doi: 10.1038/323533a0, Sheu, C. Y., and Schmit, L. A. Jr. (1972). The removal sequence of members when the maximum score is recorded is illustrated in Figure 5. doi: 10.1109/TNN.1998.712192, Tamura, T., Ohsaki, M., and Takagi, J. arXiv:1704.01212. High-speed calculation in structural analysis by reinforcement learning, in the 32nd Annual Conference of the Japanese Society for Artificial Intelligence, JSAI2018:3K1OS18a01 (in Japanese), (Kagoshima). Topological design of truss structures using simulated annealing. ] is a concatenation operator of two vectors in the column direction. Lecun, Y., Bottou, L., Bengio, Y., and Haffner, P. (1998). Blast-induced ground vibration prediction using support vector machine. An Introduction to Genetic Algorithms. doi: 10.1016/S0045-7825(97)00215-6, Perozzi, B., Al-Rfou', R., and Skiena, S. (2014). Two pin-supports are randomly chosen; one from nodes 1 and 2 and the other from nodes 4 and 5. Learn. 1, NIPS'12 (Tahoe, CA: Curran Associates Inc.), 10971105. Cambridge, MA: MIT Press. Struct. The maximum test score of 43.5 is obtained at the 2,100th episode, and the parameter at this episode is dealt as the best parameters and used hereafter. Rev. Nature 550, 354359. Although nload! Struct. Arxiv:1801.05463. Methods Appl. Example 1: 4 4-grid truss (V = 0.0853 [m3]). The parameters are tuned using a method based on 1-step Q-learning method, which is a frequently used RL method. Gradient-based learning applied to document recognition, in Proceedings of the IEEE, 22782324. Goodfellow, I. J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., et al.

In the first boundary condition B1, as shown in Figure 8A, left tip nodes 1 and 3 are pin-supported and bottom-right nodes 7 and 10 are subjected to downward unit load of 1 kN separately as different loading cases. The number of training episodes is set as 5,000. Deep learning for topology optimization design. Neural message passing for quantum chemistry. 47, 783794. The edge length of each grid is 1 m also for this Example 2. Symmetry properties in structural optimization: Some extensions. In the same manner as neural networks, a back-propagation method (Rumelhart et al., 1986), which is a gradient based method to minimize the loss function, can be used for solving Equation (11). It is verified from the numerical examples that the trained agent acquired a policy to reduce total structural volume while satisfying the stress and displacement constraints. Ringertz, U. T. (1986). A., Veness, J., Bellemare, M. G., et al. Comput. 3, 2552. During the test, nodes 1 and 5 are pin-supported, and loads are applied at node 23 in positive x and negative y direction separately as different loading conditions, which is denoted as loading condition L1. At the 18th step, a tower-like symmetric topology is created with extending members from upper tips to loaded nodes. In the loading condition L2, the left two corners are again pin-supported, and the outward unit load is separately applied at nodes 4 and 46, as shown in Figure 12A. 6 Articles, This article is part of the Research Topic, https://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf, Creative Commons Attribution License (CC BY), Department of Architecture and Architectural Engineering, Graduate School of Engineering, Kyoto University, Kyoto, Japan. (2013) explained that solving the quasi-convex symmetric optimization problem may yield highly asymmetric solution. To investigate performance of the agent for another loading condition, the structure with the loads as shown in Figure 6A, denoted as loading condition L2, is also optimized using the same agent. Figure 7. MO contributed to problem formulation and interpretation of data, and assisted in the preparation of the manuscript. The above examples using the proposed method are further investigated in view of efficiency and accuracy through comparison with genetic algorithm (GA). Therefore, the topology just before the terminal state shall be a sub-optimal topology, which is a truss with 12 members as shown in Figure 5B. Softw. The score rapidly improves in the first 1,000 episodes and the score mostly keeps above 35.0 after 2,000 episodes. doi: 10.1007/s00158-012-0877-2, Hagishita, T., and Ohsaki, M. (2009). The initial GS is illustrated in Figure 3.

Figure 4 plots the history of cumulative rewards in the test simulation recorded at every 10 episodes. Comput. Boundary condition B1 of Example 2; (A) initial GS, (B) removal sequence of members. -relaxed approach in structural topology optimization. This applicability was demonstrated through both smaller-scale and larger-scale trusses and sparse sub-optimal topologies were obtained for both cases. doi: 10.1007/BF01197454, Chou, J.-S., and Pham, A.-D. (2013). Load and support conditions are randomly provided according to a rule so that the agent can be trained to have good performance for various boundary conditions. 56, 11571168. Guo, X., Du, Z., Cheng, G., and Ni, C. (2013). Moreover, all the trained RL agents with the best parameters led to the same 12-member sub-optimal solution as Figure 5B for loading condition L1. Optimising the load path of compression-only thrust networks through independent sets. Topping, B., Khan, A., and Leite, J. Eng. Note that the total CPU time t[s] for obtaining this removal sequence of members includes initialization of the truss structure, import of the trained RL agent, and computing the removal sequence. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. The topology two steps before the terminal state contains successive V-shaped braces and is stable and statically determinate. History of cumulative reward of each test measured every 10 episodes. Human-level control through deep reinforcement learning. In the second boundary condition B2, the bottom center nodes 4 and 7 are pin-supported and upper tip nodes 3 and 12 are subjected to outward unit loads along x axis as shown in Figure 9A. Imagenet classification with deep convolutional neural networks, in Proceedings of the 25th International Conference on Neural Information Processing Systems - Vol. When a function approximator is not utilized, the action value is updated using state s, chosen action a, observed next state s and reward r as: where > 0 is a learning rate that has an effect on convergence of the training. Ohsaki, M. (1995). Struct. Right tip nodes are candidates to apply loading, and a horizontal or a vertical load with the fixed magnitude of 1.0 kN is applied at a randomly chosen node. doi: 10.1007/s00366-010-0190-x, Kirsch, U. In utilizing the trained agent in Example 1, nload! arXiv:1702.05532. Nodes 1 and 5 are pin-supported and nodes 22 and 24 are subjected to 1 kN load in positive x direction separately as two load cases. J. Mecan. Comput. The proposed method for training agent is expected to become a supporting tool to instantly feedback the sub-optimal topology and enhance our design exploration. (2018). Structural optimization using evolution strategies and neural networks.