近场波动有限元模拟的应力型时域人工边界条件及其应用
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摘要
我国的重大工程(高拱坝、核电站、高层建筑和大跨度桥梁等)建设规模已位居世界前列。重大工程在地震或爆炸等动力荷载作用下的时域数值模拟对于其设计及安全评价具有重要意义。该类结构的分析模型需要考虑无限域地基的波动辐射效应,形成近场波动问题,也称土-结动力相互作用问题。其解法是人为引入人工边界将整个开放系统划分为近场有限域和远场无限域两部分。前者包含结构以及可能的非均匀和非线性因素,采用有限元法模拟;后者简化为线弹性介质并满足辐射条件,分析中被截去,通过在有限域的人工边界处施加人工边界条件(或称无反射、透射、吸收、辐射边界条件)模拟其波动辐射效应。人工边界问题也存在于声学、电磁学、流体动力学和气象学等物理和工程领域,属于新兴的跨学科的数值计算科学问题。有效的人工边界条件应该与有限元法结合形成稳定、精确、高效并且容易实现的近场波动分析方法。目前已经出现了多种基于不同数学、物理和力学方法的人工边界条件,但仍然没有一种公认的最优方法。
本文研究应力型精确时域人工边界条件及其工程应用。首先,时域方法可以考虑近场有限域的非均匀和非线性因素。其次,应力型指人工边界条件是有限域和无限域间相互作用应力的表达式,其作为自然边界条件较好地兼容有限元法,易于形成稳定的近场波动分析方法。最后,随着计算机发展水平和工程精度要求的不断提高,精确模拟方法应该是未来的发展趋势;并且精确方法可以比近似方法设置在距离结构或者辐射源更近的位置,导致有限域的计算成本显著降低。
本文采用变量分离法求解远场无限域的定解问题,获得时间全局的人工边界条件,即其某一时刻的响应与该时刻以及之前全部时刻的响应相关。为了降低计算成本,采用由有理近似和高阶弹簧-阻尼-质量模型构成的卷积核压缩技术对人工边界条件进行时间局部化处理,获得稳定、精确、兼顾效率和易实现性的人工边界条件。本文采用的远场无限域模型包括出平面波动问题的波导模型和外域模型以及一维弹性波辐射模型。基于前者的人工边界条件可以精确地模拟出平面波动问题,但应用于重大工程结构的近场波动分析仍需开展大量的研究工作;基于后者的人工边界条件可以直接应用于工程实践,近似地模拟一般非对称弹性波在外域模型中的远场辐射问题。本文的具体研究工作如下:
一、力和位移时间卷积变换的卷积核压缩技术
1.基于线性系统的稳定性理论,提出了无限域频率响应函数有理近似的充分必要稳定性条件;采用罚函数法和遗传-单纯形优化算法建立了通过强加稳定性约束预先保证稳定的有理近似识别方法;讨论了稳定系统中的共振现象及其消除方法。
2.提出了三种在时域内实现有理近似的高阶弹簧-阻尼-质量模型;建立了用于计算模型参数的有理函数连分式展开技术;给出了高阶弹簧-阻尼-质量模型的外源激励输入方法。
3.通过分析几个典型的基础振动问题,验证了卷积核压缩技术的有效性。
二、出平面波动问题的应力型精确时域人工边界条件
1.根据波导模型和外域模型的物理边界条件选择人工边界模态函数,采用傅里叶级数描述空间全局性。
2.将改进的卷积核压缩技术应用于模态频率响应函数,进行时间局部化处理,形成辅助变量的时间二阶对称常微分方程组。
3.基于空间和时间处理,建立了可以直接进行矩阵装配的人工边界条件有限元公式,实现了精确人工边界条件与有限元法的无缝结合。
三、基于一维弹性波辐射的应力型时域人工边界条件及其工程应用
1.将改进的卷积核压缩技术分别应用于一维柱面和球面弹性波辐射模型,建立了相应的应力型精确时域人工边界条件,粘弹性边界为其低阶形式。
2.完善了粘弹性边界体系,发展了二维平面内切向和三维切向边界。
3.基于粘弹性边界给出平面地震波斜入射和竖直入射的简化输入方法,进行了小湾拱坝地震反应分析,并与透射边界结果进行了比较,分析中还初步考虑了坝体混凝土材料非线性的影响。
四、其他研究工作
1.研究了基于有理近似的时域递归算法。提出采用双线性变换从连续时间有理近似获得离散时间有理近似的变换方法;应用状态空间变换建立了含有辅助变量的单步时域递归公式。
2.分析了弹性地基表面半无限杆和波导模型中波动的截止频率和频散特性,指出基于一维外行波法向透射建立的人工边界条件难于处理该类问题,并通过数值试验证明了这一推断。
3.推导了出平面运动的直角坐标和极坐标集中质量有限元方程以及标准单元的单元矩阵;讨论了一维和二维均匀时空离散网格中波动的周期特性、截止频率和频散特性。
The scale of the major construction projects (such as high arch dam, nuclearstation, high-rise building, long-span bridge and so on) in our country has leaped intothe front ranks of the world. The time-domain numerical simulation of the majorconstructions under the seismic or blast load is an effective method for their designand safety evaluation. The analysis models of the major constructions requireconsidering the energy radiation of infinite foundation, forming the near-field wavemotion problem or called dynamic soil-structure interaction problem. The commonnumerical method to analyze the near-field wave motion problem is as follows. Afictitious boundary usually called artificial boundary is introduced to partition thewhole opening system into two parts, i.e. near-field finite domain and far-field infinitedomain. The former includes structure, and inhomogeneity and nonlinearity of media,which can be modeled by finite element method. On the other hand, the latter issimplified as linear elastic media, and satisfies the radiation condition at the artificialboundary. In practical computation, the far-field infinite domain is truncated. Aboundary condition is imposed on the artificial boundary to model the energyradiation effect of the truncated infinite domain, which is called artificial boundarycondition (ABC) or nonreflecting, transmitting, absorbing, radiation boundarycondition. Actually, the artificial boundary problems are ubiquitous in many fields ofphysics and engineering, such as acoustics, electromagnetics, computational fluiddynamics, meteorology and so on, which belonging a new interdisciplinarynumerically computational science subject. An effective ABC should combine withfinite element method to form a stable, accurate, efficient and easily implementednear-field wave motion analysis method. At present, many ABCs have been proposedbased on the different mathematical, physical and mechanical methods. However,there is still no consensus on the optimal ABC.
This dissertation studies the stress-type exact time-domain ABC and itsengineering application. First, the time-domain method can consider theinhomogeneity and nonlinearity in the near-field finite domain. Second, thestress-type ABC indicates that the ABC is an expression of interaction stress betweenthe near-field finite domain and the far-field infinite domain. It can compatible withthe finite element method very well as a natural stress boundary condition, leading tothe numerically stable near-field wave motion method. Last, with the computer leveland engineering accuracy requirement improving, the exact method should be a trend.Although the exact method has lower efficiency than the approximate method, it canbe placed nearer source or structure than the latter, leading to the lower computational cost in the near-field finite domain.
The problem statement of the far-field infinite domain is first solved by theseparation of variables. The temporal global ABC is obtained, where the response at aconstant is related with the current and all before constants. To decrease thecomputational cost, the convolution kernel compression technique, combining therational approximation and high-order spring-dashpot-mass model, is then applied tolocalize the obtained ABC. A stable, accurate, efficient and easily implemented ABCis finally obtained. In this dissertation, the far-field infinite domains include thewaveguide and exterior models of out-of-plane wave motion problem and theone-dimensional elastic wave radiation model. The ABC from the former can exactlymodel the out-of-plane wave motion problem, but many works still need to be furtherstudied for the application to the major construction. On the other hand, the ABC fromthe latter can be directly applied to engineering practice to approximately model theradiation of the general non-symmetric elastic waves in the exterior model. Theconcrete works in this dissertation are as follows.
1. The convolution kernel compression to localize the convolution transformbetween force and displacement.
(1) The necessary and sufficient stability condition for the rational approximationof frequency response function of infinite domain is presented based on the stabilitytheory of linear system. A parameter identification method guaranteeing stability apriori by enforcing the stability constraint condition is further developed based on thepenalty function method and the genetic-simplex optimization algorithm. Theresonance phenomenon in a stable system is also discussed, and a method avoidingresonance is proposed.
(2) Three new high-order spring-dashpot-mass models are proposed, as therealizations of the rational approximation into the time domain. Thecontinued-fraction expansion of rational function is developed to calculate the modelparameters. A seismic input method for the high-order spring-dashpot-mass models isalso presented.
(3) The effectiveness of the modified convolution kernel compression techniqueis demonstrated by analyzing several typical foundation vibration problems.
2. Stress-type exact time-domain ABC for out-of-plane wave motionproblem.
(1) The modal function of artificial boundary is first chosen according to thephysical boundary conditions of the waveguide and exterior models. The Fourierseries expansion is then applied to represent the spatial globality.
(2) The modified convolution kernel compression technique is applied to themodal frequency response functions to localize the exact ABC, leading to a symmetricsystem of second-order ordinary differential equations of the auxiliary variables intime.
(3) Based on the spatial and temporal treatments mentioned above, thefinite-element formulas of exact ABC are developed. They can assemble directly andcouple seamless with the finite-element model of the near-field finite domain.
3. Stress-type time-domain ABC based on radiation of one-dimensionalelastic waves and its engineering application.
(1) Based on the modified convolution kernel compression technique, the exacttime-domain ABC is developed to model the radiation of cylindrical and sphericalelastic waves from the near-field finite domain into the far-field infinite domain. Theviscous-spring boundary is the low-order form of this ABC.
(2) The viscous-spring boundary is further improved. The two-dimensionalin-plane tangential boundary and the three-dimensional tangential boundary aredeveloped.
(3) The simplified input methods of the plane seismic waves of inclined andvertical incidence are proposed for the viscous-spring boundary. The seismicresponses of Xiaowan arch dam are calculated and compared with the results by usingthe multi-transmitting formula. The effect of material nonlinearity of concrete is alsoconsidered simply.
4. Other research works.
(1) The time-domain recursive evaluation based on the rational approximation isdiscussed. The bilinear transform is used to obtain the discrete-time rationalapproximation from the continuous-time one. The single-step time-domain recursiveformulas are constructed by applying the state space transform.
(2) The cutoff frequency and dispersive property of wave propagation in thesemi-infinite rod on an elastic foundation and in the waveguide model are studied. Aconclusion that the local ABC based on the one-dimensional outgoing wave can notsolve such problems satisfactorily is drawn, which is demonstrated by numericalexperiments.
(3) The lumped-mass finite-element equations for out-of-plane wave motion arepresented in Cartesian and polar coordinates, respectively. The element matrices ofthe normal finite elements are also derived. The periodic property, cutoff frequencyand dispersive property of the wave propagation in the one- and two-dimensionalfinite-element mesh of spatially and temporally uniform discretization are discussed.
本文研究应力型精确时域人工边界条件及其工程应用。首先,时域方法可以考虑近场有限域的非均匀和非线性因素。其次,应力型指人工边界条件是有限域和无限域间相互作用应力的表达式,其作为自然边界条件较好地兼容有限元法,易于形成稳定的近场波动分析方法。最后,随着计算机发展水平和工程精度要求的不断提高,精确模拟方法应该是未来的发展趋势;并且精确方法可以比近似方法设置在距离结构或者辐射源更近的位置,导致有限域的计算成本显著降低。
本文采用变量分离法求解远场无限域的定解问题,获得时间全局的人工边界条件,即其某一时刻的响应与该时刻以及之前全部时刻的响应相关。为了降低计算成本,采用由有理近似和高阶弹簧-阻尼-质量模型构成的卷积核压缩技术对人工边界条件进行时间局部化处理,获得稳定、精确、兼顾效率和易实现性的人工边界条件。本文采用的远场无限域模型包括出平面波动问题的波导模型和外域模型以及一维弹性波辐射模型。基于前者的人工边界条件可以精确地模拟出平面波动问题,但应用于重大工程结构的近场波动分析仍需开展大量的研究工作;基于后者的人工边界条件可以直接应用于工程实践,近似地模拟一般非对称弹性波在外域模型中的远场辐射问题。本文的具体研究工作如下:
一、力和位移时间卷积变换的卷积核压缩技术
1.基于线性系统的稳定性理论,提出了无限域频率响应函数有理近似的充分必要稳定性条件;采用罚函数法和遗传-单纯形优化算法建立了通过强加稳定性约束预先保证稳定的有理近似识别方法;讨论了稳定系统中的共振现象及其消除方法。
2.提出了三种在时域内实现有理近似的高阶弹簧-阻尼-质量模型;建立了用于计算模型参数的有理函数连分式展开技术;给出了高阶弹簧-阻尼-质量模型的外源激励输入方法。
3.通过分析几个典型的基础振动问题,验证了卷积核压缩技术的有效性。
二、出平面波动问题的应力型精确时域人工边界条件
1.根据波导模型和外域模型的物理边界条件选择人工边界模态函数,采用傅里叶级数描述空间全局性。
2.将改进的卷积核压缩技术应用于模态频率响应函数,进行时间局部化处理,形成辅助变量的时间二阶对称常微分方程组。
3.基于空间和时间处理,建立了可以直接进行矩阵装配的人工边界条件有限元公式,实现了精确人工边界条件与有限元法的无缝结合。
三、基于一维弹性波辐射的应力型时域人工边界条件及其工程应用
1.将改进的卷积核压缩技术分别应用于一维柱面和球面弹性波辐射模型,建立了相应的应力型精确时域人工边界条件,粘弹性边界为其低阶形式。
2.完善了粘弹性边界体系,发展了二维平面内切向和三维切向边界。
3.基于粘弹性边界给出平面地震波斜入射和竖直入射的简化输入方法,进行了小湾拱坝地震反应分析,并与透射边界结果进行了比较,分析中还初步考虑了坝体混凝土材料非线性的影响。
四、其他研究工作
1.研究了基于有理近似的时域递归算法。提出采用双线性变换从连续时间有理近似获得离散时间有理近似的变换方法;应用状态空间变换建立了含有辅助变量的单步时域递归公式。
2.分析了弹性地基表面半无限杆和波导模型中波动的截止频率和频散特性,指出基于一维外行波法向透射建立的人工边界条件难于处理该类问题,并通过数值试验证明了这一推断。
3.推导了出平面运动的直角坐标和极坐标集中质量有限元方程以及标准单元的单元矩阵;讨论了一维和二维均匀时空离散网格中波动的周期特性、截止频率和频散特性。
The scale of the major construction projects (such as high arch dam, nuclearstation, high-rise building, long-span bridge and so on) in our country has leaped intothe front ranks of the world. The time-domain numerical simulation of the majorconstructions under the seismic or blast load is an effective method for their designand safety evaluation. The analysis models of the major constructions requireconsidering the energy radiation of infinite foundation, forming the near-field wavemotion problem or called dynamic soil-structure interaction problem. The commonnumerical method to analyze the near-field wave motion problem is as follows. Afictitious boundary usually called artificial boundary is introduced to partition thewhole opening system into two parts, i.e. near-field finite domain and far-field infinitedomain. The former includes structure, and inhomogeneity and nonlinearity of media,which can be modeled by finite element method. On the other hand, the latter issimplified as linear elastic media, and satisfies the radiation condition at the artificialboundary. In practical computation, the far-field infinite domain is truncated. Aboundary condition is imposed on the artificial boundary to model the energyradiation effect of the truncated infinite domain, which is called artificial boundarycondition (ABC) or nonreflecting, transmitting, absorbing, radiation boundarycondition. Actually, the artificial boundary problems are ubiquitous in many fields ofphysics and engineering, such as acoustics, electromagnetics, computational fluiddynamics, meteorology and so on, which belonging a new interdisciplinarynumerically computational science subject. An effective ABC should combine withfinite element method to form a stable, accurate, efficient and easily implementednear-field wave motion analysis method. At present, many ABCs have been proposedbased on the different mathematical, physical and mechanical methods. However,there is still no consensus on the optimal ABC.
This dissertation studies the stress-type exact time-domain ABC and itsengineering application. First, the time-domain method can consider theinhomogeneity and nonlinearity in the near-field finite domain. Second, thestress-type ABC indicates that the ABC is an expression of interaction stress betweenthe near-field finite domain and the far-field infinite domain. It can compatible withthe finite element method very well as a natural stress boundary condition, leading tothe numerically stable near-field wave motion method. Last, with the computer leveland engineering accuracy requirement improving, the exact method should be a trend.Although the exact method has lower efficiency than the approximate method, it canbe placed nearer source or structure than the latter, leading to the lower computational cost in the near-field finite domain.
The problem statement of the far-field infinite domain is first solved by theseparation of variables. The temporal global ABC is obtained, where the response at aconstant is related with the current and all before constants. To decrease thecomputational cost, the convolution kernel compression technique, combining therational approximation and high-order spring-dashpot-mass model, is then applied tolocalize the obtained ABC. A stable, accurate, efficient and easily implemented ABCis finally obtained. In this dissertation, the far-field infinite domains include thewaveguide and exterior models of out-of-plane wave motion problem and theone-dimensional elastic wave radiation model. The ABC from the former can exactlymodel the out-of-plane wave motion problem, but many works still need to be furtherstudied for the application to the major construction. On the other hand, the ABC fromthe latter can be directly applied to engineering practice to approximately model theradiation of the general non-symmetric elastic waves in the exterior model. Theconcrete works in this dissertation are as follows.
1. The convolution kernel compression to localize the convolution transformbetween force and displacement.
(1) The necessary and sufficient stability condition for the rational approximationof frequency response function of infinite domain is presented based on the stabilitytheory of linear system. A parameter identification method guaranteeing stability apriori by enforcing the stability constraint condition is further developed based on thepenalty function method and the genetic-simplex optimization algorithm. Theresonance phenomenon in a stable system is also discussed, and a method avoidingresonance is proposed.
(2) Three new high-order spring-dashpot-mass models are proposed, as therealizations of the rational approximation into the time domain. Thecontinued-fraction expansion of rational function is developed to calculate the modelparameters. A seismic input method for the high-order spring-dashpot-mass models isalso presented.
(3) The effectiveness of the modified convolution kernel compression techniqueis demonstrated by analyzing several typical foundation vibration problems.
2. Stress-type exact time-domain ABC for out-of-plane wave motionproblem.
(1) The modal function of artificial boundary is first chosen according to thephysical boundary conditions of the waveguide and exterior models. The Fourierseries expansion is then applied to represent the spatial globality.
(2) The modified convolution kernel compression technique is applied to themodal frequency response functions to localize the exact ABC, leading to a symmetricsystem of second-order ordinary differential equations of the auxiliary variables intime.
(3) Based on the spatial and temporal treatments mentioned above, thefinite-element formulas of exact ABC are developed. They can assemble directly andcouple seamless with the finite-element model of the near-field finite domain.
3. Stress-type time-domain ABC based on radiation of one-dimensionalelastic waves and its engineering application.
(1) Based on the modified convolution kernel compression technique, the exacttime-domain ABC is developed to model the radiation of cylindrical and sphericalelastic waves from the near-field finite domain into the far-field infinite domain. Theviscous-spring boundary is the low-order form of this ABC.
(2) The viscous-spring boundary is further improved. The two-dimensionalin-plane tangential boundary and the three-dimensional tangential boundary aredeveloped.
(3) The simplified input methods of the plane seismic waves of inclined andvertical incidence are proposed for the viscous-spring boundary. The seismicresponses of Xiaowan arch dam are calculated and compared with the results by usingthe multi-transmitting formula. The effect of material nonlinearity of concrete is alsoconsidered simply.
4. Other research works.
(1) The time-domain recursive evaluation based on the rational approximation isdiscussed. The bilinear transform is used to obtain the discrete-time rationalapproximation from the continuous-time one. The single-step time-domain recursiveformulas are constructed by applying the state space transform.
(2) The cutoff frequency and dispersive property of wave propagation in thesemi-infinite rod on an elastic foundation and in the waveguide model are studied. Aconclusion that the local ABC based on the one-dimensional outgoing wave can notsolve such problems satisfactorily is drawn, which is demonstrated by numericalexperiments.
(3) The lumped-mass finite-element equations for out-of-plane wave motion arepresented in Cartesian and polar coordinates, respectively. The element matrices ofthe normal finite elements are also derived. The periodic property, cutoff frequencyand dispersive property of the wave propagation in the one- and two-dimensionalfinite-element mesh of spatially and temporally uniform discretization are discussed.
引文
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