Particle Methods For Multi-scale And Multi-physics

M B Liu, G R Liu

• Publisher: WSPC
• Year: 2015
• eBook ISBN: 9789814571715
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Multi-scale and multi-physics modeling is useful and important for all areas in engineering and sciences. Particle Methods for Multi-Scale and Multi-Physics systematically addresses some major particle methods for modeling multi-scale and multi-physical problems in engineering and sciences. It contains different particle methods from atomistic scales to continuum scales, with emphasis on molecular dynamics (MD), dissipative particle dynamics (DPD) and smoothed particle hydrodynamics (SPH).

This book covers the theoretical background, numerical techniques and many interesting applications of the particle methods discussed in this text, especially in: micro-fluidics and bio-fluidics (e.g., micro drop dynamics, movement and suspension of macro-molecules, cell deformation and migration); environmental and geophysical flows (e.g., saturated and unsaturated flows in porous media and fractures); and free surface flows with possible interacting solid objects (e.g., wave impact, liquid sloshing, water entry and exit, oil spill and boom movement). The presented methodologies, techniques and example applications will benefit students, researchers and professionals in computational engineering and sciences.

Multi-scale and multi-physics modeling is useful and important for all areas in engineering and sciences. Particle Methods for Multi-Scale and Multi-Physics systematically addresses some major particle methods for modeling multi-scale and multi-physical problems in engineering and sciences. It contains different particle methods from atomistic scales to continuum scales, with emphasis on molecular dynamics (MD), dissipative particle dynamics (DPD) and smoothed particle hydrodynamics (SPH).

This book covers the theoretical background, numerical techniques and many interesting applications of the particle methods discussed in this text, especially in: micro-fluidics and bio-fluidics (e.g., micro drop dynamics, movement and suspension of macro-molecules, cell deformation and migration); environmental and geophysical flows (e.g., saturated and unsaturated flows in porous media and fractures); and free surface flows with possible interacting solid objects (e.g., wave impact, liquid sloshing, water entry and exit, oil spill and boom movement). The presented methodologies, techniques and example applications will benefit students, researchers and professionals in computational engineering and sciences.

Readership: Undergraduates, graduates, researchers, and professionals studying/dealing with fluid mechanics, numerical analysis and computational mathematics, engineering mechanics, ocean engineering, mechanical engineering.
Key Features:

• The authors have many years of experience in meshfree and particle methods, and are renowned scientists in related areas, with highly cited publications. This can greatly attracts fellow researchers from all around the world to probe the latest development on current major particle methods
• The authors have authored numerous technical publications, and many popular books. They truly understand what the fellow researchers think and want
• The authors have extensive network in academics and research. It is comparatively easy to introduce the book to professional organizations, international conferences, and different academic bodies such as universities and research institutes

Chapter 1

Introduction

1.1 Computer modeling

1.1.1 Computer modeling and its general solution procedure

Computer modeling (or numerical simulation using computers) has increasingly become a very important approach for solving and analyzing complex practical problems in engineering and sciences. A general procedure of computer modeling includes translating important phenomena of a physical problem into a discrete form of mathematical description, recasting the problem in discrete numerical equations, solving the equations on a computer, and then revealing the phenomena virtually according to the requirements of the analysts.

Computer modeling follows a similar procedure to serve a practical purpose. There are in principle some necessary steps in the procedure, as shown in Figure 1.1. From the physical phenomena observed, mathematical models are established with some possible simplifications and assumptions. These mathematical models are generally expressed in the form of governing equations defined in the problem domain with proper boundary conditions (BC) and/or initial conditions (IC). The governing equations may be a set of ordinary differential equations (ODE), partial differential equations (PDE), integral equations or equations in any other possible forms of physical laws. Boundary and/or initial conditions are necessary for determining the field variables in space and/or time.

To numerically solve the governing equations, the involved geometry of the problem domain needs to be divided into discrete finite number of parts, for which numerical approximations can be easily made. A computational frame is then formed known traditionally as a set of mesh, which consists of cells, grids or nodes. The grids or nodes are the locations where the field variables are evaluated, and their relations are defined by some kind of nodal connectivity defined by the mesh. Accuracy of the numerical approximation is closely related to the mesh density and pattern.

Figure 1.1 Procedure of conducting a computer modeling.

Numerical discretization provides means to change the spatial (integral or derivative) operators in the governing equations to discrete representations at the grids or nodes. Such a numerical discretization is based on one of the theories of function approximations (Liu, 2002). After the numerical discretization, the original physical equations are changed into a set of algebraic equations or ordinary differential equations, which can be solved using the existing numerical routines. In the process of establishing the algebraic or ODE equations, the so-called strong or weak forms (Liu and Gu, 2003), or weakened weak form (Liu, 2009) formulation can be used These forms of formulation can also be combined together to take the full advantages of both weak and strong form formulations.

Implementation of a numerical simulation involves translating the domain decomposition and numerical algorithms into a computer code in some programming language(s). In coding a computer program, the accuracy, and efficiency (speed and storage) are two very important considerations. Other considerations include robustness of the code (consistency check, error trap), user-friendliness of the code (easy to read, use and even to modify), and etc. Before performing a practical numerical simulation, the code should be tested against theoretical solutions, or the exact results from other established methods for benchmark problems, or the experimental data from actual engineering problems. In other words, a computer modeling needs verification and validation (V&V), as will be further discussed in Section 1.1.3.

For numerical simulations of problems in fluid mechanics, the governing equations can be established from the conservation laws, which state that field variables such as the mass, momentum and energy must be conserved during the evolution process of the flow. These three fundamental principles of conservation, together with additional information concerning the specification of the nature of the material/medium, conditions at the boundary, and conditions at the initial stage determine the behavior of the fluid system.

Figure 1.2 Domain and numerical discretization for computer modeling of a field function f(x) defined in one-dimensional space.

Except for a few circumstances of very simple settings, it is very difficult to obtain analytical solution of these integral equations or partial differential equations. Computational fluid dynamics (CFD) deals with the techniques of spatially approximating the integral or the differential operators in the integral or differential equations into a set of simple algebraic summations (or ODEs with respect to time only), which can be solved to obtain numerical values for field functions (such as density, pressure, velocity, etc.) at discrete points in space and/or time Figure 1.2). A typical computer modeling of a CFD problem deals with

1. governing equations,

2. proper boundary conditions and/or initial conditions,

3. domain discretization technique,

4. numerical discretization technique,

5. numerical technique to solve the resultant algebraic equations or ordinary differential equations.

1.1.2 Computer modeling, theory and experiment

Rather than adopting the traditional theoretical practice of constructing layers of assumptions and approximations, computer modeling attacks the original problems in detail with minimum assumptions, with the help of the increasing computer power. It provides an alternative tool of scientific investigation, instead of carrying out expensive, time-consuming or even dangerous experiments in laboratories or on site. The numerical tools are often more useful than the traditional experimental methods in terms of providing insightful and complete information that cannot be directly measured or observed, or difficult to acquire via other means. Computer modeling plays a valuable role in providing verifications for theories, offers insights to the experimental results and assists in the interpretation or even the discovery of new phenomena. It acts also as a bridge between the experimental models and the theoretical predictions

Figure 1.3 shows the connection between the computer modeling, theory and experiment. With the rapid development of computer hardware and software, computer modeling is increasingly playing a more and more important role in conducting scientific investigations. However, this does not mean we do not need experimental and theoretical works any more. It must be clearly pointed out that experimental phenomena and theoretical analyses are usually the fundaments of computer modeling and the modeling results also need to be verified and validated.

Figure 1.3 Connection between computer modeling, theory and experiment.

1.1.3 Verification and validation

Computer modeling today can server both as a research and a design tool for many important engineering and scientific projects. One typical example is the computational fluid dynamics, which is a branch of fluid mechanics that uses numerical methods to solve and analyze fluid mechanics problems. With the advent of high performance computers together with advanced numerical algorithms, open source codes and commercial CFD software are easily accessible. As such, CFD now plays a more and more important role in understanding fluid flows. The accuracy of CFD codes need to be demonstrated so that the CFD codes may be used with confidence for practical applications and the results can be considered credible for decision making in design.

Early in 1979, the Society of Computer Simulation (SCS) first defined the term “verification” and “validation” (Schlesinger, 1979), and provided two related terms, i.e., computerized model and conceptual model. In 1998, the American Institute of Aeronautics and Astronautics (AIAA) provided a guide for the verification and validation of computational fluid dynamics simulations (Reston, 1998). The guide clearly defined the key terms, discussed fundamental concepts, and specified general procedures for conducting verification and validation of CFD simulations. In 2002, Oberkampf and Trucano presented an extensive review of the literature in V&V from members of the operations research, statistics, and CFD communities and discussed methods and procedures for assessing V&V in CFD (Oberkampf and Trucano, 2002).

According to SCS’s definition, model verification substantiates that a computerized model represents a conceptual model within specified limits of accuracy, and model validation subs...

Table of contents

• Cover page
• Title page
• Copyright page
• Dedication
• Preface
• Acknowledgments
• About the Authors
• Contents
• 1 Introduction
• 2 Molecular Dynamics
• 3 Dissipative Particle Dynamics — Methodology
• 4 Dissipative Particle Dynamics — Applications
• 5 Smoothed Particle Hydrodynamics — Methodology
• 6 Smoothed Particle Hydrodynamics — Applications
• 7 Three Typical Particle Methods
• Index