一维准晶波动方程系数矩阵对称化及其SBP-SAT模拟

刘泰玉; 周月娥; 蒋关希曦; 张剑伟; 孙铖

doi:10.21656/1000-0887.450324

一维准晶波动方程系数矩阵对称化及其SBP-SAT模拟

doi: 10.21656/1000-0887.450324

刘泰玉^1,2,
周月娥^1,2,
蒋关希曦³,
张剑伟^4,5,6,
孙铖^1,2,4,5, ,

1. 广西民族大学建筑工程学院，南宁 530006;
2. 广西民族大学工程运维智能监测检测工程技术研究中心，南宁 530006;
3. 南京工业大学数理科学学院，南京 211816;
4. 哈尔滨工程大学航天与建筑工程学院，哈尔滨 150080;
5. 河北省双介质动力技术重点实验室，河北邯郸 056000;
6. 沈阳航空航天大学安全工程学院，沈阳 110000

基金项目:

广西高校中青年教师科研基础能力提升项目（2022KY0155）；广西民族大学科研基金资助项目(2021KJQD24)

详细信息

作者简介:
刘泰玉（1989—），男，讲师，硕士(E-mail: 20195041@gxmzu.edu.cn);孙铖（1992—），男，讲师，博士(通信作者. E-mail: suncheng007@126.com).

通讯作者:
孙铖（1992—），男，讲师，博士(通信作者. E-mail: suncheng007@126.com).

中图分类号: O343
计量
- 文章访问数: 22
- HTML全文浏览量: 4
- PDF下载量: 8
- 被引次数: 0
出版历程
- 收稿日期: 2024-12-04
- 修回日期: 2025-04-15
- 网络出版日期: 2026-04-01
- 刊出日期: 2026-03-01

The 1D Quasicrystal Wave Equation Coefficient Matrix Symmetrization and Its SBP-SAT Simulation

LIU Taiyu^1,2,
ZHOU Yuee^1,2,
JIANG Guanxixi³,
ZHANG Jianwei^4,5,6,
SUN Cheng^{1,2,4,5
, ,}

1. School of Civil Engineering and Architecture, Guangxi Minzu University,Nanning 530006, P.R.China;
2. Engineering Research Center for Intelligent Monitoring and Testing of Engineering Operation and Maintenance, Guangxi Minzu University, Nanning 530006, P.R.China;
3. School of Physical and Mathematical Sciences, Nanjing University of Technology, Nanjing 211816, P.R.China;
4. School of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150080, P.R.China;
5. Hebei Key Laboratory of Dual Media Power Technology, Handan, Hebei 056000, P.R.China;
6. School of Safety Engineering, Shenyang Aerospace University,Shenyang 110000, P.R.China

摘要

摘要: 研究波在准晶中的传播对深入理解准晶体的独特物理特性具有重要价值，但其数值模拟面临较大挑战.通过对波动方程系数矩阵进行对称化，可以有效整合不同类别的波动方程并降低波传播模拟的难度.研究推导了一维准晶波动方程的系数矩阵对称形式，并应用迎风格式SBPSAT差分方法对波动方程进行了离散化，同时通过能量法评估了其稳定性.数值仿真结果表明，所提出的离散框架具有较高的整合度、良好的稳定性及较强的拓展性.此外，该方法能够稳定模拟曲线域中的波传播，降低实现成本，显示了波动方程系数矩阵对称化及其离散框架在波传播模拟中的广泛应用前景.
- 准晶波动方程 /
- 系数矩阵对称式 /
- SBP-SAT /
- 有限差分方法 /
- 能量法
Abstract: The study of wave propagation in quasicrystals is of significant value for gaining a deeper understanding of the unique physical properties of quasicrystals, however, numerical simulations of such wave behaviors pose considerable challenges. Through symmetrization of the wave equation coefficient matrix, it is possible to effectively integrate different types of wave equations and reduce the complexity of wave propagation simulations. The symmetrized form of the coefficient matrix for the 1D quasicrystal wave equation was derived and the wave equation was discretized with the upwind scheme SBPSAT finite difference method, and the stability was then assessed with the energy method. Numerical simulations demonstrate that the proposed discretization framework exhibits high integration, good stability, and strong scalability. Furthermore, the method can stably simulate wave propagation in curved domains while reducing the implementation cost, indicating the broad application potential of the symmetrization technique and its discretization framework in wave propagation simulations.
- quasicrystal wave equation /
- coefficient matrix symmetrization /
- SBP-SAT /
- finite difference method /
- energy method

HTML全文

参考文献(35)

[2]钱晨, 汪久根. 准晶体及其性能研究进展[J]. 哈尔滨工业大学学报, 2017,49(7): 1-11. (QIAN Chen, WANG Jiugen. Progress in quasicrystals and their properties research[J]. Journal of Harbin Institute of Technology,2017,49(7): 1-11. (in Chinese))

SHECHTMAN D, BLECH I, GRATIAS D, et al. Metallic phase with long-range orientational order and no translational symmetry[J]. Physical Review Letters,1984,53(20): 1951-1953.

[3]MERLIN R, BAJEMA K,CLARKE R, et al. Quasiperiodic GaAs-AlAs heterostructures[J]. Physical Review Letters,1985,55(17): 1768-1770.

[4]BAK P. Phenomenological theory of icosahedral incommensurate (“quasiperiodic”) order in Mn-Al alloys[J]. Physical Review Letters,1985,54(14): 1517-1519.

[5]DING DH, YANG W, HU C, et al. Generalized elasticity theory of quasicrystals[J]. Physical Review B: Condensed Matter,1993,48(10): 7003-7010.

[6]FAN T. Mathematical Theory of Elasticity of Quasicrystals and Its Applications[M]. Heidelberg: Springer, 2011.

[7]杨娟, 徐燕, 师金华, 等. 功能梯度一维六方准晶中裂纹对SH波的散射[J]. 振动与冲击, 2023,42(12): 249-255. (YANG Juan, XU Yan, SHI Jinhua, et al. Scattering of the SH wave by a crack in functionally graded onedimensional hexagonal piezoelectric quasicrystal[J]. Journal of Vibration and Shock,2023,42(12): 249-255. (in Chinese))

[8]童鸣, 肖俊华. 一维六方准晶纳米尺度孔边周期Ⅲ型裂纹断裂力学分析[J]. 固体力学学报, 2024,45(5): 610-621. (TONG Ming, XIAO Junhua. Fracture mechanics of periodic type-Ⅲ cracks emanating from a nano-hole in one-dimensional hexagonal quasicrystals[J]. Chinese Journal of Solid Mechanics,2024,45(5): 610-621. (in Chinese))

[9]卢绍楠, 赵雪芬, 马园园. 一维六方压电准晶双材料界面共线裂纹问题[J]. 应用数学和力学, 2023,44(7): 809-824. (LU Shaonan, ZHAO Xuefen, MA Yuanyuan. Research on interfacial collinear cracks between 1D hexagonal piezoelectric quasicrystal bimaterials[J]. Applied Mathematics and Mechanics,2023,44(7): 809-824. (in Chinese))

[10]王程颜, 刘官厅. 一维六方压电准晶中唇形孔口次生四条裂纹的反平面问题[J]. 应用数学和力学, 2024,45(7): 886-897. (WANG Chengyan, LIU Guanting. The antiplane problem of a lip-shaped orifice with 4 edge cracks in 1D hexagonal piezoelectric quasicrystal[J]. Applied Mathematics and Mechanics,2024,45(7): 886-897. (in Chinese))

[11]张炳彩, 丁生虎, 张来萍. 一维六方准晶双材料中圆孔边共线界面裂纹的反平面问题[J]. 应用数学和力学, 2022,43(6): 639-647. (ZHANG Bingcai, DING Shenghu, ZHANG Laiping. The anti-plane problem of collinear interface cracks emanating from a circular hole in 1D hexagonal quasicrystal bi-materials[J]. Applied Mathematics and Mechanics,2022,43(6): 639-647. (in Chinese))

[12]胡克强, 高存法, 付佳维, 等. 无限大压电准晶介质中两个圆柱夹杂的干涉作用[J]. 内蒙古工业大学学报(自然科学版), 2023,42(3): 230-236. (HU Keqiang, GAO Cunfa, FU Jiawei, et al. Interference of two cylindrical inclusions in an infinite piezoelectric quasicrystal medium[J]. Journal of Inner Mongolia University of Technology (Natural Science Edition), 2023,42(3): 230-236. (in Chinese))

[13]CHEN W Q, MA Y L, DING H J. On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies[J]. Mechanics Research Communications,2004,31(6): 633-641.

[14]WANG X. The general solution of one-dimensional hexagonal quasicrystal[J]. Mechanics Research Communications,2006,33(4): 576-580.

[15]FAN T Y, MAI Y W. Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials[J]. Applied Mechanics Reviews,2004,57(5): 325-343.

[16]YASLAN H . On three-dimensionalelastodynamic problems of one-dimensional quasicrystals[J]. Waves in Random and Complex Media,2019,29(4): 614-630.

[17]YAKHNO V. Derivation of a solution of dynamic equations of motion for quasicrystals[J]. Journal of Engineering Mathematics,2019,118(1): 63-72.

[18]ALTUNKAYNAK M. Solving elastodynamic problems of 2D quasicrystals in inhomogeneous media[J]. Applications of Mathematics,2024,69(3): 289-309.

[19]GAO Y, ZHAO B S. General solutions of three-dimensional problems for two-dimensional quasicrystals[J]. Applied Mathematical Modelling,2009,33(8): 3382-3391.

[20]GAO Y, ZHAO B S. A general treatment of three-dimensional elasticity of quasicrystals by an operator method[J]. Physica Status Solidi B: Basic Research,2006,243(15): 4007-4019.

[21]王会苹, 王桂霞, 陈德财. 含椭圆孔有限大二十面体准晶板平面弹性问题的边界元分析[J]. 应用数学和力学, 2024,45(4): 400-415. (WANG Huiping, WANG Guixia, CHEN Decai. Boundary element analysis for the plane elasticity problems of finite icosahedral quasicrystal plates containing elliptical holes[J]. Applied Mathematics and Mechanics,2024,45(4): 400-415. (in Chinese))

[22]赵颖涛, 杨宇威, 章伟. 准晶问题多项式应力函数及其在有限元中的应用[J]. 北京理工大学学报, 2024,44(9): 887-894. (ZHAO Yingtao, YANG Yuwei, ZHANG Wei. Polynomial stress function of quasicrystal problems and their applications in FEM[J]. Transactions of Beijing Institute of Technology,2024,44(9): 887-894.(in Chinese))

[23]SUN C, YANG Z L, JIANG G X X, et al. Multiblock SBP-SAT methodology of symmetric matrix form of elastic wave equations on curvilinear grids[J]. Shock and Vibration,2020,2020: 8401537.

[24]杨在林, 魏夕杰, 孙铖, 等. TI介质中P-SV波传播的SBP-SAT模拟[J]. 振动与冲击, 2023,42(20): 91-97. (YANG Zailin, WEI Xijie, SUN Cheng, et al. P-SV wave propagation in transversely isotropic media by SBP-SAT modeling[J]. Journal of Vibration and Shock,2023,42(20): 91-97. (in Chinese))

[25]孙铖, 杨在林, 蒋关希曦, 等. 完全匹配层在矩阵式波动方程SBP-SAT方法应用[J]. 振动与冲击, 2024,43(13): 53-60. (SUN Cheng, YANG Zailin, JIANG Guanxixi, et al. Application of perfect matching layer in SBP-SAT method of matrix wave equation[J]. Journal of Vibration and Shock,2024,43(13): 53-60. (in Chinese))

[26]李航, 孙宇航, 李佳慧, 等. 基于波动方程的地震波数值模拟研究综述[J]. 吉林大学学报(地球科学版), 2025,55(2): 627-645. (LI Hang, SUN Yuhang, LI Jiahui, et al. A comprehensive review of numerical simulation of seismic waves based on wave equation[J]. Journal of Jilin University (Earth Science Edition), 2025,55(2): 627-645. (in Chinese))

[27]FANG C J, QIU H K, ZHANG Z G. Non-uniform grid finite-difference seismic wave simulation using multiblock grids by adding positive and negative singularity pairs[J]. Geophysical Journal International,2025,241(2): 1173-1185.

[28]MASSON Y. Distributional finite-difference modelling of seismic waves[J]. Geophysical Journal International,2023,233(1): 264-296.

[29]LYU C, MASSON Y, ROMANOWICZ B, et al. Introduction to the distributional finite difference method for 3D seismic wave propagation and comparison with the spectral element method[J]. Journal of Geophysical Research: Solid Earth,2024,129(4): e2023JB027576.

[30]SVRD M, NORDSTRM J. Review of summation-by-parts schemes for initial-boundary-value problems[J]. Journal of Computational Physics,2014,268: 17-38.

[31]DEL REY FERNNDEZ D C, HICKEN J E, ZINGG D W. Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations[J]. Computers & Fluids,2014,95: 171-196.

[32]STRAND B. Summation by parts for finite difference approximations for d/dx[J]. Journal of Computational Physics,1994,110(1): 47-67.

[33]RICHTMYER R D, DILL E H. Difference methods for initial-value problems[J]. Physics Today,1959,12(4): 50.

[34]杨在林, 孙铖, 蒋关希曦, 等. SBP-SAT方法及其在波动领域的应用[J]. 振动与冲击, 2020,39(12): 150-157. (YANG Zailin, SUN Cheng, JIANG Guanxixi, et al. A SBP-SAT method for the numerical solution and its application in the wave motion[J]. Journal of Vibration and Shock,2020,39(12): 150-157. (in Chinese))

[35]RYDIN Y, MATTSSON K, WERPERS J. High-fidelity sound propagation in a varying 3D atmosphere[J]. Journal of Scientific Computing,2018,77(2): 1278-1302.

施引文献

资源附件(0)

访问统计

计量

文章访问数: 22
HTML全文浏览量: 4
PDF下载量: 8
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

留言板

一维准晶波动方程系数矩阵对称化及其SBP-SAT模拟

doi: 10.21656/1000-0887.450324

作者简介:
刘泰玉（1989—），男，讲师，硕士(E-mail: 20195041@gxmzu.edu.cn);孙铖（1992—），男，讲师，博士(通信作者. E-mail: suncheng007@126.com).

通讯作者:
孙铖（1992—），男，讲师，博士(通信作者. E-mail: suncheng007@126.com).

计量

The 1D Quasicrystal Wave Equation Coefficient Matrix Symmetrization and Its SBP-SAT Simulation

计量

目录

留言板

一维准晶波动方程系数矩阵对称化及其SBP-SAT模拟

doi: 10.21656/1000-0887.450324

作者简介: 刘泰玉（1989—），男，讲师，硕士(E-mail: 20195041@gxmzu.edu.cn);孙铖（1992—），男，讲师，博士(通信作者. E-mail: suncheng007@126.com).

通讯作者: 孙铖（1992—），男，讲师，博士(通信作者. E-mail: suncheng007@126.com).

计量

出版历程

The 1D Quasicrystal Wave Equation Coefficient Matrix Symmetrization and Its SBP-SAT Simulation

计量

出版历程

目录

作者简介:
刘泰玉（1989—），男，讲师，硕士(E-mail: 20195041@gxmzu.edu.cn);孙铖（1992—），男，讲师，博士(通信作者. E-mail: suncheng007@126.com).

通讯作者:
孙铖（1992—），男，讲师，博士(通信作者. E-mail: suncheng007@126.com).