Longquan Yong

Reviewed the research progress of Newton’s iterative methods for nonlinear equation. Convergence with second order, third order, fourth order, fifth order, sixth order, seventh order, eighth order and ninth order Newton’s iterative methods are given in turn, and the efficiency index of these iterative methods are analyzed. Numerical experiments show that the convergence process of the three Newton iterative methods, and the results indicate that the convergence of the higher-order Newton’s method can be well demonstrated only when the initial point is close to the root.

To address the issues of uneven initial distribution and limited search accuracy with the traditional divergent quantum-inspired differential search (DCS) algorithm, a hybrid multi-strategy variant, termed DQDCS, is proposed. This improved version overcomes these limitations by integrating the refined set strategy and clustering process for population initialization, along with the double Q-learning model to balance exploration and exploitation This enhanced version replaces the conventional pseudo-random initialization with a refined set generated through a clustering process, thereby significantly improving population diversity. A novel position update mechanism is introduced based on the original equation, enabling individuals to effectively escape from local optima during the iteration process. Additionally, the table reinforcement learning model (double Q-learning model) is integrated into the original algorithm to balance the probabilities between exploration and exploitation, thereby accelerating the convergence towards the global optimum. The effectiveness of each enhancement is validated through ablation studies, and the Wilcoxon rank-sum test is employed to assess the statistical significance of performance differences between DQDCS and other classical algorithms. Benchmark simulations are conducted using the CEC2019 and CEC2022 test functions, as well as two well-known constrained engineering design problems. The comparison includes both recent state-of-the-art algorithms and improved optimization methods. Simulation results demonstrate that the incorporation of the refined set and clustering process, along with the table reinforcement learning model (double Q-learning model) mechanism, leads to superior convergence speed and higher optimization precision.

To address the issues of uneven initial distribution and limited search accuracy in the traditional Divergent Quantum-inspired Differential Search (DCS) algorithm, a hybrid multi-strategy variant, termed DQDCS, is proposed. This improved version overcomes these limitations by integrating the refined set strategy and clustering process for population initialization, along with the Double Q-learning model to balance exploration and exploitation This enhanced version replaces the conventional pseudo-random initialization with a refined set generated through a clustering process, thereby significantly improving population diversity. A novel position update mechanism is introduced based on the original Equation, enabling individuals to effectively escape from local optima during the iteration process. Additionally, the Tables Reinforcement Learning model (Double Q-learning model) is integrated into the original algorithm to balance the probabilities between exploration and exploitation, thereby accelerating the convergence towards the global optimum. The effectiveness of each enhancement is validated through ablation studies, and the Wilcoxon rank-sum test is employed to assess the statistical significance of performance differences between DQDCS and other classical algorithms. Benchmark simulations are conducted using the CEC2019 and CEC2022 test functions, as well as two well-known constrained engineering design problems. The comparison includes both recent state-of-the-art algorithms and improved optimization methods. Simulation results demonstrate that the incorporation of the refined set and clustering process, along with the Tables Reinforcement Learning model (Double Q-learning model) mechanism, leads to superior convergence speed and higher optimization precision.

This paper present seven uniformly smooth approximating function for absolute value function: five of them approximate absolute value function from above, and the others approximate absolute value function from below. The properties of these uniformly smooth approximating functions are studied, and approximation degree are analyzed in theory and demonstrated by images. Finally, application prospect of uniformly smooth approximating function is pointed out.

This paper presents an efficient numerical manifold method for solving the Burgers’ equation. To improve accuracy and streamline the solution process, we apply a nonlinear function transformation technique that reformulates the original problem into a linear diffusion equation. We utilize a dual cover mesh along with an explicit multi-step time integration method for spatial and temporal discretization, respectively. Constant cover functions are employed across mathematical covers, interconnected by a linear weight function for each manifold element. The full discretization formulation is derived using the Galerkin weak form. To efficiently compute the inverse of the symmetric positive definite mass matrix, we employ the Crout algorithm. The performance and convergence of our method are rigorously evaluated through several benchmark numerical tests. Extensive comparisons with exact solutions and alternative methods demonstrate that our approach delivers an accurate, stable, and efficient computational scheme for the Burgers’ equation.

In this study, an improved discontinuous deformation analysis method with sub-block strategy is introduced to numerically simulate mixed-mode fractures. This approach partitions the material domain into continuum and potential discontinuum regions, applying specialized modeling techniques to each. In the continuum region, penalty-like bonding springs are employed to glue the sub-blocks together to capture the elastic behavior of the material. In the potential discontinuum region, the cohesive springs with the stiffness based on the cohesive zone model are implemented between sub-blocks to reproduce the process of crack nucleation and propagation. The primary advantage of this method is its capability to effectively model the transition of quasi-brittle solids from a continuous to a discontinuous stage through the degradation of cohesive springs. This accurately represents material failure while maintaining stability and consistency along uncracked interfaces. Another significant benefit is the method’s efficiency, as it avoids complex contact operations along sub-block interfaces before the cohesive spring between them fails. Validation through various benchmark numerical examples, such as cantilever beam-bending and diverse fracture simulations, demonstrates the method’s accuracy and robustness by comparing the results with analytical solutions. These comparisons show that the proposed method effectively captures the interplay between tensile and shear traction components in the mixed-mode crack propagation process.

This paper presents a nonstandard numerical manifold method (NMM) for solving Burgers’ equation. Employing the characteristic Galerkin method, we initially apply the Crank–Nicolson method for temporal discretization along the characteristic. Subsequently, utilizing the Taylor expansion, we transform the semi-implicit formula into a fully explicit form. For spacial discretization, we construct the NMM dual-cover system tailored to Burgers’ equation. We choose constant cover functions and first-order weight functions to enhance computational efficiency and exactly import boundary constraints. Finally, the integrated computing scheme is derived by using the standard Galerkin method, along with a Thomas algorithm-based solution procedure. The proposed method is verified through six benchmark numerical examples under various initial boundary conditions. Extensive comparisons with analytical solutions and results from alternative methods are conducted, demonstrating the accuracy and stability of our approach, particularly in solving Burgers’ equation at high Reynolds numbers.

This paper presents a novel variant of the teaching–learning-based optimization algorithm, termed BLTLBO, which draws inspiration from the blended learning model, specifically designed to tackle high-dimensional multimodal complex optimization problems. Firstly, the perturbation conditions in the “teaching” and “learning” stages of the original TLBO algorithm are interpreted geometrically, based on which the search capability of the TLBO is enhanced by adjusting the range of values of random numbers. Second, a strategic restructuring has been ingeniously implemented, dividing the algorithm into three distinct phases: pre-course self-study, classroom blended learning, and post-course consolidation; this structural reorganization and the random crossover strategy in the self-learning phase effectively enhance the global optimization capability of TLBO. To evaluate its performance, the BLTLBO algorithm was tested alongside seven distinguished variants of the TLBO algorithm on thirteen multimodal functions from the CEC2014 suite. Furthermore, two excellent high-dimensional optimization algorithms were added to the comparison algorithm and tested in high-dimensional mode on five scalable multimodal functions from the CEC2008 suite. The empirical results illustrate the BLTLBO algorithm’s superior efficacy in handling high-dimensional multimodal challenges. Finally, a high-dimensional portfolio optimization problem was successfully addressed using the BLTLBO algorithm, thereby validating the practicality and effectiveness of the proposed method.