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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2026, Volume 47, Issue 7
publish date：July 01 2026
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2026, 47(7): 825-844.
doi: 10.21656/1000-0887.460219
Abstract:
Core analysis is the cornerstone of reservoir evaluation. The digital rock physics (DRP) enables the quantitative characterization of reservoir microstructures and flow properties through 3D imaging and numerical simulation. The evolution of DRP paradigms was reviewed from the 1stgeneration “What you see is what you get” (geometric characterization) and the 2ndgeneration “What you calculate is what you get” (numerical simulation), to the emerging 3rdgeneration “What you learn is what you get” (intelligent prediction). The 1st 2 paradigms face 2 core bottlenecks: the difficulty in determining the representative elementary volume (REV) and the prohibitive computational costs of numerical simulations. The progress of the 3rdgeneration intelligent prediction paradigm was systematically summarized, with the focus on structural properties, singlephase permeability, and multiphase flow properties (relative permeability, capillary pressure, and wettability). Centered on deep learning, this paradigm offers revolutionary pathways to overcome traditional limitations: on one hand, the generative adversarial networks (GANs) and the superresolution (SR) techniques were employed for datadriven reconstruction to mitigate REV challenges; on the other hand, the efficient surrogate models, such as the convolutional neural networks (CNNs), were constructed to accelerate property prediction by orders of magnitude. Finally, the current challenges regarding data dependence, model generalization, and physical interpretability were discussed to outlines future directions, including integrating physical mechanisms into AI models (physicsinformed AI), promoting multiscale data fusion, and constructing reservoir “digital twins” to support the development of smart oilfields.
Core analysis is the cornerstone of reservoir evaluation. The digital rock physics (DRP) enables the quantitative characterization of reservoir microstructures and flow properties through 3D imaging and numerical simulation. The evolution of DRP paradigms was reviewed from the 1stgeneration “What you see is what you get” (geometric characterization) and the 2ndgeneration “What you calculate is what you get” (numerical simulation), to the emerging 3rdgeneration “What you learn is what you get” (intelligent prediction). The 1st 2 paradigms face 2 core bottlenecks: the difficulty in determining the representative elementary volume (REV) and the prohibitive computational costs of numerical simulations. The progress of the 3rdgeneration intelligent prediction paradigm was systematically summarized, with the focus on structural properties, singlephase permeability, and multiphase flow properties (relative permeability, capillary pressure, and wettability). Centered on deep learning, this paradigm offers revolutionary pathways to overcome traditional limitations: on one hand, the generative adversarial networks (GANs) and the superresolution (SR) techniques were employed for datadriven reconstruction to mitigate REV challenges; on the other hand, the efficient surrogate models, such as the convolutional neural networks (CNNs), were constructed to accelerate property prediction by orders of magnitude. Finally, the current challenges regarding data dependence, model generalization, and physical interpretability were discussed to outlines future directions, including integrating physical mechanisms into AI models (physicsinformed AI), promoting multiscale data fusion, and constructing reservoir “digital twins” to support the development of smart oilfields.
2026, 47(7): 845-857.
doi: 10.21656/1000-0887.460076
Abstract:
The non-Fourier heat transfer characteristics of couple stress fluid in a microchannel were investigated, and effects of the couple stress parameters, the Hartmann number, the Joule heating effect and the 2-phase lag time parameters on the temperature distributions and heat transfer characteristics of the fluid were analyzed through establishment of the relevant governing equations combined with the non-Fourier heat transfer model. The results show that, the above parameters significantly influence the fluidity and heat diffusion characteristics of the couple stress fluid. The increase of the couple stress parameter aggravates the temperature gradient; the Hartmann number enhances the magnetic field confinement and inhibits the heat conduction; the Joule heating effect promotes the temperature gradient and highlights the non-Fourier effect; and the change of the biphasic hysteresis time parameter has a significant modulation effect on the temperature peak and the response speed.
The non-Fourier heat transfer characteristics of couple stress fluid in a microchannel were investigated, and effects of the couple stress parameters, the Hartmann number, the Joule heating effect and the 2-phase lag time parameters on the temperature distributions and heat transfer characteristics of the fluid were analyzed through establishment of the relevant governing equations combined with the non-Fourier heat transfer model. The results show that, the above parameters significantly influence the fluidity and heat diffusion characteristics of the couple stress fluid. The increase of the couple stress parameter aggravates the temperature gradient; the Hartmann number enhances the magnetic field confinement and inhibits the heat conduction; the Joule heating effect promotes the temperature gradient and highlights the non-Fourier effect; and the change of the biphasic hysteresis time parameter has a significant modulation effect on the temperature peak and the response speed.
2026, 47(7): 858-868.
doi: 10.21656/1000-0887.460162
Abstract:
With the widespread implementation of zero-output transformation of low-pressure cylinders in heating units, the low-pressure cylinders of steam turbines often need to operate at extremely low flow rates. To get a deeper understanding of the flow behavior under this working condition, the low-pressure cylinder of a steam turbine in a certain power plant was studied, a numerical calculation model for the final-stage flow channel was constructed, and the structure of the final-stage flow field and the temperature changes on the surface of the moving blades were inspected through simulation analysis of the flow state and aerodynamic performance in the cylinder under variable working conditions, especially at low flow rates. The research results indicate that, at low flow rates, the gas separation and backflow will occur in the final stage of the low-pressure cylinder. The separation initially occurs at the moving blade root and gradually spreads to the blade top as the flow rate decreases. Local vortices emerge in the channel, and a negative attack angle appears at the moving blade inlet, which significantly hinders the normal steam flow. When the load drops to 15%THA, a local high-temperature zone will appear at the steam outlet edge on the top of the last stage stator blade, showing the blower heating effect. As the flow rate further decreases, the high-temperature area will keep expanding and the maximum temperature will continue to rise. When the load decreases to 10%THA, the maximum surface temperature of the moving blade will rise by 40.29% compared with the rated working condition. This research provides a reference basis for the safe operation of the low-pressure cylinder of the heating unit after the zero-output transformation.
With the widespread implementation of zero-output transformation of low-pressure cylinders in heating units, the low-pressure cylinders of steam turbines often need to operate at extremely low flow rates. To get a deeper understanding of the flow behavior under this working condition, the low-pressure cylinder of a steam turbine in a certain power plant was studied, a numerical calculation model for the final-stage flow channel was constructed, and the structure of the final-stage flow field and the temperature changes on the surface of the moving blades were inspected through simulation analysis of the flow state and aerodynamic performance in the cylinder under variable working conditions, especially at low flow rates. The research results indicate that, at low flow rates, the gas separation and backflow will occur in the final stage of the low-pressure cylinder. The separation initially occurs at the moving blade root and gradually spreads to the blade top as the flow rate decreases. Local vortices emerge in the channel, and a negative attack angle appears at the moving blade inlet, which significantly hinders the normal steam flow. When the load drops to 15%THA, a local high-temperature zone will appear at the steam outlet edge on the top of the last stage stator blade, showing the blower heating effect. As the flow rate further decreases, the high-temperature area will keep expanding and the maximum temperature will continue to rise. When the load decreases to 10%THA, the maximum surface temperature of the moving blade will rise by 40.29% compared with the rated working condition. This research provides a reference basis for the safe operation of the low-pressure cylinder of the heating unit after the zero-output transformation.
2026, 47(7): 869-885.
doi: 10.21656/1000-0887.460093
Abstract:
The actuation hysteresis effect caused by cable-hole sliding friction is one of the critical challenges limiting the multi-functional and multi-domain applications of cable-driven continuum robots. During trajectory tracking, the driving cables undergo frequent switching between winding and unwinding, leading to repeated reversals of sliding friction within the guiding holes. As a result, the current configuration of the flexible arm varies significantly depending on the actuation history. A quasi-static model for the cable-driven continuum robot was developed based on the dynamics theory of flexible multibody systems. A nodal force equilibrium equation with decoupled rigid and flexible motions was established, and the tangent stiffness matrices for both nodal forces and cable driving forces were rigorously derived to enhance the convergence rate of the equilibrium equation. The trajectory tracking problem was formulated as a servo constraint problem in multibody systems. A nonlinear least-squares solver incorporating the Newton-Raphson iterative algorithm was employed to conduct path tracking simulations under both loaded and unloaded end-effector conditions. The configuration evolution law of the continuum robot was thereby obtained.
The actuation hysteresis effect caused by cable-hole sliding friction is one of the critical challenges limiting the multi-functional and multi-domain applications of cable-driven continuum robots. During trajectory tracking, the driving cables undergo frequent switching between winding and unwinding, leading to repeated reversals of sliding friction within the guiding holes. As a result, the current configuration of the flexible arm varies significantly depending on the actuation history. A quasi-static model for the cable-driven continuum robot was developed based on the dynamics theory of flexible multibody systems. A nodal force equilibrium equation with decoupled rigid and flexible motions was established, and the tangent stiffness matrices for both nodal forces and cable driving forces were rigorously derived to enhance the convergence rate of the equilibrium equation. The trajectory tracking problem was formulated as a servo constraint problem in multibody systems. A nonlinear least-squares solver incorporating the Newton-Raphson iterative algorithm was employed to conduct path tracking simulations under both loaded and unloaded end-effector conditions. The configuration evolution law of the continuum robot was thereby obtained.
2026, 47(7): 886-894.
doi: 10.21656/1000-0887.460155
Abstract:
Based on the Biot-type wave equations for unsaturated soils, the transient responses of 1D unsaturated soil columns subjected to dynamic loading were investigated, to analyze wave propagation characteristics in unsaturated soils and derive analytical solutions to provide a theoretical basis for relevant engineering problems. The effects of saturation on both the permeability coefficient and the dynamic shear modulus were incorporated into the governing equations. With the separation of variables method plus the state space method, a systematic analytical solution of the transient response was obtained. The validity of the solution was verified through comparison with the finite element results from COMSOL’s PDE module. The results indicate that, only 1 compression wave occurs in unsaturated soil due to its relatively low permeability. As the intrinsic permeability increases, the peak transient pore water pressure response will gradually decreases. Furthermore, an increase in the saturation leads to a significant rise in the amplitude of the pore water pressure response. Both the saturation and the permeability considerably influence the wave-induced responses in unsaturated soils. The proposed analytical method effectively predicts transient response characteristics and provides a theoretical support for analyzing the dynamic behaviors of unsaturated soils.
Based on the Biot-type wave equations for unsaturated soils, the transient responses of 1D unsaturated soil columns subjected to dynamic loading were investigated, to analyze wave propagation characteristics in unsaturated soils and derive analytical solutions to provide a theoretical basis for relevant engineering problems. The effects of saturation on both the permeability coefficient and the dynamic shear modulus were incorporated into the governing equations. With the separation of variables method plus the state space method, a systematic analytical solution of the transient response was obtained. The validity of the solution was verified through comparison with the finite element results from COMSOL’s PDE module. The results indicate that, only 1 compression wave occurs in unsaturated soil due to its relatively low permeability. As the intrinsic permeability increases, the peak transient pore water pressure response will gradually decreases. Furthermore, an increase in the saturation leads to a significant rise in the amplitude of the pore water pressure response. Both the saturation and the permeability considerably influence the wave-induced responses in unsaturated soils. The proposed analytical method effectively predicts transient response characteristics and provides a theoretical support for analyzing the dynamic behaviors of unsaturated soils.
2026, 47(7): 895-911.
doi: 10.21656/1000-0887.460129
Abstract:
The physical information neural networks (PINNs) are an important tool for solving partial differential equations. In the numerical solution of partial differential equations, equations with discontinuous solutions are currently a research challenge. The PINNs and their existing improved algorithms often cannot capture the characteristics of discontinuities. However, many discontinuous problems need to be considered in the field of fluid mechanics. In response to the shortcomings of the PINNs in handling equations with discontinuous solutions, an adaptive gradient-annihilated PINNs (AGA-PINNs) method was proposed to solve the Burgers equations and the Allen-Cahn equations with discontinuous solutions. The gradient related weight functions were utilized to optimize the loss function, and the training points were dynamically adjusted based on the current residual distribution during the training process to gradually enhance the model’s learning ability in discontinuous regions. The experimental results show that, the AGA-PINNs method significantly improves the accuracy of predicted physical quantities compared to the traditional PINNs, the gradient-enhanced physics-informed neural networks (gPINNs), the gradient-annihilated physics-informed neural networks (GA-PINNs) method, and the high-order deep Galerkin （DG）method. In solving the Burgers equation and the Allen-Cahn equation, the mean square errors decrease by approximately 1 order of magnitude, accurately reproducing the characteristics of shock waves in the Burgers equation and the phase separation phenomena in the Allen-Cahn equation.
The physical information neural networks (PINNs) are an important tool for solving partial differential equations. In the numerical solution of partial differential equations, equations with discontinuous solutions are currently a research challenge. The PINNs and their existing improved algorithms often cannot capture the characteristics of discontinuities. However, many discontinuous problems need to be considered in the field of fluid mechanics. In response to the shortcomings of the PINNs in handling equations with discontinuous solutions, an adaptive gradient-annihilated PINNs (AGA-PINNs) method was proposed to solve the Burgers equations and the Allen-Cahn equations with discontinuous solutions. The gradient related weight functions were utilized to optimize the loss function, and the training points were dynamically adjusted based on the current residual distribution during the training process to gradually enhance the model’s learning ability in discontinuous regions. The experimental results show that, the AGA-PINNs method significantly improves the accuracy of predicted physical quantities compared to the traditional PINNs, the gradient-enhanced physics-informed neural networks (gPINNs), the gradient-annihilated physics-informed neural networks (GA-PINNs) method, and the high-order deep Galerkin （DG）method. In solving the Burgers equation and the Allen-Cahn equation, the mean square errors decrease by approximately 1 order of magnitude, accurately reproducing the characteristics of shock waves in the Burgers equation and the phase separation phenomena in the Allen-Cahn equation.
2026, 47(7): 912-923.
doi: 10.21656/1000-0887.460125
Abstract:
Physics-informed neural network (PINN) are widely applied for solving both forward and inverse problems of differential equations. However, traditional PINNs exhibit limitations when solving differential equations over large domains, including insufficient accuracy, susceptibility to local optima, and a lack of theoretical guarantees of convergence. To address these issues, a residual-based Fourier neural network (Res-FNN) was proposed. This network integrates residual Fourier layers into the traditional PINN framework, leveraging the periodicity property of trigonometric functions to enhance the model’s accuracy in solving large-domain problems effectively. In particular, a theoretical convergence analysis was established for the Res-FNN applied to linear elliptic differential equations. Numerical experiments demonstrate that, the Res-FNN outperforms the traditional PINN in solving both forward and inverse problems, achieving higher accuracy and faster convergence speed.
Physics-informed neural network (PINN) are widely applied for solving both forward and inverse problems of differential equations. However, traditional PINNs exhibit limitations when solving differential equations over large domains, including insufficient accuracy, susceptibility to local optima, and a lack of theoretical guarantees of convergence. To address these issues, a residual-based Fourier neural network (Res-FNN) was proposed. This network integrates residual Fourier layers into the traditional PINN framework, leveraging the periodicity property of trigonometric functions to enhance the model’s accuracy in solving large-domain problems effectively. In particular, a theoretical convergence analysis was established for the Res-FNN applied to linear elliptic differential equations. Numerical experiments demonstrate that, the Res-FNN outperforms the traditional PINN in solving both forward and inverse problems, achieving higher accuracy and faster convergence speed.
2026, 47(7): 924-935.
doi: 10.21656/1000-0887.460107
Abstract:
The modified Allen-Cahn equation with the mean curvature source term can effectively model curvature-driven physical processes. However, the nonlinear term and gradient magnitude make it difficult to design efficient numerical schemes. An efficient numerical scheme was proposed for solving the equation. Based on the Strang operator splitting method, the original equation was discretized in time into three subproblems: the nonlinear equation was solved analytically; the mean curvature equation was discretized using the second-order Runge-Kutta method combined with central differences to establish a fully discrete explicit scheme, while the heat equation was discretized with the Crank-Nicolson method and solved via the alternating direction implicit (ADI) fast scheme. Theoretical analysis shows that, the proposed scheme achieves second order convergence accuracy. Finally, numerical experiments were presented to validate the convergence rate and effectiveness of the scheme.
The modified Allen-Cahn equation with the mean curvature source term can effectively model curvature-driven physical processes. However, the nonlinear term and gradient magnitude make it difficult to design efficient numerical schemes. An efficient numerical scheme was proposed for solving the equation. Based on the Strang operator splitting method, the original equation was discretized in time into three subproblems: the nonlinear equation was solved analytically; the mean curvature equation was discretized using the second-order Runge-Kutta method combined with central differences to establish a fully discrete explicit scheme, while the heat equation was discretized with the Crank-Nicolson method and solved via the alternating direction implicit (ADI) fast scheme. Theoretical analysis shows that, the proposed scheme achieves second order convergence accuracy. Finally, numerical experiments were presented to validate the convergence rate and effectiveness of the scheme.
2026, 47(7): 936-958.
doi: 10.21656/1000-0887.460121
Abstract:
The parameterized dual-continuum model plays a significant theoretical and practical role in various subsurface geological modeling applications. This model effectively captures the highly heterogeneous, high-contrast, and multiscale structural features of geological formations, while also exhibiting considerable uncertainty. To address the numerical challenges posed by such complex models, adopting appropriate model reduction techniques has become a key approach to improving computational efficiency while maintaining solution accuracy. Herein an iterative solution strategy was proposed based on the uncoupled generalized multiscale finite element method (GMsFEM) for the parameterized dual-continuum model. The original parameter-dependent model was reformulated as a new one consisting of multiscale diffusion coefficients and transfer functions (both parameter-independent), along with a parameter-dependent source term. The proposed iterative method was divided into offline and online stages. In the offline stage, multiscale basis functions were constructed in each coarse grid block based on deterministic multiscale parameters to generate a reduced-order space. In the online stage, the model was solved efficiently within the reduced space with an iterative scheme. A major advantage of this method lies in the fact that, once the multiscale basis space is constructed offline, each online iteration can leverage efficient direct solvers and reuse matrix inverses, thus significantly reducing computational costs. Furthermore, the convergence of the proposed iterative method was analyzed. Finally, numerical experiments on the parameterized dual-continuum model were conducted to demonstrate the effectiveness and efficiency of the multiscale-based iterative approach, and validate the theoretical convergence results.
The parameterized dual-continuum model plays a significant theoretical and practical role in various subsurface geological modeling applications. This model effectively captures the highly heterogeneous, high-contrast, and multiscale structural features of geological formations, while also exhibiting considerable uncertainty. To address the numerical challenges posed by such complex models, adopting appropriate model reduction techniques has become a key approach to improving computational efficiency while maintaining solution accuracy. Herein an iterative solution strategy was proposed based on the uncoupled generalized multiscale finite element method (GMsFEM) for the parameterized dual-continuum model. The original parameter-dependent model was reformulated as a new one consisting of multiscale diffusion coefficients and transfer functions (both parameter-independent), along with a parameter-dependent source term. The proposed iterative method was divided into offline and online stages. In the offline stage, multiscale basis functions were constructed in each coarse grid block based on deterministic multiscale parameters to generate a reduced-order space. In the online stage, the model was solved efficiently within the reduced space with an iterative scheme. A major advantage of this method lies in the fact that, once the multiscale basis space is constructed offline, each online iteration can leverage efficient direct solvers and reuse matrix inverses, thus significantly reducing computational costs. Furthermore, the convergence of the proposed iterative method was analyzed. Finally, numerical experiments on the parameterized dual-continuum model were conducted to demonstrate the effectiveness and efficiency of the multiscale-based iterative approach, and validate the theoretical convergence results.
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Founded in 1980 ( monthly )
Supervisor：Chongqing Jiaotong University
Founder：CHIEN Weizang
Editor-in-Chief：LU Tianjian
ISSN 1000-0887
CN 50-1060/O3
