R.W. Hockney and J.W. Eastwood. Computer Simulation Using Particles. McGraw-Hill, New York, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... rms size of the k th slice; # me is the rms size of a macro electron, which is assumed to have finite size (typically a tenth of the beam size) F is expressed by the Bassetti Erskine formula [14] The technique of using macro particles with finite size is not new in the frames of plasma physics [15, 16] and beam beam simulations for linear colliders [17] where its purpose is to avoid singularities which may invalidate the simulations while retaining the long range behaviour of the fields that is responsible for collective effects. We have chosen to use elliptical macroelectrons, and the ....

R. W. Hockney and S. W Eastwood, "Computer Simulation Using Particles", Hilger Bristol U.K. (1988).

....to reduce computational errors in the self electric field calculation and to accurately describe beam line conducting boundaries. For arbitrary beam distributions and boundaries, the charge densities and fields are evaluated on 3D grids during step by step integration of the trajectories [1]. Computational economy may be achieved by upgrading structural elements such as charge density redistribution block, space charge field solvers, trajectory integrators, and the calculational flow with parallel processing [2] Another way to reduce 3D PIC computational demands is to develop a new ....

....the particle trajectory, sc ext F F F = are the external focusing and the space charge forces, and x = dx ds. Note, the space charge force, F sc , is from the ensemble of N p beam particles. Integration schemes such as leapfrog retain accuracy to nd order. For details see Chapter 4 of [1]) The memory requirements are modest, but small integration steps are necessary to reduce round off error, making them computationally slow. A faster higher order integrator for Eq. 1) is obtained by a modification of the balance method [8] Omitting for sake of brevity the indices x in F and ....

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R.W. Hockney and J.W. Eastwood, "Computer Simulation Using Particles", McGraw-Hill Inc., 1981.

....If N is larger than several tens, the calculation of the right hand side of equa tion (1) dominates the total calculation cost. If N is very large, we could use sophisticated algorithms such as the Barnes Hut treecode[5] FMM[6] or the particle mesh or particle particle particle mesh method[7]. However, with treecode or FMM, the direct evaluation of gravitational interaction between near neighbor particles still dominates the total cost. Even in the case of the particle mesh method, if we want high accuracy, near neighbor interaction must be calculated directly. The calculation itself ....

R. W. Hockney, J. W. Eastwood, Computer Simulation Using Particles, IOP Publishing, Ltd., Bristol, 1988.

....to zero by a switching function such as (3. 21) There is evidence that cutoffs can cause undesirable artifacts [70,71] A review of this issue is found in [72] For this reason attention has turned to the development of summation schemes, such as fast multipole [1,73 75] and Ewald summation [76 78], for faster evaluation of electrostatic energies and forces without distance cutoffs. It is often the case that fast summation methods enjoy computational speedups, as compared to direction summation, only for sufficiently large systems. Large periodic water systems were used for testing fast ....

R.W. Hockney and J.W. Eastwood, Computer simulation using particles, McGraw-Hill, New York, 1981

....by the mesh size and the interpolation scheme, which makes the choice of optimal parameters more difficult. At present, there exist several implementations based on this idea, but they differ in detail. In [27] three essential methods are compared and summarized: particle particle particle mesh [54] (P M) particle mesh Ewald [26] PME) and smooth particle mesh Ewald [33] SPME) Unfortunately, the mesh based Ewald methods are affected by errors when performing interpolation, FFT, and differentiation [90] However, it would be misleading to infer that these methods sacrifice accuracy in ....

....over the grid since the density of the Gaussian screening function is highly localized. In [100] the Poisson equation is solved by a multi grid approach (O(N) The particle particle particle mesh multi pole expansion [104] P M PME, O(N log N) is basically an extension of the P M [54] method for periodic boundary conditions using multi pole expansions. The cell multi pole method [28] CMM, O(N) shares the key features of the multi pole methods, but it uses Cartesian coordinates only, unlike the other formulations that use spherical harmonics. The reduced cell multi pole ....

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R. W. Hockney and J. W. Eastwood. Computer Simulation Using Particles. McGraw-Hill, New York, 1981.

....To circumvent this problem, the position of the particles is periodically reinitialized on a uniform grid and the properties of the old particles are interpolated onto the new particle locations. This kind of interpolation has been implemented in a number of calculations involving particle methods [10, 19, 23, 24] but it has not been reported before, to the best of our knowledge, in the context of SPH, in conjunction with the approximation of the diffusion operator. For the remeshing procedure two types of interpolation are implemented: a second order ordinary interpolation and a third order smoothing ....

....operator. For the remeshing procedure two types of interpolation are implemented: a second order ordinary interpolation and a third order smoothing interpolation, which are further discussed in the following sections. 3.2.1. Ordinary Interpolation The second order ordinary interpolation formula [10, 24] is conserving the interpolated quantity (zero order moment) as well as its first (impulse) and second moment (angular impulse) This interpolation # 2 , which is used for the vortex particle simulations [25] can be expressed in one dimension as # 2 (x, h) # # # # ....

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (Inst. Phys. Publ. Bristol, 1988).

.... by a simple truncation, fast methods have been devised such as the Barnes Hut [3] and the fast multipole method [7] as well as hybrid mesh based algorithms including the particle mesh (PM) and the particle particle particle mesh (PPPM) algorithm originally proposed by Hockney and Eastwood [9]. While fast multipole methods offer an operational cost of (9(N) and an exact enforcement of the freespace boundary condition, hybrid particle mesh algorithms with an operational cost of (9 (N log N) or (9 (N) are often found to be computationally superior for problems in simple geometries and ....

....Elsevier Science B.V. All rights reserved. PII: S0021 9991 (02)00035 9 resolve these scales resulting in the PPPM algorithm. The sub grid scales are generally anisotropic as they depend on the relative position of the particles on the mesh. The PPPM algorithm presented by Hockney and Eastwood [9] reduces this anisotropy by applying an optimized Green s function in Fourier space, taking into account the projection steps and the differential operators involved in the particle mesh algorithm. Moreover, the modification and use of the Fourier components of the Green s function allows ....

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R.W. Hockney, J.W. Eastwood, Computer Simulation Using Particles, second ed., IOP, 1988.

....allows the algorithm to solve problems which may be discretized in a very irregular fashion in nearly optimal time. Numerous algorithms exist for the n body problem of evaluating the potential of a set of charges at all the other charge points, such as the particle mesh methods (see [20] for extensive references) the fast multipole method (FMM) 15] and multigrid methods [21] The various algorithms differ in the way the long range potential is approximated and in the way local interactions are treated. We have attempted to develop an algorithm which, like particle mesh ....

....accurate calculation of interactions between nearby panels is needed, but it is also necessary to remove or avoid the inaccurate contribution from the use of the grid. This is a general difficulty with grid based potential calculation methods and a variety of correction methods have been proposed [20], 28] 31] the details of which usually depend on the problem being solved, the interpolation scheme, and the nature of the grid solver. Because our algorithm works directly with the Green function, and because the iterative solver requires that many (a) b) Fig. 6. a) A sphere discretized ....

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles. New York: Adam Hilger, 1988.

....and not an explicit representation of the matrix. For example, note that for P in (4) computing Pq is equivalent to com puting n potentials due to n sources. The potentials can be computed approximately in nearly order n operations by using an implicitly defined sparse representation of P [7, 8, 9]. Several researchers simultaneously observed the power of combining discretized integral formulations, Krylov subspace methods, and implicit sparsiftcation [10, 11] Early generaI 3 D codes using sparsification were applied to capacitance extraction, and were based on the fast multipole ....

....efficiency for high accuracy [15, 16, 17] In addition, much recent work has focussed on allowing for more general Greens functions like those associated with layered me dia. There is the panel clustering idea [11] a multigrid style method [18] projection to a uniform grid combined with the FFT [9, 19] a technique based on the singularvalue decomposition [4] and approaches based on using wavelet like methods [20, 21, 22] 4 Model Order reduction The now standard approach to efficient circuitinterconnect simulation is to represent the interconnect with moment matching based reduced order ....

R. W. Hockney and J. W. Eastwood, Computer simulation using particles. New York: Adam Hilger, 1988.

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R.W. Hockney and J.W. Eastwood. Computer Simulation Using Particles. McGraw-Hill, New York, 1981.

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R.W. Hockney and J.W. Eastwood. Computer Simulation Using Particles. Institut of Physics Publishing, Bristol, 1992.

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R. W. Hackney and J. W. Eastwood. Computer Simulation Using Particles. McGraw Hill, 1981.

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R.W. Hockney and J.W. Eastwood. Computer Simulation Using Particles. McGraw Hill, New York, 1981.

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R.W. Hockney and J.W. Eastwood. Computer Simulation Using Particles. McGraw Hill, 1981. 16

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R. W. Hockney and J. W. Eastwood. Computer Simulation Using Particles. McGraw-Hill, New York, 1981.

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R. W. Hockney & J. W. Eastwood, Computer Simulation Using Particles, Adam Hilger, Bristol, New York, 1988.

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Hockney, R.W., Eastwood, J.W.: Computer Simulation Using Particles. McGrawHill, New York (1981)

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R. W. Hockney, J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York, 1981.

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R.W. Hockney and J.W. Eastwood. Computer Simulations Using Particles. Institute of Physics Publisher, Bristol and Philadelphia, 1988.

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Hockney, R.W., Eastwood, J.W., "Computer Simulation Using Particles", IOP Publishing, Bristol, 1988.

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R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles. McGraw-Hill, NY, (1981).

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R. W. Hockney, J. W. Eastwood, Computer Simulation using Particles, Adam Hilger.

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R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles. Taylor & Francis, 1988.

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R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles. (IOP, London, 1988).

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R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles, Adam Hilger, New York, 1988.

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