Three parallel algorithms for classical molecular dynamics are presented. The first assigns each processor a fixed subset of atoms; the second assigns each a fixed subset of inter-atomic forces to compute; the third assigns each a fixed spatial region. The algorithms are suitable for molecular dynamics models which can be difficult to parallelize efficiently—those with short-range forces where the neighbors of each atom change rapidly. They can be implemented on any distributed-memory parallel machine which allows for message-passing of data between independently executing processors. The algorithms are tested on a standard Lennard-Jones benchmark problem for system sizes ranging from 500 to 100,000,000 atoms on several parallel supercomputers--the nCUBE 2, Intel iPSC/860 and Paragon, and Cray T3D. Comparing the results to the fastest reported vectorized Cray Y-MP and C90 algorithm shows that the current generation of parallel machines is competitive with conventional vector supercomputers even for small problems. For large problems, the spatial algorithm achieves parallel efficiencies of 90% and a 1840-node Intel Paragon performs up to 165 faster than a single Cray C9O processor. Trade-offs between the three algorithms and guidelines for adapting them to more complex molecular dynamics simulations are also discussed.

Fast parallel algorithms for short-range molecular dynamics

Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and numerical techniques for deriving SPH formulations for partial differential equations such as the Navier-Stokes (N-S) equations, (3) the role of the smoothing kernel functions and a general approach to construct smoothing kernel functions, (4) kernel and particle consistency for the SPH method, and approaches for restoring particle consistency, (5) several important numerical aspects, and (6) some recent applications of SPH. The paper ends with some concluding remarks.

Smoothed Particle Hydrodynamics (SPH): an Overview and Recent Developments

An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non-smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi-field problems. Copyright © 2010 John Wiley & Sons, Ltd.

The extended/generalized finite element method: An overview of the method and its applications

We present two new Lagrangian methods for hydrodynamics, in a systematic comparison with moving-mesh, smoothed particle hydrodynamics (SPH), and stationary (non-moving) grid methods. The new methods are designed to simultaneously capture advantages of both SPH and grid-based/adaptive mesh refinement (AMR) schemes. They are based on a kernel discretization of the volume coupled to a high-order matrix gradient estimator and a Riemann solver acting over the volume ‘overlap’. We implement and test a parallel, second-order version of the method with self-gravity and cosmological integration, in the code gizmo:1 this maintains exact mass, energy and momentum conservation; exhibits superior angular momentum conservation compared to all other methods we study; does not require ‘artificial diffusion’ terms; and allows the fluid elements to move with the flow, so resolution is automatically adaptive. We consider a large suite of test problems, and find that on all problems the new methods appear competitive with moving-mesh schemes, with some advantages (particularly in angular momentum conservation), at the cost of enhanced noise. The new methods have many advantages versus SPH: proper convergence, good capturing of fluid-mixing instabilities, dramatically reduced ‘particle noise’ and numerical viscosity, more accurate sub-sonic flow evolution, and sharp shock-capturing. Advantages versus non-moving meshes include: automatic adaptivity, dramatically reduced advection errors and numerical overmixing, velocity-independent errors, accurate coupling to gravity, good angular momentum conservation and elimination of ‘grid alignment’ effects. We can, for example, follow hundreds of orbits of gaseous discs, while AMR and SPH methods break down in a few orbits. However, fixed meshes minimize ‘grid noise’. These differences are important for a range of astrophysical problems.

A new class of accurate, mesh-free hydrodynamic simulation methods

We present a new shape representation, the multi-level partition of unity implicit surface, that allows us to construct surface models from very large sets of points. There are three key ingredients to our approach: 1) piecewise quadratic functions that capture the local shape of the surface, 2) weighting functions (the partitions of unity) that blend together these local shape functions, and 3) an octree subdivision method that adapts to variations in the complexity of the local shape.Our approach gives us considerable flexibility in the choice of local shape functions, and in particular we can accurately represent sharp features such as edges and corners by selecting appropriate shape functions. An error-controlled subdivision leads to an adaptive approximation whose time and memory consumption depends on the required accuracy. Due to the separation of local approximation and local blending, the representation is not global and can be created and evaluated rapidly. Because our surfaces are described using implicit functions, operations such as shape blending, offsets, deformations and CSG are simple to perform.

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Multi-level partition of unity implicits

In this paper, we present a meshless discretization technique for instationary convection-diffusion problems. It is based on operator splitting, the method of characteristics, and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h-version or a p-version. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convection-diffusion, instationary diffusion, linear advection, and elliptic problems.

A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs

ESPResSo 3.1 - Molecular Dynamics Software for Coarse-Grained Models: A. Arnold, O. Lenz, S. Kesselheim, R. Weeber, F. Fahrenberger, D. Roehm, P. Kosovan, C. Holm.- On the Rate of Convergence of the Hamiltonian Particle-Mesh Method Onno Bokhove, Vladimir Molchanov, Marcel Oliver, Bob Peeters.- Peridynamics: A Nonlocal Continuum Theory.- Etienne Emmrich, Richard B. Lehoucq, Dimitri Puhst.- Immersed Molecular Electrokinetic Finite Element Method for Nano-Devices in Biotechnology and Gene Delivery: Wing Kam Liu, Adrian M. Kopacz, Tae-Rin Lee, Hansung Kim, Paolo Decuzzi.- Corrected Stabilized Non-conforming Nodal Integration in Meshfree Methods: Marcus Ruter, Michael Hillman, Jiun-Shyan Chen.- Multilevel Partition of Unity Method for Elliptic Problems with strongly Discontinuous Coefficients: Marc Alexander Schweitzer.- HOLMES: Convergent Meshfree Approximation Schemes of Arbitrary Order and Smoothness: Agustin Bompadre, Luigi E. Perotti, Christian J. Cyron, Michael Ortiz.- A Meshfree Splitting Method for Soliton Dynamics in Nonlinear Schrodinger Equations: Marco Caliari, Alexander Ostermann, Stefan Rainer.- A Meshless Discretization Method for Markov State Models Applied to Explicit Water Peptide Folding Simulations: Konstantin Fackeldey, Alexander Bujotzek, Marcus Weber.- Kernel-based Collocation Methods versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations: Gregory E. Fasshauer, Qi Ye.- A Meshfree Method for the Analysis of Planar Flows of Inviscid Fluids: Vasily N. Govorukhin.- Some Regularized Versions of the Method of Fundamental Solutions: Csaba Gaspar.- A Characteristic Particle Method for Traffic Flow Simulations on Highway Networks: Yossi Farjoun, Benjamin Seibold.- Meshfree Modeling in Laminated Composites: Daniel C. Simkins, Jr., Nathan Collier, Joseph B. Alford

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Meshfree methods for partial differential equations

In this sequel to [12, 13, 14, 15] we focus on the implementation of Dirichlet boundary conditions in our partition of unity method. The treatment of essential boundary conditions with meshfree Galerkin methods is not an easy task due to the non-interpolatory character of the shape functions. Here, the use of an almost forgotten method due to Nitsche from the 1970’s allows us to overcome these problems at virtually no extra computational costs. The method is applicable to general point distributions and leads to positive definite linear systems. The results of our numerical experiments, where we consider discretizations with several million degrees of freedom in two and three dimensions, clearly show that we achieve the optimal convergence rates for regular and singular solutions with the (adaptive) h-version and (augmented) p-version.

A Particle-Partition of Unity Method Part V: Boundary Conditions

In this paper we present a meshfree discretization technique based only on a set of irregularly spaced points $x_i \in \mathbb R^d$ and the partition of unity approach. In this sequel to [M. Griebel and M. A. Schweitzer, SIAM J. Sci. Comput., 22 (2000), pp. 853--890] we focus on the cover construction and its interplay with the integration problem arising in a Galerkin discretization. We present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme. Here, we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency. The use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly. Now the overall computational costs are dominated by the number of the integration cells. We present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency. In fact, the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method, yet the presented method is completely independent of a mesh. Moreover, our method is applicable to general domains and allows for the construction of approximations of any order and regularity.

A Particle-Partition of Unity Method--Part II: Efficient Cover Construction and Reliable Integration

We present an algebraic multigrid (AMG) method for the efficient solution of linear block-systems stemming from a discretization of a system of partial differential equations (PDEs). It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise fashion. We apply this approach to linear elasticity and show that the block interpolation, described in this paper, reproduces the rigid body modes, i.e., the kernel elements of the discrete linear elasticity operator. It is well known from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain convergence rates which are independent of the problem size. We furthermore present results of various numerical experiments in two and three dimensions. They confirm that the method is robust with respect to variations of the Poisson ratio $\nu$. We obtain rates $\rho<0.4$ for $\nu<0.4$. These measured rates clearly show that the method provides fast convergence for a large variety of discretized elasticity problems.

Marc Alexander Schweitzer

Papers

A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs

Meshfree methods for partial differential equations

A Particle-Partition of Unity Method Part V: Boundary Conditions

A Particle-Partition of Unity Method--Part II: Efficient Cover Construction and Reliable Integration

An Algebraic Multigrid Method for Linear Elasticity