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14
Hyperbolic Divergence Cleaning for Smoothed Particle Magnetohydrodynamics
"... We present a constrained formulation of Dedner et al’s hyperbolic/parabolic divergence cleaning scheme for enforcing the ∇ · B = 0 constraint in Smoothed Particle Magnetohydrodynamics (SPMHD) simulations. The constraint we impose is that energy removed must either be conserved or dissipated, such t ..."
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We present a constrained formulation of Dedner et al’s hyperbolic/parabolic divergence cleaning scheme for enforcing the ∇ · B = 0 constraint in Smoothed Particle Magnetohydrodynamics (SPMHD) simulations. The constraint we impose is that energy removed must either be conserved or dissipated, such that the scheme is guaranteed to decrease the overall magnetic energy. This is shown to require use of conjugate numerical operators for evaluating ∇·B and ∇ψ in the SPMHD cleaning equations. The resulting scheme is shown to be stable at density jumps and free boundaries, in contrast to an earlier implementation by Price & Monaghan (2005). Optimal values of the damping parameter are found to be σ = 0.2–0.3 in 2D and σ = 0.8– 1.2 in 3D. With these parameters, our constrained Hamiltonian formulation is found to provide an effective means of enforcing the divergence constraint in SPMHD, typically maintaining average values of h∇·B/B to 0.1–1%, up to an order of magnitude better than artificial resistivity without the associated dissipation in the physical field. Furthermore, when applied to realistic, 3D simulations we find an improvement of up to two orders of magnitude in momentum conservation with a corresponding improvement in numerical stability at essentially zero additional computational expense. 1.
On the maximum time step in weakly compressible SPH
"... In the SPH method for viscous fluids, the time step is subject to empirical stability criteria. We proceed to a stability analysis of the Weakly Compressible SPH equations using the von Neumann approach in arbitrary space dimension for unbounded flow. Considering the continuous SPH interpolant base ..."
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In the SPH method for viscous fluids, the time step is subject to empirical stability criteria. We proceed to a stability analysis of the Weakly Compressible SPH equations using the von Neumann approach in arbitrary space dimension for unbounded flow. Considering the continuous SPH interpolant based on integrals, we obtain a theoretical stability criterion for the time step, depending on the kernel standard deviation, the speed of sound and the viscosity. The stability domain appears to be almost independent of the kernel choice for a given space discretization. Numerical tests show that the theory is very accurate, despite the approximations made. We then extend the theory in order to study the influence of the method used to compute the density, of the gradient and divergence SPH operators, of background pressure, of the model used for viscous forces and of a constant velocity gradient. The influence of time integration scheme is also studied, and proved to be prominent. All of the above theoretical developments give excellent agreement against numerical results. It is found that velocity gradients almost do not affect stability, provided some background pressure is used. Finally, the case of bounded flows is briefly addressed from numerical tests in three cases: a laminar Poiseuille flow in a pipe, a liddriven cavity and the collapse of a water column on a wedge.
Unified semianalytical wall boundary conditions applied to 2D incompressible SPH
"... This work aims at improving the 2D incompressible SPH model (ISPH) by adapting it to the unified semianalytical wall boundary conditions proposed by Ferrand et al. [10]. The ISPH algorithm considered is as proposed by Lind et al. [25], based on the projection method with a divergencefree velocity ..."
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This work aims at improving the 2D incompressible SPH model (ISPH) by adapting it to the unified semianalytical wall boundary conditions proposed by Ferrand et al. [10]. The ISPH algorithm considered is as proposed by Lind et al. [25], based on the projection method with a divergencefree velocity field and using a stabilising procedure based on particle shifting. However, we consider an extension of this model to ReynoldsAveraged NavierStokes equations based on the k − turbulent closure model, as done in [10]. The discrete SPH operators are modified by the new description of the wall boundary conditions. In particular, a boundary term appears in the Laplacian operator, which makes it possible to accurately impose a von Neumann pressure wall boundary condition that corresponds to impermeability. The shifting and freesurface detection algorithms have also been adapted to the new boundary conditions. Moreover, a new way to compute the wall renormalisation factor in the frame of the unified semianalytical boundary conditions is proposed in order to decrease the computational time. We present several verifications to the present approach, including a liddriven cavity, a water column collapsing on a wedge and a periodic schematic fishpass. Our results are compared to Finite Volumes methods, using Volume of Fluids in the case of freesurface flows. We briefly investigate the convergence of the method and prove its ability to model complex freesurface and turbulent flows. The results are generally improved when compared to a weakly compressible SPH model with the same boundary conditions, especially
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FIELD OF PHYSICS
, 2014
"... This is brought to you for free and open access by the Theses and Dissertations at Research Showcase @ CMU. It has been accepted for inclusion in Dissertations by an authorized administrator of Research Showcase @ CMU. For more information, please contact research ..."
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This is brought to you for free and open access by the Theses and Dissertations at Research Showcase @ CMU. It has been accepted for inclusion in Dissertations by an authorized administrator of Research Showcase @ CMU. For more information, please contact research
On the consistency of MPS
"... The consistency of Moving Particle Semiimplicit (MPS) method in reproducing the gradient, divergence and Laplacian differential operators is discussed in the present paper. Its relation to the Smoothed Particle Hydrodynamics (SPH) method is rigorously established. The application of the MPS method ..."
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The consistency of Moving Particle Semiimplicit (MPS) method in reproducing the gradient, divergence and Laplacian differential operators is discussed in the present paper. Its relation to the Smoothed Particle Hydrodynamics (SPH) method is rigorously established. The application of the MPS method to solve the NavierStokes equations using a fractional step approach is treated, unveiling inconsistency problems when solving the Poisson equation for the pressure. A new corrected MPS method incorporating boundary terms is proposed. Applications to one dimensional boundary value Dirichlet and mixed NeumannDirichlet problems and to twodimensional freesurface flows are presented.
A Switch for Artificial Resistivity and Other Dissipation Terms
"... Abstract—We describe a new switch to reduce dissipation from artificial resistivity in Smoothed Particle Magnetohydrodynamics simulations. The switch utilises the gradient of the magnetic field to detect shocks, setting αB = h∇B/B. This measures the relative degree of discontinuity, and the swit ..."
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Abstract—We describe a new switch to reduce dissipation from artificial resistivity in Smoothed Particle Magnetohydrodynamics simulations. The switch utilises the gradient of the magnetic field to detect shocks, setting αB = h∇B/B. This measures the relative degree of discontinuity, and the switch is not dependent on the absolute field strength. We present results comparing the new resistivity switch to the switch of Price & Monaghan (2005), showing that it is more robust in capturing shocks (especially in weak fields), while leading to less overall dissipation. The design of this switch is generalised to create similar switches for artificial viscosity and thermal conduction, with proof of concept tests conducted on a Sod shock tube and KelvinHelmholtz instabilities. I.
EFFICIENT AND SCALABLE ALGORITHMS FOR SMOOTHED PARTICLE HYDRODYNAMICS ON HYBRID SHARED/DISTRIBUTEDMEMORY ARCHITECTURES
"... Abstract. This paper describes a new fast and implicitly parallel approach to neighbourfinding in multiresolution Smoothed Particle Hydrodynamics (SPH) simulations. This new approach is based on hierarchical cell decompositions and sorted interactions, within a taskbased formulation. It is shown ..."
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Abstract. This paper describes a new fast and implicitly parallel approach to neighbourfinding in multiresolution Smoothed Particle Hydrodynamics (SPH) simulations. This new approach is based on hierarchical cell decompositions and sorted interactions, within a taskbased formulation. It is shown to be faster than traditional treebased codes, and to scale better than domain decompositionbased approaches on hybrid shared/distributedmemory parallel architectures, e.g. clusters of multicores, achieving a 40 × speedup over the Gadget2 simulation code.
SPH Entropy Errors and the Pressure Blip
"... The spurious pressure jump at a contact discontinuity, in SPH simulations of the compressible Euler equations is investigated. From the spatiotemporal behaviour of the error, the SPH pressure jump is likened to entropy errors observed for artificial viscosity based finite difference/volume schemes. ..."
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The spurious pressure jump at a contact discontinuity, in SPH simulations of the compressible Euler equations is investigated. From the spatiotemporal behaviour of the error, the SPH pressure jump is likened to entropy errors observed for artificial viscosity based finite difference/volume schemes. The error is observed to be generated at startup and dissipation is the only recourse to mitigate it’s effect. We show that similar errors are generated for the Lagrangian plus remap version of the Piecewise Parabolic Method (PPM) finite volume code (PPMLR). Through a comparison with the direct Eulerian version of the PPM code (PPMDE), we argue that a lack of diffusion across the material wave (contact discontinuity) is responsible for the error in PPMLR. We verify this hypothesis by constructing a more dissipative version of the remap code using a piecewise constant reconstruction. As an application to SPH, we propose a hybrid GSPH scheme that adds the requisite dissipation by utilizing a more dissipative Riemann solver for the energy equation. The proposed modification to the GSPH scheme, and it’s improved treatment of the anomaly is verified for flows with strong shocks in one and two dimensions. The result that dissipation must act across the density and energy equations provides a consistent explanation for many of the hitherto proposed “cures ” or “fixes ” for the problem.
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"... EPJ manuscript No. (will be inserted by the editor) Recent advances in the simulation of particleladen flows ..."
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EPJ manuscript No. (will be inserted by the editor) Recent advances in the simulation of particleladen flows