This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials are derived first. Shifted Chebyshev polynomials are used to approximate the deflection function, and the nonlinear fractional order governing equation is expressed in the form of operator matrices. Next, the collocation method is used to discretize the equations into the form of algebraic equations for solution. It simplifies the calculation. MATLAB software was used to program this algorithm to simulate the numerical solution of the deflection. The effectiveness and accuracy of the algorithm are verified by the numerical example. Finally, numerical simulations are carried out on the viscoelastic micro-beams. It is found that the viscous damping coefficient is inversely proportional to the deflection, and the length scale parameter of the micro-beam is also inversely proportional to the deflection. In addition, the stress and strain of micro-beam, the change of deflection under different simple harmonic loads, and potential energy of micro-beam are discussed. The results of the study fully demonstrated that the shifted Chebyshev polynomial algorithm is effective for the numerical simulations of viscoelastic micro-beams.

This paper primarily investigates the dynamics response of viscoelastic orthotropic plates under a fractional-order derivative model, which is efficiently simulated numerically using the FKV (Fractional Kelvin–Voigt) model and the shifted Legendre polynomial algorithm. By establishing the fractional-order governing equation and directly solving it in the time domain using a shifted Legendre polynomial, the approach achieves low error and high accuracy. The analysis shows that the load, plate thickness, and creep time all affect the plate displacement, and the fractional-order model outperforms the integer-order model to better capture the dynamics response of the material.

A numerical scheme utilizing shifted Bernstein polynomials is developed to address the variable fractional-order governing equation in viscoelastic fluid-conveying pipes. The pipe’s mechanical response is characterized through a variable fractional-order Kelvin–Voigt (FKV) model, which effectively captures the time-dependent and memory properties of viscoelastic materials. By coupling the FKV constitutive model with the motion equation, the governing equation for a viscoelastic pipe is obtained. The deformation field is approximated using shifted Bernstein trial functions, leading to the construction of a derivative matrix with a variable fractional order. The obtained governing relation is expressed in matrix form, and after discretization, an algebraic system is formulated that is solvable in the time domain to evaluate the pipe displacement. Moreover, a convergence investigation is carried out to examine the reliability and effectiveness of the proposed framework. Computational results demonstrate that the introduced method delivers outstanding precision and performance, while the viscoelastic pipe’s response under different scenarios—including applied loads and fluid flow rates—is comprehensively investigated.

This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the Fractional Kelvin–Voigt (FKV) model with the Caputo fractional derivative and the theory of nonlocal elasticity. Then, the shifted Legendre polynomial is used to approximate the displacement function, and the governing equations are transformed into algebraic equations to facilitate the numerical solution in the time domain. Moreover, the systematic convergence analysis is carried out to verify the convergence of the ternary displacement function and its fractional derivatives in the equation, ensuring the rigor of the mathematical model. Finally, a dimensionless numerical example is given to verify the feasibility of the proposed algorithm, and the effects of material parameters on plate displacement are analyzed for double-layer plates with different materials.

This study presents a shifted Bernstein polynomial-based method for numerically solving the variable fractional order control equation governing a viscoelastic bar. Initially, employing a variable order fractional constitutive relation alongside the equation of motion, the control equation for the viscoelastic bar is derived. Shifted Bernstein polynomials serve as basis functions for approximating the bar’s displacement function, and the variable fractional derivative operator matrix is developed. Subsequently, the displacement control equation of the viscoelastic bar is transformed into the form of a matrix product. Substituting differential operators into the control equations, the control equations are discretized into algebraic equations by the method of matching points, which in turn allows the numerical solution of the displacement of the variable fractional viscoelastic bar control equation to be solved directly in the time domain. In addition, a convergence analysis is performed. Finally, algorithm precision and efficacy are confirmed via computation.

Against the backdrop of China’s dual-carbon goals—peaking carbon emissions by 2030 and achieving carbon neutrality by 2060—capacity optimization in the steel industry can no longer rely solely on aggregate output reduction. Instead, governance must shift toward a multidimensional approach that combines scale control, structural adjustment, and coordinated regional allocation. This study develops a quantifiable and interpretable assessment model for capacity-adjustment pressure. Monthly provincial crude steel output is used as a high-frequency proxy for capacity utilization and production adjustment. Additive time-series decomposition is applied to extract three components from monthly output—trend, residual, and volatility—which respectively represent structural evolution, short-term deviations, and exposure to shocks. The model further incorporates multidimensional variables, including downstream steel demand, resource and transport constraints, scrap steel ratio, and policy constraints. On this basis, a two-stage hybrid assessment framework is developed. In the first stage, extreme gradient boosting (XGBoost) is used to learn nonlinear relationships and derive data-driven feature importance. In the second stage, a composite pressure index is constructed and transformed into a standardized 0–100 score through a robust rank-based mapping mechanism. Dual thresholds are then used to generate three policy recommendations: maintaining current capacity, structural optimization, and capacity reduction. The results show that production trends and volatility intensity are the primary drivers of capacity-adjustment pressure, while pronounced spatial heterogeneity requires highly localized strategies. The classification assigns 25 provinces to maintaining current capacity, 3 to structural optimization, and 3 to targeted capacity reduction. Finally, integrating SHapley Additive exPlanations (SHAP) enhances model interpretability and provides a quantitative basis for shifting from indiscriminate capacity suppression toward differentiated, region-specific capacity governance, thereby supporting the sustainable low-carbon development of the global steel industry.