High-order Integration Schemes Research Articles

Multiscale Analysis of Sandwich Beams with Polyurethane Foam Core: A Comparative Study of Finite Element Methods and Radial Point Interpolation Method.

This study presents a comprehensive multiscale analysis of sandwich beams with a polyurethane foam (PUF) core, delivering a numerical comparison between finite element methods (FEMs) and a meshless method: the radial point interpolation method (RPIM). This work aims to combine RPIM with homogenisation techniques for multiscale analysis, being divided in two phases. In the first phase, bulk PUF material was modified by incorporating circular holes to create PUFs with varying volume fractions. Then, using a homogenisation technique coupled with FEM and four versions of RPIM, the homogenised mechanical properties of distinct PUF with different volume fractions were determined. It was observed that RPIM formulations, with higher-order integration schemes, are capable of approximating the solution and field smoothness of high-order FEM formulations. However, seeking a comparable field smoothness represents prohibitive computational costs for RPIM formulations. In a second phase, the obtained homogenised mechanical properties were applied to large-scale sandwich beam problems with homogeneous and approximately functionally graded cores, showing RPIM's capability to closely approximate FEM results. The analysis of stress distributions along the thickness of the beam highlighted RPIM's tendency to yield lower stress values near domain edges, albeit with convergence towards agreement among different formulations. It was found that RPIM formulations with lower nodal connectivity are very efficient, balancing computational cost and accuracy. Overall, this study shows RPIM's viability as an alternative to FEM for addressing practical elasticity applications.

The equivalent source method is an indirect boundary element method with an implicit Voronoi mesh

The Equivalent Source Method (ESM) consists of modeling the acoustical field radiated or scattered from an object by a set of radiating monopole sources located below its surface. While highly efficient for modeling the scattered field by a smooth object, ESM becomes less effective in representing acoustic diffraction from sharp edges. In this article, ESM is understood as an approximation to the single layer integral equation. In this framework, ESM belongs to the family of Indirect Boundary Element Methods (BEM) with an implicit surface mesh slightly below the object surface, and a zero order integration scheme on each element of the mesh. This implicit mesh is well approximated by the Voronoi diagram associated to the source distribution. This formalism establishes a strong link between these two widely used methods (ESM and BEM) and allows the construction of hybrid methods that can combine performances of both methods. On the first hand, a new hybrid method able to model edge diffraction while remaining as computationally efficient as ESM is developed. In this method, labeled “BEM Voronoi”, equivalent sources are positioned on the surface’s object rather than below it and singularities are handled with the integral formulation. On the other hand, a set of new hybrid methods which combines ESM with higher order integration scheme is constructed. Performances of the new methods developed in this paper are numerically compared to each other and to Finite Element simulations (used as a reference) for modeling scattering from an object excited by a plane wave. Each hybrid model is tested on multiple configurations involving two sets of boundary conditions (rigid and elastic objects respectively) successively immersed in two different external environments (water and air) and for both smooth and a sharp-edged objects. The results show that the hybrid method “BEM Voronoi” better handles edge diffraction than conventional ESM, at a much lower computational cost (implementation of BEM Voronoi is 50 times faster than conventional BEM implementation).

Propagators for molecular dynamics in a magnetic field

ABSTRACT Ab initio molecular dynamics in a magnetic field requires solving equations of motion with velocity-dependent forces – namely, the Lorentz force arising from the nuclear charges moving in a magnetic field and the Berry force arising from the shielding of these charges from the magnetic field by the surrounding electrons. In this work, we revisit two existing propagators for these equations of motion, the auxiliary-coordinates-and-momenta (ACM) propagator and the Tajima propagator (TAJ), and compare them with a new exponential (EXP) propagator based on the Magnus expansion. Additionally, we explore limits (for example, the zero-shielding limit), the implementation of higher-order integration schemes, and series truncation to reduce computational cost by carrying out simulations of a HeH + model system for a wide range of field strengths. While being as efficient as the TAJ propagator, the EXP propagator is the only propagator that converges to both the schemes of Spreiter and Walter (derived for systems without shielding of the Lorentz force) and to the exact cyclotronic motion of a charged particle. Since it also performs best in our model simulations, we conclude that the EXP propagator is the recommended propagator for molecules in magnetic fields.

Verification and accuracy check of simulations with PoPe and iPoPe

The theoretical background of the PoPe and iPoPe verification scheme is presented. Verification is performed using the simulation output of production runs. The computing overhead is estimated to be at most 10%. PoPe or iPoPe calculations can be done offline provided the necessary data is stored, for example additional time slices, or online where iPoPe is more effective. The computing overhead is mostly that of storing the necessary data. The numerical error is determined and split into a part proportional to the operators, which are combined to form the equations to be solved, thus modifying their control parameters, completed by a residual error orthogonal to these operators. The accuracy of the numerical solution is determined by this modification of the control parameters. The PoPe and iPoPe methods are illustrated in this paper with simulations of a simple mechanical system with chaotic trajectories evolving into a strange attractor with sensitivity to initial conditions. We show that the accuracy depends on the particular simulation both because the properties of the numerical solution depend on the values of the control parameter, and because the target accuracy will depend on the problem that is addressed. One shows that for a case close to bifurcations between different states, the accuracy is determined by the level of detail of the bifurcation phenomena one aims at describing. A unique verification index, the PoPe index, is proposed to characterise the accuracy, and consequently the verification, of each production run. The PoPe output allows one to step beyond verification and analyse for example the numerical scheme efficiency. For the chosen example at fixed PoPe index, therefore at fixed numerical error, one finds that the higher order integration scheme, comparing order 4 to order 2 Runge-Kutta time stepping, reduces the computation cost by a factor 4.

Hybrid symplectic integrators for planetary dynamics

Abstract Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high-order integration scheme for the duration of the encounter while otherwise using a standard second-order Wisdom–Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high-order integrator. We show that the original motivation for choosing a polynomial switching function in MERCURY is not correct. We explain the nevertheless excellent performance of the MERCURY integrator and then explore a wide range of different switching functions including an infinitely differentiable function and a Heaviside function. We find that using a Heaviside function leads to a significantly simpler scheme compared to MERCURY , while maintaining the same accuracy in short-term simulations.

Stable finite element analysis of viscous dusty plasma

PurposeThe purpose of this paper is to provide an analysis of dust acoustic (solitary) waves including viscosity. Specifically, the authors consider a dusty unmagnetized plasma system consisting of negatively charged dust and Boltzmann electrons and ions.Design/methodology/approachIn this paper, a Petrov–Galerkin weak form with upwinding is adopted. Nonlinearity of ion and electron number density in terms of an electrostatic potential is included. A fully implicit time integration is used (backward Euler method), which requires the first derivative of the weak form. A three-field formulation is proposed, with the dust number density, the electrostatic potential and the dust velocity being the unknown fields.FindingsIn this study, two numerical examples are introduced and results show great promise for the proposed formulation as a predictive tool in viscous dusty plasmas. Presence of solitary waves was demonstrated. Dusty plasma vortices are predicted in 2D and 3D, as mentioned in the specialized literature.Research limitations/implicationsWe observed some dependence on step size, which is due to the simple time-stepping scheme. This can be solved with a higher order integration scheme, which implies an added cost to the solution.Practical implicationsDusty plasmas are found in astrophysics (Saturn rings) and electronics industry at several scales and have high impact as a contaminant.Originality/valueTo the authors’ knowledge, this is the first paper with a simulation of dusty plasma including vortices.

High-order integration scheme for relativistic charged particle motion in magnetized plasmas with volume preserving properties

A numerical algorithm for solving full gyro orbit of relativistic charged particle motion in magnetized plasmas is presented. The algorithm developed here achieves the following features simultaneously. (1) The time advancement is explicit, and (2) the integration is performed with respect to the observation time in a laboratory frame. (3) It is suitable for accurate long time integration with its volume preserving properties in the phase space. (4) The algorithm can properly treat the E×B drift velocity in electromagnetic fields for large relativistic factors, and (5) can be extended to arbitrary high orders with the aid of symmetric composition methods. Because our algorithm is formulated in the Lorentz covariant form, explicit conservation of the Minkowski norm is not assumed. Nevertheless, no secular growth of the numerical errors in the norm does occur, and its influence can be minimized up to the levels of round-off errors when a high-order scheme is applied. Numerical results are compared with explicit and implicit Runge–Kutta methods. The numerical accuracy and the computational efficiency are discussed for long time integration in toroidal magnetic field configuration.

A new mathematical model for studying fully ionized plasma flows in MPD thrusters

The paper describes a novel mathematical model to study the physics of fully ionized plasma flow through MPD thrusters otherwise known as magnetoplasmadynamic thrusters. The developed model consists of a set of differential equations obtained by coupling the Navier–Stokes equations with Maxwell’s equations. The model developed was tested for various 1D and 1.5D cases. These simulations have been carried out for a special case of geometries called Tikhonov fitted geometries for alleviating the instability phenomenon commonly known as onset phenomenon in self-field MPD thrusters. The work was also extended to applied field MPD thrusters. A new equation for magnetic field was introduced to incorporate a virtual cathode. All the computations were carried out using a higher order integration scheme. The results obtained were found to be in good agreement with the previous work done on similar models.

An accurate time integration scheme for arbitrary rotation motion: application to metal forming simulation

Within the frame of implicit velocity based formulations with solid elements, usual time integration schemes often turn out unsatisfactory when the movement has large rotations, especially in metal forming applications such as ring rolling or cross-wedge rolling. These rotations generally require using a much higher order integration scheme with inherent difficulties in implementing such schemes. For pure rotation motions, it is possible to use a low order integration scheme by rewriting the motion equations in the cylindrical frame that is supported by the rotation axis. Accordingly, a first order scheme is sufficient to accurately integrate the movement but it is restricted to specific problems. In the more general case, it is possible to derive parts of the domain where rotations are predominant along with the governing rotation axis from the velocity field gradient. The motion equations are then rewritten in the resulting local cylindrical frame. Performances of this first order scheme are first evaluated and highlighted over simple analytical problems, before being applied to the finite element simulation of the torsion test, and then to more complex metal forming problems involving large rotations. The accuracy and efficiency of this scheme is so numerically demonstrated.

On the optimal numerical time integration for DEM using Hertzian force models

The numerical accuracy of a selection of different time integration techniques used to solve particle motion is investigated using a normal collision employing the non-linear Hertzian contact force. The findings are compared against the linear force model where it has been found that the expected order of accuracy of higher-order integration schemes is not realised (Tuley et al., 2010). The proposed mechanism for this limitation has been cited as the errors in integration which occur across the force profile discontinuity. By investigating the characteristics of both the non-linear elastic and the non-linear damped Hertzian contact models, it has been found that higher orders of accuracy are recoverable and depends on the degree of the governing non-linear equation. The numerical errors of the linear and non-linear force models are however markedly different in character.

Diffusion tensor imaging (DT I) and Transcranial magnetic stimulation (TMS): an integrated approach

In present study we adopted an innovative approach, integrating the “Diffusion Tensor Imaging tractography (DTI)” technique with navigated transcranial magnetic stimulation (nTMS). In this way it is possible to improve accuracy of motor fiber reconstruction by positioning the fiducial seeding from DTI reconstruction, over the motor areas localized by nTMS. This will allow a direct comparison between the density of the motor map obtained with nTMS with the related axonal density obtained with DTI. DTI uses anisotropic water diffusion like test to study in vivo white matter anatomy in order to reconstruct tri-dimensional fiber bundles improving the practical 1 mm. MRI scanning resolution. Therefore we used the appropriate acquisition and reconstruction DTI techniques in order to study local cerebral pathways. Related acquisition must be done following resonance protocol and varying the magnetic gradient, in order to have the following steps: DWI acquisition; tensor calculation; scalar maps; 3d visualization; fiducial seeding. DTI technique can exploit two different classes of fiber tracking algorithms: “deterministic” or “probabilistic” tracking higherorder integration schemes. The deterministic one, allows to calculate the directions of streamline propagation, and reveals only, presence or absence of a connection. On the other hand a probabilistic tracking approach is achieved: starting from a basic point, fibers are propagated multiple times through the tensor field while varying, in a stochastic way, the estimated fiber orientation along the traversed voxels. In addition nTMS is a newly evolving technique for in vivo investigating human motor system combining spatial information from high-resolution MRI with the functionality of non-invasive cortical stimulation. In this way it is possible to target motor areas more precisely obtaining a discrete motor maps of facial, hand and leg muscles. This innovative approach will indeed provide a new tool in neuroscience for an in vivo direct correlation between anatomical circuits and electrophysiological data.

Time-Step Considerations in Particle Simulation Algorithms for Coulomb Collisions in Plasmas

The accuracy of first-order Euler and higher-order time-integration algorithms for grid-based Langevin equations collision models in a specific relaxation test problem is assessed. We show that statistical noise errors can overshadow time-step errors and argue that statistical noise errors can be conflated with time-step effects. Using a higher-order integration scheme may not achieve any benefit in accuracy for examples of practical interest. We also investigate the collisional relaxation of an initial electron-ion relative drift and the collisional relaxation to a resistive steady-state in which a quasi-steady current is driven by a constant applied electric field, as functions of the time step used to resolve the collision processes using binary and grid-based, test-particle Langevin equations models. We compare results from two grid-based Langevin equations collision algorithms to results from a binary collision algorithm for modeling electron-ion collisions. Some guidance is provided on how large a time step can be used compared to the inverse of the characteristic collision frequency for specific relaxation processes.

A Linear Surface Characteristics Approximation for Neutron Transport in Unstructured Meshes

The method of characteristics (MOC) in unstructured meshes has become a standard for reactor physics applications. One of the major drawbacks of the MOC is the difficulty to implement higher-order integration schemes to improve spatial convergence. In this paper we present a high-order MOC spatial discretization that uses linear interpolation on surface values for the collision source. This conservative linear surface (CLS) scheme exhibits parabolic convergence with the mesh size but lacks positivity. Numerical results for the well-known Stepanek benchmark and for more realistic boiling water reactor assemblies show CLS faster convergence over the standard step characteristics scheme. A generalization of the synthetic DPN acceleration scheme provides an efficient method to accelerate the internal transport iterations.

High order integration schemes on the unit sphere

A method for refining high order numerical integration schemes is described. Particular focus is on integration schemes over the unit sphere with octahedral symmetry. The method is powerful enough that new integration schemes can be found from rough intuitive guesses. New schemes up to order 59 are presented. © 1996 by John Wiley & Sons, Inc.

High order integration schemes on the unit sphere

A method for refining high order numerical integration schemes is described. Particular focus is on integration schemes over the unit sphere with octahedral symmetry. The method is powerful enough that new integration schemes can be found from rough intuitive guesses. New schemes up to order 59 are presented. © 1996 by John Wiley & Sons, Inc.

An O(N log N) integration scheme for collisional stellar systems

view Abstract Citations (59) References (32) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS An O(N N) Integration Scheme for Collisional Stellar Systems McMillan, Stephen L. W. ; Aarseth, Sverre J. Abstract A novel tree-based, high-order integration scheme for the numerical simulation of collisional N-body systems is presented. In the present implementation the tree is allowed to deform with time, and the properties of each component cell are not calculated afresh at each step but instead are predicted only as needed. Cells are represented by a multipole expansion that can extend as far as the octupole term, depending on the relative costs of the multipole and individual-particle calculations. A block time step algorithm is used to simplify scheduling and to allow full vectorization of the code. The integrator conserves energy to a few parts per million per crossing time, and its measured O(N exp 1.5) scaling suggests that it will become competitive with NBODY5, the 'standard' high-precision integrator for small systems, for N greater than about 10,000. It is concluded that tree-based schemes can be confidently applied to collisional problems without excessive concern that the tree algorithm will seriously disturb the relaxation processes that drive the evolution. Publication: The Astrophysical Journal Pub Date: September 1993 DOI: 10.1086/173068 Bibcode: 1993ApJ...414..200M Keywords: Celestial Mechanics; Collisions; Galactic Clusters; Globular Clusters; Star Clusters; Stellar Systems; Algorithms; Cosmology; Digital Simulation; Numerical Integration; Astrophysics; CELESTIAL MECHANICS; STELLAR DYNAMICS; METHODS: NUMERICAL full text sources ADS |

On the dynamics of light quarks in QCD

We describe recent results concerning the behavior of lattice QCD with light dynamical Wilson and Staggered quarks. We show that it is possible to reach regions of parameter space with light pions m π ≈ 0.2/ a using Wilson fermions. If the Hybrid Molecular Dynamics (HMD) algorithm is used with the same parameters it gives incorrect results. We also present preliminary results using a higher-order integration scheme.

Integration errors in a three-dimensional model for canopy albedos

A three-dimensional model of the radiative transfer within a vegetation canopy gives albedo and bidirectional reflectance for climate and remote-sensing applications. Major sources of numerical error in this model are evaluated in the single-scattering limit by comparison with an analytic solution, and procedures for the amelioration of these errors are developed. The model consists of a three-dimensional array of cells and treats multiple scattering by individual cells through integration over discrete angular sections and iteration of the scattering. Two distinct sources of error are identified: (i) errors in bidirectional reflectance from the integration of absorption and emission along the path of the radiation, and (ii) errors in integrating the bidirectional reflectances over solid angle to synthesize the albedo. Path integration errors may result from neglect of self-absorption by individual cells upon emission, inclusion of insufficient number of cell interactions, and inclusion of too few cells, i.e., unless cells near the outside of the canopy have leaf area index LAI ⪡ 1, the implied gradients of light attenuation are too sharp to be resolved. These errors are minimized by including a self-absorption term, ensuring that the number of interactions are sufficient, selecting enough vertical layers, and providing most resolution in LAI near the outside of the canopy. The spherical integration error results from the coarseness and arrangement of angular sectors and from the procedure used to sum their contributions. Error may be reduced by adding more spherical resolution and by changing the angular integration to one with equal increments in the cosine of the zenith angle. Higher-order integration schemes may also be helpful.

Evaluation of the computational errors of strapdown navigation systems.

The results of an evaluation of the computational errors for conventional, restrained, strapdown sensor mechanizations are presented. The analytic models of the computational errors (truncation, word length, integration scheme, integration interval, and sensor output resolution) presented were developed through detailed digital simulation of the navigation sensors and different data processing schemes. (Both the processes of computing vehicle attitude and transforming the accelerometer data from the vehicle to the inertial frame are considered.) These error models demonstrate the techniques by which computational processes can be properly designed for different applications and the magnitude of the computational errors controlled. Test data obtained during the centrifuge testing of a strapdown system are also presented, and demonstrate good agreement with the attitude computational error models. Of additional significance are the development and presentation herein of a gyro data processing technique that eliminates the commutativity error and permits the use of higher-order integration schemes for determining attitude; a fourth-order scheme is presented as an example.