应用数学和力学

2023 Vol. 44, No. 11

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Special Topics on Thermal Stresses
Dynamics and Control
Applied Mathematics

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Special Topics on Thermal Stresses

Evaluation of Fracture and its Effects on Energy Conversion Performance of Porous Foam Thermoelectric Generators

CUI Youjiang, WANG Baolin, WANG Kaifa

2023, 44(11): 1291-1298. doi: 10.21656/1000-0887.440147

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Abstract:
The thermoelectric generator could convert the waste heat into electricity and reduce carbon dioxide emissions. This meets the national development needs for energy conservation and emission reduction, and finally helps realize the carbon neutrality. The heat and electric conduction model was established to explain the internal mechanism of high output power of the porous thermoelectric foam. The effects of geometry and porosity on the fracture failure of the porous thermoelectric foam were also discussed. Then, the influential mechanism of fracture on the energy conversion performance of the porous thermoelectric foam was revealed. The results show that, the interfacial shear stress between the thermoelectric foam and the metal layer will decrease with the porosity. As long as an internal crack of the porous thermoelectric foam starts to extend, it will not stop until the device is completely broken. Moreover, the output power will first increase to the peak value and then decrease with the crack propagation. This is because the crack propagation indirectly raises the porosity of the thermoelectric foam and the contact area between the thermoelectric foam and the waste heat, and in turn promotes the output power of the thermoelectric device. With further crack propagation, both the thermal conductivity and the electrical conductivity of the thermoelectric foam will weaken, and the output power of the thermoelectric foam will decrease.

Topology Optimizations of Integrated Thermal Protection Systems in Multiple Reentry Load Cases

DENG Tongxiang, KUANG Geping, HU Zhaocai, TANG Guangwu, YANG Qiang, GAO Bo

2023, 44(11): 1299-1310. doi: 10.21656/1000-0887.440163

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Abstract:
The integrated thermal protection system (ITPS) needs to meet both load-bearing and heat-insulating requirements. In terms of the ITPS with a corrugated sandwich structure, this requires the connection structure of the ITPS to have high mechanical properties with low thermal conductivity. However, the re-entry environment is severe, how to reasonably design the connection structure is key to improve the performances of the ITPS. To solve this problem, 2 extreme load conditions corresponding to the maximum aerodynamic heat load and the maximum aerodynamic pressure load during the reentry process were comprehensively considered, the objective function was constructed with the minimized strain energy and the net heat transfer rate, the mass was used as a constraint, and the topology optimization of the ITPS connection structure was carried out. Then, the configuration obtained through the topology optimization was reconstructed and the thermal mechanical coupling analysis was carried out. The results show that, the maximum displacement of the top panel, the temperature of the bottom panel and the mass of the optimized connection structure were reduced effectively compared with those of the initial corrugated sandwich configuration and the topology optimization configuration in single load cases in the literatures. Due to the reduction of material consumption and the increase of the structural complexity, the stress level of the connection structure increases, but it still meets the requirements of use. This means that, the topology optimization strategy considering multiple reentry load cases can effectively improve the stiffness and the insulation capacity of the ITPS and alleviate the thermal short-circuiting of the structure. With the development of additive manufacturing and other related technologies, the topology optimization method has broad prospects in the design of the connection structures for the ITPS and other thermal structures.

Surface Effects on Thermal Stresses Around the Nanohole in Thermoelectric Material

ZHAO Jieyan, YANG Haibing

2023, 44(11): 1311-1324. doi: 10.21656/1000-0887.440151

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Abstract:
Based on the complete Gurtin-Murdoch (G-M) low-order surface energy model, the surface effects at nanoscale were further explored. The transition from macroscale to microscale was achieved through construction of reasonable stress boundary conditions in view of the change of hole geometry configuration. With the series expansion techniques and complex variable methods, the semi-analytic solutions for the electric field, the temperature field, and the full stress field in the vicinity of the nanohole within the thermoelectric matrix were derived eventually with a built thermal-electrical-force theoretical framework model at nanoscale. Numerical results show that, compared with the complete G-M model, the simplified G-M model (neglecting the effects of nanohole geometry changes) would overestimate the surface effects and far-field thermoelectric loading effects on the thermal stress distributions. In addition, the surface effects can relieve the thermal stress concentration around the nanohole to some extent.

Dispersion and Attenuation Characteristics of Fractional-Order Thermoelastic Guided Waves in Functionally Graded Piezoelectric Hollow Cylinders

YU Jiangong, WANG Kai, REN Xiaoqiang, WANG Xianhui, ZHANG Bo

2023, 44(11): 1325-1340. doi: 10.21656/1000-0887.440144

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Abstract:
Based on the fractional-order thermo-electric-elastic theory and the Legendre polynomial series method, a mathematical model for guided wave propagation in functionally graded hollow cylinders was established. The effects of the fractional order, the piezoelectric effect, and the radius-thickness ratio on the wave propagation, especially on its attenuation, were discussed. The numerical analysis results indicate that, the piezoelectric effect on attenuation mainly concentrates near the cutoff frequency and the mutation frequency, and causes the mutation frequency to shift forward. The fractional order has a great impact on the phase velocity and attenuation of the thermal wave mode, and has an opposite impact on the phase velocity around the crossover frequency point where the crossover mode occurs with the thermal wave velocity. But the thermal wave attenuation gradually decreases with the fractional order. Meanwhile, the 1st longitudinal mode attenuation is suppressed by the piezoelectric effect. However, the attenuation of other modes significantly increases, and the impact of the electrical open circuit is greater than that of the electrical short circuit.

Dynamics and Control

A Magnetoelastic Coupling Dynamical Model for Functional Gradient Shells Under Magnetic Field Actions

HU Yuda, LIAO Feng

2023, 44(11): 1341-1353. doi: 10.21656/1000-0887.440048

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Abstract:
For metal-ceramic functional gradient cylindrical shells in electromagnetic fields, the nonlinear constitutive relations were determined based on the geometry and Hooke's law on the physical neutral surface. According to the Kirchhoff-Love theory, the strain energy expression and the kinetic energy expression with its variational operator were given for the heterogeneous elastic shell. The model of the eddy current Lorentz force and the magnetization force for ferromagnetic functional gradient shells under electromagnetic field actions, was derived with the electromagnetic elasticity theory. The magnetoelastic coupling nonlinear vibration equations for the shell were obtained by means of Hamilton's variational principle, and the dynamical model describing the coupling characteristics of the deformation field and the electromagnetic field was established for functional gradient structures. Through numerical examples for natural vibrations of functional gradient shells, the characteristic equation and the natural frequency variation law were obtained. The results show that, the natural frequency decreases with the magnetic induction intensity and the material volume fraction index, and the phenomenon of minimum frequency will occur in the circumferential wave number influence curves. This study provides a reference for the theoretical modeling and dynamic analysis of multi-field coupling systems.

Free Vibration Analysis of Porous 2D Functionally Graded Material Microbeams on Winkler's Foundation

ZHAO Yingzhi, TANG Huaiping, LAI Zedong, ZHANG Jiajie

2023, 44(11): 1354-1365. doi: 10.21656/1000-0887.440050

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Abstract:
Based on the modified couple stress theory and the Timoshenko beam theory, the governing equations for free vibration of porous 2D functional graded material (FGM) on Winkler's foundation were derived under Hamilton's principle. The differential quadrature method was used to obtain the numerical solutions of the vibration frequencies and fundamental mode shapes of microbeams with both ends clamped (C-C) and simply supported (S-S). The improved stiffness matrix was used to greatly improve the calculation efficiency. The proposed model was degenerated to the macro and micro 2D-FGM models, which were compared with those in previous literatures for validation. The results show that, the present mathematical model is suitable for different types of 2D material distributions. The dimensionless frequencies increase with the dimensionless elastic modulus of Winkler's foundation. Under a certain dimensionless elastic foundation modulus, the dimensionless frequencies decrease with the functionally graded index, the axial functionally graded index and the porosity. The effect of the material variation on the mode shape increases with the mode number. For the same parameter, the dimensionless frequencies of the beam with uniform porosity distribution are slightly lower than those with linear porosity distribution.

Dynamics Analysis of Large-Deformation Flexible Multibody Systems Based on the Adaptive Modal Selection Method

LIU Zebin, LI Haiyan, ZHAN Hongyuan, LIANG Guiming

2023, 44(11): 1366-1377. doi: 10.21656/1000-0887.430368

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Abstract:
During the modal truncation to reduce the model order of flexible multibody systems, the inappropriate modal selection would impair the precision of dynamic responses, or even cause divergent solutions. Thus, an efficient method of adaptive modal selection based on the absolute nodal coordinate formulation (ANCF) was proposed for large-deformation flexible multibody systems. The dynamic model for the system was established with the ANCF beam elements. The full modal sparse representation was used for the coordinates of the interior region. The sampling matrix was built through the Latin hypercube sampling to reduce the number of dynamic equations. The sparse modal coordinates' norm optimization problem was constructed with the sampling dynamic equations as constraints, to which the solution could give modes of significant contribution. Two examples show that, the numerical results are very close to the results of common methods and the computation efficiency markedly improves.

Applied Mathematics

Iterative Methods for Random Generalized Quasi Variational Inequalities With Applications

WANG Yannan, ZENG Jiaqin, HUANG Nanjing

2023, 44(11): 1378-1388. doi: 10.21656/1000-0887.440199

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Abstract:
To obtain the iterative methods for solving a class of random generalized quasi variational inequalities (RGQVIs) in the Hilbert spaces, the measurability of the projection operators on varying-constraint sets depicted by the mapping from points to random-value sets with closed (convex) values, was proved. Moreover, the random iterative algorithm was proposed for solving RGQVIs, and the convergence of the random sequences generated with the random iterative algorithm was obtained under some suitable conditions of monotony and Lipschitz continuity. Finally, 2 applications were given with depicting results of the random generalized Nash games and random Walrasian equilibrium problems, respectively.

Transmission Eigenvalues for Helmholtz Equation Perturbation Problems of Isotropic Media With Voids

LI Shixuan, LIU Lihan

2023, 44(11): 1389-1397. doi: 10.21656/1000-0887.440221

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Abstract:
Transmission eigenvalues are of major interests in the inverse scattering theory for uniqueness. For the Helmholtz equation of isotropic inhomogeneous media with voids, the existence of transmission eigenvalues was studied for the Helmholtz equation under the refractive index perturbation of the media. Firstly, through construction of the Neumann-Dirichlet operator, the equivalent form of the transmission eigenvalue problem was obtained. Then, the eigenvalue function was built to transform the perturbation problem for transmission eigenvalues into the perturbation problem for zero eigenvalues of operators. Finally, the perturbation method based on the implicit function theorem was used to prove the existence of transmission eigenvalues.

A 4th-Order WENO-Type Entropy Stable Scheme for Ideal Magnetohydrodynamic Equations

ZHANG Chengzhi, ZHENG Supei, CHEN Xue, ZHANG Rui

2023, 44(11): 1398-1412. doi: 10.21656/1000-0887.440178

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Abstract:
A 4th-order entropy stable semi-discrete finite volume scheme was constructed for ideal magnetohydrodynamic equations. This scheme combines the high-order entropy conservative flux with the dissipation term reconstructed with the WENO scheme in the spatial direction. With a switching function added to the dissipation term, the numerical flux has lower dissipation and the WENO reconstruction satisfies the sign property. The source term used to control the divergence of the magnetic field is discretized with the center difference scheme to obtain high-order accuracy consistent with the entropy conservative flux. Several 1D and 2D cases show that, the scheme has no oscillation and strong robustness, and can accurately capture discontinuities.