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      • 粒子法のいま、そして未来へ >
        • 第1回 粒子法のいま
        • 第2回 SPH法におけるカーネル近似とカーネル関数の条件
        • 第3回 SPH法における空間離散化
      • 粒子法の非圧縮条件とは
      • 粒子法入門 >
        • 第1回 粒子法って何?
        • 第2回 粒子法は、他の方法とどう違うか
        • 第3回 粒子法の大きさと質量について
        • ​第4回 「粒子の動かし方」と「加速度の求め方」について
        • ​第5回 計算時間を短縮する方法について
    • Technical Column >
      • Growing the particle method, and its present state >
        • 1. Present State of the Particle Method
        • 2. Kernel Approximation and Kernel Function Conditions in the SPH Method (Preparation for Spatial Discretization)
      • Incompressibility of the particle method
      • Introduction to the particle method >
        • 1. What is a particle method?
        • 2. In what ways is the particle method different from other methods?
        • 3. Mass and volume of particles
        • 4. How to move particles and how to calculate accelerations of particles
        • 5. How to shorten the simulation time
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2. In what ways is the particle method different from other methods?

In the last column, I explained that the particle method can easily solve many difficult problems. Why can the particle method do that? In this column, I would like to explain the reason by comparing the particle method with a conventional mesh-based method. ​

1. Finite Difference Method

The finite difference method (FDM) is a traditional mesh-based method for simulating fluid dynamics. Figure 1 shows a fluid flow, and Fig. 2 shows the conceptual image of the simulated flow by FDM. A simulation domain is discretized by mesh. Calculation points, which are like observation points, are set on certain places in the mesh, such as centers or sides of mesh cells. These calculation points have parameters such as pressure, velocity, fluid density or temperature. The parameters are updated every time step. Basically, calculation mesh is neither moved nor deformed in FDM*1. Therefore, the calculation points also do not move. That is to say, the flow is observed at the same points which are fixed to the space. For example, when we simulate a time change of temperature by FDM, temperatures are evaluated at fixed places. Every time step, temperature of a different fluid element*2 is observed at a place because fluid elements pass through the fixed observation point.
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Figure 1 Fluid flow in a pipe
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Figure 2 Fluid flow in a pipe expressed by calculation mesh in the finite difference method
Arrow directions and lengths show their velocity direction and absolute values, respectively. Although each side of a mesh cell has a parameter of velocity, which expresses the absolute value of velocity in vertical direction to the side, only representative arrows are drawn to simplify the figure.

Note 1:

There are some exemptions. For example, the Arbitrary Lagrangian Eulerian (ALE) method moves and deforms the mesh.

Note 2:

An element means substance whose volume is very small. Although elements are not handled directly in FDM, we need to consider the equation of momentum for every fluid element because fluid consists of microscopic elements.

2. Particle method

In the particle method, the fluid flow in a pipe shown in Fig. 1 is expressed by particles as shown in Fig.3. A mesh is not necessary. Fluid is expressed by a group of calculation points. Each calculation point is called a particle in the particle method. Each fluid particle expresses small domain of fluid and has parameters for pressure, velocity etc. In contrast to FDM, calculation points in the particle method move in space. The movement velocities of calculation points are the same as their fluid velocities. Because of the movement of calculation points, the particle method is able to be applied to various complex flows such as Fig.4. Movie 1 shows an example of a simulation result obtained by the particle method.
The particle method is simple and easy to implement because we do not need to discretize the advection term of the Navier-Stokes equations. The advection term is called the "convective rate of change" in some books. The advection term is non-linear and complex. The advection term appears when we calculate a substantial derivative*3 by using parameters which are observed at points fixed to the space. In the particle method, we use parameters which follow particles' movement. Therefore, the advection term does not appear in the particle method.
Let's compare the particle method with FDM in a case where time change of fluid temperature is calculated. In FDM, the temperature is measured at fixed points. That is to say, the temperature of different fluid particles is calculated every time step. On the other hand, in the particle method the temperature of the same particle is evaluated. Because of this difference, we do not need to calculate the complicated advection term of temperature. In the same manner, the time change of fluid velocity is also easily solved in the particle method without considering the advection term of velocity. That is why the MPS method can easily solve many difficult problems.​

Note 3:

Also called Lagrangian derivative. A time derivative of a physical quantity. The time derivative is evaluated by following a designated substance. That is to say, the evaluation point is moved with the designated substance.
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Fig. 3 Fluid flow in a pipe expressed by particles in the particle method
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Fig. 4  Schematic diagram of complex fluid flow simulation made possible by the particle method
 Movie 1  Example of complex fluid behavior simulated by the particle method 
  • 1. WHAT IS A PARTICLE METHOD?
  • ​2. IN WHAT WAYS IS THE PARTICLE METHOD DIFFERENT FROM OTHER METHODS?
  • 3. MASS AND VOLUME OF PARTICLES
  • 4. HOW TO MOVE PARTICLES AND HOW TO CALCULATE ACCELERATIONS OF PARTICLES​​​
  • 5. HOW TO SHORTEN THE SIMULATION TIME
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Kazuya Shibata, Ph.D.
Assistant Professor at Department of System Innovation, Graduate School of Engineering, The University of Tokyo.
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2007年 東京大学大学院 工学系研究科
           システム量子工学専攻 博士課程修了 博士(工学)
2007年 (独)海上技術安全研究所 入所
           ​海の10モードプロジェクトチーム研究員
2009年 東京大学大学院 工学系研究科 システム創成学専攻 助教
2013年 東京大学大学院 工学系研究科 システム創成学専攻 講師
2017年 東京大学大学院 工学系研究科 システム創成学専攻 准教授
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  • Home
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    • Particleworks解析事例
    • Particleworks Case Examples
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    • World-Wide User Case Studies
  • Features
    • What is MPS?
    • Simulation Flow
    • Pre- and Post-Processing
    • Physics
    • Multiphysics Solution
    • GPU High Performance Computing
    • Operation environment
  • Particleworks for Ansys
  • News
  • Seminar, Event
  • Technical
    • 粒子法・MPS法
    • 技術コラム >
      • DX時代の製品開発プロセスとCAEの重要性 >
        • 第1回 序 略歴とコラム紹介
        • 第2回 DXとデジタルエンジニアリング
        • 第3回 製品開発プロセスの目指す姿
        • 第4回 DX時代のCAE
        • 第5回 評価CAEの概要と課題
        • 第6回 評価CAEの課題解決手法
        • 第7回 企画CAEの概要と課題
        • 第8回 企画CAEの運用と応用
        • 第9回 設計CAEの概要と課題
        • 第10回 設計CAEの課題解決の進め方
        • 第11回 開発プロセス運用の仕組み作り
        • 第12回 まとめと変革の時代に求められるエンジニア像
      • 粒子法のいま、そして未来へ >
        • 第1回 粒子法のいま
        • 第2回 SPH法におけるカーネル近似とカーネル関数の条件
        • 第3回 SPH法における空間離散化
      • 粒子法の非圧縮条件とは
      • 粒子法入門 >
        • 第1回 粒子法って何?
        • 第2回 粒子法は、他の方法とどう違うか
        • 第3回 粒子法の大きさと質量について
        • ​第4回 「粒子の動かし方」と「加速度の求め方」について
        • ​第5回 計算時間を短縮する方法について
    • Technical Column >
      • Growing the particle method, and its present state >
        • 1. Present State of the Particle Method
        • 2. Kernel Approximation and Kernel Function Conditions in the SPH Method (Preparation for Spatial Discretization)
      • Incompressibility of the particle method
      • Introduction to the particle method >
        • 1. What is a particle method?
        • 2. In what ways is the particle method different from other methods?
        • 3. Mass and volume of particles
        • 4. How to move particles and how to calculate accelerations of particles
        • 5. How to shorten the simulation time
    • 粒子法用語集
    • Particle Method Glossary
    • 参考文献・ウェブサイト
    • Reference Book/URL
    • 論文・講演
  • Contact
    • 導入の流れとライセンス形態
    • Particleworks / GranuleworksプリインストールGPU搭載ワークステーション
    • 開発元・パートナー
    • Developers, Partners
    • お問い合わせ
    • Contact Us