摘要
基于时域有限元方法,为增材制造技术中常涉及的声波—弹性波耦合问题构建理论框架。在三维直角坐标系下引入流体介质所满足的关于压力场的时域声波方程和固体介质所满足的关于位移场的时域弹性波方程。将时域波动方程转换到频域进行复坐标系拉伸,再对拉伸后的频域波动方程进行波场分裂。复拉伸因子是表征声波/弹性波衰减的物理量,在声波方程/弹性波方程中体现为近乎于复介声常数/复介弹常数的反映特性,与介声常数/介弹常数、声导率/弹性导率、声波/弹性波传播速度等参数构成了环环相扣的有机整体,但表现形式与电磁波理论中复介电常数的情形不同。提出一种通过引入过渡变量求解卷积的处理策略,将分裂后的频域波动方程转化回时域,得到带有完全匹配层(PML)的时域波动方程,其中声波方程和弹性波方程中最大未知量个数分别仅为7个和27个。推导带有PML的时域波动方程在有限元方法实现中所需的等效积分弱形式,进而基于声波和弹性波在流体和固体区域交界面的双向转换机制,导出带有PML的声波—弹性波耦合方程的等效积分弱形式。为节约PML计算成本,根据波动衰减方位数量的不同,对角点采取分门别类的处理方法,使得通常情况下带有PML模型的计算资源消耗相比模型不带有PML时仅增加1倍左右。波动方程及其PML内在机理的深入探究和计算效率的深度优化,为增材制造技术中涉及声波—弹性波耦合问题的应用提供了前提和保障,而带有PML的声波—弹性波耦合方程的等效积分弱形式,则是促进基于时域有限元方法的声波—弹性波耦合在增材制造技术中得以应用的理论精髓所在。
Based on the Finite-Element Time-Domain (FETD) method, a theoretical framework is established for the acoustic-elastic coupling issue often involved in the additive manufacturing technology. The time-domain acoustic wave equation about the pressure field satisfied by the fluid medium and the time-domain elastic wave equation about the displacement field satisfied by the solid medium are introduced in the 3D Cartesian coordinate system. These time-domain wave equations are converted to the frequency domain to carry out the complex coordinate stretching, and then the stretched wave equation in the frequency domain is split. The complex stretching factor is a physical quantity that characterizes the attenuation of the acoustic/elastic wave. In the acoustic/elastic wave equation, the complex stretching factor holds reflection characteristics close to the complex acoustic permittivity/ complex elastic permittivity, and it constitutes an organic whole with parameters including acoustic permittivity/elastic permittivity, acoustic conductivity/elastic conductivity and acoustic/elastic wave propagating velocity linked with each other. However, its expressing form is different from that of the complex electric permittivity in the electromagnetic wave theory. A processing strategy to cope with the convolution is proposed by introducing the transition variables. The split wave equation is converted from the frequency domain back to the time domain, so that the wave equation with Perfectly Matched Layer (PML) in the time domain is obtained. The maximum number of unknown coefficients in the acoustic wave equation and the elastic wave equation are only 7 and 27, respectively. The equivalent integral weak form of the wave equation with PML in the time domain is derived for the realization of the finite element method, and further, the equivalent integral weak form of the acoustic-elastic coupling equation with PML is derived based on the bidirectional conversion mechanism of the acoustic wave and the elastic wave at the interface between the fluid region and the solid region. In order to save the computing cost of PML, cornered points are classified according to the number of wave attenuation originations, which makes the computing resource consumption of model with PML in most cases increase by only about 1 time compared with that without PML. The thorough investigation of the internal mechanism and the indepth optimization of the computing efficiency of the wave equation and its PML, are the premise and the guarantee for the application of acoustic-elastic coupling issue involved in the additive manufacturing technology. Meanwhile, the equivalent integral weak form of the acoustic-elastic coupling equation with PML, is exactly the theoretical essence that promotes the FETD method based acoustic-elastic coupling to be applied in the additive manufacturing technology.
作者
张阔
ZHANG Kuo(China Center for Information Industry Development, Beijing 100048, China;Beijing CCID Publishing & Media Co., Ltd., Beijing 100048, China)
出处
《工业技术创新》
2019年第5期74-85,90,共13页
Industrial Technology Innovation
关键词
增材制造
时域有限元
声波-弹性波耦合
复拉伸因子
复介声常数
复介弹常数
声导率
弹性导率
完全匹配层
等效积分弱形式
计算资源消耗
Additive Manufacturing Technology
Finite-Element Time-Domain
Acoustic-Elastic Coupling
Complex Stretching Factor
Complex Acoustic Permittivity
Complex Elastic Permittivity
Acoustic Conductivity
Elastic Conductivity
Perfectly Matched Layer
Equivalent Integral Weak Form
Computing Resource Consumption
作者简介
通信作者:张阔(1987-),男,博士,出版专业技术人员(中级),《工业技术创新》责任编辑。研究方向:出版学、信息与计算科学、关键共性技术创新。E-mail:274739214@qq.com,ORCID:0000-0002-4874-0003.
