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. NavierStokes 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 particle、Neighboring 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):335354. doi:http://dx.doi.org/10.1016/00219991(92)90240Y. 
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 NavierStokes 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. Nabla、Gradient） 
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, NavierStokes equations） 
Equation of motion
An equation which expresses a motion. For example, the NavierStokes equations are representative equations of motion for fluid.
（cf. NavierStokes 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. Semiimplicit 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/m^{3}.
（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 stepbystep.
(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.

Freeslip boundary
A boundary condition for walls. The effect of viscosity is neglected between fluid and a freeslip boundary. i.e. the fluid slips on a freeslip boundary.
（cf. Noslip 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 NavierStokes 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. NavierStokes 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, Semiimplicit 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.

Meshbased 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 meshbased methods.
(cf. Particle method) 
Metaball
A technique for visualization. In metaball, a contour of the density distribution is drawn. Freesurfaces 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 sharedmemory parallel computers.
（cf. OpenMP） 
Multiphysics
Phenomena for which multiple physical aspects are important for accurate modeling. Fluidelastic interactions and fluidrigid interactions are examples of multiphysical phenomena.

Nabla
An operator expressed by the symbol ∇, where
（cf. Gradient, Gradient model, Discretization, Laplacian, Divergence） 
NavierStokes 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） 
Noslip boundary
A boundary condition for walls. Fluid sticks to a noslip boundary because of viscosity, and the fluid velocity on a noslip boundary is the same as the velocity of the boundary.
（cf. Freeslip 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 sharedmemory 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 semiimplicit (MPS) method. (Note: There are some other particle methods such as discrete element method (DEM) which simulates noncontinuums.)
(cf. Meshbased 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 freesurface 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 NavierStokes 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. NavierStokes 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.

Semiimplicit method
A numerical method which uses both explicit and implicit procedures. The MPS method is a semiimplicit 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 halffilled 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 OverRelaxation 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 NavierStokes equation is a law of conservation of momentum for fluid.
（cf. NavierStokes equations） 
the MKS system of units
A system of unis. Distance, mass and time are expressed by meter, kilogram 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 NavierStokes equations. The viscous term expresses the effect of viscosity on fluid. The viscous term diffuses the momentum of fluid.
（cf. NavierStokes 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）