Finite pointset method
Encyclopedia
In applied mathematics
, the Finite Pointset Method (FPM) is a method for the solution of the equations governing viscous
fluid flows, including the effects of heat
and mass transfer
. FPM models problems in continuum mechanics
. The method solves not only fluid flows, but also problems with elastic or plastic deformations. More generally spoken, FPM considers all the viscous as well as elastic/plastic stress tensor
s, and any mixture of it.
, which represents the continuum or in other words a continuum domain (fluid or solid) is replaced by a discrete number of points, which are referred to as finite points. Each finite point carries all fluid information, like density, velocity, pressure, temperature. Finite points can move with fluid velocity (Lagrangian approach) or the flow information runs through the finite points if they are located constant in space (Eulerian approach). Also a mixture approach (Arbitrary Lagrangian Eulerian ALE) is possible. This is useful in case of using the Eulerian approach in combination with free surfaces or moving parts. Therefore, finite points themselves can be considered as geometrical grids of the fluid domain.
The finite point density is prescribed by a smoothing length defined locally. FPM does not use a rigid neighbourhood list for a certain finite point as it is required in a mesh based method. All neighbours are allowed to move and the neighbourhood list is re-computed each time step. Thus the simple idea of FPM is to use a dynamical discretization method, but the finite points itself are carrying all fluid information.
This method has various advantages over grid-based techniques; for example, it can handle fluid domains, which change naturally, whereas grid based techniques require additional computational effort. The finite points have to completely cover the whole flow domain, i.e. the point cloud has to fulfill certain quality criteria (finite points are not allowed to form “holes” which means finite points have to find sufficiently numerous neighbours; also, finite points are not allowed to cluster; etc.).
The finite point cloud is a geometrical basis, which allows for a numerical formulation making FPM a general finite difference idea applied to continuum mechanics. That especially means, if the point cloud would reduce to a regular cubic point grid, then FPM would reduce to a classical finite difference method. The idea of general finite differences also means that FPM is not based on a weak formulation like Galerkin’s approach. Rather, FPM is a strong formulation which models differential equations by direct approximation of the occurring differential operators. The method used is a moving least squares idea which was especially developed for FPM.
et al. 1965, Hirt et al. 1981, Kelecy et al. 1997, Kothe at el. 1992, Maronnier et al. 1999, Tiwari et al. 2000). A classical grid free Lagrangian method is Smoothed Particle Hydrodynamics (SPH), which was originally introduced to solve problems in astrophysics (Lucy 1977, Gingold et al. 1977).
It has since been extended to simulate the compressible Euler equations in fluid dynamics and applied to a wide range of problems, see (Monaghan 92, Monaghan et al. 1983, Morris et al. 1997). The method has also been extended to simulate inviscid incompressible free surface flows (Monaghan 94). The implementation of the boundary conditions is the main problem of the SPH method.
Another approach for solving fluid dynamic equations in a grid free framework is the moving least squares or least squares method (Belytschko et al. 1996, Dilts 1996, Kuhnert 99, Kuhnert 2000, Tiwari et al. 2001 and 2000). With this approach boundary conditions can be implemented in a natural way just by placing the finite points on boundaries and prescribing boundary conditions on them (Kuhnert 99). The robustness of this method is shown by the simulation results in the field of airbag deployment in car industry. Here, the membrane (or boundary) of the airbag changes very rapidly in time and takes a quite complicated shape (Kuhnert et al. 2000).
In (Tiwari et al. 2000) we have performed simulations of incompressible flows as the limit of the compressible Navier–Stokes equations with some stiff equation of state. This approach was first used in (Monaghan 92) to simulate incompressible free surface flows by SPH. The incompressible limit is obtained by choosing a very large speed of sound in the equation of state such that the Mach number becomes small. However the large value of the speed of sound restricts the time step to be very small due to the CFL-condition
.
The projection method
of Chorin
(Chorin 68) is a widely used approach to solve problems governed by the incompressible Navier–Stokes equation in a grid based structure. In (Tiwari et al. 2001), this method has been applied to a grid free framework with the help of the weighted least squares method. The scheme gives accurate results for the incompressible Navier–Stokes equations. The occurring Poisson equation for the pressure field is solved by a grid free method. In (Tiwari et al. 2001), it has been shown that the Poisson equation can be solved accurately by this approach for any boundary conditions. The Poisson solver can be adapted to the weighted least squares approximation procedure with the condition that the Poisson equation and the boundary condition must be satisfied on each finite point. This is a local iteration procedure.
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
, the Finite Pointset Method (FPM) is a method for the solution of the equations governing viscous
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
fluid flows, including the effects of heat
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...
and mass transfer
Mass transfer
Mass transfer is the net movement of mass from one location, usually meaning a stream, phase, fraction or component, to another. Mass transfer occurs in many processes, such as absorption, evaporation, adsorption, drying, precipitation, membrane filtration, and distillation. Mass transfer is used...
. FPM models problems in continuum mechanics
Continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...
. The method solves not only fluid flows, but also problems with elastic or plastic deformations. More generally spoken, FPM considers all the viscous as well as elastic/plastic stress tensor
Stress tensor
Stress tensor may refer to:* Stress , in classical physics* Stress-energy tensor, in relativistic theories* Maxwell stress tensor, in electromagnetism...
s, and any mixture of it.
Description
FPM is a meshfree method. The basis of the computations is a point cloudPoint cloud
A point cloud is a set of vertices in a three-dimensional coordinate system. These vertices are usually defined by X, Y, and Z coordinates, and typically are intended to be representative of the external surface of an object....
, which represents the continuum or in other words a continuum domain (fluid or solid) is replaced by a discrete number of points, which are referred to as finite points. Each finite point carries all fluid information, like density, velocity, pressure, temperature. Finite points can move with fluid velocity (Lagrangian approach) or the flow information runs through the finite points if they are located constant in space (Eulerian approach). Also a mixture approach (Arbitrary Lagrangian Eulerian ALE) is possible. This is useful in case of using the Eulerian approach in combination with free surfaces or moving parts. Therefore, finite points themselves can be considered as geometrical grids of the fluid domain.
The finite point density is prescribed by a smoothing length defined locally. FPM does not use a rigid neighbourhood list for a certain finite point as it is required in a mesh based method. All neighbours are allowed to move and the neighbourhood list is re-computed each time step. Thus the simple idea of FPM is to use a dynamical discretization method, but the finite points itself are carrying all fluid information.
This method has various advantages over grid-based techniques; for example, it can handle fluid domains, which change naturally, whereas grid based techniques require additional computational effort. The finite points have to completely cover the whole flow domain, i.e. the point cloud has to fulfill certain quality criteria (finite points are not allowed to form “holes” which means finite points have to find sufficiently numerous neighbours; also, finite points are not allowed to cluster; etc.).
The finite point cloud is a geometrical basis, which allows for a numerical formulation making FPM a general finite difference idea applied to continuum mechanics. That especially means, if the point cloud would reduce to a regular cubic point grid, then FPM would reduce to a classical finite difference method. The idea of general finite differences also means that FPM is not based on a weak formulation like Galerkin’s approach. Rather, FPM is a strong formulation which models differential equations by direct approximation of the occurring differential operators. The method used is a moving least squares idea which was especially developed for FPM.
History
In order to overcome the disadvantages of the classical methods many approaches have been developed to simulate such flows (Hansbo 92, HarlowFrancis H. Harlow
Francis Harvey Harlow is an American theoretical physicist known for his work in the field of fluid dynamics. He was a researcher at Los Alamos National Laboratory, Los Alamos, New Mexico...
et al. 1965, Hirt et al. 1981, Kelecy et al. 1997, Kothe at el. 1992, Maronnier et al. 1999, Tiwari et al. 2000). A classical grid free Lagrangian method is Smoothed Particle Hydrodynamics (SPH), which was originally introduced to solve problems in astrophysics (Lucy 1977, Gingold et al. 1977).
It has since been extended to simulate the compressible Euler equations in fluid dynamics and applied to a wide range of problems, see (Monaghan 92, Monaghan et al. 1983, Morris et al. 1997). The method has also been extended to simulate inviscid incompressible free surface flows (Monaghan 94). The implementation of the boundary conditions is the main problem of the SPH method.
Another approach for solving fluid dynamic equations in a grid free framework is the moving least squares or least squares method (Belytschko et al. 1996, Dilts 1996, Kuhnert 99, Kuhnert 2000, Tiwari et al. 2001 and 2000). With this approach boundary conditions can be implemented in a natural way just by placing the finite points on boundaries and prescribing boundary conditions on them (Kuhnert 99). The robustness of this method is shown by the simulation results in the field of airbag deployment in car industry. Here, the membrane (or boundary) of the airbag changes very rapidly in time and takes a quite complicated shape (Kuhnert et al. 2000).
In (Tiwari et al. 2000) we have performed simulations of incompressible flows as the limit of the compressible Navier–Stokes equations with some stiff equation of state. This approach was first used in (Monaghan 92) to simulate incompressible free surface flows by SPH. The incompressible limit is obtained by choosing a very large speed of sound in the equation of state such that the Mach number becomes small. However the large value of the speed of sound restricts the time step to be very small due to the CFL-condition
Courant–Friedrichs–Lewy condition
In mathematics, the Courant–Friedrichs–Lewy condition is a necessary condition for convergence while solving certain partial differential equations numerically by the method of finite differences. It arises when explicit time-marching schemes are used for the numerical solution...
.
The projection method
Projection method (fluid dynamics)
The projection method is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin in 1967 and independently by Roger Temam as an efficient means of solving the incompressible Navier-Stokes equations...
of Chorin
Alexandre Chorin
Alexandre J. Chorin is a professor of mathematics at the University of California, Berkeley who works in applied mathematics. He is known for his contributions to the field of Computational fluid dynamics....
(Chorin 68) is a widely used approach to solve problems governed by the incompressible Navier–Stokes equation in a grid based structure. In (Tiwari et al. 2001), this method has been applied to a grid free framework with the help of the weighted least squares method. The scheme gives accurate results for the incompressible Navier–Stokes equations. The occurring Poisson equation for the pressure field is solved by a grid free method. In (Tiwari et al. 2001), it has been shown that the Poisson equation can be solved accurately by this approach for any boundary conditions. The Poisson solver can be adapted to the weighted least squares approximation procedure with the condition that the Poisson equation and the boundary condition must be satisfied on each finite point. This is a local iteration procedure.