Energy Conserving Time Integration Research Articles

Momentum and near-energy conserving/decaying time integrator for beams with higher-order interpolation on SE(3)

This study aims to develop a simple yet robust time-integration scheme for configuration-interpolated beam finite elements. Geometrically exact theory is employed to model the beams. It utilises the rotations which belong to a non-commutative Lie group and thus require special attention. The configuration is interpolated using a two-node SE(3) interpolation or its generalised implicit variant which enables higher orders. Such interpolation treats position and orientation as a unit and a member of the SE(3) Lie group. This unified approach allows for elegant mathematical manipulation. By adapting the Lie midpoint rule appropriately, it becomes possible to express energy changes in terms of inertial and internal forces, thus enabling the derivation of a momentum conserving and almost energy conserving time integration algorithm. The precision of energy conservation depends solely on the length of the finite elements. With further modification, this algorithm can also become an energy-decaying algorithm. Despite the configuration-dependent nature of the interpolation, the need to deal with its derivatives is avoided, which simplifies the implementation. The method is tested using 3D numerical example with finite rotations which confirms, that the method indeed has the conservation properties and is stable and robust.

Temporal stability of collocation, Petrov–Galerkin, and other non-symmetric methods in elastodynamics and an energy conserving time integration

Non-symmetric matrices may arise in the discretization of self-adjoint problems when a Petrov–Galerkin, collocation, or finite-volume method is employed. While these methods have been widely applied, in this paper it is shown that the use of these non-symmetric matrices is incompatable with the conservation of energy in elastodynamics. First, the consistency between the continuous forms of the momentum equation and the energy equation is examined. It is shown that the conservation of linear momentum is equivalent to conservation of energy provided the solution is sufficiently smooth. The semi-discrete counterparts are then analyzed, where it is demonstrated that they are also equivalent, but only conditionally: the mass and stiffness matrices must be symmetric. As a result, employing a non-symmetric method in elastodynamics may artificially generate or dissipate energy. The fully discrete forms with Newmark time integration are then examined where it is shown that unconditionally unstable algorithms may arise. An energy-conserving time integration algorithm is then proposed which provides stability in the solutions of non-symmetric systems. The collocation and finite-volume methods are employed in numerical examples to demonstrate stability issues and the effectiveness of the proposed time integration methodology.

A vibro-impact acoustic black hole for passive damping of flexural beam vibrations

Nonlinear flexural vibrations of slender beams holding both an Acoustic Black Hole termination and a contact non-linearity are numerically studied. The Acoustic Black Hole (ABH) effect is a passive vibration mitigation technique, which has shown attractive properties above a given cut-on frequency. In this contribution, a vibro-impact acoustic black hole (VI-ABH) is introduced, the contact nonlinearity being used as a mean to transfer energy from low to high frequencies. A numerical model of a VI-ABH is derived from a Euler-Bernoulli beam. The contact law is handled with a penalization approach, the visco-elastic layer with a Ross-Kerwin-Ungard model and the problem is solved with a modal approach combined with an energy-conserving time integration scheme. Numerical results show that the VI-ABH brings about important modifications, and changes the nature of more traditional black holes, by redistributing all the vibrational energy. It can lead to a strong decrease of the resonance magnitude at low frequencies. Under steady state noise excitation, parametric studies are realised in the cases of a single contact, a grid of contacts and bilateral contacts layouts, in order to find some optimal designs. Transient dynamics is also studied through the analysis of displacement signal envelope and energy decay time. All the numerical results constantly show that the combination of the ABH effect and an energy transfer provided by contact nonlinearity leads to very attractive mitigation template including low frequencies.

An algorithmic stress formula for energy conserving time integration in a mixed framework for polyconvex large strain elasticity

AbstractThe present work deals with a mixed variational formulation of elastodynamics along with an Energy‐Momentum consistent (EMC) discretization in space and time. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Improvement of the acoustic black hole effect by using energy transfer due to geometric nonlinearity

Acoustic Black Hole effect (ABH) is a passive vibration damping technique without added mass based on flexural waves properties in thin structures with variable thickness. A common implementation is a plate edge where the thickness is locally reduced with a power law profile and covered with a viscoelastic layer. The plate displacement in the small thickness region is large and easily exceeds the plate thickness. This is the origin of geometric nonlinearity which can generate couplings between linear eigenmodes of the structure and induce energy transfer between low and high frequency regimes. This phenomenon may be used to increase the efficiency of the ABH treatment in the low frequency regime where it is usually inefficient. An experimental investigation evidenced that usual ABH implementation gives rise to measurable geometric nonlinearity and typical nonlinear phenomena. In particular, strongly nonlinear regime and wave turbulence are reported. The nonlinear ABH beam is then modeled as a von Kármán plate with variable thickness. The model is solved numerically by using a modal method combined with an energy-conserving time integration scheme. The effects of both the thickness profile and the damping layer are then investigated in order to improve the damping properties of an ABH beam. It is found that a compromise between the two effects can lead to an important gain of efficiency in the low frequency range.

Towards a pancreatic surgery simulator based on model order reduction

In this work a pancreatic surgery simulator is developed that provides the user with haptic feedback. The simulator is based on the use of model order reduction techniques, particularly Proper Generalized Decomposition methods. The just developed simulator presents some notable advancements with respect to existing works in the literature, such as the consideration of non-linear hyperelasticity for the constitutive modeling of soft tissues, an accurate description of contact between organs and momentum and energy conserving time integration schemes. Pancreas, liver, gall bladder, and duodenum are modeled in the simulator, thus providing with a very realistic and immersive perception to the user.

Conservative fourth-order time integration of non-linear dynamic systems

Abstract An energy conserving time integration algorithm with fourth-order accuracy is developed for dynamic systems with nonlinear stiffness. The discrete formulation is derived by integrating the differential state-space equations of motion over the integration time increment, and then evaluating the resulting time integrals of the inertia and stiffness terms via integration by parts. This process introduces the time derivatives of the state space variables, and these are then substituted from the original state-space differential equations. The resulting discrete form of the state-space equations is a direct fourth-order accurate representation of the original differential equations. This fourth-order form is energy conserving for systems with force potential in the form of a quartic polynomial in the displacement components. Energy conservation for a force potential of general form is obtained by addition of a higher order secant-type correction term. The formulation leads to a consistent representation of the motion within a time increment corresponding to cubic Hermite interpolation in time. This in turn leads to excellent phase representation with only a small fourth-order error, permitting integration of oscillatory systems with only a few integration points per period. Three numerical examples demonstrate the high accuracy of the algorithm.

A Fokker–Planck based kinetic model for diatomic rarefied gas flows

A Fokker–Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker–Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau–Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.

Investigating a flexible wind turbine using consistent time‐stepping schemes

PurposeWind turbines are of growing importance for the production of renewable energy. The kinetic energy of the blowing air induces a rotary motion and is thus converted into electricity. From the mechanical point of view the complex dynamics of wind turbines become a matter of interest for structural optimization and optimal control in order to improve stability and energy efficiency. The purpose of this paper therefore is to present a mechanical model of a three‐blade wind turbine with a momentum and energy conserving time integration of the system.Design/methodology/approachThe authors present a mechanical model based upon a rotationless formulation of rigid body dynamics coupled with flexible components. The resulting set of differential‐algebraic equations will be solved by using energy‐consistent time‐stepping schemes. Rigid and orthotropic‐elastic body models of a wind turbine show the robustness and accuracy of these schemes for the relevant problem.FindingsNumerical studies prove that physically consistent time‐stepping schemes provide reliable results, especially for hybrid wind turbine models.Originality/valueThe application of energy‐consistent methods for time discretization is intended to provide computational robustness and to reduce the computational costs of the dynamical wind turbine systems. The model is aimed to give a first access into the investigation of fluid‐structure interaction for wind turbines.

Energy-conserving Runge–Kutta methods for the incompressible Navier–Stokes equations

Energy-conserving methods have recently gained popularity for the spatial discretization of the incompressible Navier–Stokes equations. In this paper implicit Runge–Kutta methods are investigated which keep this property when integrating in time. Firstly, a number of energy-conserving Runge–Kutta methods based on Gauss, Radau and Lobatto quadrature are constructed. These methods are suitable for convection-dominated problems (such as turbulent flows), because they do not introduce artificial diffusion and are stable for any time step. Secondly, to obtain robust time-integration methods that work also for stiff problems, the energy-conserving methods are extended to a new class of additive Runge–Kutta methods, which combine energy conservation with L-stability. In this class, the Radau IIA/B method has the best properties. Results for a number of test cases on two-stage methods indicate that for pure convection problems the additive Radau IIA/B method is competitive with the Gauss methods. However, for stiff problems, such as convection-dominated flows with thin boundary layers, both the higher order Gauss and Radau IIA/B method suffer from order reduction. Overall, the Gauss methods are the preferred method for energy-conserving time integration of the incompressible Navier–Stokes equations.

Conservative integration of rigid body motion by quaternion parameters with implicit constraints

SUMMARY An angular momentum and energy-conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four-component conjugate momentum vector via Hamilton's equations. The introduction of an extended mass matrix leads to a symmetric set of eight state-space equations of motion. The extra inertial parameter serves as a multiplier on the kinematic constraint, and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments. External loads enter these equations via the set of momentum equations. Initially, the normalization of the quaternion array is introduced via a Lagrange multiplier. However, this Lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential, and elimination of the Lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient. An algorithm is developed by forming a finite increment of the Hamiltonian. This procedure identifies the proper selection of increments and mean values, and leads to an algorithm with conservation of momentum and energy. Implementation, conservation properties, and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top. Copyright © 2012 John Wiley & Sons, Ltd.

Optimal size, shape, and control design in dynamics of planar frame structures under large displacements and rotations

Size, shape, and drive optimization procedures are combined with an energy-conserving time-integration scheme for the dynamic analysis of planar geometrically non-linear frame structures undergoing large overall motions. The solution method is based on the finite-element formulation, employing the classical displacement-based planar beam finite elements described in an inertial frame. Finite axial, bending, and shear strains are taken into account. If the system is conservative, the energy and momenta conservation in the discrete system during motion is guaranteed. Size, shape, and drive design variables are introduced into the model. Shape parameterization is achieved by the design element technique, using Bezier patches. The sensitivity analysis is performed by the discrete approach and the analytical direct differentiation method. A gradient-based optimization method, using an automatically adjustable convex approximation technique, is employed. The efficiency and the applicability of the approach are demonstrated via numerical examples. The shape and the driving function of a load-moving robot arm are optimized to reduce oscillations in its final position. The shape of a steel frame is optimized to reduce oscillations after an idealized ground motion jerk.

Stable long‐term simulation of dynamically loaded elastomers

AbstractA well–known problem in long–term simulations of flexible solid bodies is the restriction to small time steps of standard time integrators in order to obtain a stable simulation. One approach is to achieve exact energy conservation while simulating a nonlinear elastic body (see Reference [1] and the references therein). Additionally, total linear and total angular momentum is conserved for a free motion. This approach leads to a qualitatively improved solution, because the approximated time evolution exactly fulfills the same physical laws as the exact time evolution. Incorporating energy dissipation, the energy conserving time integration is extended to an energy consistent time integration. It turned out that such an energy consistent time integration is also of great advantage when computing finite motions of flexible solid bodies with material dissipation (see References [2,3]). This paper points out that an energy consistent time discretisation is also advantageous for dynamic finite deformation thermoviscoelasticity under dynamic loads. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The role of geometric stiffness in momentum and energy conserving time integration

AbstractA momentum and energy conserving time integration algorithm is developed for the motion of elastic bodies described in terms of the quadratic Green strain. Momentum conserving algorithms are formulated from an integral of the equations of motion and energy conservation has traditionally been obtained by evaluating the contribution from the internal forces by use of a combined mean value of stresses and virtual strains on the element level. It is here demonstrated that momentum and energy conservation can be obtained from the classic central difference formulation by including an extra global term in the form of the increment of the geometric stiffness matrix over the current time step, usually directly available in global form in existing finite element programmes. The theory is derived by the use of a state‐space formulation, where this extra term is located in the same position as the viscous damping matrix, indicating that the effect of the extra incremental geometric stiffness term in the non‐linear algorithm is equivalent to a variable damping term depending on the change of the state of stress over a time increment. In the actual numerical algorithm, the new value of the velocity vector is eliminated, leaving a non‐linear equation for the displacement increment alone, followed by an explicit vector update of the velocity increment. Copyright © 2006 John Wiley & Sons, Ltd.

Energy conserving time integration scheme for geometrically exact beam

An energy conserving finite-element formulation for the dynamic analysis of geometrically non-linear beam-like structures undergoing large overall motions has been developed. The formulation uses classical displacement-based planar beam finite elements described in an inertial frame. It takes into account finite axial, bending and shear strains. A theoretically consistent approach is used to derive a novel and simple energy conserving scheme, using the unconventional incremental strain update rather than the standard strong form. Numerical examples demonstrate perfect energy and momenta conservation, stability and robustness of the scheme, and good convergence properties in terms of both the Newton–Raphson method and time step size.

Vector finite-element modeling of the full-wave Maxwell equations to evaluate power loss in bent optical fibers

The loss of power incurred by the bending of step-indexed optical fibers is measured is calculated in this paper using vector finite-element modeling of the full-wave Maxwell equations in the optical regime. It is demonstrated that fewer grid elements can be used to model light transmission in longer fiber lengths by using high-order basis functions in conjunction with a high-order energy-conserving time-integration method. The power in the core is measured at several points to determine the percentage loss. The effect of bending on light polarization is also demonstrated.

Fluid structure interaction problems in large deformation

The present article deals with the simulation of fluid structure interaction problems in large deformation, and discusses two aspects of their numerical solution: (i) the derivation of energy conserving time integration schemes in presence of fluid structure coupling, moving grids, and nonlinear kinematic constraints such as incompressibility and contact, (ii) the introduction of adequate preconditioners efficiently chaining local fluid and structure solvers. Solutions are proposed, analyzed and tested using nonlinear energy correcting terms, and added mass based Dirichlet Neumann preconditioners. Numerical applications include nonlinear impact problems in elastodynamics and blood flows predictions within flexible arteries. To cite this article: P. Le Tallec et al., C. R. Mecanique 333 (2005).

Fluid structure interaction with large structural displacements

The calculation of viscous flows inside deformable structures faces major difficulties when enforcing the kinematic compatibility at the fluid structure interface and updating the geometry of the domain. This paper proposes to overcome these problems by considering the fluid and the structure as a unique continuous medium, to be studied in a fixed reference configuration. The resulting problem is then split into a fluid and a structural part through an additive decomposition of the space of kinematically admissible test functions. This approach treats the structure in a fully Lagrangian way and uses an associated arbitrary Lagrangian Eulerian (ALE) formulation for the fluid. Such a strategy has three advantages which are detailed in the present paper. • it can be discretised in time by implicit, stable, energy conserving time integration schemes, and solved by simple, iterative uncoupled algorithms; • the fluid and the structural problems can be approximated by independent finite element spaces; • the most recent geometrically exact nonlinear shell models can be used for the structures. The full strategy has been implemented into an industrial CFD code. Results obtained in the simulation of industrial hydraulic shock absorbers are presented at the end of the paper.

Kinetic energy conservative algorithm in moving grid system using segregated finite element formulation

Kinetic energy conservation for fixed and moving grids is examined in time-accurate finite element computation of fully unsteady inviscid flows. As numerical algorithms, fractional step method (FSM) and modified SIMPLE are used. To simulate the flow in moving grid system, arbitrary Lagrangian-Eulerian (ALE) method is adopted. In the present study, the energy conserving time integration rule for finite element algorithm is proposed and discussed schematically. It is shown that the discretization by Crank-Nicolson in time and Galerkin (central difference) in space must be used to ensure energy conservation. The developed code has been tested for a standing vortex in fixed or moving grid system, sloshing in a tank and propagation of a solitary wave, and has been shown to be a completely energy conserving algorithm.

Dynamics of flexible beams for multibody systems: A computational procedure

A computational procedure suitable for the solution of equations of motions for flexible multibody systems has been developed. The flexible beams are modeled using a nonlinear rod-type theory which accounts for both finite rotations and large deformations. The present formulation incorporates physical measures of conjugate Cauchy stress and covariant strain increments referenced with respect to a convected coordinate system. As a consequence, the beam model can easily be interfaced with real-time strain measurements and feedback control systems. A distinct feature of the present work is the computational preservation of total energy for undamped systems; this is obtained via an objective strain increment/stress update procedure combined with an energy-conserving time integration algorithm which contains an accurate update of angular orientations. The procedure is demonstrated via several example problems.