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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2024, Volume 45, Issue 12
publish date：December 01 2024
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2024, 45(12): 1455-1472.
doi: 10.21656/1000-0887.450027
Abstract:
An accurate phase field method for 2-phase flow with soluble surfactants was developed based on the phase field theory. The key point of this method was the utilization of consistent and conservative mass flux to ensure the conservation of momentum transport across the interface. The finite-volume method was used to discretize the governing equations in their conservative form. The 5th-order WENO scheme was chosen to effectively handle the convective terms, aimed to enhance accuracy and robustness in addressing steep variations in the interfacial region. Furthermore, various 2D difference templates were designed to optimize gradient discretization in the surface tension term. Particularly, with the template corresponding to the lattice Boltzmann D2Q9 model, a notable reduction of the spurious velocity and a significant improvement of the accuracy of surfactant concentration prediction were achieved. Various examples such as static droplets, fusion of 2 droplets, bubble rise with a large density ratio, deformation and breakage of individual droplets in shear flow demonstrate the accuracy, conservative properties, and robustness of the proposed method.
An accurate phase field method for 2-phase flow with soluble surfactants was developed based on the phase field theory. The key point of this method was the utilization of consistent and conservative mass flux to ensure the conservation of momentum transport across the interface. The finite-volume method was used to discretize the governing equations in their conservative form. The 5th-order WENO scheme was chosen to effectively handle the convective terms, aimed to enhance accuracy and robustness in addressing steep variations in the interfacial region. Furthermore, various 2D difference templates were designed to optimize gradient discretization in the surface tension term. Particularly, with the template corresponding to the lattice Boltzmann D2Q9 model, a notable reduction of the spurious velocity and a significant improvement of the accuracy of surfactant concentration prediction were achieved. Various examples such as static droplets, fusion of 2 droplets, bubble rise with a large density ratio, deformation and breakage of individual droplets in shear flow demonstrate the accuracy, conservative properties, and robustness of the proposed method.
2024, 45(12): 1473-1482.
doi: 10.21656/1000-0887.440347
Abstract:
With the high-precision and high-resolution numerical method, the dynamics of double-diffusive convection of different liquid metals in a long cavity under external magnetic fields in 2 directions was directly simulated, to reveal the influences of the fluid property parameter of Prandtl number Pr, the magnetic field direction and the magnetic field intensity on the flow and the heat and mass transfers. The results indicate that, within the range of the considered Pr values, the flow transitions from periodic to steady with the increase of Pr for weak magnetic fields. Specifically, when the Pr is 0.03, the convective system will have unsteady solution, and the flow will be periodic. The efficiency of heat and mass transfers initially increase rapidly, then slowly. For the moderately strong magnetic field, the flow remains steady, and the growth rate of the heat and mass transfer efficiency slows down further with Pr increasing. For the strong magnetic field, the flow is always steady, and the efficiency of the heat and mass transfer hardly changes with the Pr. Under the same magnetic field intensity, compared with the inclined magnetic field with a direction of 45° and the horizontal magnetic field, the vertical magnetic field has a weaker suppressive effect on the heat and mass transfer efficiency.
With the high-precision and high-resolution numerical method, the dynamics of double-diffusive convection of different liquid metals in a long cavity under external magnetic fields in 2 directions was directly simulated, to reveal the influences of the fluid property parameter of Prandtl number Pr, the magnetic field direction and the magnetic field intensity on the flow and the heat and mass transfers. The results indicate that, within the range of the considered Pr values, the flow transitions from periodic to steady with the increase of Pr for weak magnetic fields. Specifically, when the Pr is 0.03, the convective system will have unsteady solution, and the flow will be periodic. The efficiency of heat and mass transfers initially increase rapidly, then slowly. For the moderately strong magnetic field, the flow remains steady, and the growth rate of the heat and mass transfer efficiency slows down further with Pr increasing. For the strong magnetic field, the flow is always steady, and the efficiency of the heat and mass transfer hardly changes with the Pr. Under the same magnetic field intensity, compared with the inclined magnetic field with a direction of 45° and the horizontal magnetic field, the vertical magnetic field has a weaker suppressive effect on the heat and mass transfer efficiency.
2024, 45(12): 1483-1493.
doi: 10.21656/1000-0887.440304
Abstract:
Based on the smooth kernel approximation and the Taylor series expansion of the smooth particle hydrodynamics (SPH) method, the kernel function moment was used to revise the KDF-SPH (kernel derivative free SPH) method. To prove the applicability and feasibility of the proposed revised scheme, the scheme was applied to the numerical simulations of 1D shock tube problems under different conditions, and the simulation results were analyzed. The results show that, the revised method can well capture the positions and strengths of shock waves and contact discontinuities. The revised method does not require the derivability of the kernel function, does not calculate the kernel function moment, and has a smaller computation cost with a higher calculation efficiency.
Based on the smooth kernel approximation and the Taylor series expansion of the smooth particle hydrodynamics (SPH) method, the kernel function moment was used to revise the KDF-SPH (kernel derivative free SPH) method. To prove the applicability and feasibility of the proposed revised scheme, the scheme was applied to the numerical simulations of 1D shock tube problems under different conditions, and the simulation results were analyzed. The results show that, the revised method can well capture the positions and strengths of shock waves and contact discontinuities. The revised method does not require the derivability of the kernel function, does not calculate the kernel function moment, and has a smaller computation cost with a higher calculation efficiency.
2024, 45(12): 1494-1505.
doi: 10.21656/1000-0887.440291
Abstract:
There is a highly nonlinear coupling effect between liquid pressure pulsation and pipeline structure in the infusion pipeline system. Severe coupling vibration will occur under external excitation, which will lead to the failure of the infusion pipeline and connection structure. In view of the lack of analytical methods for the fluid-solid coupling vibration responses of the infusion pipeline, the dynamic characteristics and vibration response theory for the infusion pipeline system with complex supports were given based on the differential transformation method (DTM), and the fluid-solid coupling vibration differential equation for the straight infusion pipeline with complex elastic supports within the span was established based on the Bernoulli-Euler beam theory. The expressions based on the DTM for calculating natural frequencies, displacement responses and support constraint reactions of the pipeline system with the simple support and additional elastic supports were derived in detail. The influences of the internal pressure, the flow rate, the additional support stiffness and the support position on natural frequencies and support constraint reactions of the pipeline system were studied, and the calculation accuracy of the DTM was verified with the structural finite element analysis method. The research shows that, the application of the DTM in the calculation of fluid-solid coupling vibration characteristics and responses of the infusion pipeline system with complex supports has high accuracy and good applicability, especially in the calculation of vibration responses of the pipeline system with complex boundaries and additional supports within the span. Based on the DTM, the mechanical responses of the fluid-solid coupling pipeline system can be conveniently calculated under forced vibration. It provides a theoretical basis for the design of pipelines and connected structures.
There is a highly nonlinear coupling effect between liquid pressure pulsation and pipeline structure in the infusion pipeline system. Severe coupling vibration will occur under external excitation, which will lead to the failure of the infusion pipeline and connection structure. In view of the lack of analytical methods for the fluid-solid coupling vibration responses of the infusion pipeline, the dynamic characteristics and vibration response theory for the infusion pipeline system with complex supports were given based on the differential transformation method (DTM), and the fluid-solid coupling vibration differential equation for the straight infusion pipeline with complex elastic supports within the span was established based on the Bernoulli-Euler beam theory. The expressions based on the DTM for calculating natural frequencies, displacement responses and support constraint reactions of the pipeline system with the simple support and additional elastic supports were derived in detail. The influences of the internal pressure, the flow rate, the additional support stiffness and the support position on natural frequencies and support constraint reactions of the pipeline system were studied, and the calculation accuracy of the DTM was verified with the structural finite element analysis method. The research shows that, the application of the DTM in the calculation of fluid-solid coupling vibration characteristics and responses of the infusion pipeline system with complex supports has high accuracy and good applicability, especially in the calculation of vibration responses of the pipeline system with complex boundaries and additional supports within the span. Based on the DTM, the mechanical responses of the fluid-solid coupling pipeline system can be conveniently calculated under forced vibration. It provides a theoretical basis for the design of pipelines and connected structures.
2024, 45(12): 1506-1514.
doi: 10.21656/1000-0887.440375
Abstract:
The analysis of the stochastic bifurcation behaviors of stochastic nonlinear systems often requires artificial judgment based on the joint probability density and cannot be automated. A new calculation method for automatic calculation of random bifurcation points was proposed. The bi-stable Van der Pol system under strong noise excitation was taken as an example, the influences of damping coefficient changes on stochastic dynamic responses were analyzed. The research results show that, the joint probability density of the system bifurcates for 3 times with the increase of the damping coefficient, exhibiting 4 different types of geometric features. The proposed method can hopefully be applied to the study of stochastic bifurcation behaviors of other stochastic nonlinear systems.
The analysis of the stochastic bifurcation behaviors of stochastic nonlinear systems often requires artificial judgment based on the joint probability density and cannot be automated. A new calculation method for automatic calculation of random bifurcation points was proposed. The bi-stable Van der Pol system under strong noise excitation was taken as an example, the influences of damping coefficient changes on stochastic dynamic responses were analyzed. The research results show that, the joint probability density of the system bifurcates for 3 times with the increase of the damping coefficient, exhibiting 4 different types of geometric features. The proposed method can hopefully be applied to the study of stochastic bifurcation behaviors of other stochastic nonlinear systems.
2024, 45(12): 1515-1529.
doi: 10.21656/1000-0887.450201
Abstract:
The oscillating buoy wave energy converter is a kind of core working unit of wave energy power generation system, and its development is of significances for breaking the power supply bottlenecks in coastal area development and offshore platform construction of in China. Then the coupled motion model for multi-DOF wave energy conversion structures was built to investigate its mechanical configuration, parameter design and energy capture mechanism. Through optimization of the intelligent algorithms such as the particle swarm algorithm, the problems of the too large multi-DOF iteration scale and the local optimal solution dilemma were overcome, the functions of the algorithm were enriched, and the oscillation and energy capture effects of the wave energy converter based on 2D/4D, linear/nonlinear damping and structure dimension parameters in 2 scenarios, were qualitatively and quantitatively evaluated. The capture energy advantages of vibration control conditions of multi-DOF and nonlinear damping, were verified, and the dynamic behavior law, the parameter optimization design and the efficient path of energy capture mechanism were explored simultaneously. Random loads were introduced to optimize the accuracy of the model, and the effects of noise differences on the energy capture were refined. The work provides a new idea for the effective application mode of wave energy conversion structure in practical engineering.
The oscillating buoy wave energy converter is a kind of core working unit of wave energy power generation system, and its development is of significances for breaking the power supply bottlenecks in coastal area development and offshore platform construction of in China. Then the coupled motion model for multi-DOF wave energy conversion structures was built to investigate its mechanical configuration, parameter design and energy capture mechanism. Through optimization of the intelligent algorithms such as the particle swarm algorithm, the problems of the too large multi-DOF iteration scale and the local optimal solution dilemma were overcome, the functions of the algorithm were enriched, and the oscillation and energy capture effects of the wave energy converter based on 2D/4D, linear/nonlinear damping and structure dimension parameters in 2 scenarios, were qualitatively and quantitatively evaluated. The capture energy advantages of vibration control conditions of multi-DOF and nonlinear damping, were verified, and the dynamic behavior law, the parameter optimization design and the efficient path of energy capture mechanism were explored simultaneously. Random loads were introduced to optimize the accuracy of the model, and the effects of noise differences on the energy capture were refined. The work provides a new idea for the effective application mode of wave energy conversion structure in practical engineering.
2024, 45(12): 1530-1540.
doi: 10.21656/1000-0887.450028
Abstract:
With droplets on the surface of soft matter or micro/nanostructures, wetting-induced elastocapillary deformation should be considered. Based on a new wetting equation, the relationship between the droplet spreading radius and its surface curvature was determined, and the necessary conditions for the droplet to hold the spherical cap without the influence of gravity were obtained. Combined with the Winkler foundation model, the elastocapillary deformation of soft material microbeam on elastic substrate was calculated. The analytical solution for the deflection of the microbeam was given. With polystyrene and polyethylene beams as examples, the influences of the droplet spreading radius, the beam material elastic modulus and the Winkler foundation parameter on the microbeam deflections and detachments from the substrate, were analyzed and discussed.
With droplets on the surface of soft matter or micro/nanostructures, wetting-induced elastocapillary deformation should be considered. Based on a new wetting equation, the relationship between the droplet spreading radius and its surface curvature was determined, and the necessary conditions for the droplet to hold the spherical cap without the influence of gravity were obtained. Combined with the Winkler foundation model, the elastocapillary deformation of soft material microbeam on elastic substrate was calculated. The analytical solution for the deflection of the microbeam was given. With polystyrene and polyethylene beams as examples, the influences of the droplet spreading radius, the beam material elastic modulus and the Winkler foundation parameter on the microbeam deflections and detachments from the substrate, were analyzed and discussed.
2024, 45(12): 1541-1554.
doi: 10.21656/1000-0887.450161
Abstract:
Based on the specified stress method, a new spatial finite element formulation for concrete cracking was derived according to the linear elasticity theory. This formulation was used to create a calculation program with the C++ language. The accuracy of the proposed cracking algorithm was validated through 3 numerical examples, where the theoretical results were compared with the ABAQUS XFEM calculations. Beyond conventional cracking algorithms, the proposed cracking algorithm has the advantage that once the stress at the cracking integration point is specified as zero (in the cracking state), it will remain zero in subsequent calculations. There is no need for an iterative process to adjust it to zero, which means significant reduction of the number of iterations and the amount of data processing required in each iteration. In comparison to the ABAQUS XFEM algorithm, which is limited to the 1st-order elements, the proposed cracking algorithm can utilize the 2nd-order elements for crack calculation, and allows for a more accurate determination of the cracked regions and states under the same computational conditions. This work provides a new approach and algorithm for commercial finite element software to conduct more refined crack calculations with the 2nd-order elements.
Based on the specified stress method, a new spatial finite element formulation for concrete cracking was derived according to the linear elasticity theory. This formulation was used to create a calculation program with the C++ language. The accuracy of the proposed cracking algorithm was validated through 3 numerical examples, where the theoretical results were compared with the ABAQUS XFEM calculations. Beyond conventional cracking algorithms, the proposed cracking algorithm has the advantage that once the stress at the cracking integration point is specified as zero (in the cracking state), it will remain zero in subsequent calculations. There is no need for an iterative process to adjust it to zero, which means significant reduction of the number of iterations and the amount of data processing required in each iteration. In comparison to the ABAQUS XFEM algorithm, which is limited to the 1st-order elements, the proposed cracking algorithm can utilize the 2nd-order elements for crack calculation, and allows for a more accurate determination of the cracked regions and states under the same computational conditions. This work provides a new approach and algorithm for commercial finite element software to conduct more refined crack calculations with the 2nd-order elements.
2024, 45(12): 1555-1566.
doi: 10.21656/1000-0887.440324
Abstract:
Based on the spherical cavity expansion theory, combined with the spatially mobilized plane (SMP) criterion and the critical state concept, the Wheeler model describing the anisotropy of soft clay was applied to analyze the spherical cavity expansion problem in split grouting. A dual interval analysis model was used to divide the soil into elastic and plastic zones, and the stress-strain and displacement fields of the soil around the cavity were derived under the elastic-plastic interval boundary conditions. Theoretical calculations of the soil expansion radius, the critical fracturing pressure, and the plastic volumetric strain after split grouting were carried out and demonstrated by several examples. The results show that, the radial and circumferential stresses decrease with parameter r/rp;the critical fracturing pressure increases with the internal friction angle; the internal friction angle is an important influential factor in the elastic-plastic analysis of the soil, and the larger the internal friction angle is, the smaller the plastic circumferential strain, the plastic radius and the plastic displacement, and the larger the splitting pressure, will be.
Based on the spherical cavity expansion theory, combined with the spatially mobilized plane (SMP) criterion and the critical state concept, the Wheeler model describing the anisotropy of soft clay was applied to analyze the spherical cavity expansion problem in split grouting. A dual interval analysis model was used to divide the soil into elastic and plastic zones, and the stress-strain and displacement fields of the soil around the cavity were derived under the elastic-plastic interval boundary conditions. Theoretical calculations of the soil expansion radius, the critical fracturing pressure, and the plastic volumetric strain after split grouting were carried out and demonstrated by several examples. The results show that, the radial and circumferential stresses decrease with parameter r/rp;the critical fracturing pressure increases with the internal friction angle; the internal friction angle is an important influential factor in the elastic-plastic analysis of the soil, and the larger the internal friction angle is, the smaller the plastic circumferential strain, the plastic radius and the plastic displacement, and the larger the splitting pressure, will be.
2024, 45(12): 1567-1576.
doi: 10.21656/1000-0887.440345
Abstract:
Based on the ideal Mohr Coulomb (M-C) yield criterion, the concepts and formulas of equivalent stresses in tension, compression, and shear, as well as the corresponding three strength evaluation conditions were presented. According to the equivalent principle of plastic work, the equivalent plastic strain and the equivalent plastic shear strain in tension and compression were derived, which are conjugated with the equivalent stress mentioned above. For different friction coefficients, the variation characteristics of these equivalent strains were discussed. Differing from the Mises equivalent strain, the M-C equivalent strain can reflect the influence of the hydrostatic pressure, and can also degenerate into a simple stress state. These concepts of equivalent stress and equivalent strain have clear physical meanings and can be applied to more accurately and effectively evaluate the strengths of materials with different tensile and compressive properties. They also have direct application values for calibrating constitutive model parameters under complex stress states through experiments under simple stress states.
Based on the ideal Mohr Coulomb (M-C) yield criterion, the concepts and formulas of equivalent stresses in tension, compression, and shear, as well as the corresponding three strength evaluation conditions were presented. According to the equivalent principle of plastic work, the equivalent plastic strain and the equivalent plastic shear strain in tension and compression were derived, which are conjugated with the equivalent stress mentioned above. For different friction coefficients, the variation characteristics of these equivalent strains were discussed. Differing from the Mises equivalent strain, the M-C equivalent strain can reflect the influence of the hydrostatic pressure, and can also degenerate into a simple stress state. These concepts of equivalent stress and equivalent strain have clear physical meanings and can be applied to more accurately and effectively evaluate the strengths of materials with different tensile and compressive properties. They also have direct application values for calibrating constitutive model parameters under complex stress states through experiments under simple stress states.
2024, 45(12): 1577-1588.
doi: 10.21656/1000-0887.440344
Abstract:
In view of the absorption and diffusion characteristics of laser in the paint layer and the base layer, heat transfer model Ⅰ and thermal connection boundary conditions for the paint layer and the metal base layer was established. As a comparison, corresponding heat transfer models Ⅱ and Ⅲ were established, with the thermal connection conditions at the interface between the paint layer and the base layer ignored. In model Ⅲ, both the paint layer and the base layer were connected to the bulk heat source. A semianalytical solution to the problem was obtained through the Laplacian transformation. Corresponding numerical solution examples were given through the Laplacian numerical inverse transformation. The results show that, model Ⅰ gives continuous temperature distributions, while model Ⅱ and model Ⅲ give discontinuous temperature distributions, which goes against physical reality. Secondly, based on model Ⅰ, the predicted time and energy density for cleaning and removing the paint layer are between those of model Ⅱ and model Ⅲ. The work provides a theoretical reference for improving the laser cleaning process.
In view of the absorption and diffusion characteristics of laser in the paint layer and the base layer, heat transfer model Ⅰ and thermal connection boundary conditions for the paint layer and the metal base layer was established. As a comparison, corresponding heat transfer models Ⅱ and Ⅲ were established, with the thermal connection conditions at the interface between the paint layer and the base layer ignored. In model Ⅲ, both the paint layer and the base layer were connected to the bulk heat source. A semianalytical solution to the problem was obtained through the Laplacian transformation. Corresponding numerical solution examples were given through the Laplacian numerical inverse transformation. The results show that, model Ⅰ gives continuous temperature distributions, while model Ⅱ and model Ⅲ give discontinuous temperature distributions, which goes against physical reality. Secondly, based on model Ⅰ, the predicted time and energy density for cleaning and removing the paint layer are between those of model Ⅱ and model Ⅲ. The work provides a theoretical reference for improving the laser cleaning process.
2024, 45(12): 1589-1592.
Abstract:
After reading Discussion on the Modified Reciprocal Theorem of Works (Discussion for short), there are 2 main judgments about Discussion: ① the understanding of the proposition for the reciprocal theorem of Betti’s works is incomplete and inaccurate; ② the assertion that the corrected reciprocal theorem of works is actually another manifestation of Betti’s reciprocal theorem, is wrong.
After reading Discussion on the Modified Reciprocal Theorem of Works (Discussion for short), there are 2 main judgments about Discussion: ① the understanding of the proposition for the reciprocal theorem of Betti’s works is incomplete and inaccurate; ② the assertion that the corrected reciprocal theorem of works is actually another manifestation of Betti’s reciprocal theorem, is wrong.
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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
