非線形弾性材料からなるトラスの釣合い解析法への加速勾配法の適用

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  • APPLICATION OF ACCELERATED GRADIENT METHOD TO EQUILIBRIUM ANALYSIS OF TRUSSES WITH NONLINEAR ELASTIC MATERIALS

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<p> In recent years, optimization methods are used for many computational design methods in the field of architecture and architectural engineering. Optimization methods are roughly classified into heuristic methods typified by genetic algorithms and mathematical programming methods. Although the latter is inferior in versatility, it has an advantage that the calculation is fast. Along with the development of computers, mathematical programming methods have been developed dramatically and are used in various fields as a solution to optimization problems. Among them, the gradient method, which is the most classical method, is a method devised at the dawn of mathematical programming method, has been used for many years in various fields because of ease of algorithm implementation. Since the convergence is slow, a simple gradient method is rarely used nowadays. However, since the acceleration scheme of gradient methods greatly improve the convergence speed of the above solution, it has attracted attention in recent years as an effective solution method for solving a large scale problem. In fact, in fields other than the architectural engineering, accelerated gradient methods (AGM) are extensively used, for example, in machine learning for big data. In this paper, the computational performance of AGM is verified using a general benchmark problem. Furthermore, AGM is applied to equilibrium analysis of trusses with nonlinear elastic materials. Since solving the stiffness equation is synonymous with minimizing the total potential energy, equilibrium analysis can be formulated as a total potential energy minimization problem. AGM is used for solving such a problem and the effectiveness of the proposed method is discussed.</p><p> First of all, the minimization problem of Rosenbrock function is solved as a benchmark problem to verify the computational performance of AGM. A quasi-Newton method based on the BFGS formula (BFGS), a truncated Newton method (TNC), a conjugate gradient method (CG) are compared with AGM. As a result of the verification of the benchmark problem, it was shown that AGM can solve large-scale problems at the fastest speed.</p><p> Based on the result of the benchmark, using AGM for large-scale structural analysis with material nonlinearity is considered in this paper. Since solving the stiffness equation is equivalent to minimizing the total potential energy, equilibrium analysis can be formulated as a total potential energy minimization problem. The gradient of the total potential energy is equivalent to the unbalanced nodal force vector. Therefore, equilibrium analysis of trusses with nonlinear elastic materials is performed by applying AGM by utilizing unbalanced force as a gradient in this paper.</p><p> Equilibrium displacements of some three-dimensional truss structures which have a total of three restoring force characteristics of linear, bilinear, and high-order nonlinear are calculated by minimizing the total potential energy. As a result, it was observed that the AGM quickly converged to the optimum solution stably regardless of material properties. According to a comparison experiment between BFGS, TNC, and CG, it was confirmed that AGM is the fastest in calculation time, especially for large scale truss structures.</p><p> In this paper, the effectiveness of AGM and its applicability to equilibrium analysis of trusses with nonlinear elastic materials were confirmed.</p>

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