Applied Mathematics and Mechanics was founded in 1980 by CHIEN Wei-zang, a celebrated Chinese scientist in mechanics and mathematics. The current editor in chief is Professor LU Tianjian from Nanjing University of Aeronautics and Astronautics. The Journal was a quarterly in the beginning, a bimonthly the next year, and then a monthly ever since 1985. It carries original research papers on mechanics, mathematical methods in mechanics and interdisciplinary mechanics based on artificial intelligence mathematics. It also strengthens attention to mechanical issues in interdisciplinary fields such as mechanics and information networks, system control, life sciences, ecological sciences, new energy, and new materials, making due contributions to promoting the development of new productive forces.
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Display Method:
2025, Volume 46, Issue 1
publish date：January 01 2025
Display Method:
2025, 46(1).
doi: 10.21656/1000-0887.450340
Abstract:
2025, 46(1): 1-11.
doi: 10.21656/1000-0887.450036
Abstract:
For the elements of degree m(>1), simplified form solution u* based on the element energy projection (EEP) method has at least 1-order higher accuracy than conventional finite element solution uh.As a result, the EEP element, with simplified form EEP solution uh in as the final solution, was proposed, and a corresponding adaptive finite element analysis strategy for EEP elements was developed. By means of the developed algorithm, the 1D 2-point boundary value problem was analyzed, and the computation results are in good agreement with theoretical solutions, verifying the effectiveness and reliability of the proposed adaptivity strategy. The theoretical study and numerical experiments show that, the proposed method provides an EEP element solution satisfying the preset error tolerances in the maximum norm with fewer elements and less adaptive steps compared to conventional finite elements.
For the elements of degree m(>1), simplified form solution u* based on the element energy projection (EEP) method has at least 1-order higher accuracy than conventional finite element solution uh.As a result, the EEP element, with simplified form EEP solution uh in as the final solution, was proposed, and a corresponding adaptive finite element analysis strategy for EEP elements was developed. By means of the developed algorithm, the 1D 2-point boundary value problem was analyzed, and the computation results are in good agreement with theoretical solutions, verifying the effectiveness and reliability of the proposed adaptivity strategy. The theoretical study and numerical experiments show that, the proposed method provides an EEP element solution satisfying the preset error tolerances in the maximum norm with fewer elements and less adaptive steps compared to conventional finite elements.
2025, 46(1): 12-28.
doi: 10.21656/1000-0887.450038
Abstract:
The application of stress-based bidirectional evolutionary structural optimization (BESO) in the context of the modified couple stress elasticity theory was investigated. This methodology allows for structure optimization of homogenized continuum with a microstructural composition of size effects. The classical BESO technique was extended through the introduction of a novel formulation of couple stress based non-classical equivalent stress, and the minimization design was conducted under the constraint of volume criterion. The iterative update of design variables relies on the sensitivity analysis involving direct derivation of the enriched p-norm global stress with couple stress contributions. Since the high-order elasticity is involved, the FEM implementation requires at least the C1 nodal continuity. Thus, a Lagrangian finite element complemented by additional integration functions was implemented. The method was validated with 3 distinct cases through investigation of the size effects on the stress optimization and the subsequent structure design. The impacts of other parameters including the norm p value and the material volume fraction, were explored. The results demonstrate the potential of the proposed stress-based BESO method in addressing structural optimization of problems involving size effects.
The application of stress-based bidirectional evolutionary structural optimization (BESO) in the context of the modified couple stress elasticity theory was investigated. This methodology allows for structure optimization of homogenized continuum with a microstructural composition of size effects. The classical BESO technique was extended through the introduction of a novel formulation of couple stress based non-classical equivalent stress, and the minimization design was conducted under the constraint of volume criterion. The iterative update of design variables relies on the sensitivity analysis involving direct derivation of the enriched p-norm global stress with couple stress contributions. Since the high-order elasticity is involved, the FEM implementation requires at least the C1 nodal continuity. Thus, a Lagrangian finite element complemented by additional integration functions was implemented. The method was validated with 3 distinct cases through investigation of the size effects on the stress optimization and the subsequent structure design. The impacts of other parameters including the norm p value and the material volume fraction, were explored. The results demonstrate the potential of the proposed stress-based BESO method in addressing structural optimization of problems involving size effects.
2025, 46(1): 29-39.
doi: 10.21656/1000-0887.450071
Abstract:
Based on the modal space time-domain precise integration, a dynamic parameter identification method was proposed. Firstly, an identification model was constructed based on the time-domain measurement signals and the theoretical prediction model with the time-domain precise integration method in the modal space. Secondly, the quadratic function of the unconstrained vector was derived through the Kronecker product of matrices, and the mathematical expressions of the mode shapes were analyzed and given. Finally, through mathematical transformations of the identification optimization problem, only the dynamics spectrum parameters (frequencies and damping ratios) need be identified, to greatly reduce the dimensionality of the identification parameters. In numerical examples, the dynamic parameter identification for the spring-mass system and the high-speed pantograph system were studied. The identified natural frequencies and damping ratios have errors less than 8% compared to the theoretical values. The cosine of the angle between the identified and the theoretical mode shapes is close to 1, which verifies the accuracy of the identification results. The proposed method can effectively achieve the separation of dynamic spectral parameters (frequencies, damping ratios) and spatial parameters (modal shapes), and has better solving efficiency and application prospects.
Based on the modal space time-domain precise integration, a dynamic parameter identification method was proposed. Firstly, an identification model was constructed based on the time-domain measurement signals and the theoretical prediction model with the time-domain precise integration method in the modal space. Secondly, the quadratic function of the unconstrained vector was derived through the Kronecker product of matrices, and the mathematical expressions of the mode shapes were analyzed and given. Finally, through mathematical transformations of the identification optimization problem, only the dynamics spectrum parameters (frequencies and damping ratios) need be identified, to greatly reduce the dimensionality of the identification parameters. In numerical examples, the dynamic parameter identification for the spring-mass system and the high-speed pantograph system were studied. The identified natural frequencies and damping ratios have errors less than 8% compared to the theoretical values. The cosine of the angle between the identified and the theoretical mode shapes is close to 1, which verifies the accuracy of the identification results. The proposed method can effectively achieve the separation of dynamic spectral parameters (frequencies, damping ratios) and spatial parameters (modal shapes), and has better solving efficiency and application prospects.
2025, 46(1): 92-104.
doi: 10.21656/1000-0887.450066
Abstract:
The transport infrastructure is the foundation of modern social and economic development, where the asphalt pavement plays an important role as a key component. Accurate prediction of asphalt pavement conditions is of great significance to guide pavement maintenance work. Rutting is an important indicator for evaluating the health condition of asphalt pavement. Existing asphalt pavement condition prediction models are mainly based on mechanical experience models or machine learning technologies. However, these methods lack interpretability and cannot provide relevant information on the extent to which the input features affect rutting. Herein, an interpretable integrated learning framework (FI-EL-SHAP) was established, in which the FI module filters features with the entropy weight method and the Pareto analysis, the EL module evaluates the performances of different models and selects the optimal model, and the SHAP module performs visual analysis on the relationship between input features and model outputs to reveal the impacts of different features on model prediction results. This study realizes a quantitative analysis of the rut formation mechanism while ensuring the model accuracy.
The transport infrastructure is the foundation of modern social and economic development, where the asphalt pavement plays an important role as a key component. Accurate prediction of asphalt pavement conditions is of great significance to guide pavement maintenance work. Rutting is an important indicator for evaluating the health condition of asphalt pavement. Existing asphalt pavement condition prediction models are mainly based on mechanical experience models or machine learning technologies. However, these methods lack interpretability and cannot provide relevant information on the extent to which the input features affect rutting. Herein, an interpretable integrated learning framework (FI-EL-SHAP) was established, in which the FI module filters features with the entropy weight method and the Pareto analysis, the EL module evaluates the performances of different models and selects the optimal model, and the SHAP module performs visual analysis on the relationship between input features and model outputs to reveal the impacts of different features on model prediction results. This study realizes a quantitative analysis of the rut formation mechanism while ensuring the model accuracy.
2025, 46(1): 105-113.
doi: 10.21656/1000-0887.450122
Abstract:
Physics-informed neural networks (PINNs) were applied in combination with generalized Miura transformations to investigate 3 KdV-type equations. Several novel soliton solutions, including the kink-bell solution of the mKdV equation, were derived analytically with the improved PINN method; a single-soliton-like solution of the KdV equation, was achieved through the Miura transformation; and a dark-soliton solution of a strongly nonlinear KdV equation, was obtained by means of both the generalized Miura transformation and the PINN methods. Comparison of the numerical results obtained under the PINN framework with the exact solutions from theoretical analysis shows that, the proposed algorithm effectively uncovers new numerical solutions of partial differential equations and offers valuable insights for theoretical research.
Physics-informed neural networks (PINNs) were applied in combination with generalized Miura transformations to investigate 3 KdV-type equations. Several novel soliton solutions, including the kink-bell solution of the mKdV equation, were derived analytically with the improved PINN method; a single-soliton-like solution of the KdV equation, was achieved through the Miura transformation; and a dark-soliton solution of a strongly nonlinear KdV equation, was obtained by means of both the generalized Miura transformation and the PINN methods. Comparison of the numerical results obtained under the PINN framework with the exact solutions from theoretical analysis shows that, the proposed algorithm effectively uncovers new numerical solutions of partial differential equations and offers valuable insights for theoretical research.
2025, 46(1): 114-128.
doi: 10.21656/1000-0887.450013
Abstract:
A finite element (FE) method based on the generalized backward differentiation θ formula (generalized BDF2-θ) was presented to solve nonlinear distributed-order time-fractional hyperbolic wave equations. The temporal direction was approximated with the generalized BDF2-θ to get the FE fully discrete scheme. The proposed model with high-order temporal derivatives was transformed into a coupled system including 2 lower-order equations. The stability of the proposed FE scheme and the optimal error estimates for 2 functions u and p were discussed. Several numerical examples indicate the feasibility and efficiency of the schemes.
A finite element (FE) method based on the generalized backward differentiation θ formula (generalized BDF2-θ) was presented to solve nonlinear distributed-order time-fractional hyperbolic wave equations. The temporal direction was approximated with the generalized BDF2-θ to get the FE fully discrete scheme. The proposed model with high-order temporal derivatives was transformed into a coupled system including 2 lower-order equations. The stability of the proposed FE scheme and the optimal error estimates for 2 functions u and p were discussed. Several numerical examples indicate the feasibility and efficiency of the schemes.
2025, 46(1): 40-54.
doi: 10.21656/1000-0887.450167
Abstract:
Due to low computational power consumption and high efficiency, GPUs/TPUs/NPUs with single/half-precision computing units make the main computing mode for artificial intelligence, but they can’t be directly applied to solve differential equations requiring high floating-point accuracy, nor can they directly replace double-precision units. With the combined advantages of single and double precisions, a mixed-precision solution scheme balancing efficiency and accuracy, was proposed for large sparse linear equations. The sparse GMRES-IR algorithm for large sparse matrices was developed. Firstly, the characteristics of matrix data distributions in fluid dynamics simulation problems were analyzed. With double precision for pre-processing and single precision for detailed iteration, the single precision calculation was applied to the main time-consuming part of the algorithm, to enhance computational efficiency. Solutions of 33 linear equation systems from open-source datasets validate the accuracy and efficiency of the proposed method. The results show that, on a single-core CPU, under the same accuracy requirements, the proposed mixed-precision algorithm can achieve an acceleration effect of up to 2.5 times, and the effect is more prominent for large-scale matrices.
Due to low computational power consumption and high efficiency, GPUs/TPUs/NPUs with single/half-precision computing units make the main computing mode for artificial intelligence, but they can’t be directly applied to solve differential equations requiring high floating-point accuracy, nor can they directly replace double-precision units. With the combined advantages of single and double precisions, a mixed-precision solution scheme balancing efficiency and accuracy, was proposed for large sparse linear equations. The sparse GMRES-IR algorithm for large sparse matrices was developed. Firstly, the characteristics of matrix data distributions in fluid dynamics simulation problems were analyzed. With double precision for pre-processing and single precision for detailed iteration, the single precision calculation was applied to the main time-consuming part of the algorithm, to enhance computational efficiency. Solutions of 33 linear equation systems from open-source datasets validate the accuracy and efficiency of the proposed method. The results show that, on a single-core CPU, under the same accuracy requirements, the proposed mixed-precision algorithm can achieve an acceleration effect of up to 2.5 times, and the effect is more prominent for large-scale matrices.
2025, 46(1): 55-71.
doi: 10.21656/1000-0887.450098
Abstract:
In recent years, the deep learning method incorporating physical information provided a new idea for solving partial differential equations. However, most of the studies so far has low computational accuracy and poor time extrapolation for problems with discontinuities in the solution space. To address the above 2 problems, the ROE-PIGNN model was proposed for fusing equations or data information with the graph neural networks and the ROE scheme in computational fluid dynamics. Numerical experiments show that, the model achieves a computational accuracy comparable to that of the ROE scheme in solving the shock tube problem controlled by the Eulerian equation, and has the ability of extrapolation over a certain time range. Finally, the 2D cylindrical bypass flow traditional problem controlled by the Navier-Stokes (NS) equations was solved. The experimental results show that, the model can predict the subsequent periodic flow and reproduce the flow structure more accurately at some key positions, with an error reduction of 60% compared to the purely data-driven approach.
In recent years, the deep learning method incorporating physical information provided a new idea for solving partial differential equations. However, most of the studies so far has low computational accuracy and poor time extrapolation for problems with discontinuities in the solution space. To address the above 2 problems, the ROE-PIGNN model was proposed for fusing equations or data information with the graph neural networks and the ROE scheme in computational fluid dynamics. Numerical experiments show that, the model achieves a computational accuracy comparable to that of the ROE scheme in solving the shock tube problem controlled by the Eulerian equation, and has the ability of extrapolation over a certain time range. Finally, the 2D cylindrical bypass flow traditional problem controlled by the Navier-Stokes (NS) equations was solved. The experimental results show that, the model can predict the subsequent periodic flow and reproduce the flow structure more accurately at some key positions, with an error reduction of 60% compared to the purely data-driven approach.
2025, 46(1): 72-83.
doi: 10.21656/1000-0887.450137
Abstract:
Under the adjustment of solution pH values and background salt concentrations, the electroosmotic flows of the Powell-Eyring fluids in parallel plate nanochannels were studied with the homotopic perturbation method, and approximate solutions were obtained. The accuracy of the obtained approximate solution was verified with the Chebyshev spectrum configuration method. On this basis, the effects of dimensionless pressure gradient G, background salt concentration MKCI, the pH value, and the viscosity ratio γ of the Powell-Eyring fluid and the Newtonian fluid, on velocity profile u and volume flow rate (average velocity)Q, were studied. The results demonstrate that, the homotopy perturbation method converges rapidly, requiring only an expansion up to the 1st-order solution to perfectly match the numerical solution. Meanwhile,MKCI,pH,γ and G have significant effects on the charge density and the electroosmotic flow velocity of the Powell-Eyring fluid in the nanochannel.
Under the adjustment of solution pH values and background salt concentrations, the electroosmotic flows of the Powell-Eyring fluids in parallel plate nanochannels were studied with the homotopic perturbation method, and approximate solutions were obtained. The accuracy of the obtained approximate solution was verified with the Chebyshev spectrum configuration method. On this basis, the effects of dimensionless pressure gradient G, background salt concentration MKCI, the pH value, and the viscosity ratio γ of the Powell-Eyring fluid and the Newtonian fluid, on velocity profile u and volume flow rate (average velocity)Q, were studied. The results demonstrate that, the homotopy perturbation method converges rapidly, requiring only an expansion up to the 1st-order solution to perfectly match the numerical solution. Meanwhile,MKCI,pH,γ and G have significant effects on the charge density and the electroosmotic flow velocity of the Powell-Eyring fluid in the nanochannel.
2025, 46(1): 84-91.
doi: 10.21656/1000-0887.450010
Abstract:
Aimed at the anomalous diffusion behavior in porous medium, a time fractional diffusion model with variable coefficients was proposed to simulate the anomalous diffusion of methane in coal medium. The time fractional fractal diffusion model with constant coefficients was extended to the case with variable coefficients, the numerical scheme based on graded meshing of the fractional model with variable coefficients was established. Based on the numerical solution and experimental data, an efficient cuckoo search algorithm was proposed to estimate several important parameters in the model. Finally, the effectiveness of the fractional diffusion model with variable coefficients and the cuckoo search algorithm in studying the direct and inverse problems of anomalous diffusion in porous media, was verified with numerical experiments.
Aimed at the anomalous diffusion behavior in porous medium, a time fractional diffusion model with variable coefficients was proposed to simulate the anomalous diffusion of methane in coal medium. The time fractional fractal diffusion model with constant coefficients was extended to the case with variable coefficients, the numerical scheme based on graded meshing of the fractional model with variable coefficients was established. Based on the numerical solution and experimental data, an efficient cuckoo search algorithm was proposed to estimate several important parameters in the model. Finally, the effectiveness of the fractional diffusion model with variable coefficients and the cuckoo search algorithm in studying the direct and inverse problems of anomalous diffusion in porous media, was verified with numerical experiments.
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Founded in 1980 ( monthly )
Supervisor：Chongqing Jiaotong University
Founder：CHIEN Weizang
Honorary Chief Editor：ZHONG Wanxie
Editor-in-Chief：LU Tianjian
ISSN 1000-0887
CN 50-1060/O3
