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）
A condition which is given to a boundary, such as a wall or a free surface.
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）
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）
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.
An iterative solver for solving simultaneous equations.
（cf. Gaussian elimination）
A law of mass conservation. Conservation of mass for a fluid is expressed by the equation of continuity.
（cf. Equation of continuity）
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）
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:http://dx.doi.org/10.1016/0021-9991(92)90240-Y.
The meaning of dam break is the same as that of collapse of a water column.
（cf. Collapse of water column）
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.
An equation which expresses the nature of diffusive phenomena.
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）
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）
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)
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）
An equation which expresses a motion. For example, the Navier-Stokes equations are representative equations of motion for fluid.
（cf. Navier-Stokes equations）
The equation which expresses the relationship between pressure and density of a fluid.
An expression for describing physical quantities by using parameters whose positions are fixed to a space.
（cf. Lagrangian description）
A method for obtaining unknown values by solving an equation which only contains known values in time.
（cf. Semi-implicit method）
Friction due to fluid viscosity.
（cf. Viscous term）
A time integration scheme which solves a differential equation step-by-step.
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.
A particle on a free surface.
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）
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.
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）
Fluid whose viscosity is very high.
(cf. Viscous term)
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）
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）
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.
The Lagrangian derivative is the same as the substantial derivative.
（cf. Substantial derivative）
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）
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）
The mass of a particle. In the MPS method, each particle has its own mass.
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)
A library for parallel computing. MPI enable us to carry out parallel computing on distributed memory or shared-memory parallel computers.
Phenomena for which multiple physical aspects are important for accurate modeling. Fluid-elastic interactions and fluid-rigid interactions are examples of multi-physical phenomena.
Equations of motion for fluid. The equations express the relationship between the acceleration and the force acting on fluid.
（cf. Equation of motion）
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）
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）
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.
A library for parallel computing. OpenMP is executed on shared-memory computers.
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）
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)
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）
The identification number of each particle.
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.
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）
A radius for calculating physical quantities such as density and pressure gradient at a local point.
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）
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 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）
The right hand side of the Poisson equation.
（cf. Poisson equation）
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. Sound waves travel at the speed of sound.
（cf. Equation of state）
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）
A tensile force acting on a fluid surface. Surface tension is induced because of intermolecular forces.
A law which expresses momentum conservation. The Navier-Stokes equation is a law of conservation of momentum for fluid.
（cf. Navier-Stokes equations）
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.
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.
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.
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）
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）
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）