Variational formulation for the smooth particle hydrodynamics (SPH) simulation of fluid and solid problems

SUMMARYSmoothed particle hydrodynamics (SPH) is becoming increasingly common in the numerical simulation of complex fluid flows and an understanding of the errors is necessary. Recent advances have established techniques for ensuring completeness conditions (low-order polynomials are interpolated exactly) are enforced when estimating property gradients, but the consequences on errors have not been investigated. Here, we present an expression for the error in an SPH estimate, accounting for completeness, an expression that applies to SPH generally. We revisit the derivation of the SPH equations for fluids, paying particular attention to the conservation principles. We find that a common method for enforcing completeness violates a property required of the kernel gradients, namely that gradients with respect the two position variables be equal and opposite. In such models this means conservation principles are not enforced and we present results that show this. As an aside we show the summation interpolant for density is a solution of, and may be used in the place of, the discretized, symmetrized continuity equation. Finally, we examine two examples of discretization errors, namely numerical boundary layers and the existence of crystallized states.

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“…In one such method the kernel is modified so that for particle a the corrected gradient of the kernel at b is given by [8] …”

mentioning

confidence: 99%

Completeness, conservation and error in SPH for fluids

Vaughan

Healy

Bryan

et al. 2007

Numerical Methods in Fluids

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“…(22), (23) are satisfied when a corrected smoothing function, e.g. the Shepard function (Bonet et al, 2004) is adopted.…”

Section: J-th Order Consistencymentioning

confidence: 99%

Simulating Flows with SPH: Recent Developments and Applications

Viccione¹,

Bovolin²,

Carratelli³

2011

Hydrodynamics - Optimizing Methods and Tools

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“…Variational consistency of equations is crucial in SPH and arbitrary combinations of mass and momentum discretizations introduce numerical noise in the results [3,8]. It has been demonstrated that variational consistency can be more important than devising higher order SPH approximation schemes [24].…”

Section: Comparison To Traditional Sph Schemesmentioning

confidence: 99%