Momentum Preserving Research Articles

Variational integrators for mechanical control systems with symmetries

Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems. &nbsp An interesting family of geometric integrators can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators, being one of their main properties the preservation of geometric features as the symplecticity, momentum preservation and good behavior of the energy. We construct variational integrators for higher-order mechanical systems on trivial principal bundles and their extension for higher-order constrained systems, paying particular attention to the case of underactuated mechanical systems.

Spectral variational integrators

In this paper, we present a new variational integrator for problems in Lagrangian mechanics. Using techniques from Galerkin variational integrators, we construct a scheme for numerical integration that converges geometrically, and is symplectic and momentum preserving. Furthermore, we prove that under appropriate assumptions, variational integrators constructed using Galerkin techniques will yield numerical methods that are arbitrarily high-order. In particular, if the quadrature formula used is sufficiently accurate, then the resulting Galerkin variational integrator has a rate of convergence at the discrete time-steps that is bounded below by the approximation order of the finite-dimensional function space. In addition, we show that the continuous approximating curve that arises from the Galerkin construction converges on the interior of the time-step at half the convergence rate of the solution at the discrete time-steps. We further prove that certain geometric invariants also converge with high-order, and that the error associated with these geometric invariants is independent of the number of steps taken. We close with several numerical examples that demonstrate the predicted rates of convergence.

Longer time steps in explicit dynamics subject to small nonlinearities

AbstractAn approach that increases the critical time step in explicit time integration is presented. By derivation as a variational integrator, symplecticity and momentum preservation are ensured. The linear response is assumed to be dominant and is integrated implicitly, while the nonlinear forces are explicitly integrated. MOLLY filters the high‐frequent portions of the nonlinear forces and increases the critical time step. An SDOF example and a structural example using modal reduction verify the improved stability. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational multirate integration of constrained dynamics

AbstractMechanical systems with dynamics on varying time scales, in particular those including highly oscillatory motion, impose challenging questions for numerical integration schemes. Tiny step sizes are required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a larger time step is accurate enough. Small time steps increase integration times unnecessarily, especially for costly function evaluations. For systems comprising fast and slow dynamics, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. Main challenges are the identification of fast and slow parts (e.g. by separating the energy or by distinguishing sets of variables), the synchronisation of their dynamics and in particular the treatment of mixed parts that often appear when fast and slow dynamics are coupled by constraints. In this contribution, a multirate integrator is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Variational integrators (based on a discrete version of Hamilton's principle) lead to symplectic and momentum preserving integration schemes that also exhibit good energy behavior. The resulting multirate variational integrator has the same preservation properties. An example demonstrates the performance of the multirate integrator for constrained multibody dynamics. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Extension of fractional step techniques for incompressible flows: The preconditioned Orthomin(1) for the pressure Schur complement

The objective of this paper is to present different fractional step schemes in the algebraic context to solve the incompressible Navier–Stokes equations, test them and pick the best one in terms of efficiency and robustness. The equivalence between fractional step schemes and iterative methods for the pressure Schur complement system has been well established in the literature. For example, the classical incremental projection scheme can be associated with a Richardson iteration for the pressure Schur complement plus a correction to enforce the mass conservation. We introduce in this paper an Orthomin(1) iteration which minimizes the Schur complement residual at each solver iteration by using, in the updating step, a factor dynamically computed. Two versions are considered, namely the momentum preserving and continuity preserving versions. The method is compared to the classical Richardson method, including the continuity and momentum preserving versions. In addition, two Schur complement preconditioners are considered and compared, based on the approximation of the weak Uzawa operator. From the implementation point of view, the benefit of the method is two fold. On the one hand, it can be easily implemented starting from the global matrix of the monolithic scheme, without changing the assembly. On the other hand, it enables the use of simple algebraic solvers without the need for complex preconditioners; this is a requirement for massively parallel computers. The four methods are finally tested and compared through the solution of numerical examples. The main conclusion is that with very few additional computation, the Orthomin(1) iteration largely improves the global convergence properties of the fractional schemes here presented.

Multisymplectic box schemes for the complex modified Korteweg–de Vries equation

In this paper, two multisymplectic integrators, an eight-point Preissman box scheme and a narrow box scheme, are considered for numerical integration of the complex modified Korteweg–de Vries equation. Energy and momentum preservation of both schemes and their dispersive properties are investigated. The performance of both methods is demonstrated through numerical tests on several solitary wave solutions.

A homogeneous nonequilibrium molecular dynamics method for calculating the heat transport coefficient of mixtures and alloys

This work generalizes Evans' homogeneous nonequilibrium method for estimating heat transport coefficient to multispecies molecular systems described by general multibody potentials. The proposed method, in addition to being compatible with periodic boundary conditions, is shown to satisfy all the requirements of Evans' original method, namely, adiabatic incompressibility of phase space, equivalence of the dissipative and heat fluxes, and momentum preservation. The difference between the new equations of motion, suitable for mixtures and alloys, and those of Evans' original work are quantified by means of simulations for fluid Ar-Kr and solid GaN test systems.

Dynamic materials with formation of clots: optimal mass transport in one spatial dimension without takeover

AbstractThe concept of dynamic materials (DM) was introduced in the earlier work [1, 2, 9] and developed there for the case of non‐colliding characteristics. The present paper extends this concept toward the case when the characteristics collide. The extension is formulated for a linear continuity equation in 1D with a no takeover condition applied to the moving masses. This implies the formation of clots with the preservation of both mass and momentum, which makes the approach resemble the method of “sticky particles” known in pressureless gas dynamics. The extension introduces non‐linearity into the transportation problem that was originally missing, thus placing it closer to a non‐linear conservation law.

Modification of Weakly Compressible Smoothed Particle Hydrodynamics for Preservation of Angular Momentum in Simulation of Impulsive Wave Problems

In this work, Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) is applied for numerical simulation of impulsive wave. Properties of linear and angular momentum in WCSPH formulation are studied. Kernel gradient of viscous term in momentum equation is corrected to ensure preservation of angular momentum. Corrected WCSPH method is used to simulate solitary Scott Russell wave and applied to simulate impulsive wave generated by two-dimensional under water landslide. In each of the test cases, results of corrected WCSPH are compared with experimental results. The results of the numerical simulations and experimental works are matched and a satisfactory agreement is observed. Furthermore, vorticity contours computed by the corrected WCSPH are compared with uncorrected WCSPH so that the effect of corrective term on preservation of angular momentum is illustrated. Numerical model is also applied for simulation of water entry of half buoyant circular cylinder into initially calm water. Comparison between experimental and computational results proves applicability of the WCSPH method for simulation of these kinds of problems.

Generalization of the homogeneous nonequilibrium molecular dynamics method for calculating thermal conductivity to multibody potentials

This work provides a generalization of Evans' homogeneous nonequilibrium method for estimating thermal conductivity to molecular systems that are described by general multibody potentials. A perturbed form of the usual Nose-Hoover equations of motion is formally constructed and is shown to satisfy the requirements of Evans' original method. These include adiabatic incompressibility of phase space, equivalence of the dissipative and heat fluxes, and momentum preservation.

Hamiltonian particle‐mesh simulations for a non‐hydrostatic vertical slice model

AbstractA Lagrangian particle method is developed for the simulation of atmospheric flows in a non‐hydrostatic vertical slice model. The proposed particle method is an extension of the Hamiltonian particle mesh (HPM) [Frank J, Gottwald G, Reich S. 2002. The Hamiltonian particle‐mesh method. In Meshfree Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Vol. 26, Griebel M, Schweitzer M (eds). Springer‐Verlag: Berlin Heidelberg; 131–142] and provides preservation of mass, momentum, and energy. We tested the method for the gravity wave test in Skamarock W, Klemp J. 1994. Efficiency and accuracy of the Klemp‐Wilhelmson time‐splitting technique. Monthly Weather Review 122: 2623–2630 and the bubble experiments in Robert A. 1993. Bubble convection experiments with a semi‐implicit formulation of the Euler equations. Journal of the Atmospheric Sciences 50: 1865–1873. The accuracy of the solutions from the HPM simulation is comparable to those reported in these references. A particularly appealing aspect of the method is in its non‐diffusive transport of potential temperature. The solutions are maintained smooth largely due to a ‘regularization’ of pressure, which is controlled carefully to preserve the total energy and the time‐reversibility of the model. In case of the bubble experiments, one also needs to regularize the buoyancy contributions. The simulations demonstrate that particle methods are potentially applicable to non‐hydrostatic atmospheric flow regimes and that they lead to a highly accurate transport of materially conserved quantities such as potential temperature under adiabatic flow regimes. Copyright © 2009 Royal Meteorological Society

An extended multiscale principle of virtual velocities approach for evolving microstructure

A hierarchical multiscale approach is presented for modeling microstructure evolution in heterogeneous materials. Preservation of momentum across each scale transition is incorporated through the application of the principle of virtual velocities at the fine scale giving rise to the appropriate continuum momentum balance equations at the coarse scale. In addition to satisfying momentum balance and invariance of momentum among scales, invariance of elastic free energy, stored free energy, and dissipation between two scales of observation is regarded as crucial to the physics of each scale transition. The preservation of this energy partitioning scheme is obtained through construction of constitutive relations within the framework of internal state variable theory. Internal state variables that are directly computed from the fine scale response are introduced to augment the state equations and describe the inelastic energy storage and dissipation within the fine scale. By virtue of a second gradient kinematic decomposition, the framework naturally gives rise to couple stresses.

Momentum and energy preserving integrators for nonholonomic dynamics

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian , where Q is the configuration manifold, and a (generally nonintegrable) distribution . In the proposed method, a discretization of the constraints is not required. We show that the method preserves the discrete nonholonomic momentum map, and also that the nonholonomic constraints are preserved in average. We study, in particular, the case where Q has a Lie group structure and the discrete Lagrangian and/or nonholonomic constraints have various invariance properties, and show that the method is also energy-preserving in some important cases.

Development of CMPS Method for Accurate Water-Surface Tracking in Breaking Waves

A Corrected Moving Particle Semi-implicit (CMPS) method is proposed for the accurate tracking of water surface in breaking waves. The original formulations of standard MPS method are revisited from the view point of momentum conservation. Modifications and corrections are made to ensure the momentum conservation in a particle-based calculation of viscous incompressible free-surface flows. A simple numerical test demonstrates the excellent performance of the CMPS method in exact conservation of linear momentum and significantly enhanced preservation of angular momentum. The CMPS method is applied to the simulation of plunging breaking and post-breaking of solitary waves. Qualitative and quantitative comparisons with the experimental data confirm the high capability and precision of the CMPS method. A tensor-type strain-based viscosity is also proposed to further enhanced CMPS reproduction of a splash-up.

Corrected Incompressible SPH method for accurate water-surface tracking in breaking waves

A Corrected Incompressible SPH (CISPH) method is proposed for accurate tracking of water surface in breaking waves. Corrective terms are derived based on a variational approach to ensure the angular momentum preservation of Incompressible SPH (ISPH) formulations. The proposed CISPH method is applied to solve the Navier–Stokes equation for simulating the breaking and post-breaking of solitary waves on a plane slope. The enhanced precision (compared to the ISPH method) of the CISPH method is confirmed through both qualitative and quantitative comparisons with experimental data. The introduction of corrective terms significantly improves the capability and the accuracy of the ISPH method in the simulation of wave breaking and post-breaking.

Lie group variational integrators for the full body problem in orbital mechanics

Equations of motion, referred to as full body models, are developed to describe the dynamics of rigid bodies acting under their mutual gravitational potential. Continuous equations of motion and discrete equations of motion are derived using Hamilton’s principle. These equations are expressed in an inertial frame and in relative coordinates. The discrete equations of motion, referred to as a Lie group variational integrator, provide a geometrically exact and numerically efficient computational method for simulating full body dynamics in orbital mechanics; they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long time periods. They are also efficient in only requiring a single evaluation of the gravity forces and moments per time step. The Lie group variational integrator also preserves the group structure without the use of local charts, reprojection, or constraints. Computational results are given for the dynamics of two rigid dumbbell bodies acting under their mutual gravity; these computational results demonstrate the superiority of the Lie group variational integrator compared with integrators that are not symplectic or do not preserve the Lie group structure.

Time‐step adaptivity in variational integrators with application to contact problems

AbstractVariable time‐step methods, with general step‐size control objectives, are developed within the framework of variational integrators. This is accomplished by introducing discrete transformations similar to Poincarés time transformation. While gaining from adaptive time‐steps, the resulting integrators preserve the structural advantages of variational integrators, i.e. they are symplectic and momentum preserving. As an application, the methods are utilized for dynamic multibody systems governed by contact force laws. A suitable scaling function defining the step‐size control objective is derived.

Geometric integrators for ODEs

Geometric integration is the numerical integration of a differential equation, while preserving one or more of its 'geometric' properties exactly, i.e. to within round-off error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase-space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. In this paper we present a survey of geometric numerical integration methods for ordinary differential equations. Our aim has been to make the review of use for both the novice and the more experienced practitioner interested in the new developments and directions of the past decade. To this end, the reader who is interested in reading up on detailed technicalities will be provided with numerous signposts to the relevant literature.

On conservative and monotone one-dimensional cellular automata and their particle representation

Number-conserving (or conservative) cellular automata (CA) have been used in several contexts, in particular traffic models, where it is natural to think about them as systems of interacting particles. In this article we consider several issues concerning one-dimensional cellular automata which are conservative, monotone (specially “non-increasing”), or that allow a weaker kind of conservative dynamics. We introduce a formalism of “particle automata”, and discuss several properties that they may exhibit, some of which, like anticipation and momentum preservation, happen to be intrinsic to the conservative CA they represent. For monotone CA we give a characterization, and then show that they too are equivalent to the corresponding class of particle automata. Finally, we show how to determine, for a given CA and a given integer b, whether its states admit a b-neighborhood-dependent relabeling whose sum is conserved by the CA iteration; this can be used to uncover conservative principles and particle-like behavior underlying the dynamics of some CA. Complements at http://www.dim.uchile.cl/~anmoreir/ncca

Temperature fluctuation under bubbles of pool boiling in decompression

AbstractExperiments on pool boiling of water were performed under decompression. Temperature fluctuation was measured with three thermocouples which were equipped on the copper block surface by plating. Rapid temperature drops in this fluctuation arise at the same times. This shows the existence of evaporation of the microlayer under bubbles. Evaluation of this evaporation of the microlayer and rapid temperature drops revealed the existence of a new evaporation factor in obedience to the liquid surface overheating. Temperature drops are of two types at decompression. One is rapid drop due to cavity existence and the other is slow drop due to generation in the superheated bulk liquid on the mirror surface. This rapid temperature drop is gradual at the first stage. For verification of gradual drops, we introduced an equation concerning micro film thickness under the bubble on the basis of the viscosity and the preservation of momentum. This equation has an exponential factor for microfilms. Numerical calculations showed good agreement with the experimental data. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 567–581, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.10102