Finite pointset method

Last updated December 25, 2023

In applied mathematics, the name finite pointset method is a general approach for the numerical solution of problems in continuum mechanics, such as the simulation of fluid flows. In this approach (often abbreviated as FPM) the medium is represented by a finite set of points, each endowed with the relevant local properties of the medium such as density, velocity, pressure, and temperature.^[1]

The sampling points can move with the medium, as in the Lagrangian approach to fluid dynamics or they may be fixed in space while the medium flows through them, as in the Eulerian approach. A mixed Lagrangian-Eulerian approach may also be used. The Lagrangian approach is also known (especially in the computer graphics field) as particle method.

Finite pointset methods are meshfree methods and therefore are easily adapted to domains with complex and/or time-evolving geometries and moving phase boundaries (such as a liquid splashing into a container, or the blowing of a glass bottle) without the software complexity that would be required to handle those features with topological data structures. They can be useful in non-linear problems involving viscous fluids, heat and mass transfer, linear and non-linear elastic or plastic deformations, etc.

Description

In the simplest implementations, the finite point set is stored as an unstructured list of points in the medium. In the Lagrangian approach the points move with the medium, and points may be added or deleted in order to maintain a prescribed sampling density. The point density is usually prescribed by a smoothing length defined locally. In the Eulerian approach the points are fixed in space, but new points may be added where there is need for increased accuracy. So, in both approaches the nearest neighbors of a point are not fixed, and are determined again at each time step.

Advantages

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 reduced 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, Harlow 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).

Tiwari et al. (2000) 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.

Software

Nogrid points
MESHFREE

Related Research Articles

In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests.

<span class="mw-page-title-main">Smoothed-particle hydrodynamics</span> Method of hydrodynamics simulation

Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method, and the resolution of the method can easily be adjusted with respect to variables such as density.

Fluid mechanics is the branch of physics concerned with the mechanics of fluids and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical, and biomedical engineering, as well as geophysics, oceanography, meteorology, astrophysics, and biology.

The Advection Upstream Splitting Method (AUSM) is a numerical method used to solve the advection equation in computational fluid dynamics. It is particularly useful for simulating compressible flows with shocks and discontinuities.

Fluid–structure interaction (FSI) is the interaction of some movable or deformable structure with an internal or surrounding fluid flow. Fluid–structure interactions can be stable or oscillatory. In oscillatory interactions, the strain induced in the solid structure causes it to move such that the source of strain is reduced, and the structure returns to its former state only for the process to repeat.

In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field.

Alexandre Joel Chorin is an American mathematician known for his contributions to computational fluid mechanics, turbulence, and computational statistical mechanics.

Stokesian dynamics is a solution technique for the Langevin equation, which is the relevant form of Newton's 2nd law for a Brownian particle. The method treats the suspended particles in a discrete sense while the continuum approximation remains valid for the surrounding fluid, i.e., the suspended particles are generally assumed to be significantly larger than the molecules of the solvent. The particles then interact through hydrodynamic forces transmitted via the continuum fluid, and when the particle Reynolds number is small, these forces are determined through the linear Stokes equations. In addition, the method can also resolve non-hydrodynamic forces, such as Brownian forces, arising from the fluctuating motion of the fluid, and interparticle or external forces. Stokesian Dynamics can thus be applied to a variety of problems, including sedimentation, diffusion and rheology, and it aims to provide the same level of understanding for multiphase particulate systems as molecular dynamics does for statistical properties of matter. For $rigid particles of radius suspended in an incompressible Newtonian fluid of viscosity and density, the motion of the fluid is governed by the Navier-Stokes equations, while the motion of the particles is described by the coupled equation of motion:$

Pressure-correction method is a class of methods used in computational fluid dynamics for numerically solving the Navier-Stokes equations normally for incompressible flows.

In computational fluid dynamics, shock-capturing methods are a class of techniques for computing inviscid flows with shock waves. The computation of flow containing shock waves is an extremely difficult task because such flows result in sharp, discontinuous changes in flow variables such as pressure, temperature, density, and velocity across the shock.

Gretar Tryggvason is Department Head of Mechanical Engineering and Charles A. Miller Jr. Distinguished Professor at Johns Hopkins University. He is known for developing the front tracking method to simulate multiphase flows and free surface flows. Tryggvason was the editor-in-chief of Journal of Computational Physics from 2002–2015.

The moving particle semi-implicit (MPS) method is a computational method for the simulation of incompressible free surface flows. It is a macroscopic, deterministic particle method developed by Koshizuka and Oka (1996).

Phillip Colella is an American applied mathematician and a member of the Applied Numerical Algorithms Group at the Lawrence Berkeley National Laboratory. He has also worked at Lawrence Livermore National Laboratory. He is known for his fundamental contributions in the development of mathematical methods and numerical tools used to solve partial differential equations, including high-resolution and adaptive mesh refinement schemes. Colella is a member of the US National Academy of Sciences.

Francis Harvey Harlow was 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. Harlow is credited with establishing the science of computational fluid dynamics (CFD) as an important discipline.

The multiphase particle-in-cell method (MP-PIC) is a numerical method for modeling particle-fluid and particle-particle interactions in a computational fluid dynamics (CFD) calculation. The MP-PIC method achieves greater stability than its particle-in-cell predecessor by simultaneously treating the solid particles as computational particles and as a continuum. In the MP-PIC approach, the particle properties are mapped from the Lagrangian coordinates to an Eulerian grid through the use of interpolation functions. After evaluation of the continuum derivative terms, the particle properties are mapped back to the individual particles. This method has proven to be stable in dense particle flows, computationally efficient, and physically accurate. This has allowed the MP-PIC method to be used as particle-flow solver for the simulation of industrial-scale chemical processes involving particle-fluid flows.

Gerris is computer software in the field of computational fluid dynamics (CFD). Gerris was released as free and open-source software, subject to the requirements of the GNU General Public License (GPL), version 2 or any later.

High-order compact finite difference schemes are used for solving third-order differential equations created during the study of obstacle boundary value problems. They have been shown to be highly accurate and efficient. They are constructed by modifying the second-order scheme that was developed by Noor and Al-Said in 2002. The convergence rate of the high-order compact scheme is third order, the second-order scheme is fourth order.

The finite point method (FPM) is a meshfree method for solving partial differential equations (PDEs) on scattered distributions of points. The FPM was proposed in the mid-nineties in, and with the purpose to facilitate the solution of problems involving complex geometries, free surfaces, moving boundaries and adaptive refinement. Since then, the FPM has evolved considerably, showing satisfactory accuracy and capabilities to deal with different fluid and solid mechanics problems.

<span class="mw-page-title-main">FEATool Multiphysics</span>

FEATool Multiphysics is a physics, finite element analysis (FEA), and partial differential equation (PDE) simulation toolbox. FEATool Multiphysics features the ability to model fully coupled heat transfer, fluid dynamics, chemical engineering, structural mechanics, fluid-structure interaction (FSI), electromagnetics, as well as user-defined and custom PDE problems in 1D, 2D (axisymmetry), or 3D, all within a graphical user interface (GUI) or optionally as script files. FEATool has been employed and used in academic research, teaching, and industrial engineering simulation contexts.

References

↑ Belytschko T., Krongauz Y., Flemming M., Organ D., Liu W.K.S., Smoothing and accelerated computations in the element free Galerkin method, J. Comp. Appl. Maths,. vol. 74, 1996, p. 111-126.

Ash N., Poo J. Y., Coalescence and separation in binary collisions of liquid drops, J. Fluid Mech., vol. 221, 1990, p. 183 - 204.
Chorin A., Numerical solution of the Navier-Stokes equations, J. Math. Comput,. vol. 22, 1968, p. 745-762.
Dilts G. A., Moving least squares particle hydrodynamics. I: consistency and stability, Hydrodynamics methods group report, Los Alamos National Laboratory, 1996
Gingold R. A., Monaghan J. J., Smoothed particle hydrodynamics: theory and application to non-spherical stars, Mon. Not. R. Astron. Soc., vol. 181, 1977, p. 375-389.
Ginzburg I., Wittum G., Two-phase flows on interface refined grids modeled with VOF, staggered finite volumes, and spline interpolants, J. Comput. Phys.,. vol. 166, 2001, p. 302-335.
Hansbo P., The characteristic streamline diffusion method for the time dependent incompressible Navier-Stokes equations, Comp. Meth. Appl. Mech. Eng., vol. 99, 1992, p. 171-186.
Harlow F. H., Welch J. E., Numerical study of large amplitude free surface motions, Phys. Fluids, vol.8, 1965, p. 2182.
Hirt C. W., Nichols B. D., Volume of fluid (VOF) method for dynamic of free boundaries, J. Comput. Phys., vol. 39, 1981, p. 201.
Kelecy F. J., Pletcher R. H., The development of free surface capturing approach for multi dimensional free surface flows in closed containers, J. Comput. Phys., vol. 138, 1997, p. 939.
Kothe D. B., Mjolsness R. C., RIPPLE: A new model for incompressible flows with free surfaces, AIAA Journal, Vol. 30, No 11, 1992, p. 2694-2700.
Kuhnert J., General smoothed particle hydrodynamics, Ph.D. thesis, Kaiserslautern University, Germany, 1999.
Kuhnert J., An upwind finite pointset method for compressible Euler and Navier-Stokes equations, preprint, ITWM, Kaiserslautern, Germany, 2000.
Kuhnert J., Tramecon A., Ullrich P., Advanced Air Bag Fluid Structure Coupled Simulations applied to out-of Position Cases, EUROPAM Conference Proceedings 2000, ESI Group, Paris, France
Landau L. D., Lifshitz E. M., Fluid Mechanics, Pergamon, New York, 1959.
Lafaurie B., Nardone C., Scardovelli R., Zaleski S., Zanetti G., Modelling Merging and Fragmentation in Multiphase Flows with SURFER, J. Comput. Phys., vol. 113, 1994, p. 134 - 147.
Lucy L. B., A numerical approach to the testing of the fission hypothesis, Astron. J., vol. 82, 1977, p. 1013.
Maronnier V., Picasso M., Rappaz J., Numerical simulation of free surface flows, J. Comput. Phys. vol. 155, 1999, p. 439.
Martin J. C., Moyce M. J., An experimental study of the collapse of liquid columns on a liquid horizontal plate, Philos. Trans. Roy. Soc. London, Ser. A 244, 1952, p. 312.
Monaghan J. J., Smoothed particle hydrodynamics, Annu. Rev. Astron. Astrop, vol. 30, 1992, p. 543-574.
Monaghan J. J., Simulating free surface flows with SPH, J. Comput. Phys., vol. 110, 1994, p. 399.
Monaghan J. J., Gingold R. A., Shock Simulation by particle method SPH, J. Comput. Phys., vol. 52, 1983, p. 374-389.
Morris J. P., Fox P. J., Zhu Y., Modeling Low Reynolds Number Incompressible Flows Using SPH, J. Comput. Phys., vol. 136, 1997, p. 214-226.
Tiwari S., Kuhnert J., Grid free method for solving Poisson equation, Berichte des Fraunhofer ITWM, Kaiserslautern, Germany, Nr. 25, 2001.
Tiwari S., Kuhnert J., Finite pointset method based on the projection method for simulations of the incompressible Navier-Stokes equations, M. Griebel, M. A. Schweitzer (Eds.), Springer LNCSE: Meshfree Methods for Partial Differential Equations, Springer-Verlag, Berlin, 26, 2003, p. 373-387.
Tiwari S., Kuhnert J., Particle method for simulations of free surface flows, preprint Fraunhofer ITWM, Kaiserslautern, Germany, 2000.
Tiwari S., A LSQ-SPH approach for compressible viscous flows, Hyperbolic Problems: Theory, Numerics, Applications: Eighth International Conference in Magdeburg, February/March 2000, Volume II (International Series of Numerical Mathematics), Vol. 141, 2000, 901–910.
Tiwari S., Manservisi S., Modeling incompressible Navier-Stokes flows by LSQ-SPH, Berichte des Fraunhofer ITWM, Kaiserslautern, Germany, 2000.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Bely-1] Belytschko T., Krongauz Y., Flemming M., Organ D., Liu W.K.S., Smoothing and accelerated computations in the element free Galerkin method, J. Comp. Appl. Maths,. vol. 74, 1996, p. 111-126.

[1]