Design optimization of geometrically nonlinear structures is well known as a computationally expensive problem by using incremental-iterative solution techniques. To handle the problem effectively ...the optimization algorithm needs to ensure that the trade-off between the computational time and the quality of the solution is found. In this study, a deep neural network (DNN)-based surrogate model which integrates with differential evolution (DE) algorithm is developed and applied for solving the optimum design problem of geometrically nonlinear space truss under displacement constraints and refer to the approach as DNN-DE. Accordingly, this surrogate model, also is known as a deep neural network, is established to replace conventional finite element analyses (FEAs). Each dataset is created based on FEA which employs the total Lagrangian formulation and the arc-length procedure. Several numerical examples are given to demonstrate the efficiency and validity of the proposed paradigm. These results indicate that the proposed approach not only reduces the computational cost dramatically but also guarantees convergence.
•A machine learning-based surrogate model is proposed to integrate with DE algorithm for solving the optimum structure.•The deep neural network is capable of exactly predicting the displacement of nonlinear response.•The combining method is effective, reduces the computational time, and guarantees solution accuracy.
In this paper, an efficient deep unsupervised learning (DUL)-based framework is proposed to directly perform the design optimization of truss structures under multiple constraints for the first time. ...Herein, the members’ cross-sectional areas are parameterized using a deep neural network (DNN) with the middle spatial coordinates of truss elements as input data. The parameters of the network, including weights and biases, are regarded as decision variables of the structural optimization problem, instead of the member’s cross-sectional areas as those of traditional optimization algorithms. A new loss function of the network model is constructed with the aim of minimizing the total structure weight so that all constraints of the optimization problem via unsupervised learning are satisfied. To achieve the optimal parameters, the proposed model is trained to minimize the loss function by a combination of the standard gradient optimizer and backpropagation algorithm. As soon as the learning process ends, the optimum weight of truss structures is indicated without utilizing any other time-consuming metaheuristic algorithms. Several illustrative examples are investigated to demonstrate the efficiency of the proposed framework in requiring much lower computational cost against other conventional methods, yet still providing high-quality optimal solutions.
•A robust and simple unsupervised neural network framework for large deformation analysis of truss structures.•Training data is small in size, independent of sampling procedures, without finite ...element analysis.•Reliability and stability of the proposed approach for nonlinear analysis without incremental-iterative techniques.
In this study, a robust and simple unsupervised neural network (NN) framework is proposed to perform the geometrically nonlinear analysis of inelastic truss structures. The core idea is to employ the NN to directly estimate nonlinear structural responses without utilizing any time-consuming incremental-iterative algorithms as those done in standard finite element method (FEM). To achieve such an objective, the loss function built via the total potential energy principle under boundary conditions (BCs) is minimized in the suggested NN model whose weights and biases are considered as design variables. In our computational framework, spatial coordinates of truss nodes are treated as input data, whilst corresponding displacement degrees of freedom are taken account of output. At the beginning of each training step, feedforward is performed to get the predicted displacement field, and it is used to derive the loss function based on the physical law. Then, back-propagation is applied to update the parameters of the network. This adjustment, which is the so-called learning process, is repeated until the potential energy is minimized. Once the network is properly trained, the mechanical responses of inelastic structures can be easily obtained. The suggested methodology is also extremely simple to implement, while the unlabeled data is available, small in size, independent of sampling techniques, and without finite element analyses (FEAs). Several benchmark examples regarding geometrical and material nonlinear analysis of truss structures are tested to show the effectiveness and reliability of the proposed paradigm. Obtained outcomes indicate that the developed NN framework is robust and can be extended to apply for other structures.
In this study, a physics-informed neural energy-force network (PINEFN) framework is first proposed to directly solve the optimum design of truss structures that structural analysis is completely ...removed from the implementation of the global optimization. Herein, a loss function is constructed to guide the training network based on the complementary energy, constitutive equations, and weight of the structure. Now only neural network (NN) is used in our scheme to minimize the loss function wherein weights and biases of the network are considered as design variables. In this model, spatial coordinates of truss members are examined as input data, while corresponding cross-sectional areas and redundant forces unknown to the network are taken account of output. Accordingly, the predicted outputs obtained by feedforward are employed to establish the loss function relied on physics laws. And then, back-propagation and optimizer are applied to automatically calculate sensitivity and adjust parameters of the network, respectively. This whole process, which is the so-called training, is repeated until convergence. The optimum weight of the structure corresponding to the minimum loss function is indicated as soon as the training process ends without using any structural analyses. Several benchmark examples for sizing optimization of truss structures are examined to determine the reliability, efficiency, and applicability of the proposed model. Obtained outcomes indicated that it not only reduces the computational cost dramatically but also yields higher accuracy and faster convergence speed compared with recent literature.
Surrogate modeling techniques are widely employed in solving constrained expensive black-box optimization problems. Therein, Kriging is among the most popular surrogates in which the trend function ...is considered as a constant mean. However, it also encounters several challenges related to capturing the overall trend with a relatively limited number of function evaluations as well as searching feasible points with complex or discontinuous feasible regions. To address this above issue, this paper presents an improved surrogate blind Kriging (IBK) and a combined infill strategy to find the optimal solution. According to enhancing the prediction accuracy of metamodels of objective and constraints, the high-order effects of regression function in the blind Kriging are identified by promising a variable selection technique. In addition, an infill strategy is developed based on the probability of feasibility, penalization, and constrained expected improvement for updating blind Kriging metamodels of the objective and constraints. At each iteration, two infill sample points are allocated at the positions to achieve improvement in optimality and feasibility. The IBK metamodels are updated by the newly-added infill sample points, which leads the proposed framework search to rapidly converge to the optimal solution. The performance and applicability of the proposed model are tested on several numerical benchmark problems via comparing with other metamodel-based constrained optimization methods. The obtained results indicate that IBK generally has a greater efficiency performance and outperforms the competitors in terms of a limited number of function evaluations. Finally, IBK is successfully applied to structural design optimization. The optimization results show that IBK is able to find the best feasible design with fewer evaluation functions compared with other studies, and this demonstrates the effectiveness and practicality of the proposed model for solving the constrained expensive black-box engineering design optimization problems.
The share of photovoltaic (PV) systems in the distribution networks is rapidly growing, leading to the common issue of overvoltage at the end of distribution feeders during the periods when peak ...generation is surplus to consumption. In this study, a hierarchical control is proposed to mitigate overvoltage at the point of connection of PV systems in physical low-voltage microgrids. The proposed mechanism is comprised of primary and secondary control layers to tackle the overvoltage problems given the communication capability is available. This mechanism employs a multi-objective optimisation approach to effectively coordinate curtailed active power and absorbed reactive power of the PV inverters with the aim of minimising the active power curtailment. The feasibility of the proposed control approach is successfully verified through simulations on a simplified LV network.
In this work, a direct physics-informed neural network (DPINN) is first proposed to analyze the stability of truss structures that incremental-iterative algorithm is completely removed from the ...implementation process. Instead of resolving of nonlinear equations as in conventional numerical methods, a neural network (NN) is employed to minimize the loss function which is designed to guide the training network based on the structural instability information. In our computational framework, the parameters including weights and biases of the network are considered as design variables. In addition, spatial coordinates of joints are examined as input data, while corresponding displacements and load factor unknown to the network are taken account of output. To address this challenge, the predicted outputs obtained by feedforward are utilized to establish the loss function relied on the residual load and stiffness characteristics of the structure as the first stage. And then, back-propagation and optimizer are applied to automatically calculate sensitivity and adjust parameters of the network, respectively. This entire process known as training is repeated until convergence. To that end, the position of the critical point is indicated as soon as the training ends by our network without using any time-consuming incremental-iterative algorithms as well as structural analyses. Several benchmark examples of truss structures associated with the geometric nonlinearity influence are investigated to evaluate the efficiency of the proposed scheme. The obtained results reveal that the present framework is extremely simple to implement and also yields the strong robustness as well as higher accuracy.
•A direct PINN framework for predicting the first critical point.•The lost function involving the instability information is minimized.•The stability analysis is replaced by an optimization problem.•Simplicity, efficiency, and robustness of the proposed approach for structural stability analysis.
In this paper, a robust deep neural network (DNN)-based parameterization framework is proposed to directly solve the optimum design for geometrically nonlinear trusses subject to displacement ...constraints. The core idea is to integrate DNN into Bayesian optimization (BO) to find the best optimum structural weight. Herein, the design variables of the structure are parameterized by weights and biases of the network with the spatial coordinates of all joints as the training data. A loss function of the network is built based on the predicted cross-sectional areas and deflection constraints obtained by supporting finite element analysis (FEA) and arc-length method. Accordingly, the optimum weight corresponding to the minimum loss function is indicated as soon as the complete training process. And then it is also serving as an objective of the BO for performing the hyperparameter optimization (HPO) to find the best optimum structural weight. Several illustrative numerical examples for geometrically nonlinear space trusses are examined to determine the efficiency and reliability of the proposed approach. The obtained results demonstrate that our framework can overcome the drawbacks of applications of machine learning in computational mechanics.
•A deep neural network-based parameterization framework for structural optimization.•Automatic tuning of hyperparameters of the network by using Bayesian optimization.•The best optimum weight is found by training without any other algorithms.•The suggested model provides high-quality solutions and avoids the local optimum.
Supplementation of oils rich in polyunsaturated fatty acids and condensed tannin has been known as a feeding approach to improve healthy fatty acids in ruminant milk, but it can cause an adverse ...effect on feed intake and animal performance. This study aimed to investigate the effect of feeding oil alone or in combination with grape seed tannin extract (GSTE) on feed intake, milk yield and composition of dairy cows. Sixteen low production dairy cows in mid-lactation fed a basal diet based on agro-industrial by-products were arranged to a completely randomized design for a 6-week duration. Animals were fed basal diet without oil and GSTE inclusion (CON), 2.5% DM soybean oil (SBO), 2.5% DM blend of soybean oil and tuna fish oil at 3:2 w:w (SFO), or SFO plus 0.4% DM GSTE (OCT). The results showed that DM intake was reduced (P < 0.05) by 14.4% in OCT relative to CON. Milk yield was not affected by oil and GSTE supplementation, but SFO and OCT strongly depressed milk fat, protein and total solids (P < 0.001). In conclusion, in a low production cow diet based on agro-industrial by-products containing high lipid, supplementation of oil and GSTE should be considered in the aspects of feed intake and milk composition.
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