基于SPH方法的冲击动力学若干问题研究
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摘要
光滑粒子流体动力学(SPH)方法是一种求解偏微分方程的数值方法,属于无网格法的一种。由于SPH方法彻底摆脱了计算网格的约束,采用任意分布的粒子来表示求解域,不会遇到网格变形过大或网格畸变的问题,所以特别适合求解冲击动力学中的大变形问题。同时由于SPH法是具有拉格朗日性质的动力学方法,所以可以方便的跟踪物质的运动轨迹,适合描述流体界面的大变形运动过程以及流体与固体之间的相互作用。SPH方法作为一种具有无网格、自适应、稳定以及拉格朗日性质的动力学求解算法,已经成为冲击动力学研究的一个热点问题,并在工程实践中有着广泛的应用前景。本文基于国家自然科学基金委员会和中国工程物理研究院共同设立的“NSAF联合基金”项目“舰艇复杂板壳结构在冲击载荷作用下的破坏机理研究”的研究背景,详细讨论了SPH法的基本理论及其数值特性,并对船体壳板的穿甲问题、剪切式吸能器的性能研究、充液容器的跌落碰撞、双层充液壳体结构的冲击安全性以及水力阻力器的水阻力特性等若干冲击动力学问题进行了研究。
本文对SPH方法的基本理论进行了全面总结,介绍了SPH法的基本思想,重点对SPH法的两个重要步骤,即核函数近似过程和粒子近似过程,进行了详细说明,推导出任意函数及其导数的SPH近似公式,并讨论了光滑函数的常见形式和基本性质。随后采用SPH法对连续介质力学的基本控制方程组进行离散近似,推导出描述流体或固体运动的常用SPH公式,并对人工粘性、人工热流、状态方程、本构关系、积分方法等一系列相关数值问题进行了讨论。在此基础上,用FORTRAN语言编写相应的三维SPH计算程序,为进一步开展数值研究奠定基础。
详细研究了SPH法的数值特性。通过用SPH法求解一般函数的数值算例,详细讨论了光滑函数的形式、光滑长度、粒子的分布对SPH法计算精度的影响,并指出传统的SPH算法本身存在着边界粒子计算精度低和张力不稳定的固有缺陷。随后对几种改进传统SPH法的数值方法,如改进边界粒子精度的SPH法、CSPH法和DSPH法,分别进行了理论分析和数值研究,说明这些算法可以一定程度上改善边界粒子的计算精度,并修正不连续处的计算误差。同时介绍了SPH方法中固定边界条件的一般处理方法,并通过求解三个典型数值算例,验证了自编SPH程序的可靠性,说明SPH法在冲击等问题上有广泛的应用前景。
采用SPH法研究冲击碰撞时固体材料的断裂破坏机理。在船体壳板的穿甲问题中,采用SPH法模拟靶板材料在弹丸撞击下的断裂过程。在弹丸低速或亚弹速撞击靶板时,SPH法的计算结果和有限元法计算结果、靶板穿甲实验的试验结果都非常一致;在弹丸高速或超高速撞击靶板时,SPH法的计算结果和相关文献中的实验结果也基本吻合,说明了SPH方法在求解高速冲击中材料大变形问题上的优越性。在剪切式吸能器的性能研究中,首次引入SPH法模拟了吸能器的剪切碰撞过程,成功克服了有限元法模拟剪切式吸能器时需要单元数巨大、计算时间过长的困难。建立了剪切式吸能器的SPH数值模型,分析了其多次分解、逐次吸收冲击能量的吸能特性,讨论了碰撞速度、被剪切板厚度和材质等因素对吸能特性的影响,并将数值仿真结果和台车碰撞实验的试验结果进行了比较。研究表明,剪切式吸能器在碰撞速度小于40 km/h的情况下能发挥较好的吸能效果。
提出了一种采用SPH法耦合FE法求解流固耦合问题的新方法,即采用SPH粒子模拟具有大变形的流体,FE单元模拟具有不规则形状的固体,通过SPH粒子和FE单元之间的耦合算法模拟流体和固体之间的相互作用。这种方法既利用了SPH法求解大变形问题的优势,又利用了FE法模拟复杂形状以及处理边界条件上的优势,为求解流固耦合问题提供了一条崭新的途径。
采用SPH法耦合FE法,对充液容器跌落碰撞过程中的动力学响应进行了分析,并和ALE法的计算结果比较,说明SPH法模拟的自由液面大变形更加真实。同时将SPH法引入双层充液壳体结构冲击响应的研究,建立了双层充液壳体结构的简化SPH计算模型,通过模拟弹丸击穿充液箱体的过程,分析讨论了夹层流体对箱体中前后两层壁板变形、应力和应变水平的影响,说明流体的流固耦合作用是双层充液壳体结构安全性设计中不可忽视的因素。此外,还首次采用SPH法耦合FE法的方法,模拟了水力阻力器高速冲入水槽后的减速制动过程,研究了水力阻力器的阻力特性问题,讨论了冲击初速度和实际浸水深度对阻力器阻力大小的影响,并总结出水阻力的简单经验公式。研究表明,水力阻力器在流体流动充分发展之前,会出现局部的高水压和高应力,这是结构破坏的主要因素。
最后,对本文的研究工作进行了总结,对未来的发展进行了展望。本文的研究工作为SPH数值算法的工程应用打下了基础,不但将SPH法成功应用于模拟固体材料的断裂破坏,而且提出了SPH法耦合FE法解决流固耦合问题的全新方法,为SPH法在冲击动力学中的应用开辟了一个新的领域。
Smoothed particle hydrodynamics (SPH) method is a new numerical method solving partial differential equations, which is one of meshless methods. The most attractive feature of SPH method is that the field variable approximation is performed at each time step based on a current local set of arbitrarily distributed particles, not on nodes of elements, therefore it can naturally handle problems with extremely large deformation. Since the SPH particle is connected to the moving material similar to the Lagrangian description, the entire time history of all the field variables at a material point can be naturally obtained and thus it is easy to trace material boundaries, free surfaces and moving fluid-structure interfaces of fluids. SPH method, as a meshless, adaptive, stable and Lagrangian solver for dynamic problems, has distinct advantages over the conventional grid-based numerical methods, so SPH method is one of most attractive and hopeful numerical method in impact dynamics. In this thesis, the theory of SPH method has been studied and applied to different complex engineering problems, such as penetration of ship hull plates, shearing action of energy absorber, dropping emulation of fluid-filled tank, penetration of submerged double shells and fluid-structure interaction of hydrodynamic damper, which were supported by fund project of national natural science foundation of China combined with china academy of engineering physics.
The basic concept and the essential formulations of SPH method are introduced firstly. The SPH approximations are studied, which include the strategy of the SPH method, the continuous kernel approximation and the discrete particle approximation. The description and major properties of the smoothing functions are also discussed. The detailed SPH formulations is derived by discretizing the partial differential equations in continuity mechanics spatially, leading to a set of ordinary differential equations with respect to time. The numerical aspects in implementing the SPH formulations, including artificial viscosity, artificial heat, equation of state, constitutive relations and integration method, are also discussed. A three-dimensional SPH source code written by FORTRAN is provided to solve the problems, which is a base in the further research.
The capability of the SPH method has been demonstrated through solving a number of example functions. It is noted that the calculation accuracy is influenced by smoothing function, smoothing length and initial distribution of particles, and boundary deficiency problem and tensile instability problem are inevitable in the traditional SPH method. Some derivative SPH methods, such as improved SPH, CSPH and DSPH, are discussed to improve the traditional SPH. Numerical studies show that the derivative SPH methods not only remedy the boundary deficiency problem but also well simulate the discontinuity of field variable functions. Some special boundary treatments are needed in the SPH method. Three classical physical problems are solved by my SPH codes and it can be seen that the SPH method is suitable for solving the impact problems.
SPH method has been applied to simulate the fracture and failure of solid materials in the impact successfully. In simulation of ship hull plates penetrated by projectile with low velocity, the results of SPH method are consistent with results of finite element method or ballistic experiment. In simulation of ship hull plates penetrated by projectile with high velocity, the large deformation of ship hull plates and the debris cloud of materials same as experimental literatures are easily obtained with the SPH method, which is the most outstanding advantage over other grid-based methods. Subsequently, SPH method is applied to simulate the shearing actions of new type energy absorber, because the large number of elements and the large CPU time are unacceptable if using finite element method. By reason of structure design, energy absorber can switch the impact to thousands of shearing actions among thin ring plates inside the absorber and the impact energy will be decentralized and dissipated gradually. The results of SPH method show that the impact velocity, thickness and material of ring plates are important related factors of energy absorption ability and the energy absorber is effective for collision that impact velocity is lower than 40 km/h. The sled crash test is carried out to validate the result of simulations.
SPH method has been applied to simulate the large deformation of fluid and the fluid-structure interaction successfully. A new idea of SPH method coupled FE method is proposed, which creates SPH particles in fluid domain, divides FE elements in solid domain and calculates the fluid-structure interaction through the contact force between SPH particles and FE elements. The coupled method makes full use of SPH method to solve large deformation problem and FE method to treat complex shapes or boundary conditions, which is an attractive and promising numerical algorithm in engineering.
SPH method coupled with FE method is used to simulate the dropping of rectangular fluid-filled tank firstly. Compared with ALE method, the slosh of fluid and the deformation of tank simulated by SPH method are more close to the truth. Secondly, penetration of submerged double shells is simulated using SPH method coupled with FE method. When the projectile at high speed penetrates the fluid-filled tank, the water hammer effect will obviously change the deformation or stress level of the frontal and back walls along the impact direction, so the interaction between the water and the double shells can not be neglected in structural security analysis. Subsequently, the dynamic characteristics of hydrodynamic damper during the rush into the water channel with high velocity are successfully simulated by SPH method coupled with FE method. The water resistance, the pressure in the interface and the stress of structure are investigated, and the relationship among the peak of water resistance, initial velocity and actual draught is also discussed. The pressure or stress concentration is the main risk during the impact and an empirical formula is put forward to predict the water resistance of hydrodynamic damper in engineering.
In this thesis, the fracture and failure of structures is simulated by SPH method and some fluid-structure coupling problems are successfully solved by SPH method coupled with FE method. It can be concluded that the SPH method is a kind of new potential numerical method and has a bright future in the research of impact dynamics.
本文对SPH方法的基本理论进行了全面总结,介绍了SPH法的基本思想,重点对SPH法的两个重要步骤,即核函数近似过程和粒子近似过程,进行了详细说明,推导出任意函数及其导数的SPH近似公式,并讨论了光滑函数的常见形式和基本性质。随后采用SPH法对连续介质力学的基本控制方程组进行离散近似,推导出描述流体或固体运动的常用SPH公式,并对人工粘性、人工热流、状态方程、本构关系、积分方法等一系列相关数值问题进行了讨论。在此基础上,用FORTRAN语言编写相应的三维SPH计算程序,为进一步开展数值研究奠定基础。
详细研究了SPH法的数值特性。通过用SPH法求解一般函数的数值算例,详细讨论了光滑函数的形式、光滑长度、粒子的分布对SPH法计算精度的影响,并指出传统的SPH算法本身存在着边界粒子计算精度低和张力不稳定的固有缺陷。随后对几种改进传统SPH法的数值方法,如改进边界粒子精度的SPH法、CSPH法和DSPH法,分别进行了理论分析和数值研究,说明这些算法可以一定程度上改善边界粒子的计算精度,并修正不连续处的计算误差。同时介绍了SPH方法中固定边界条件的一般处理方法,并通过求解三个典型数值算例,验证了自编SPH程序的可靠性,说明SPH法在冲击等问题上有广泛的应用前景。
采用SPH法研究冲击碰撞时固体材料的断裂破坏机理。在船体壳板的穿甲问题中,采用SPH法模拟靶板材料在弹丸撞击下的断裂过程。在弹丸低速或亚弹速撞击靶板时,SPH法的计算结果和有限元法计算结果、靶板穿甲实验的试验结果都非常一致;在弹丸高速或超高速撞击靶板时,SPH法的计算结果和相关文献中的实验结果也基本吻合,说明了SPH方法在求解高速冲击中材料大变形问题上的优越性。在剪切式吸能器的性能研究中,首次引入SPH法模拟了吸能器的剪切碰撞过程,成功克服了有限元法模拟剪切式吸能器时需要单元数巨大、计算时间过长的困难。建立了剪切式吸能器的SPH数值模型,分析了其多次分解、逐次吸收冲击能量的吸能特性,讨论了碰撞速度、被剪切板厚度和材质等因素对吸能特性的影响,并将数值仿真结果和台车碰撞实验的试验结果进行了比较。研究表明,剪切式吸能器在碰撞速度小于40 km/h的情况下能发挥较好的吸能效果。
提出了一种采用SPH法耦合FE法求解流固耦合问题的新方法,即采用SPH粒子模拟具有大变形的流体,FE单元模拟具有不规则形状的固体,通过SPH粒子和FE单元之间的耦合算法模拟流体和固体之间的相互作用。这种方法既利用了SPH法求解大变形问题的优势,又利用了FE法模拟复杂形状以及处理边界条件上的优势,为求解流固耦合问题提供了一条崭新的途径。
采用SPH法耦合FE法,对充液容器跌落碰撞过程中的动力学响应进行了分析,并和ALE法的计算结果比较,说明SPH法模拟的自由液面大变形更加真实。同时将SPH法引入双层充液壳体结构冲击响应的研究,建立了双层充液壳体结构的简化SPH计算模型,通过模拟弹丸击穿充液箱体的过程,分析讨论了夹层流体对箱体中前后两层壁板变形、应力和应变水平的影响,说明流体的流固耦合作用是双层充液壳体结构安全性设计中不可忽视的因素。此外,还首次采用SPH法耦合FE法的方法,模拟了水力阻力器高速冲入水槽后的减速制动过程,研究了水力阻力器的阻力特性问题,讨论了冲击初速度和实际浸水深度对阻力器阻力大小的影响,并总结出水阻力的简单经验公式。研究表明,水力阻力器在流体流动充分发展之前,会出现局部的高水压和高应力,这是结构破坏的主要因素。
最后,对本文的研究工作进行了总结,对未来的发展进行了展望。本文的研究工作为SPH数值算法的工程应用打下了基础,不但将SPH法成功应用于模拟固体材料的断裂破坏,而且提出了SPH法耦合FE法解决流固耦合问题的全新方法,为SPH法在冲击动力学中的应用开辟了一个新的领域。
Smoothed particle hydrodynamics (SPH) method is a new numerical method solving partial differential equations, which is one of meshless methods. The most attractive feature of SPH method is that the field variable approximation is performed at each time step based on a current local set of arbitrarily distributed particles, not on nodes of elements, therefore it can naturally handle problems with extremely large deformation. Since the SPH particle is connected to the moving material similar to the Lagrangian description, the entire time history of all the field variables at a material point can be naturally obtained and thus it is easy to trace material boundaries, free surfaces and moving fluid-structure interfaces of fluids. SPH method, as a meshless, adaptive, stable and Lagrangian solver for dynamic problems, has distinct advantages over the conventional grid-based numerical methods, so SPH method is one of most attractive and hopeful numerical method in impact dynamics. In this thesis, the theory of SPH method has been studied and applied to different complex engineering problems, such as penetration of ship hull plates, shearing action of energy absorber, dropping emulation of fluid-filled tank, penetration of submerged double shells and fluid-structure interaction of hydrodynamic damper, which were supported by fund project of national natural science foundation of China combined with china academy of engineering physics.
The basic concept and the essential formulations of SPH method are introduced firstly. The SPH approximations are studied, which include the strategy of the SPH method, the continuous kernel approximation and the discrete particle approximation. The description and major properties of the smoothing functions are also discussed. The detailed SPH formulations is derived by discretizing the partial differential equations in continuity mechanics spatially, leading to a set of ordinary differential equations with respect to time. The numerical aspects in implementing the SPH formulations, including artificial viscosity, artificial heat, equation of state, constitutive relations and integration method, are also discussed. A three-dimensional SPH source code written by FORTRAN is provided to solve the problems, which is a base in the further research.
The capability of the SPH method has been demonstrated through solving a number of example functions. It is noted that the calculation accuracy is influenced by smoothing function, smoothing length and initial distribution of particles, and boundary deficiency problem and tensile instability problem are inevitable in the traditional SPH method. Some derivative SPH methods, such as improved SPH, CSPH and DSPH, are discussed to improve the traditional SPH. Numerical studies show that the derivative SPH methods not only remedy the boundary deficiency problem but also well simulate the discontinuity of field variable functions. Some special boundary treatments are needed in the SPH method. Three classical physical problems are solved by my SPH codes and it can be seen that the SPH method is suitable for solving the impact problems.
SPH method has been applied to simulate the fracture and failure of solid materials in the impact successfully. In simulation of ship hull plates penetrated by projectile with low velocity, the results of SPH method are consistent with results of finite element method or ballistic experiment. In simulation of ship hull plates penetrated by projectile with high velocity, the large deformation of ship hull plates and the debris cloud of materials same as experimental literatures are easily obtained with the SPH method, which is the most outstanding advantage over other grid-based methods. Subsequently, SPH method is applied to simulate the shearing actions of new type energy absorber, because the large number of elements and the large CPU time are unacceptable if using finite element method. By reason of structure design, energy absorber can switch the impact to thousands of shearing actions among thin ring plates inside the absorber and the impact energy will be decentralized and dissipated gradually. The results of SPH method show that the impact velocity, thickness and material of ring plates are important related factors of energy absorption ability and the energy absorber is effective for collision that impact velocity is lower than 40 km/h. The sled crash test is carried out to validate the result of simulations.
SPH method has been applied to simulate the large deformation of fluid and the fluid-structure interaction successfully. A new idea of SPH method coupled FE method is proposed, which creates SPH particles in fluid domain, divides FE elements in solid domain and calculates the fluid-structure interaction through the contact force between SPH particles and FE elements. The coupled method makes full use of SPH method to solve large deformation problem and FE method to treat complex shapes or boundary conditions, which is an attractive and promising numerical algorithm in engineering.
SPH method coupled with FE method is used to simulate the dropping of rectangular fluid-filled tank firstly. Compared with ALE method, the slosh of fluid and the deformation of tank simulated by SPH method are more close to the truth. Secondly, penetration of submerged double shells is simulated using SPH method coupled with FE method. When the projectile at high speed penetrates the fluid-filled tank, the water hammer effect will obviously change the deformation or stress level of the frontal and back walls along the impact direction, so the interaction between the water and the double shells can not be neglected in structural security analysis. Subsequently, the dynamic characteristics of hydrodynamic damper during the rush into the water channel with high velocity are successfully simulated by SPH method coupled with FE method. The water resistance, the pressure in the interface and the stress of structure are investigated, and the relationship among the peak of water resistance, initial velocity and actual draught is also discussed. The pressure or stress concentration is the main risk during the impact and an empirical formula is put forward to predict the water resistance of hydrodynamic damper in engineering.
In this thesis, the fracture and failure of structures is simulated by SPH method and some fluid-structure coupling problems are successfully solved by SPH method coupled with FE method. It can be concluded that the SPH method is a kind of new potential numerical method and has a bright future in the research of impact dynamics.
引文
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