Propagation Of Discontinuity Waves Research Articles

Dynamic modeling and analysis of discontinuous wave propagation in a rod

The wave propagation in an impacting rod is instant and nearly discontinuous, which remains a challenging problem for both dynamic modeling and experimental test. This paper presents an Enriched Finite Element Method (Enriched FEM) with controllable dispersion error to accurately and efficiently model the discontinuous wave propagation. The Enriched FEM approximates the displacement solutions based on a linear interpolation enriched with a series of harmonic functions. The paper focuses on the dispersion error of the enriched finite element from spatial-temporal discretization and the sampling theorem of the enriched finite element with different cutoff numbers, and then gives a simple quantitative criterion on the interdependent relationship between the size of mesh and time step so as to reduce the numerical dispersion error. In addition, the paper presents the experimental study on the wave propagation of an elastic rod colliding with a rigid wall. The experimental results explicitly show the incidence and reflection of the stress wave in the rod in the contact duration. They also indicate that the amplitude of the strain almost does not decay with the wave propagation along the rod. The numerical results based on the Hertzian contact model are in good agreement with the experimental observations, especially at wave interfaces.

Explicit asynchronous time scheme with local push-forward stepping for discontinuous elastic wave propagation: One-dimensional heterogeneous cases and Hopkinson bar experiment

This is a presentation of robust and accurate explicit time-stepping strategy for finite element modeling of elastic discontinuous wave propagation in strongly heterogeneous, multi-material and graded one-dimensional media. One of the major issues in FEM modeling is the existence of spurious numerical stress oscillations close to theoretical wave fronts due to temporal-spatial dispersion behavior of FE discretization. The numerical strategy presented for modeling of 1D discontinuous elastic waves is based on (a) pushforward-pullback local stepping — ensuring the elimination of dispersion due to different critical time step sizes of finite elements, (b) domain decomposition via localized Lagrange multipliers — to satisfy coupling kinematics and dynamic equations , (c) asynchronous time scheme — ensuring the correct information transfer of quantities for the case of integer ratios of time step size for all domain pairs. Dispersion behaviors, existence of spurious stress oscillations, and sensitivity of the dispersion to time step size are then suppressed. The proposed method is numerically tested with regard to the rectangular step pulse elastic propagation problem considering in-space varying Young’s modulus. To prove robustness and accuracy, a comparison with results from commercial software, an analytical solution, and experimental data from partial assembly of a split Hopkinson pressure bar (SHPB) setup is provided.

Towards shock absorbing hyperelastic metamaterial design. (I) Macroscopic scale: Computational shock-capturing

The work explores the computational modeling of propagating shocks in hyperelastic materials to be considered in the context of numerical analysis and design of mechanical energy absorbing materials. The computational approach can be classified as shock-capturing, since it aims at capturing the effects of the shock formation and propagation on the mechanical response in the solid, as opposed to the shock-fitting approach, an alternative focusing on determining the exact shock wave-front position in the propagation domain. Numerical simulation of the onset and propagation of discontinuous strain waves (mechanical shocks) across a solid is performed in that specific approach, and it is computationally assessed in different loading situations. The concept of extrinsic dissipation, arisen by breaking some of the polyconvexity aspects in hyperelastic materials, is recalled and exemplified via a large strains neo-Hookean polyconvex hyperelastic model which is perturbed in a number of different formats. The obtained mechanical responses, exhibiting propagating mechanical shocks, and the corresponding extrinsic dissipation, are explored together with the inherited computational issues, by means of numerical examples selected as representative for the aimed purposes. In a second stage, the problem of the dynamic impact of a rigid solid on low-density shock-absorbing specimens made of these perturbed hyperelastic materials is analyzed, and the amount and time-evolution of the resulting extrinsic dissipation are evaluated to characterize their performance for shock-absorbing purposes. Also, the exhibited property of shape recovery after impact is highlighted as a distinctive feature in front of alternative shock-absorbing materials. In the work, the analysis and numerical modeling are made at the single (macroscopic) scale, but the objective of the research is to constitute the first exploration of a more general setting, to be developed in future works: the multiscale design of shock-absorbing metamaterials.

Explicit multistep time integration for discontinuous elastic stress wave propagation in heterogeneous solids

SummaryA multistep explicit time integration algorithm is presented for tracking the propagation of discontinuous stress waves in heterogeneous solids whose subdomain‐to‐subdomain critical time step ratios range from tens to thousands. The present multistep algorithm offers efficient and accurate computations for tracking discontinuous waves propagating through such heterogeneous solids. The present algorithm, first, employs the partitioned formulation for representing each subdomain, whose interface compatibility is enforced via the method of the localized Lagrange multipliers. Second, for each subdomain, the governing equations of motion are decomposed into the extensional and shear components so that tracking of waves of different propagation speeds is treated with different critical step sizes to significantly reduce the computational dispersion errors. Stability and accuracy analysis of the present multistep time integration is performed with one‐dimensional heterogeneous bar. Analyses of the present algorithm are also demonstrated as applied to the stress wave propagation in one‐dimensional heterogeneous bar and in heterogeneous plain strain problems.

B-spline based finite element method in one-dimensional discontinuous elastic wave propagation

The B-spline variant of the finite element method (FEM) is tested in one-dimensional discontinuous elastic wave propagation. The B-spline based FEM (called Isogeometric analysis IGA) uses spline functions as testing and shape functions in the Galerkin continuous content. Here, the accuracy of stress distribution and spurious oscillations of the B-spline based FEM are studied in numerical modeling of one-dimensional propagation of stress discontinuities in a bar, where the analytical solution is known. For time integration, the Newmark method, implicit form of the generalized-α method, the central difference method and the predictor/multi-corrector method are tested and compared. The use of lumped and consistent mass matrices in explicit time integration is discussed. Due to accuracy, the consistent mass matrix is preferred in explicit time integration in IGA.

Transient SU/PG modelling of discontinuous wave propagation

This study proposed a transient 2-D hydrodynamic model adopting streamline upwind/Petrov-Galerkin (SU/PG) scheme for the analysis of discontinuous wave propagations by the sudden collapse of circular cylinder as well as by the dam break. The computation of sudden collapse in the circular cylinder problem identified both the outward-propagating circular shock waves and inward-propagating circular rarefaction waves. It was also confirmed that the advancing front and the depressive wave preserve the cylindrical symmetry, and the bore wave without oscillations was computed. The simulation results of the dam break shows that the bore and depressive waves which are propagating downstream and upstream, respectively are effectively developed and the shock front is sharply captured. The SU/PG scheme needs no tuning of adjustable parameters such as the artificial viscosity coefficient to be a viable option for accurate prediction of the flow characteristics in a discontinuous free surface case.

Reducing spurious oscillations in discontinuous wave propagation simulation using high-order finite elements

In this paper, a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves. The spatial domain is discretized by the finite element method. To obtain accurate results, the standard finite element method requires a very fine mesh, which is why the computational effort can be very time consuming. The use of high-order finite element methods–such as the spectral element method based on Lagrange polynomials through Gauss–Lobatto–Legendre points or the iso-geometric analysis using non-uniform rational B-splines–can reduce the enormous computational costs significantly, compared to the standard finite element method. However, explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave. The procedure proposed here is basically similar to the explicit method of Noh and Bathe, but the semi-discrete equation of motion is modified by introducing a damping parameter to suit the high-order FEM analysis of the discontinuous wave propagation. The performance due to this modification is tested for one-dimensional wave propagation problems using both lumped and consistent mass matrices. Then, the idea of a combination of the consistent and row sum lumped mass matrices is evaluated. The proposed method is studied also in two-dimensions by considering the Lamb problem of wave propagation. The results are promising enough to provide a better simulation of the discontinuous wave propagation problem.

A method for multidimensional wave propagation analysis via component‐wise partition of longitudinal and shear waves

SUMMARYAn explicit integration algorithm for computations of discontinuous wave propagation in two‐dimensional and three‐dimensional solids is presented, which is designed to trace extensional and shear waves in accordance with their respective propagation speeds. This has been possible by an orthogonal decomposition of the total displacement into extensional and shear components, leading to two decoupled equations: one for the extensional waves and the other for shear waves. The two decoupled wave equations are integrated with their CFL time step sizes and then reconciled to a common step size by employing a previously developed front‐shock oscillation algorithm that is proven to be effective in mitigating spurious oscillations. Numerical experiments have demonstrated that the proposed algorithm for two‐dimensional and three‐dimensional wave propagation problems traces the stress wave fronts with high‐fidelity compared with existing conventional algorithms. Copyright © 2013 John Wiley & Sons, Ltd.

Shielding of elastic nonstationary waves by interfaces

The ray method is used to solve the problem of the propagation of discontinuous (weak shock) waves in inhomogeneous elastic media. A procedure for drawing the fronts of reflected and refracted waves at interfaces and calculating their intensities is proposed. The effect of shielding discontinuous waves by one or two interfaces is studied. The cases of slipping and non-slipping contact are examined

A method for computation of discontinuous wave propagation in heterogeneous solids: basic algorithm description and application to one‐dimensional problems

SUMMARYAn explicit integration algorithm for computations of discontinuous wave propagation in heterogeneous solids is presented, which is aimed at minimizing spurious oscillations when the wave fronts pass through several zones of different wave speeds. The essence of the present method is a combination of two wave capturing characteristics: a new integration formula that is obtained by pushforward–pullback operations in time designed to filter post‐shock oscillations, and the central difference method that intrinsically filters front‐shock oscillations. It is shown that a judicious combination of these two characteristics substantially reduces both spurious front‐shock and post‐shock oscillations. The performance of the new method is demonstrated as applied to wave propagation through a uniform bar with varying courant numbers, then to heterogeneous bars. Copyright © 2012 John Wiley & Sons, Ltd.

TVD scheme for computing open channel wave flows

For the shallow water equations in the first approximation (Saint-Venant equations), a TVD scheme is developed for shock-capturing computations of open channel flows with discontinuous waves. The scheme is based on a special nondivergence approximation of the total momentum equation that does not involve integrals related to the cross-section pressure force and the channel wall reaction. In standard divergence difference schemes, most of the CPU time is spent on the computation of these integrals. Test computations demonstrate that the discontinuity relations reproduced by the scheme are accurate enough for actual discontinuous wave propagation to be numerically simulated. All the qualitatively distinct solutions for a dam collapsing in a trapezoidal channel with a contraction in the tailwater area are constructed as an example.

Open-channel waves generated by propagation of a discontinuous wave over a bottom step

This paper presents the results of theoretical and experimental studies of open-channel waves generated by the propagation of a discontinuous dam-break wave over a bottom step. The cases where the initial tailwater level is higher than the step height (the step is under water) and where this value is smaller than the step height (at the initial time, water is absent on the step) are considered. Exact solutions are constructed using modified first-approximation equations of shallow-water theory, which admit the propagation of discontinuous waves in a dry channel. On the stationary hydraulic jump formed above the bottom step, the total free-stream energy is assumed to be conserved. These solutions agree with experimental data on various parameters (types of waves, wave propagation velocity, asymptotic depths behind the wave fronts).

Modified shallow water equations which admit the propagation of discontinuous waves over a dry bed

A method for modeling the propagation of discontinuous waves over a dry bed using the first approximation of shallow water theory is proposed. The method is based on a modified conservation law of total momentum that takes into account the concentrated momentum losses due to the formation of local turbulent vortex structures in the fluid surface layer at a discontinuous-wave front. A quantitative estimate of these losses is obtained by deriving the shallow water equations from the Navier-Stokes equations with allowance for viscosity, which has a rapidly increasing effect in the turbulent flow regions described by discontinuous waves. The stability of the discontinuous waves admitted by the modified system of conservation laws of shallow water theory is examined. As an example, a comparative analysis is performed of the solutions of the dam-break problem obtained for the classical and modified shallow water models.

Propagation of discontinuouswaves along a dry bed

The possibility of simulating the process of propagation of discontinuous waves along a dry bed on the basis of the equations of the first approximation of shallow water theory is studied. It is shown that the consistent losses of the free-stream total momentum and energy can be found from the mass conservation law within the framework of the shallow water equations. As an example, solutions of the problem of dam break with a dry bed in the lower pool and the problem of impingement of a discontinuous wave on a coastal shelf are constructed. These exact solutions are compared with the results of laboratory experiments.

Numerical simulation of discontinuous waves propagating over a dry bed

A numerical algorithm is proposed for simulating the propagation of discontinuous waves over a dry bed governed by the shallow water equations in the first approximation. The algorithm is based on a modified conservation law of total momentum that takes into account the concentrated momentum loss associated with the formation of local eddy structures within the framework of the long-wave approximation. The modified conservation law involves a heuristic parameter that is chosen so as to agree with laboratory experiments. Numerical results are presented for the formation, propagation, and transformation of a discontinuous wave arising in a complete or partial (in the planned case) collapse of a dam over a bed with a horizontal or sloping bottom or a bottom with a local obstacle in the tailwater area.

An Embedded Boundary Method for the Wave Equation with Discontinuous Coefficients

A second order accurate embedded boundary method for the two-dimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is allowed to intersect the mesh in an arbitrary fashion. By using ghost points on either side of the interface, previous embedded boundary techniques for the Neumann and Dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy. The resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points, resulting in a fully explicit time-integration method. We prove that the one-dimensional restriction of the method is stable without damping for arbitrary locations of the interface relative to the grid. For the two-dimensional case, the previously developed fourth order $A^T A$-dissipation is generalized to handle jump conditions. We demonstrate that this operator provides sufficient stabilization to enable long-time simulations while being weak enough to preserve the accuracy of the solution. Numerical examples are given where the method is used to study electromagnetic scattering of a plane wave by a dielectric cylinder. The numerical solutions are evaluated against the analytical solution due to Mie, and pointwise second order accuracy is confirmed.

A diffusive-hyperbolic model for heat conduction

The present paper faces the problem of heat conduction within the framework of thermodynamics with internal state variables. A model, in which the heat flux vector depends both on the gradient of the absolute temperature and the gradient of a scalar internal variable, is proposed. Such a model leads to a diffusive-hyperbolic system which in general is parabolic, but also allows to shift to the hyperbolic regime. In the hyperbolic case the propagation of weak discontinuity waves is investigated. The Rankine-Hugoniot and Lax conditions for the propagation of strong shock waves are analyzed as well.

On the axi-symmetric equations of a rigid heat conductor: discontinuity waves in a non-constant state

By requiring the invariance with respect to a stretching group of transformations the non-linear system of equations governing the heat propagation in a rigid conductor is mapped in a form which has a constant state solution in terms of suitable similarity variables. It has been shown that constant state solutions become special non-constant state solutions in the original variables. In particular, we prove that the temperature dependence of the propagation of discontinuity waves along a solid in an equilibrium state well reproduces the temperature behaviour of the second sound speed.

Dissipation Improvement of MUSCL Scheme for Computational Aeroacoustics

ABSTRACTAn upwind finite-volume scheme is studied for solving the solutions of two dimensional Euler equations. It based on the MUSCL (Monotone Upstream Scheme for Conservation Laws) approach with the Roe approximate Riemann solver for the numerical flux evaluation. First, dissipation and dispersion relation, and group velocity of the scheme are derived to analyze the capability of the proposed scheme for capturing physical waves, such as acoustic, entropy, and vorticity waves. Then the scheme is greatly enhanced through a strategy on the numerical dissipation to effectively handle aeroacoustic computations. The numerical results indicate that the numerical dissipation strategy allows that the scheme simulates the continuous waves, such as sound and sine waves, at fourth-order accuracy and captures the discontinuous waves, such a shock wave, sharply as well as most of upwind schemes do. The tested problems include linear wave convection, propagation of a sine-wave packet, propagation of discontinuous and sine waves, shock and sine wave interaction, propagation of acoustic, vorticity, and density pulses in an uniform freestream, and two-dimensional traveling vortex in a low-speed freestream.

On Discontinuity Waves in Linear Piezoelectricity

A body composed of a linear piezoelectric medium is considered. It is shown that the condition of local propagation for a singular hypersurface S of any given order r, with r≥1, can be expressed in terms of a suitable acoustic tensor. This tensor does not depend on the order r and coincides with the one used for plane progressive waves in the homogeneous case. Thus, just as in Linear Elasticity, the laws of propagation of such discontinuity waves are the same as those for plane progressive waves. For any r≥1 singular hypersurfaces are characteristic for the linear piezoelectric partial differential equations, whereas for r=0 singular hypersurfaces may be non-characteristic for such equations. A condition is written which characterizes the strong waves of order 0 that are characteristic. For the latter waves the aforementioned acoustic tensor can be used to express the condition of local propagation.