Particle Method Glossary

  • Advection term

    A term which appears when we express a substantial derivative in the Euler expression. The advection term expresses the effect of position change due to flow in a time derivative. The advection term is called “convective rate of change” in some books.
    (cf. Navier-Stokes equation, Pressure term, Viscosity term, External force

  • Boundary condition

    A condition which is given to a boundary, such as a wall or a free surface.

  • Bucket

    A technique used for reducing the computational cost of finding neighboring particles. A simulation domain is divided into many small subdomains called buckets. Neighboring particles are only searched among particles located within the surrounding buckets.
    (cf. Neighboring particleNeighboring particle search

  • Collapse of a water column

    A water column, whose initial velocity is zero, is collapsed by gravity. This is called a dam break or collapse of a water column.
    (cf. Dam break

  • Compiler

    A program which converts a source code written in a programming language into an executable file written in a machine language. Visual Studio, gcc, intel compiler, Xcode are representative compilers.

  • Conjugate gradient method

    An iterative solver for solving simultaneous equations.
    (cf. Gaussian elimination

  • Conservation of mass

    A law of mass conservation. Conservation of mass for a fluid is expressed by the equation of continuity.
    (cf. Equation of continuity

  • Courant number

    The Courant number is a dimensionless number defined by c=uΔt/Δx, where u is the maximum fluid velocity, Δx is the spatial resolution and Δt is the time increment. In an explicit method, Δt is limited by the courant number.
    (cf. Diffusion number, Time increment

  • CSF model

    A model of surface tension. CSF is the abbreviation of Continuum Surface Force. The concept of the CSF model was proposed in the following paper:
    Brackbill JU, Kothe DB, Zemach C (1992) A continuum method for modeling surface tension. Journal of Computational Physics 100 (2):335-354. doi:

  • Dam break

    The meaning of dam break is the same as that of collapse of a water column.
    (cf. Collapse of water column

  • Dummy wall particle

    A type of particle which do not have a pressure parameter. Dummy wall particles are installed behind wall particles, and are used to calculate the particle number density of fluid or wall particles. In the calculations of pressure Poisson equation and pressure gradient, dummy wall particles are neglected because they do not have a pressure parameter. This expresses that the pressure gradient on the dummy wall particle is a zero vector.

  • Differential operator

    Operators which differentiate a function. Nabla and Laplacians are representative differential operators.
    (cf. Nabla, Laplacian, Divergence, Laplacian model

  • Diffusion equation

    An equation which expresses the nature of diffusive phenomena.

  • Diffusion number

    Dimensionless number D defined by D=νΔt/(Δx)2., where ν is the kinematic viscosity coefficient, Δx is the spatial resolution and Δt is the time increment. When we simulate the Navier-Stokes equations, Δt should be small so that D becomes smaller than a certain value.
    (cf. Courant number, Time increment

  • Dirichlet boundary condition

    A condition which is given to a boundary for solving a differential equation. A Dirichlet boundary condition directory fixes the boundary value of a function. In the particle method, a Dirichlet bounday condition is given to a free surface.
    (cf. Neumann boundary condition, Boundary condition, Free surface

  • Discretization

    To divide a space or time into small segments and to represent each small segment by a few representative values (e.g. an average value). Computers have to solve equations by using the discretized values because computers cannot deal with continuous functions.
    (cf. Gradient model, Laplacian model)

  • Distance function

    Mesh data of which each mesh node has the value of distance between the node and a target object, such as a wall or a rigid body.
    (cf. Polygon wall boundary

  • Divergence

    A mathematical operator expressed by an inner product with nabla (∇∙ ).
    (cf. NablaGradient

  • Equation of continuity

    The equation which expresses the conservation of mass. The equation of continuity is one of the governing equations of fluid mechanics.
    (cf. Governing equation, Navier-Stokes equations

  • Equation of motion

    An equation which expresses a motion. For example, the Navier-Stokes equations are representative equations of motion for fluid.
    (cf. Navier-Stokes equations

  • Equation of state

    The equation which expresses the relationship between pressure and density of a fluid.

  • Euler description

    An expression for describing physical quantities by using parameters whose positions are fixed to a space.
    (cf. Lagrangian description

  • Explicit method

    A method for obtaining unknown values by solving an equation which only contains known values in time.
    (cf. Semi-implicit method

  • External force

    A force which is caused from the outside of the fluid. Gravity is a representative external force.
    (cf. Pressure term, viscosity term, Advection term

  • Fluid density

    Density of fluid. Mass of fluid per unit volume. The unit of fluid density is kg/m3.
    (cf. Particle number density, Equation of state

  • Fluid friction

    Friction due to fluid viscosity.
    (cf. Viscous term

  • Four basic operations of arithmetic

    Addition, subtraction, multiplication and division.
    (cf. Gradient model, Laplacian model

  • Fractional step method

    A time integration scheme which solves a differential equation step-by-step.

    (cf. Time marching, SMAC method)

  • Free surface

    A boundary which freely changes its shape on the basis of a governing equation. For example, a boundary between liquid and air represents a free surface.

  • Free surface particle

    A particle on a free surface.

  • Free-slip boundary

    A boundary condition for walls. The effect of viscosity is neglected between fluid and a free-slip boundary. i.e. the fluid slips on a free-slip boundary.
    (cf. No-slip boundary

  • Gaussian elimination

    A method for solving simultaneous equations. This can be called row reduction. This is not an iterative method but a direct method.
    (cf. Conjugate gradient method, SOR method

  • Ghost particle

    A particle which does not affect the simulation. Fluid particles which have gone out from the simulation domain are changed to ghost particles.
    (cf. Wall particle, Dummy wall particles

  • Governing equation

    An equation which expresses a rule of physical phenomena. The governing equations of the particle method are the equation of the continuity and the Navier-Stokes equations.

  • Gradient

    A vector which expresses a slope and the steepness of a function.
    (cf. Nabla, Gradient model, Laplacian, Divergence

  • Gradient model

    Models for calculating nabla (∇) on a computer. In general, nabla can be expressed by the four basic operations of arithmetic.
    (cf. Nabla, Laplacian, Laplacian model, Four basic operations of arithmetic

  • Gravity term

    The term which expresses the effect of gravity in an equation of motion.
    (cf. Navier-Stokes equations, External force, Pressure term, viscosity term, viscosity term, Advection term

  • Highly viscous fluid

    Fluid whose viscosity is very high.
    (cf. Viscous term)

  • Implicit method

    A method for obtaining unknown future values by solving mathematical relations which includes unknown future values. In general, the mathematical relations are expressed as simultaneous equation. Compared with explicit methods, implicit methods are able to use a longer time increment.
    (cf. Explicit method, Semi-implicit method

  • Initial distance between particles

    In the initial state, particles are arranged at regular intervals in the x, y and z directions. The interval is called the initial distance between particles.
    (cf. Spatial resolution

  • Kernel function

    A function used in Smoothed Particle Hydrodynamics (SPH). A derivative of a kernel function has something in common with the weight function of the MPS method.

  • Lagrangian derivative

    The Lagrangian derivative is the same as the substantial derivative.
    (cf. Substantial derivative

  • Lagrangian description

    An expression for describing physical quantities by using parameters whose evaluation points are moved following the motion and deformation of the substance.
    (cf. Euler description

  • Laplacian

    Name of the operator ∇^2. The operator is expressed by
    (cf. Nabla, Divergence, Laplacian model

  • Laplacian model

    A numerical model for ∇2. The Laplacian model discretizes the Laplacian of a function and converts the differential operator to operators which are executable on computers, such as the four basic operations of arithmetic.
    (cf. Laplacian, Gradient model, Discretization

  • Marching cube

    A technique for visualization. A contour of the density distribution is expressed by a great amount of polygons. Marching cube is one of the techniques of metaball.
    (cf. Metaball, Visualization)

  • Mass of particle

    The mass of a particle. In the MPS method, each particle has its own mass.

  • Mesh-based method

    A simulation method which discretizes a space by computing a mesh within it. The finite difference method (FDM), the finite volume method (FVM) and the finite element method (FEM) are representative mesh-based methods.
    (cf. Particle method)

  • Metaball

    A technique for visualization. In metaball, a contour of the density distribution is drawn. Free-surfaces simulated by a particle method are often drawn by metaball.
    (cf. Visualization, Marching cube

  • MPI

    A library for parallel computing. MPI enable us to carry out parallel computing on distributed memory or shared-memory parallel computers.
    (cf. OpenMP

  • Multi-physics

    Phenomena for which multiple physical aspects are important for accurate modeling. Fluid-elastic interactions and fluid-rigid interactions are examples of multi-physical phenomena.

  • Nabla

    An operator expressed by the symbol ∇, where
    (cf. Gradient, Gradient model, Discretization, Laplacian, Divergence

  • Navier-Stokes equations

    Equations of motion for fluid. The equations express the relationship between the acceleration and the force acting on fluid.
    (cf. Equation of motion

  • Neighboring particles

    The particles which are near the particle under consideration.
    (cf. Bucket, Neighboring particle search

  • Neighboring particle search

    The procedure of finding neighboring particles. Domain decomposition using buckets is often used for reducing the computation cost of neighboring particle search.
    (cf. Neighboring particle, Bucket

  • Neumann boundary condition

    A condition which is given to a boundary for solving a differential equation. A Neumann boundary condition fixes the value of a derivative of a function on a boundary.
    (cf. Dirichlet boundary condition

  • No-slip boundary

    A boundary condition for walls. Fluid sticks to a no-slip boundary because of viscosity, and the fluid velocity on a no-slip boundary is the same as the velocity of the boundary.
    (cf. Free-slip boundary

  • Normalization

    A numerical operation which divides an integral by a certain value so that the integral after the division becomes 1.
    A numerical operation which multiplies a constant to a vector so that the absolute value of the multiplied vector becomes 1.

  • OpenMP

    A library for parallel computing. OpenMP is executed on shared-memory computers.
    (cf. MPI

  • Particle interaction model

    Models which approximate and discretize differential operators such as nabla and the Laplacian. The discretized operators are expressed in the form of interactions between particles.
    (cf. Discretization, Gradient model, Laplacian model

  • Particle method

    A simulation method which discretize a continuum by moving calculation points called particles. In this website, the particle method means the moving particle semi-implicit (MPS) method. (Note: There are some other particle methods such as discrete element method (DEM) which simulates non-continuums.)
    (cf. Mesh-based method)

  • Particle number density

    A parameter which expresses the density of particles about a point. In general, the particle number density is calculated by a summation of weight function among neighboring particles. The particle number density is used to evaluate the rate of change of fluid density in pressure calculation, or is used to detect free-surface particles.
    (cf. Weight function, free surface, Equation of state

  • Particle’s ID

    The identification number of each particle.

  • Poisson equation

    A partial differential equation expressed in a form of ∇2 P=b, where P and b are functions. Function b is a source term which does not contain P.
    (cf. Source term, Laplacian

  • Polygon wall boundary

    A technique which expresses walls as polygons. By using the polygon wall boundary, we do not need to distribute wall particles on wall boundaries, and therefore we can reduce the number of particles. Another benefit is that polygon wall boundaries can express smooth boundary shapes, which conventional wall particle boundaries cannot.

  • Potential model

    A surface tension model based on the potential force between particles.
    (cf. Surface tension, CSF model

  • Pressure term

    A term which appears in the Navier-Stokes equations and the Euler equations. The pressure term expresses the influence on fluid motion due to pressure. The pressure term can be called “pressure gradient term”.
    (cf. Navier-Stokes equations, viscosity term, external force, viscosity term, Advection term

  • Radius of the supporting domain

    A radius for calculating physical quantities such as density and pressure gradient at a local point.

  • Semi-implicit method

    A numerical method which uses both explicit and implicit procedures. The MPS method is a semi-implicit method because pressure is solved implicitly while the other parts are solved explicitly.
    (cf. Implicit method, Explicit method

  • Sloshing

    A fluid phenomenon in which fluid is moved in a periodical manner causing the fluid surface to also deform periodically. For example, the fluid oscillation in a half-filled tank due to an earthquake is a representative sloshing.

  • SMAC method

    SMAC is the abbreviation of a Simplified Marker and Cell method. The time integration method used in the SMAC method is called SMAC.
    (cf. Time marching method

  • SOR method (Successive Over-Relaxation method)

    An iterative method for solving simultaneous equations.
    (cf. Conjugate gradient method, Gaussian elimination

  • Source term

    The right hand side of the Poisson equation.
    (cf. Poisson equation

  • Spatial resolution

    The size of a discretized part of space. In a particle method, the size of a particle expresses the spatial resolution.
    (cf. Initial distance between particles

  • Speed of sound

    Speed of sound. Sound waves travel at the speed of sound.
    (cf. Equation of state

  • Substantial derivative

    The substantial derivative, also called the Lagrangian derivative, expresses 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.
    (cf. Lagrangian expression

  • Surface tension

    A tensile force acting on a fluid surface. Surface tension is induced because of intermolecular forces.

  • The law of conservation of momentum

    A law which expresses momentum conservation. The Navier-Stokes equation is a law of conservation of momentum for fluid.
    (cf. Navier-Stokes equations)

  • the MKS system of units

    A system of unis. Distance, mass and time are expressed by meter, kilo-gram and second, respectively in the MKS system of units. The MKS system is a subset of the SI system.

  • Time increment

    The time difference between two time steps. Time increment is expressed by Δt.
    (cf. Courant number, Diffusion number, Explicit method, Implicit method

  • Time marching method

    A method of integrating a system of equations in time through finite time steps ∆t.
    (cf. Fractional step method, SMAC method

  • Turbulent flow

    Flow which contains irregular disturbances. Compared to the laminar flows, which does not contain irregular disturbances, parts of turbulent flows actively exchange momentum and heat.

  • V&V

    The abbreviation of verification and validation. In the field of simulation, verification means to compare simulation results with analytical results or other simulation results. Validation means to compare simulation results with experimental results.

  • Viscous term

    One of the terms in the Navier-Stokes equations. The viscous term expresses the effect of viscosity on fluid. The viscous term diffuses the momentum of fluid.
    (cf. Navier-Stokes equations, Pressure term, External force

  • Visualization

    To display data graphically in a figure or movie.
    (cf. Marching cube, Metaball

  • Wall particle

    A particle which composes a wall. A wall particle has a pressure parameter and is expressed by a fluid particle whose motion is fixed in space.
    (cf. Dummy wall particle, Ghost particle, Boundary condition

  • Weight function

    A function which expresses the influence of a neighboring particle. The function is also used for a weighted average.
    (cf. Particle number density, Gradient model, Laplacian model

  • Weight function for wall

    A function which is used in a polygon wall boundary. The weight function expresses the influence of a wall on surrounding particles.
    (cf. Polygon wall boundary

  • Wetness

    An affinity of fluid for wall boundary. In general, the wetness is expressed by a contact angle.
    (cf. Surface tension, Potential model