The Finite Mass Method on Domains with Boundary

Klingler, Markus; Leinen, Peter; Yserentant, Harry

doi:10.1137/s1064827502420483

Cited by 6 publications

(5 citation statements)

References 8 publications

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“…The previous meshfree method most similar to RRM is the Finite Mass Method (FMM) [34] , [35] . FMM divides the fluid into finite-sized cells (called mass packets in FMM papers) with an internal distribution typically described by third-order B-splines.…”

Section: Discussionmentioning

confidence: 99%

The Repeated Replacement Method: A Pure Lagrangian Meshfree Method for Computational Fluid Dynamics

Walker

2012

PLoS ONE

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In this paper we describe the repeated replacement method (RRM), a new meshfree method for computational fluid dynamics (CFD). RRM simulates fluid flow by modeling compressible fluids’ tendency to evolve towards a state of constant density, velocity, and pressure. To evolve a fluid flow simulation forward in time, RRM repeatedly “chops out” fluid from active areas and replaces it with new “flattened” fluid cells with the same mass, momentum, and energy. We call the new cells “flattened” because we give them constant density, velocity, and pressure, even though the chopped-out fluid may have had gradients in these primitive variables. RRM adaptively chooses the sizes and locations of the areas it chops out and replaces. It creates more and smaller new cells in areas of high gradient, and fewer and larger new cells in areas of lower gradient. This naturally leads to an adaptive level of accuracy, where more computational effort is spent on active areas of the fluid, and less effort is spent on inactive areas. We show that for common test problems, RRM produces results similar to other high-resolution CFD methods, while using a very different mathematical framework. RRM does not use Riemann solvers, flux or slope limiters, a mesh, or a stencil, and it operates in a purely Lagrangian mode. RRM also does not evaluate numerical derivatives, does not integrate equations of motion, and does not solve systems of equations.

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Section: Discussionmentioning

confidence: 99%

The Repeated Replacement Method: A Pure Lagrangian Meshfree Method for Computational Fluid Dynamics

Walker

2012

PLoS ONE

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“…The Finite Mass Method (FMM) published by Gauger et al [ 33 ] and Klingler et al [ 34 ] is a meshfree method that uses extended pieces of material in a way somewhat similar to the cells of RRM. FMM divides the material being simulated into finite-sized “mass packets”, which are modeled by differentiable functions with compact support such as cubic B-splines.…”

Section: Overview Of the Repeated Replacement Methodsmentioning

confidence: 99%

A Lagrangian meshfree method applied to linear and nonlinear elasticity

Walker

2017

PLoS ONE

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The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and Sedov-Taylor solvers for them, and detail the numerical techniques needed to embody those solvers in code.

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“…Their computation is a core algorithmic issue. Let us recall the strategy for the computation of the pressure forces [9,15]: The crucial distinction is between Fig. 2 Computation of the discrete force integrals of internal pressure data associated with particles (like m i , S i , q i , H i , q i , and H i ), and data required at the particle quadrature points like ρ, s, ∂ ε ∂ρ , and ∂ ε ∂s .…”

Section: Pressure Forcesmentioning

confidence: 99%

“…Finally, in [9] the model is supplemented such that the particles can even change their shape by linear deformations. Extensions and special variants are described in [14,15].…”

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confidence: 99%

The finite mass mesh method

Bubeck

Hiptmair

Yserentant

2005

Comput. Visual Sci.

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The finite mass method is a purely Lagrangian scheme for the spatial discretisation of the macroscopic phenomenological laws that govern the flow of compressible fluids. In this article we investigate how to take into account long range gravitational forces in the framework of the finite mass method. This is achieved by incorporating an extra discrete potential energy of the gravitational field into the Lagrangian that underlies the finite mass method. The discretisation of the potential is done in an Eulerian fashion and employs an adaptive tensor product mesh fixed in space, hence the name finite mass mesh method for the new scheme. The transfer of information between the mass packets of the finite mass method and the discrete potential equation relies on numerical quadrature, for which different strategies will be proposed. The performance of the extended finite mass method for the simulation of two-dimensional gas pillars under self-gravity will be reported. IntroductionThe finite mass method gives a discrete macroscopic description of the flow of a compressible fluid. It is a purely Lagrangian approach based on first principles of fluid mechanics. The main idea is to discretise the fluid by means of a finite number of mass packets which are called "particles". These particles have finite extension, move independently,

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The Finite Mass Method on Domains with Boundary

Cited by 6 publications

References 8 publications

The Repeated Replacement Method: A Pure Lagrangian Meshfree Method for Computational Fluid Dynamics

The Repeated Replacement Method: A Pure Lagrangian Meshfree Method for Computational Fluid Dynamics

A Lagrangian meshfree method applied to linear and nonlinear elasticity

The finite mass mesh method

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