应用数学和力学

ZHONG Wanxie, WU Feng, SUN Yan, YAO Zheng

2018, 39(8): 855-874. doi: 10.21656/1000-0887.390062

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Abstract:
Here, an elementary introduction to the displacement method-based water wave dynamics theory was presented. The periodic travelling wave solutions of the linear water waves and shallow water waves were given. The symplectic perturbation method was proposed to analyze the periodic travelling wave solution for the water system with a general depth. Numerical tests were given to demonstrate the correctness of the proposed method. The present research emphasizes the dynamics property of water waves. By means of the proposed theory, the particle trajectory can be obtained directly, and the periodic travelling wave with sharp surface can be simulated.

A Symplectic Superposition Method for Bending Problems of Free-Edge Rectangular Thick Plates Resting on Elastic Foundations

LI Rui, TIAN Yu, ZHENG Xinran, WANG Bo

2018, 39(8): 875-891. doi: 10.21656/1000-0887.390186

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Abstract:
Based on the symplectic superposition method proposed in recent years, the bending problems of free-edge rectangular thick plates resting on elastic foundations were analytically solved. The original problem was split into 3 subproblems corresponding to the bending problems of rectangular thick plates with 2 opposite edges slidingly clamped and resting on elastic foundations, which were solved with the symplectic geometry method. The analytic solution of the original problem was then obtained through superposition. Compared to the conventional analytic approaches such as the semi-inverse method, the symplectic superposition method has the advantages of both rationality of the symplectic method and regularity of the superposition method. The solution procedure starts from the basic equations of elasticity, and a rigorous derivation yields the analytic solutions, thus extending the scope of problems to be solved. The present method can serve as an effective analytic approach to complex boundary value problems of high-order partial differential equations in elasticity, as represented by the rectangular plate problems.

A Reduced-Order Model Method for Blade Vibration Due to Upstream Wake Based on the Harmonic Balance Method

LUO Xiao, ZHANG Xinyan, ZHANG Jun, LI Lizhou, YANG Minglei, YUAN Meini

2018, 39(8): 892-899. doi: 10.21656/1000-0887.380294

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Abstract:
In aero-engines, downstream blades are forced to vibrate by the upstream wake, which often severely affects the flutter and fatigue performance of blades. Hence, an efficient method is needed for the analysis of this complex fluid-structure interaction phenomenon. A reduced-order model (ROM) method for blade vibration was proposed based on the harmonic balance method. The upstream wake was firstly decomposed into a number of harmonic waves by FFT (fast Fourier transform), and the aerodynamic forces on the blade were obtained through calculation of the amplitudes of aerodynamic forces on the blade due to each harmonic wave; then the blade vibration was fast analyzed by means of the structure dynamic equations for the blade coupled with the aerodynamic ROM. The results show that this method can analyze the flutter characteristics of the blade under wake excitation quickly and accurately.

Study on Prolate Spheroid Pitching Oscillation in Viscous Stratified Flow

XIONG Ying, GUAN Hui, WU Chuijie

2018, 39(8): 900-912. doi: 10.21656/1000-0887.390041

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Abstract:
The numerical calculation model for the continuous stratified flow was established through the study on the decaying process of the free pitching oscillation of the prolate spheroid in viscous stratified flow. The correctness of the numerical model was verified through numerical simulation of the viscous flow field of a sphere and calculation of its increasing drag coefficient. Under the 45° prolate spheroid initial angle the pitching oscillation process was investigated, with the Aitken sub-relaxation self-adaptive algorithm-based 2-way fluid-solid coupling method, and numerical simulation of the flow field around the prolate spheroid in pitching decaying oscillation at different values of Froude number Fr_i was performed. The numerical results show that, the pitching up and down agitates the surrounding fluid, and forms 4 symmetric density vortex rings on both sides of the prolate spheroid; the vertical density stratification limits vertical propagation of the vortex rings and accelerates disappearance of the vortex rings, and this limitation contributes to the development of horizontal motion. At higher Fr_i and Reynolds number Re,the 2-way coupling method suppresses numerical oscillations. The research also finds that, with the increase of the incoming velocity, the drag coefficient decreases, which means that for the prolate spheroid with free pitching oscillation, the phenomenon of negative drag still appears.

A Peridynamic Model for Heterogeneous Concrete Materials

LI Tianyi, ZHANG Qing, XIA Xiaozhou, GU Xin

2018, 39(8): 913-924. doi: 10.21656/1000-0887.380274

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Abstract:
The classical continuum mechanics and some related numerical methods often fail in analysis of the damage evolution and progressive failure process of concrete materials, due to the difficulties in handling discontinuity and considering the intrinsic microstructure, which is of significance to the damage evolution. A peridynamic concrete model in view of the heterogeneity was established. The digital image processing (DIP) technology was firstly adopted to extract the information of aggregate shape and distribution in a concrete specimen, and then the bondbased peridynamic model reflecting 3 different bonds was applied to analyze the tension and compression failure of standard concrete specimens. The proposed analysis tool was further developed with the FORTRAN language in the framework of ABAQUS. The numerical results are in good agreement with reported experimental data, indicating that the proposed peridynamic model for heterogeneous concrete can simulate the complex failure process of concrete well, and paves a way to further exploration of the failure mechanism of concrete materials.

An Optimization Algorithm for CAE Design of Carbon Fiber Reinforced Composite Chassis Longitudinal Arms

ZHU Di, YAO Yuan, PENG Xiongqi

2018, 39(8): 925-934. doi: 10.21656/1000-0887.390001

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Abstract:
The rear longitudinal arm is one of the main structures of the automobile chassis. Design of the rear longitudinal arm with carbon fiber reinforced polymer (CFRP) can reduce its weight effectively. However, the application of composite materials also brings great challenges to the optimization design process, such as complex multiple conditions and a large number of design variables. The secondary development of ABAQUS was conducted with Python to fulfill the global ergodic search for thickness ratios of different ply angles to find the effective range and the optimum solution. In order to reduce the long running time under multi working conditions, the treebased algorithms, such as XGBoost, DART and random forest, were introduced into the thickness ratio calculation. In view of both the running time and the computation accuracy, for 0 or 10 cases of calculation under the new condition, the accuracy rate of the TsaiWu factor can reach 96.3% and 98.3% (compared with failure value 1). If the number of cases under new working conditions increases to 40 while existing working conditions decreases by half, the accuracy rate can reach 95.0%. The developed algorithm provides a useful reference for reducing the running time of optimization design of composite parts under multi working conditions.

An Entropy Stable Scheme for Shallow Water Equations With Source Terms

ZHANG Haijun, FENG Jianhu, CHENG Xiaohan, LI Xue

2018, 39(8): 935-945. doi: 10.21656/1000-0887.380195

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Abstract:
An entropy stable scheme was developed for the shallow water equations with source terms, and a 1st-order entropy stable scheme and a high-order entropy conservation scheme were combined with a flux limiter function. The new entropy scheme preserves advantages of both the entropy conservation scheme and the entropy stable scheme, having higher accuracy in the regions of the smooth solutions and capturing shocks accurately while avoiding non-physical phenomena in the regions of the discontinuous solutions, thus achieves high resolution. The new scheme was successfully applied to calculate the classical 1D and 2D problems. The numerical results show that the new scheme does be an ideal method to simulate the shallow water equations with source terms.

Precision Analysis of the 3rd-Order WENO-Z Scheme and Its Improved Scheme

XU Weizheng, WU Weiguo

2018, 39(8): 946-960. doi: 10.21656/1000-0887.390011

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Abstract:
Firstly, the sufficient conditions for the 3rd-order WENO scheme satisfying the convergence precision were deduced. Based on the Taylor series method, the precision of the conventional 3rd-order WENO-Z scheme in the smooth flow field was analyzed. It was found that at the critical points, the 3rd-order WENO-Z scheme fails to achieve the convergence precision. In order to improve the precision near the critical points for the 3rd-order WENO-Z scheme, an improved 3rd-order WENO-Z scheme (WENO-NZ3) was constructed in view of the balance between precision and stability to finally determine the parameters. The improvement of the precision was verified through 2 typical numerical tests. What is more, the Sod shock wave tube, the shock-entropy wave interaction, the Rayleigh-Taylor instability and the 2D Riemann problem were calculated to confirm that the WENO-NZ3 scheme performs better than the conventional WENO schemes like WENO-JS3, WENO-Z3 and WENO-N3.

Numerical Analysis and Simulation of Solutions to a Class of Boussinesq Systems With Source Terms

HE Yubo, DONG Xiaoliang, LIN Xiaoyan

2018, 39(8): 961-978. doi: 10.21656/1000-0887.380126

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Abstract:
For a class of initial value problems of the high-order Boussinesq systems with source terms, a D1Q5 nonstandard lattice Boltzmann model (LBM) with correction functions and source terms was proposed. Different local equilibrium distribution functions and correction functions were selected, and the nonlinear wave equation was recovered by means of the Chapman-Enskog multi-scale analysis and the Taylor expansion technique. Some initial boundary value problems of the Boussinesq systems with analytical solutions were simulated to verify the effectiveness of the LBM. The results show that the numerical solutions agree well with the analytical solutions and the norm errors obtained with the LBM are smaller than those with the modified finite difference method (MFDM). Furthermore, some problems without analytical solutions were numerically studied with the present method and the MFDM. The comparison shows that the numerical solutions from the LBM are in good agreement with those from the MFDM, which validates the effectiveness and stability of the proposed model.

A Semi-Analytical Method for Stress Functions Meeting Crack Opening Displacements in Fracture Process Zones

HOU Yongkang, DUAN Shujin, AN Ruimei

2018, 39(8): 979-988. doi: 10.21656/1000-0887.380296

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Abstract:
Based on the Duan-Nakagawa model, with the weighted integral method, a semi-analytical method for stress functions meeting crack opening displacements in fracture process zones was proposed. The weighted function was determined by means of the boundary selected point method and the superposition of analytical functions with the same crack length but different fracture process zone lengths, to meet the given crack opening displacement in the fracture process zone, and then the final stress function and displacement function can be obtained with the weighted integral method. As an example, a special analytical solution for a double edge notched plate under Mode-I loading was derived, and the tensile strain softening curve and the fracture energy were obtained.

2018 Vol. 39, No. 8