应用数学和力学

2023 Vol. 44, No. 7

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Solid Mechanics
Applied Mathematics

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Solid Mechanics

Digital Twin Method for Dynamic Structures Based on Reduced Order Models and Data Driving

WANG Qingshan, YAN Bo, CHEN Yan, DENG Mao, CAI Yuanbin

2023, 44(7): 757-768. doi: 10.21656/1000-0887.430384

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Abstract:
A digital twin construction method based on the reduced order model library and machine learning was proposed for structures under dynamic loads. Firstly, the high-fidelity finite element models were established according to the possible damage states occurring during the service of the physical structures. Secondly, the Krylov subspace order reduction method was used to reduce the orders of the models and the reduced order models were assembled to a library. Finally, the random forest machine learning algorithm was used to train the model selector, infer the current state of the physical structure through the sensor data from the structure, and then drive the digital twin to evolve with the physical structure. A physical frame structure was designed and manufactured to simulate the damages of different degrees at different points, and verify the proposed digital twin construction method for dynamic structures.

A Discrete Element Method for Irregular Granular Materials Based on Multi-Dilated Polyhedron Elements

LI Dianzhe, LIU Lu, JI Shunying

2023, 44(7): 769-783. doi: 10.21656/1000-0887.430152

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Abstract:
Irregular granular materials are widely available in nature and industrial fields. To construct a theoretical model closer to the real granular materials, a multi-dilated polyhedron model based on the dilated polyhedron element was developed. To verify the reliability of the multi-dilated polyhedron model, the discharge processes of the convex triangular prism particles and concave upward-downward conical particles in the flat bottom hopper were simulated and compared with experimental results, to show good consistency. Besides, the piling and discharge process of differently shaped multi-dilated polyhedron particles were simulated. Furthermore, the effects of particle shapes on the piling fractions, mass flow rates and angles of repose were discussed. The results indicate that, given a more complex particle shape, the interlocking between particles will be stronger, thereby the stability of the granular system will be higher. The effective application of multi-dilated polyhedron elements provides a new model-building method for irregular granular materials.

Thermomechanical Responses of YSZ Under Ultrashort Thermal Shock Based on the L-S Generalized Thermoelastic Theory

ZHAO Yi, TIAN Xiaogeng

2023, 44(7): 784-796. doi: 10.21656/1000-0887.430134

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Abstract:
Based on the L-S generalized thermoelastic theory and in view of the specific heat capacity change of material with temperature, the control equations for the thermoelastic coupling system with internal heating source were established. The thermomechanical responses of yttrium tetragonal zirconia (YSZ) under the action of ultrashort pulse laser were studied by means of the finite element method. The effects of the specific heat capacity change with temperature, and the pulse width of the laser on the thermomechanical responses and the mechanical wave reflections in material were obtained. The results show that, under repeated pulse laser actions, the stress and displacement of the material will undergo fluctuations, and the mechanical response will be more sensitive to heating than the thermal response. The specific heat capacity change with temperature will result in the decrease of the thermal response. The study provides an important guidance for improving the ultrashort pulse laser machining quality.

3D Fast Multipole Boundary Element Method Analysis of Heat Exchange Performance of Buried Pipe Groups

SONG Zixin, HU Zongjun, HU Bin, NIU Zhongrong

2023, 44(7): 797-808. doi: 10.21656/1000-0887.430210

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Abstract:
Based on the 3-node triangular linear element and to overcome the element cross-leaf integration problem, a new 3D fast method was formulated for 3D potential problems through combination of the fast multipole boundary element method (FMBEM) with the semi-analytical algorithm of nearly singular integral, to realize the accurate calculation of the nearly singular integral in the 3D boundary element method (BEM). This method is applicable to the heat exchange of thin-wall structures of U-type buried pipe groups. In the cooling and heating conditions, the effects of the wall thickness of the U-type buried pipe group were analyzed by means of the new FMBEM, and the thermal interaction between multiple buried pipes was discussed. The calculation results show that, for a constant thermal conductivity of the pipe wall, the thicker the pipe wall is, the greater the effect on the heat exchange between the pipe fluid and the soil will be. For a constant borehole spacing, the bigger the number of buried pipes in a group is, the stronger the thermal interference between the pipes will be. The main strategy to increase the heat exchange of the pipe group is to reduce the thermal interference between the heat exchange pipes. Due to the accurate calculation of the nearly singular integral, the proposed 3D FMBEM can effectively solve the 3D heat exchange problems of thin-thick coupled bodies. This method and the results for the provide references for the engineering application of buried pipe heat exchangers.

Research on Interfacial Collinear Cracks Between 1D Hexagonal Piezoelectric Quasicrystal Bimaterials

LU Shaonan, ZHAO Xuefen, MA Yuanyuan

2023, 44(7): 809-824. doi: 10.21656/1000-0887.430111

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Abstract:
By means of the analytic continuation, the singularity principal part analysis and the extended Liouville theorem in the complex function theory, the anti-plane elastic problem of interfacial collinear cracks between 1D hexagonal piezoelectric quasicrystal bimaterials under concentrated loads, was addressed. The closed solutions for biomaterial interface containing 1 and 2 finite-length cracks under concentrated loads were derived. At the same time, the crack tip field intensity factors (including the phonon field, the phason field stress intensity factors and the electric displacement intensity factor) were given. The effects of the ratio of the external load to the coupling coefficient on the intensity factor variation of the crack tip field were analyzed by numerical examples. The numerical results show that, the intensity factor of the crack tip field increases with the crack length and with the ratio of coupling coefficients, while the electric displacement intensity factor keeps almost unchanged. The field intensity factor of the crack tip varies with the crack length in different styles under different loads. The research results provide a theoretical reference for the design and preparation of piezoelectric quasicrystals.

The Half Space Problem of Cubic Quasicrystal Piezoelectric Materials

LI Guangfang, LIU Fangfang, YU Jing, LI Lianhe

2023, 44(7): 825-833. doi: 10.21656/1000-0887.430221

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Abstract:
The half space problem of cubic quasicrystal piezoelectric materials was considered. The governing equations of elasticity for cubic quasicrystal piezoelectric materials under anti-plane deformation and in-plane electric field were given. Combined with the surface boundary conditions in the semi-infinite region, a general solution was obtained by means of the operator theory and the complex function method. Then the analytical expressions of the displacements and stresses of the phonon field and the phason field, and the electric displacements of the half space problem under concentrated linear surface forces, were derived.

Symplectic Analysis on the Bending Problem of Decagonal Symmetric 2D Quasicrystal Plates With 2 Opposite Edges Simply Supported

FAN Junjie, LI Lianhe, ALATANCANG

2023, 44(7): 834-846. doi: 10.21656/1000-0887.430267

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Abstract:
The symplectic method for the elastic problem of decagonal symmetric 2D quasicrystal plates with 2 opposite edges simply supported, was discussed. The basic equations of the elastic theory for decagonal symmetric 2D quasicrystals were transformed into the Hamilton dual equations. With the method of separation of variables, the symplectic eigenvalues of the corresponding Hamilton operator matrix and the symplectic eigenfunction system were obtained. The completeness of the symplectic eigenfunction system of the Hamilton operator matrix in the sense of the Cauchy principal value was proved. Based on the symplectic eigenfunction expansion of the Hamilton system, the analytical solution to the bending problem of the decagonal symmetric 2D quasicrystal plate was given.

Applied Mathematics

The Random Source Inverse Method and Properties for a Class of Stochastic Differential Equations

CHEN Chen, FENG Xiaoli, CHEN Hanzhang

2023, 44(7): 847-856. doi: 10.21656/1000-0887.430170

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Abstract:
The random source inverse method and properties for a class of stochastic differential equations driven by the fractional Brownian motion with Hurst index H∈(0, 1). This problem can be obtained from the transform of many stochastic models and is a widely followed problem. For the direct problem, the mild solution to the equation was obtained by means of constant variation, and according to the statistical properties of the mild solution, the well-posedness of the direct problem was discussed. For the inverse problem, some statistics of the random source term were determined from the random data at the final moment, to prove the uniqueness of the inverse problem, and the stability of the inverse problem with a(x) in different ranges was discussed.

The l₁ Filter for Discrete-Time Switched Singular Positive Systems With Time-Varying Delays

WANG Jinling, HOU Yuxiao, LI Qiang, XIE Baoying

2023, 44(7): 857-869. doi: 10.21656/1000-0887.430125

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Abstract:
The design problem of l₁ filters was studied for a class of discrete-time switched singular systems with positive constraints and time-varying delays. Through construction of an appropriate co-positive Lyapunov function and with the average dwell time technique, sufficient conditions in the form of linear programming were provided to ensure the corresponding filtering error system to be positive, regular, causal and exponentially stable. In addition, the effect of the exogenous disturbance input on the system performance was also analyzed and discussed. The sufficient conditions and design mechanisms for the corresponding filter to ensure that the filtering error system has the prescribed l₁ disturbance attenuation performance were also given. Finally, a simulation example was provided to verify the effectiveness and feasibility of the proposed method.

Distributed Formation Maneuver Control of Networked Euler-Lagrange Systems

YANG Jikang, YU Jinwei, YANG Weihua

2023, 44(7): 870-883. doi: 10.21656/1000-0887.430130

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Abstract:
The adaptive formation maneuver control of networked Euler-Lagrange systems was studied. By means of the sliding mode control approach, an adaptive formation maneuver control algorithm was proposed. Based on the Lyapunov stability theory, the stability of the closed-loop system was proved. The remarkable feature of the algorithm is the special directed network topology introduced to describe the communication interaction behavior between agents. Hence, without the need for knowing or estimating the time-varying maneuver parameters only known to the leaders, the followers in the system can realize the changes of formation continuously, including the scale, the direction, the displacement and the shape. Numerical simulation results verify the effectiveness of the proposed control scheme.

Bifurcation Analysis of the Permanent Magnet Synchronous Motor Model Under White Gaussian Noises

YE Zhengwei, DENG Shengwen, LIANG Xiangling

2023, 44(7): 884-894. doi: 10.21656/1000-0887.430285

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Abstract:
The Gaussian white noise was introduced into the permanent magnet synchronous motor (PMSM) model, to obtain the system Itô stochastic differential equation through the polar transformation and with the stochastic average method. Hence, the probability density function of the system was calculated, and the mechanism of the P-bifurcation of the system was revealed through numerical simulation. In addition, the complex dynamics of the system in the 2-parameter space was discussed. The simulation results show that, there are lots of "fish-shaped" periodic regions in the parameter space. The regions become unstable under effects of system noises. It is worth noting that, under noises of a certain intensity, the system dynamics will switch from periodic motion to convergence, indicating the dual nature of the effects of noises on the PMSM system dynamics.