3D Wave Equation Research Articles

Critical exponent for nonlinear damped wave equations with non-negative potential in 3D

We are studying possible interaction of damping coefficients in the subprincipal part of the linear 3D wave equation and their impact on the critical exponent of the corresponding nonlinear Cauchy problem with small initial data. The main new phenomena is that certain relation between these coefficients may cause very strong jump of the critical Strauss exponent in 3D to the critical 5D Strauss exponent for the wave equation without damping coefficients.

Application of M.Riesz potentials for solving a 3D inverse problem of acoustic sounding

An inverse coefficient problem for time-dependent 3D wave equation is under consideration. We recover a spatially varying coefficient of this equation knowing special time integrals of the wave eld in an observation domain. The inverse problem has applications to the acoustic sounding, medical imaging, etc. We reduce the inverse problem to a new linear 3D Fredholm integral equation of the first kind in which the integral operator has the form of well-known M.Riesz potentials. The equation has a unique solution for a considered class of coefficients. Assuming a special scheme for recording the data of the inverse problem, we present and substantiate a numerical algorithm of solving this integral equation. The algorithm does not require significant computational resources and a long solution time. It is based on the use of fast Fourier transform. Typical results of solving 3D inverse problem in question on a personal computer for simulated data demonstrate high capabilities of the proposed algorithm.

Improved well-posedness for the quadratic derivative nonlinear wave equation in 2D

In this paper we consider the Cauchy problem for the nonlinear wave equation (NLW) with quadratic derivative nonlinearities in two space dimensions. Following Grünrock's result in 3D, we take the data in the Fourier-Lebesgue spaces Hˆsr, which coincide with the Sobolev spaces of the same regularity for r=2, but scale like lower regularity Sobolev spaces for 1<r<2. We show local well-posedness (LWP) for the range of exponents s>1+32r, 1<r≤2. On one end this recovers the sharp result on the Sobolev scale, H74+, while on the other end establishes the Hˆ521+ result, which scales like the Sobolev H32+, thus, corresponding to a 14 derivative improvement on the Sobolev scale.

On the 1d wave equation in time-dependent domains and the problem of debond initiation

Motivated by a debonding model for a thin film peeled from a substrate, we analyse the one-dimensional wave equation, in a time-dependent domain which is degenerate at the initial time. In the first part of the paper we prove existence for the wave equation when the evolution of the domain is given; in the second part of the paper, the evolution of the domain is unknown and is governed by an energy criterion coupled with the wave equation. Our existence result for such coupled problem is a contribution to the study of crack initiation in dynamic fracture.

Partitioned Finite Element Method for port-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2D wave equations

A 2D wave equation with boundary damping of impedance type can be recast into an infinite-dimensional port-Hamiltonian system (pHs) with an appropriate feedback law, where the structure operator J is formally skew-symmetric. It is known that the underlying semigroup proves dissipative, even though no dissipation operator R is to be found in the pHs model. The Partitioned Finite Element Method (PFEM) introduced in Cardoso-Ribeiro et al. (2018), is structure-preserving and provides a natural way to discretize such systems. It gives rise to a non null symmetric matrix R. Moreover, since this matrix accounts for boundary damping, its rank is very low: only the basis functions at the boundary have an influence. Lastly, this matrix can be factorized out when considering the boundary condition as a feedback law for the pHs, involving the impedance parameter. Note that pHs – as open system – is used here as a tool to accurately discretize the wave equation with boundary damping as a closed system. In the worked-out numerical examples in 2D, the isotropic and homogeneous case is presented and the influence of the impedance is assessed; then, an anisotropic and heterogeneous wave equation with space-varying impedance at the boundary is investigated.

Stability analysis of a 1D wave equation with a nonmonotone distributed damping

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation subject to a nonmonotone distributed damping. A well-posedness result is provided together with a precise characterization of the asymptotic behavior of the trajectories of the system under consideration. The well-posedness is proved in the nonstandard Lp functional spaces, with p ∈ [2, ∞], and relies mostly on some results collected in Haraux (2009). The asymptotic behavior analysis is based on an attractivity result on a specific infinite-dimensional linear time-variant system.

On the temporal basis function, stabilisation and acceleration of the boundary element method for the 3D wave equation

On the temporal basis function, stabilisation and acceleration of the boundary element method for the 3D wave equation

Long range numerical simulation of acoustical shock waves in a 3D moving heterogeneous and absorbing medium

Acoustical shock waves can be generated by numerous atmospheric sources, either natural – like thunder and volcanoes – or anthropic – like explosions, sonic boom or buzz saw noise. The prediction of their long-range propagation remains a numerical challenge at 3D because of the large propagation distance to wavelength ratio, and of the high frequency / small wavelength content associated to shocks. In this paper, an original numerical method for propagating acoustical shock waves in three-dimensional heterogeneous media is proposed. Heterogeneities can result from temperature or density gradients and also from atmospheric shear and turbulent flows. The method called FLHOWARD (for FLow and Heterogeneities in a One-Way Approximation of the nonlineaR wave equation in 3D) is based on a one-way solution of a generalized nonlinear wave equation. Even though backscattering is neglected, it does not suffer from the limitations of classical ray theory nor from the angular limitations of the popular parabolic methods. The numerical approach is based on a split-step method, which has the advantage of splitting the original equation into simpler ones associated with specific physical mechanisms: diffraction, flows, heterogeneities, nonlinearities, absorption and relaxation. The method has been developed on parallel architecture for very high demanding 3D configurations using the Single Method Multiple Data paradigm. The method is validated through several test cases. A study of the lateral cut-off of the sonic boom finally illustrates the potentialities of the method for realistic cases.

Retarded potentials for wave propagation in elastic trusses

AbstractA space‐time boundary element method for the dynamic simulation of simplified elastic truss systems is proposed. The considered truss systems consist of elastic rods whose dynamic behaviour is governed solely by the 1D wave equation. By using time domain boundary integral equations the problem reduces to the nodes of the truss system and therefore no spatial discretization is necessary. The discretization employs an adaptive mesh refinement scheme enabling the accurate resolution of non‐smooth solutions.

A New Fast Algorithm Based on Half-Step Discretization for 3D Quasilinear Hyperbolic Partial Differential Equations

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function [Formula: see text] For the derivation of the method, we use two inner half-step points each in [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.

An improved pseudospectral meshless radial point interpolation (PSMRPI) method for 3D wave equation with variable coefficients

In this paper, a pseudospectral meshless radial point interpolation (PSMRPI) technique is applied to the three-dimensional wave equation with variable coefficients subject to given appropriate initial and Dirichlet boundary conditions. The present method is a kind of combination of meshless methods and spectral collocation techniques. The point interpolation method along with the radial basis functions is used to construct the shape functions as the basis functions in the frame of the spectral collocation methods. These basis functions will have Kronecker delta function property, as well as unitary possession. In the proposed method, operational matrices of higher order derivatives are constructed and then applied. The merit of this innovative method is that, it does not require any kind of integration locally or globally over sub-domains, as it is essential in meshless methods based on Galerkin weak forms, such as element-free Galerkin and meshless local Petrov–Galerkin methods. Therefore, computational cost of PSMRPI method is low. Further, it is proved that the procedure is stable with respect to the time variable over some conditions on the 3D wave model, and the convergence of the technique is revealed. These latest claims are also shown in the numerical examples, which demonstrate that PSMRPI provides excellent rate of convergence.

Exact solution for the acoustical impulse response of a line source above an absorbing plane.

The solution of the two-dimensional (2D) wave equation excited by a Dirac delta function above an infinite plane is the 2D impulse response of the half-space. Equivalently, it can be considered as the sound field generated by an infinite line source in three-dimensional (3D) space. For a point source in 3D, it is possible to express the resulting impulse function in closed form if the infinite plane is characterized by a masslike or pure absorbing behavior. Based on the 3D impulse response above an absorbing plane, a closed form solution for a line source over absorbing ground is found. This solution consists of the direct pressure contribution plus a reflected signal. The analytical expression for the reflection factor R is the main result of the work. R is a simple algebraic function of impedance, time, and spatial coordinates. The acoustical field generated by a line source with a general time dependence can be constructed by convolution. For a Gaussian signal, the resulting pressure is computed numerically. For a rectangular pulse, an analytical solution in 3D and 2D is derived. The deviations from the numerically obtained results are very small.

Three-dimensional complex variable element-free Galerkin method

• The complex variable moving least squares (CVMLS) approximation is developed for 3D functions with 2D basis functions. • The complex variable element-free Galerkin (CVEFG) method is developed for 3D elliptic problems and 3D wave equations. • Error and convergence of the CVMLS approximation and the CVEFG method are analyzed theoretically. • Numerical results reveal that the present method has better accuracy and computational efficiency than other methods. The complex variable element-free Galerkin (CVEFG) method is an efficient meshless Galerkin method that uses the complex variable moving least squares (CVMLS) approximation to form shape functions. In the past, applications of the CVMLS approximation and the CVEFG method are confined to 2D problems. This paper is devoted to 3D problems. Computational formulas and theoretical analysis of the CVMLS approximation on 3D domains are developed. The approximation of a 3D function is formed with 2D basis functions. Compared with the moving least squares approximation , the CVMLS approximation involves fewer coefficients and thus consumes less computing times. Formulations and error analysis of the CVEFG method to 3D elliptic problems and 3D wave equations are provided. Numerical examples are given to verify the convergence and accuracy of the method. Numerical results reveal that the CVEFG method has better accuracy and higher computational efficiency than other methods such as the element-free Galerkin method.

Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

Wave propagation in elastic trusses: An approach via retarded potentials

We propose a space–time boundary element method for the dynamic simulation of elastic truss systems. The considered truss systems consist of several members, where in each elastic rod the dynamic behaviour is governed by the 1D wave equation. The time domain fundamental solution and boundary integral equations are used to establish the dynamic Dirichlet-to-Neumann map for a single rod. Thus, we are able to reduce the problem to the nodes of the truss system and therefore only a temporal discretization at the truss nodes is necessary. We introduce a stepwise solution strategy with local step size which ensures stability. Furthermore, the discretization within each of these time steps can be refined adaptively to reduce the approximation error efficiently. The optimal convergence of the method is demonstrated in numerical examples. Due to adaptive refinement, this optimal convergence rate is retained even for non-smooth solutions. Finally, the method is applied to study typical truss systems.

Vibrationally directed assembly of micro- and nanoparticle-polymer composites

In this study, we examine directed self-assembly of micro- and nanoparticles on a vibrating substrate as a viable pathway to large-scale assembly of microstructures and composite materials. We demonstrate the vibration-driven assembly of glass bead microparticles and iron oxide nanoparticles in contact with a photocurable hydrogel (PEGDA) over an area of 3000 mm2. The competition between acoustic radiation force and vibration-generated fluid flow in a viscous medium above a vibrating plate determines the particle transport characteristics. Based on a suspension balance model of this competition, we find that glass microparticles are dominated by displacement gradients and migrate towards displacement anti-nodes. Iron oxide nanoparticles that are smaller than the characteristic boundary layer generated by the flow will drive particles towards displacement nodes. We find close agreement between the observed experimental results when compared to a numerical solution to the 2D wave equation that governs this case. We also demonstrate that patterns assembled by vibration for glass microparticles or iron oxide nanoparticles dispersed in PEGDA can be immobilized by a UV light, allowing this approach to be used as a fabrication process for heterogeneously structured particle-polymer composites. The composites produced by this technique are robust and can be held by hand for application to tunable material properties for applications to bioelectronics and soft robotics.This work has been selected by the Editors as a Featured Cover Article for this issue.

Adaptive error feedback regulation problem for 1D wave equation

SummaryBy using the adaptive control approach, we solve the error feedback regulator problem for the one‐dimensional wave equation with a general harmonic disturbance anticollocated with control and with two types of disturbed measurements, ie, one collocated with control and the other anti‐collocated with control. Different from the classical error feedback regulator design, which is based on the internal mode principle, we give the adaptive servomechanism design for the system by making use of the measured tracking error (and its time derivative) and the estimation mechanism for the parameters of the disturbance and of the unknown reference. Constructing auxiliary systems and observer and applying the backstepping method for infinite‐dimensional system play important roles in the design. The control objective, which is to regulate the tracking error to zero and to keep the states bounded, is achieved.

Space-time L 2 estimates, regularity and almost global existence for elastic waves

Abstract In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .

PRACTICAL ERROR ANALYSIS FOR THE THREE-LEVEL BILINEAR FEM AND FINITE-DIFFERENCE SCHEME FOR THE 1D WAVE EQUATION WITH NON-SMOOTH DATA

We deal with the standard three-level bilinear FEM and finite-difference scheme with a weight to solve the initial-boundary value problem for the 1D wave equation. We consider the rich collection of initial data and the free term which are the Dirac δ-functions, discontinuous, continuous but with discontinuous derivatives and from the Sobolev spaces, accomplish the practical error analysis in the L2, L1, energy and uniform norms as the mesh re_nes and compare results with known theoretical error bounds.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.