Energy Conservation Properties Research Articles

PGD reduced-order modeling for structural dynamics applications

We propose in this paper a Proper Generalized Decomposition (PGD) approach for the solution of problems in linear elastodynamics. The novelty of the work lies in the development of weak formulations of the PGD problems based on the Lagrangian and Hamiltonian Mechanics, the main objective being to devise numerical methods that are numerically stable and energy conservative. We show that the methodology allows one to consider the Galerkin-based version of the PGD and numerically demonstrate that the PGD solver based on the Hamiltonian formulation offers better stability and energy conservation properties than the Lagrangian formulation. The performance of the two formulations is illustrated and compared on several numerical examples describing the dynamical behavior of a one-dimensional bar.

Analytical solutions of the energy harvesting potential from vehicle vertical vibration based on statistical energy conservation

Two kinds of approaches have been proposed to harvest the vibrational energy from a running vehicle, namely using a regenerative damper to replace a traditional hydraulic suspension damper, or adding one or more dynamic vibration absorbers (DVAs). Extensive studies have been conducted by simulation or experiment on the first approach, whereas the studies on the second approach are very limited and even the energy harvesting potential is not clear. This paper tries to present an analytical solution, to uniformly reveal the energy harvesting potential from vehicle vertical vibration by both approaches, with a focus on approaches with DVAs. A generic quarter car model is proposed to represent various vibration energy harvesting configurations. With a newly enunciated statistical energy conservation property for linear vibration systems, the analytical solution of the energy harvesting potential is derived for the proposed model subjected to random excitation. It is illustrated from these analytical expressions that the total vehicle vertical vibration energy is not directly related to DVA configurations, and the vibration energy dissipated in the tire can be transferred to the absorbers. Literature results and newly implemented statistical analyses are presented to validate and demonstrate the derived solutions. Sensitivity analysis is then performed to reveal the effect of various vehicle parameters on the energy harvesting potential.

Large–Eddy Simulation of the Flow Inside a Scour Hole Around a Circular Cylinder Using a Cut Cell Immersed Boundary Method

We present a novel symmetry-preserving cut cell finite volume method which is a three-dimensional generalisation of the method by Dröge and Verstappen (Int J Numer Method Fluids 47:979–985, 2005). A colour-coding scheme for the three-dimensional cut momentum cell faces reduces the number of possible cut cell configurations. A cell merging strategy is employed to alleviate time step constraints. We demonstrate the energy conservation property of the convective and pressure gradient terms, and the second-order spatial convergence with suitable benchmark cases. We used the scheme to perform highly resolved large–eddy simulations of the flow inside a scour hole around a circular cylinder mounted vertically in a flume. The simulation results are extensively compared to a stereoscopic particle image velocimetry experiment of the same configuration performed by Jenssen and Manhart (Exp Fluids 61:217, 2020). We demonstrate that for the investigated Reynolds numbers (20,000 and 40,000) nearly converged solutions are obtained; however at large computational efforts (up to 2.35 billion cells for the higher Reynolds number). It turns out that the flow topology of the horseshoe vortex system is strongly dependent on the grid resolution. For simulation results obtained on the finest grid, the mean flow and turbulence quantities agree well with the experiment. We investigate the shape and turbulence structure of the horseshoe vortex based on three-dimensional fields, and discuss the distribution of the mean and standard deviation of the wall shear stress in the scour hole and the implications for the physics of the scouring process over a sand bed.

Sound synthesis of a 3D nonlinear string using a covariant Lie group integrator of a geometrically exact beam model

We present a three dimensional nonlinear string model based on a geometrically exact beam. The beam model is obtained by applying a variational principle using a covariant Lagrangian formulation; in particular, the equations of motion and the boundary conditions are treated in an unified manner. Following an analogous discrete variational principle, a Lie group variational integrator is given. The energy and momentum conservation properties of the integrator are discussed and illustrated. This geometrically exact beam serves as a basis to formulate a prestressed damped string model with coupled non trivial boundary conditions. Simulation results are discussed and validated against analytical solutions obtained in the context of a small displacement hypothesis.

Two high-order energy-preserving and symmetric Gauss collocation integrators for solving the hyperbolic Hamiltonian systems

In this paper, we first derive the energy-preserving collocation integrator for solving the hyperbolic Hamiltonian systems. Then, two concrete high-order energy-preserving and symmetric integrators are presented by choosing the collocation nodes as two and three Gauss–Legendre points, respectively. The convergence and the symmetry of the constructed energy-preserving integrators are rigorously analysed. Numerical results verify the energy conservation property and the accuracy of the proposed integrators.

A new piecewise reproducing kernel function algorithm for solving nonlinear Hamiltonian systems

This paper aims to study a new reproducing kernel (RK) function-based collocation method for nonlinear Hamiltonian systems. By applying the associated functions of RK spaces W2,02, W2,03 and W2,04, we propose the second, third and fourth-order schemes, respectively. Since the coefficient matrix of the linear system obtained by our scheme is symmetric and positive definite, our approach is uniquely solvable.The numerical experiments verify that our algorithms are efficient and can simulate the long time behavior including energy conservation and symplectic structure properties.

Nonuniform 3D finite-difference elastic wave simulation on staggered grids

We have developed develop an approach to simulate the 3D isotropic elastic wave propagation using nonuniform finite-difference discretization on staggered grids. Specifically, we consider simulation domains composed of layers of uniform grids with different grid spacings, separated by nonconforming interfaces. We determine that this layer-wise finite-difference discretization has the potential to significantly reduce the simulation cost, compared to its fully uniform counterpart. Stability of such a discretization is achieved by using specially designed difference operators, which are variants of the standard difference operators with adaptations near boundaries or interfaces, and penalty terms, which are appended to the discretized wave system to weakly impose boundary or interface conditions. Combined with specially designed interpolation operators, the discretized wave system is shown to preserve the energy-conserving property of the continuous elastic wave equation, and a fortiori ensures the stability of the simulation. Numerical examples are presented to demonstrate the efficacy of the proposed simulation approach.

Contrast-independent partially explicit time discretizations for multiscale wave problems

In this work, we design and investigate contrast-independent partially explicit time discretizations for wave equations in heterogeneous high-contrast media. We consider multiscale problems, where the spatial heterogeneities are at subgrid level and are not resolved. In our previous work Chung et al. (2021) [25], we have introduced contrast-independent partially explicit time discretizations and applied to parabolic equations. The main idea of contrast-independent partially explicit time discretization is to split the spatial space into two components: contrast dependent (“fast”) and contrast independent (“slow”) spaces defined via multiscale space decomposition. Using this decomposition, our goal is to appropriately introduce time splitting such that the resulting scheme is stable and can guarantee contrast-independent discretization under some suitable conditions. In this paper, we propose contrast-independent partially explicitly scheme for wave equations, where the splitting requires a careful design. We prove that the proposed splitting is unconditionally stable under some suitable conditions formulated for the second space (“slow”). This condition requires the second space to be contrast independent. Our approaches identify local features that need implicit treatment and the latter is done on a coarse grid using appropriate splitting algorithms. We note that the proposed work extends our previous work Chung et al. (2021) [25] to wave equations. This extension consists of (1) a different discretization (both spatial and temporal), (2) energy conservation property (instead of energy decay), and (3) a different proof strategy. We present numerical results and show that the proposed methods provide results similar to implicit methods with the time step that is independent of the contrast.

Performance analysis of a tuned point absorber using SPH calm water and wave tank simulations

In this paper, two smoothed particle hydrodynamics (SPH) models, namely SPH-W and SPH-C were used to evaluate the motion response of a point absorber wave energy converter (WEC). In the SPH-W model, a long wave flume was constructed and a long simulation was performed to obtain the motion response of the WEC. In the SPH-C model, the SPH method is only used to find the hydrodynamic coefficients of the device by analysing a few seconds of free-decaying motion of WEC in calm water in a much smaller numerical flume. Then, these coefficients were inserted in the equation of motion of a heaving WEC that was solved using a 4th order Runge-Kutta (ODE45) solver in MATLAB. First, the energy conservation property of the WCSPH model was examined through a standing wave benchmark test. Then, the wave-point absorber interaction was simulated. While the simulation time for SPH-C model is much smaller than that of SPH-W, it gave almost similar results for the motion response of WEC. These two models were used to evaluate the effects of the control force and the draft of a cone-cylinder point absorber on its hydrodynamic responses. The results showed that compared to the effect of the supplementary inertia, changes in the draft of the WEC have a small influence on its hydrodynamic responses. The buoy draft has an inverse relationship with both added mass and damping coefficients. However, increasing the supplementary mass increases the added mass and decreases the hydrodynamic damping coefficients.

MPS–FEM coupled method for 3D dam-break flows with elastic gate structures

In this paper, the MPS–FEM coupled method is applied for the three-dimensional (3D) dam-break flow with elastic gate, where the Moving Particle Semi-implicit (MPS) method and the Finite Element Method (FEM) are respectively used to solve fluid flows and structure deformation. A partitioned coupling strategy of traditional Conventional Serial Staggered (CSS) strategy is employed for MPS method and FEM method coupling. An interfacial data interpolation module called three-dimensional Kernel Function Based Interpolation Technique (3D-KFBI) is developed for the three-dimensional fluid–structure interface. Based on the MPS–FEM​ coupled method, a self-developed MPSFSI solver is developed. Then, the stability, accuracy, convergence and energy conservation properties of the developed data interpolation algorithm is validated by two cases. Afterwards, the proposed coupled method is applied for the case of dam-break flow interacting with the elastic gate. A good agreement between present numerical result and published experiment data Antoci et al. (2007) demonstrates the accuracy and efficiency of the present coupled method. At last, MPSFSI solver is applied for the problem of the narrow dam-break flow with the elastic gate to investigate the three-dimensional effect.

New High‐Order Energy Preserving Method of the Fractional Coupled Nonlinear Schrödinger Equations

The symplectic structure is given for the fractional coupled nonlinear Schrödinger equations. The Fourier spectral method and the fourth‐order combination average vector field (AVF) method are applied to discretize the structure, and a new format for the fractional coupled nonlinear Schrödinger equations is obtained. The numerical experiments are showed to illustrate the property of the new format. The new scheme can maintain the energy conservation property better than the classical symplectic scheme.

A massless boundary component mode synthesis method for elastodynamic contact problems

We propose to combine the ideas of mass redistribution and component mode synthesis. More specifically, we employ the MacNeal method, which readily leads to a singular mass matrix, and an accordingly modified version of the Craig-Bampton method. Besides obtaining a massless boundary, we achieve a drastic reduction of the mathematical model order in this way compared to the parent finite element model. Contact is modeled using set-valued laws and time stepping is carried out with a semi-explit scheme. We assess the method’s computational performance by a series of benchmarks, including both frictionless and frictional contact. The results indicate that the proposed method achieves excellent energy conservation properties and superior convergence behavior. It reduces the spurious oscillations and decreases the computational effort by about 1–2 orders of magnitude compared to the current state of the art (mass-carrying component mode synthesis method). We believe that the computational performance and favorable energy conservation properties will be valuable for the prediction of vibro-impact processes and physical damping.

Gravity matching navigation algorithm based on multiscale search and Hadamard transformed difference

The accuracy and stability of navigation algorithms are crucial preconditions for underwater gravity matching aided navigation. To improve the matching accuracy and robustness of the matching algorithm, this paper presents a novel gravity matching navigation algorithm based on multiscale search and Hadamard transformed difference. The Hadamard transformation process was first time introduced in the new algorithm for the Walsh–Hadamard kernel function with the property of energy conservation. The gravity measurement sequences could be converted to the Hadamard domain; thus, the difference in numerical values, tendency, and spatial structure of the gravity measurement sequence were also a focus in the new algorithm, whereas only gravity statistical values were considered in classical matching algorithms. Therefore, with the proposed algorithm, the number of measurements necessary for matching can be effectively reduced, while improving the matching accuracy and success rate. In addition, a multiscale neighborhood search strategy based on contour constraints was designed to improve the matching efficiency, whereas a point-by-point global search was widely used in classical matching algorithms. Marine gravity maps of the South China Sea were used to construct the simulation tests. Simulation results show that under various conditions, the new algorithm has lower requirements for the number of measurements, measuring accuracy, and matching areas while exhibiting a higher navigation accuracy, matching success rate, and matching efficiency. Thus, the proposed algorithm could provide a new option for future practical applications of gravity matching aided navigation.

Optimal l∞ error estimates of the conservative scheme for two-dimensional Schrödinger equations with wave operator

In this work, we consider the numerical computation for the two-dimensional generalized nonlinear Schrödinger equations with wave operator. Based on the scalar auxiliary variable (SAV) approach, the original problem is transformed into an equivalent one, which corresponds to the energy-conservation laws. We present an energy-conserving and linearly implicit three-level scheme for the equivalent system. The energy-conserving property, boundedness of the numerical solution and convergence analysis in the discrete maximum norm are derived, which has not restricted to specific forms of cubic nonlinear term f and not needed the sharply restriction on mesh size. Finally, numerical experiments on several models confirm our theoretical results.

Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

In this paper, we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave equations. Optimal error estimates in time and superconvergent error estimates in space are established without certain time-step restrictions. The key is to estimate directly the solution bounds in the H2-norm for both the nonlinear wave equation and the corresponding fully discrete scheme, while the previous investigations rely on the temporal-spatial error splitting approach. Numerical examples are presented to confirm energy-conserving properties, unconditional convergence and optimal error estimates, respectively, of the proposed fully discrete schemes.

Convergence of an energy-conserving scheme for nonlinear space fractional Schrödinger equations with wave operator

This paper focuses on the construction and analysis of the energy-conserving numerical schemes for the generalized nonlinear space fractional Schrödinger equations with wave operator. Combining the scalar auxiliary variable (SAV) approach, we present an energy-conserving and linearly implicit scheme, while the previous conservative schemes are generally fully implicit. The energy-conserving property, boundedness and convergence of the numerical solution of the fully discrete scheme are derived for one and multi-dimensional cases. The numerical analysis is also considered. Finally, numerical examples on several fractional models illustrate that the proposed scheme can guarantee conservation of the system energy and confirm our theoretical results.

The energy-preserving time high-order AVF compact finite difference scheme for nonlinear wave equations in two dimensions

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The Lp-norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.

A conservative difference scheme with optimal pointwise error estimates for two‐dimensional space fractional nonlinear Schrödinger equations

AbstractIn this paper, a linearized semi‐implicit finite difference scheme is proposed for solving the two‐dimensional (2D) space fractional nonlinear Schrödinger equation (SFNSE). The scheme has the property of mass and energy conservation at the discrete level, with an unconditional stability and a second‐order accuracy for both time and spatial variables. The main contribution of this paper is an optimal pointwise error estimate for the 2D SFNSE, which is rigorously established for the first time. Moreover, a novel technique is proposed for dealing with the nonlinear term in the equation, which plays an essential role in the error estimation. Finally, the numerical results confirm well with the theoretical findings.

A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

Geometric electrostatic particle-in-cell algorithm on unstructured meshes

We present a geometric particle-in-cell (PIC) algorithm on unstructured meshes for studying electrostatic perturbations with frequency lower than electron gyrofrequency in magnetized plasmas. In this method, ions are treated as fully kinetic particles and electrons are described by the adiabatic response. The PIC method is derived from a discrete variational principle on unstructured meshes. To preserve the geometric structure of the system, the discrete variational principle requires that the electric field is interpolated using Whitney 1-forms, the charge is deposited using Whitney 0-forms and the electric field is computed by discrete exterior calculus. The algorithm has been applied to study the ion Bernstein wave (IBW) in two-dimensional magnetized plasmas. The simulated dispersion relations of the IBW in a rectangular region agree well with theoretical results. In a two-dimensional circular region with fixed boundary condition, the spectrum and eigenmode structures of the IBW are obtained from simulations. We compare the energy conservation property of the geometric PIC algorithm derived from the discrete variational principle with that of previous PIC methods on unstructured meshes. The comparison shows that the new PIC algorithm significantly improves the energy conservation property.