1.6 Smoothed particle hydrodynamics (SPH)

Transcription

1 26 Smoothed Particle Hydrodynamics 1.6 Smoothed particle hydrodynamics (SPH) The SPH method In the SPH method, the state of a system is represented by a set of particles, which possess individual material properties and move according to the governing conservation equations (see, Figure 1.9). Since its invention to solve astrophysical problems in three-dimensional open space (Lucy, 1977; Gingold and Monaghan, 1977), SPH has been extensively studied and extended to dynamic response with material strength as well as dynamic fluid flows with large deformations. Figure 1.9 SPH particles used in a shaped charge detonation simulation. The particles are irregularly distributed. Smoothed particle hydrodynamics, as a meshfree, Lagrangian, particle method, has its particular characteristics. It has some special advantages over the traditional grid-based numerical methods, the most significant one among

2 Introduction 27 which is the adaptive nature of the SPH method. This adaptability of SPH is achieved at the very early stage of the field variable approximation that is performed at each time step based on a current local set of arbitrarily distributed particles. Because of this adaptive nature of the SPH approximation, the formulation of SPH is not affected by the arbitrariness of the particle distribution. Therefore, it can naturally handle problems with extremely large deformation. This is, therefore, the most attractive feature of the SPH method. The meshfree nature of the SPH method is also due to the above-mentioned adaptive formulation and the use of particles to represent the problem domain and to act as the computational frame for field variable approximations. The SPH approximation does not require a pre-defined mesh to provide any connective of the particles in the process of computation, and it works well even without any particle refinement operation. This meshfree nature is very attractive for problems where the traditional FEM or FDM encounters difficulties mentioned earlier. Besides the meshfree and adaptive nature, another exciting and attractive feature of the SPH method is the harmonic combination of the Lagrangian formulation and particle approximation. Unlike the meshfree nodes in other meshfree methods, which are only used as interpolation points, the SPH particles also carry material properties, and are allowed to move in light of the internal interactions and external forces. Functioning as both approximation points and material components, the SPH particles seem to be endowed with life. The pith and marrow of the method are fully embodied in the three terms of SMOOTHED PARTICLE HYDRODYNAMICS. The first term SMOOTHED represents the smoothed approximation nature by using the weighted average over the neighboring particles for stability; the third term HYDRODYNAMICS is the right niche of the method in the application to hydrodynamics problems. It is this harmonic combination of the adaptive, Lagrangian and particle nature in the SPH method that leads to various practical applications in different areas in engineering and science Briefing on the history of the SPH method Smoothed particle hydrodynamics (SPH) is a meshfree, Lagrangian particle method for modeling fluid flows. SPH was first invented to solve astrophysical problems in three-dimensional open space, in particular polytropes (Lucy, 1977; Gingold and Monaghan, 1977), since the collective movement of those particles is similar to the movement of a liquid or gas flow, and it can be modeled by the governing equations of the classical Newtonian hydrodynamics. Although the traditional grid-based numerical methods such as the finite difference methods (FDM) and finite element methods (FEM) exist, and in some circumstances, are better developed methods than the SPH method, they have difficulties in handling some complex phenomena as discussed earlier. This

3 28 Smoothed Particle Hydrodynamics motivated researchers to seek for alternatives to solve these kinds of problems, and the SPH method has then become a good choice. Several review papers on the SPH method have been published, including those by Benz (1989; 1990) and Monaghan(1992). Unlike the famous particle-in-cell (PIC) method developed in early 1960s (Harlow, 1963; 1964; Brackbill et al., 1988; Munz et al., 1999; Cushman et al., 2000), the SPH method does not need a grid/mesh to calculate the spatial derivatives. These particles are capable of moving in the space, carry all the computational information, and thus form the computational frame for solving the partial differential equations describing the conservation laws of the continuum fluid dynamics. Invention Smoothed particle hydrodynamics (SPH) is a meshfree, adaptive, Lagrangian particle method for modeling fluid flows. SPH was first invented to solve astrophysical problems in three-dimensional open space, in particular polytropes (Lucy, 1977; Gingold and Monaghan, 1977), since the collective movement of those particles is similar to the movement of a liquid or gas flow, and it can be modeled by the governing equations of the classical Newtonian hydrodynamics. Extensions Today, the SPH method is being used in many areas, such as the simulations of binary stars and stellar collisions (Benz, 1988; 1990; Monaghan, 1992; Frederic et al., 1999), supernova (Hultman and Pharayn, 1999), collapse as well as the formation of galaxies (Monaghan and Lattanzio, 1991; Berczik and Kolesnik, 1993; 1998; Berczik, 2000), coalescence of black holes with neutron stars (Lee, 1998; 2000), single and multiple detonation of white dwarfs (Senz et al., 1999), even the evolution of the universe (Monaghan, 1990). The SPH method has also been applied extensively to a vast range of problems in either computational fluid or solid mechanics because of relatively strong ability to incorporate complicated physical effects into the SPH formulations. In some sense, the term hydrodynamics may be interpreted as mechanics in general. When the SPH approximation is used to create point-dependent shape functions (see, Chapter 3), it can be applied to other areas of mechanics rather than classical hydrodynamics. Hence in some literatures (Kum et al., 1995, Posch et al. 1995), it is called Smoothed Particle Mechanics. Applications The earliest applications of SPH were mainly focused on fluid dynamics related areas. These include elastic flow (Swegle, 1992), magneto-hydrodynamics (Morris, 1996), multi-phase flows (Monaghan and Kocharyan, 1995), quasi-

4 Introduction 29 incompressible flows (Monaghan, 1994; Morris et al., 1997), gravity currents (Monaghan, 1995a), flow through porous media (Morris et al. 1999; Zhu et al. 1999), heat conduction (Monaghan, 1995b; Chen et al., 1999a), shock simulations (Monaghan and Gingold, 1983; Monaghan, 1987; 1989; Morris and Monaghan, 1997), heat transfer and mass flow (deary, 1998), ice and cohesive grains (Gutfraind and Savage, 1998; Oger and Savage, 1999). Benz and Asphaug (1993; 1994; 1995) extended SPH to the simulation of the fracture of brittle solids. Bonet and Kulasegaram (2000) applied SPH to the simulation of metal forming. The SPH method has been very attractive in simulating large deformation and impulsive loading events. One significant application area is high (or hyper) velocity impact (HVI) problems concerning the effects of projectiles impacting upon space assets (satellites, space stations, shuttles). In HVI problems, shock waves propagate through the colliding bodies, which behave like fluids (Zukas, 1982, 1990). Libersky and his co-workers (Libersky et al., 1991; 1993; 1995; Randies and Libersky, 1996; Randies et al., 1995a, b) and Johnson et al. (1993; 1996a, b) have made outstanding contributions in the application of SPH to impact problems. Another important application of the SPH method is the explosion phenomena arising from the detonation of high explosive (HE). Swegle and Attaway (1995) have investigated the feasibility of using the SPH method for underwater explosion calculations. Recently, Liu and his co-workers have applied the SPH method to model a series of explosion phenomena including high explosive detonation, explosion, underwater shock, and water mitigation of shocks (Liu, et al. 2000; 2002a; 2003a-f). The application of SPH to a wide range of problems has led to significant extensions and improvements of the original SPH method. The numerical aspects have been gradually improved, some inherent drawbacks of SPH were identified, and modified techniques or corrective methods were also proposed. Swegle et al. (1995) identified the tensile instability problem that can be important for materials with strength; Morris (1996) identified the particle inconsistency problem that can lead to poor accuracy in the solution. Over the past years, different modifications or corrections have been tried to restore the consistency and to improve the accuracy of the SPH method. These modifications lead to various versions of the SPH methods and corresponding formulations. Monaghan (1988; 1982; 1985) proposed symmetrization formulations that were reported to have better effects. Johnson and Beissel (Johnson et al., 1996; Johnson and Beissel, 1996) gave an axis symmetry normalization formulation so that, for velocity fields that yield constant values of normal velocity strains, the normal velocity strains can be exactly reproduced. W. K. Liu et al. (Liu and Chen, 1995) presented the reproducing kernel particle method (RKPM) that can result in better accuracy in the particle approximation. Chen et al. (1999a; b; c) proposed a corrective smoothed particle method (CSPM) which improves the simulation accuracy both inside the problem domain and around the boundary area. Randies and Libersky (2000) extended

5 3 0 Smoothed Particle Hydrodynamics the stress point method (Dyka and Ingel, 1995; Dyka et al., 1997) to multidimensional space to improve the tensile instability and zero energy mode problems (Vignjevic et al., 2000). Other notable modifications or corrections of the SPH method include the moving least square particle hydrodynamics (MLSPH) by Dilts (1999; 2000), the integration kernel correction by Bonet and Kulasegaram (2000), and the correction by Belytschko et al. (1998). Presently, the SPH is a method that can simulate general fluid dynamic problems fairly well. Challenges Though the SPH method has been extensively applied to different areas, there are still a lot of issues that need to be further investigated. This is especially true in the numerical analyses of the method. Due to the meshfree particle nature of the method, it is not always straightforward to directly apply the techniques that were developed for grid-based Eulerian methods or Lagrangian methods to the SPH method. Some authors have tried to perform theoretical and numerical analyses on SPH (Gingold and Monaghan, 1982; Monaghan, 1982; Morris, 1994; 1996; Balsara, 1995; Meglicki, 1995; Ben et al., 1996a, b; Fulk, 1994; Swegle, et al., 1994; 1995). Through these studies, issues related to the stability, accuracy and convergence properties of the SPH method are gradually becoming understood. However, most of the analyses are based on uniformly distributed particles, and sometimes only for one-dimensional cases, the results obtained by such analyses are often limited to idealized circumstances. For more general cases especially those with large deformations and impulsive loadings where the particles are usually highly disordered, the obtained results may not always be reliable, as it is not yet very clear how the particle irregularity affects the accuracy of the solutions. There is still a long way for the method to become extensively applicable, practically useful and robust as the traditional grid-based methods such as FEM and FDM. This is because much work needs to be done to consolidate the theoretical foundations of the SPH method, and to remedy its inherent numerical drawbacks. Moreover, there should be a necessary process for any numerical technique to develop, advance, improve, and to be validated so as to be more efficient, robust in practical applications The SPH method in this book As can be seen in the previous discussions, there are so many kinds of meshfree methods for different applications. The purpose of this book is not to provide a comprehensive record of all the emerged meshfree methods. For such purpose, the readers are suggested to refer to the monograph on meshfree methods by Liu (2002) and some other review papers (Belytschko et al., 1996; 1998; Li and Liu, 2002). The emphasis of this book will be on the meshfree particle methods,

6 Introduction 31 specifically, the smoothed particle hydrodynamics and its different modifications and variations. Some novel applications of the SPH method will also be addressed in detail. Devoting this volume entirely to the SPH method is based on the following reasons. 1. The SPH method, as the oldest MPM, is quickly approaching to its mature stage; 2. With the continuing improvements and modifications, the accuracy, stability and adaptively of the SPH method have reached an acceptable level for practical engineering applications; 3. Applications of the SPH method are very wide, ranging from CFD to CSM, from micro-scale to macro-scale and to astronomical scale, from discrete systems to continuum systems; 4. Some commercial codes have incorporated the SPH processor into their software packages with many successful practical applications. There are a number of versions of the SPH method proposed so far. This book provides an introduction to the traditional SPH method and its variations such as the CSPM, DSPH, and ASPH. The theories related to the SPH method will be systematically discussed. A comprehensive but concise collection of numerical techniques is presented. Some important implementation issues are discussed. A SPH source code in FORTRAN is provided. The SPH code consists of most of the standard SPH techniques, and can be easily extended to other variations of SPH with modifications either on the continuous integral representation or the discretized particle approximation. Releasing the sample source code is to suit the needs of readers for an easy comprehension, understanding, quick implementation, practical applications and further development of the MPMs. Many novel and interesting applications in the areas related to CFD will be presented. These include incompressible flows, free surface flows, high compressive flows, high explosive (HE) detonation, HE explosion, underwater explosion, water mitigation of shocks, high velocity impact, penetration, and multiple scale simulations coupled with atomistic method.

7 32 Smoothed Particle Hydrodynamics As mentioned in the preceding discussions, a simulation using the SPH method involves two major steps: particle representation and particle approximation. The particle representation is an issue related to only the initial creation of the particles, and it can be solved using the existing software packages commercially available. Therefore, the effort of this book is mainly on the central issue of the SPH particle approximation. The next Chapter will be focusing on the basic ideas and essential formulations of the SPH method that are useful for all different versions of the SPH methods.