3D Wave Equation Research Articles

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough for the sound waves to decay. The computational cost is lowered by strategically truncating the computational domain with an efficient boundary treatment. One commonly used boundary treatment is the perfectly matched layer (PML), which dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable, and an optimization procedure for the damping functions is proposed to enhance the performance of the formulation.

P-304 Quantitative analysis of uterine peristalsis using optical flow for motion quantification in women with moderate/severe adenomyosis undergoing frozen-thawed embryo transfer following prolonged downregulation with GnRHanalogue

Abstract Study question What is the impact of prolonged downregulation with GnRH analogue (GnRHa) on uterine peristalsis in women with moderate/severe adenomyosis undergoing frozen-thawed embryo transfer? Summary answer Women with moderate/severe adenomyosis have lower coordination of uterine peristalsis and myometrial contractility; prolonged downregulation with GnRHa appears to offer benefit. What is known already Adenomyosis can adversely reduce the chance of pregnancy in women undergoing assisted conception, with altered uterine peristalsis proposed as a mechanism. We aimed to i) develop an objective tool to quantify uterine peristalsis on transvaginal sonography(TVS) using an algorithm based on optical flow for wave motion quantification, ii) compare uterine peristalsis between women with a normal uterus and women with moderate/severe adenomyosis, iii) evaluate the impact of prolonged downregulation with GnRH analogue for 6 weeks vs 1 week on uterine peristalsis in women with moderate/severe adenomyosis undergoing frozen-thawed embryo transfer within the MODA RCT (Modified Downregulation for Adenomyosis). Study design, size, duration Prospective observational cohort study(July 2021–Jan 2024) in women with moderate/severe adenomyosis(n = 26) and a normal uterus(n = 40) having TVS during assisted conception. Participants/materials, setting, methods We included women with moderate/severe adenomyosis or a normal uterus undergoing assisted conception.TVS was performed at baseline(BL),day 6-10 of ovarian stimulation(OS) and day of embryo transfer(ET)by three observers(A/B/C)to assess inter-/intra-observer agreement. TVS recordings(n = 146) were stored offline for optical flow analysis using 3D Slicer(Ver. 4.11). Pixel-level displacements for points at endometrial-myometrial junction(EMJ)and myometrium were used to compute frequency(f) and amplitude(u) by fitting to the 2D-wave equation, fit error was taken as a measure of coordination. Main results and the role of chance We recruited 66 women with moderate/severe adenomyosis (n = 26) or a normal uterus (n = 40) undergoing assisted conception. Uterine peristalsis at the EMJ in women with adenomyosis showed a significantly poorer fit with the ideal wave equation and a significantly higher fit error compared to women with normal uterus at all time points (BL (+ 0.649), OS (+ 0.630), ET (+ 0.685) (n = 66, p<0.001), indicating lower coordination of uterine peristalsis in women with adenomyosis compared to women with a normal uterus. Fit error of myometrial contractility was also significantly lower in women with adenomyosis at all time points (BL (+ 0.656), OS (+ 0.510) and ET (+ 0.639) (n = 66, p < 0.001). There was no significant difference in wave amplitude or frequency in women with adenomyosis compared to women with a normal uterus (p > 0.06). Women with moderate/severe adenomyosis exposed to prolonged downregulation with GnRH analogue for 6 weeks had lower fit error of uterine peristalsis (UP) and myometrial contractility (MC) on the day of embryo transfer compared to women receiving 1 week GnRH analogue (UP ET(+0.615) (n = 19, p = 0.058) (MC ET(+4.28) (n = 19, p < 0.05). Inter- and intra-observer agreement was found to be moderate to substantial across all wave features(ICC 0.58-0.79). Limitations, reasons for caution There is a need to recruit a larger cohort of patients and correlate wave features with clinical outcomes to increase the robustness of the conclusions drawn in this study. Wider implications of the findings Our algorithm for wave motion quantification offers an objective,reproducible measuring tool to quantify uterine peristalsis with potential for wide clinical application.Women with adenomyosis have higher fit error and lower coordination of uterine peristalsis and myometrial contractility,which may account for their reduction in chances of conception.Prolonged GnRHa appears to offer benefit. Trial registration number N/A

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

Adaptive identification of linear infinite-dimensional systems

We propose an adaptive algorithm for identifying the unknown parameter (possibly vector-valued) in a linear stable single-input single-output infinite-dimensional system. We assume that the transfer function of the infinite-dimensional system can be expressed as a ratio of two infinite series in s (the Laplace variable). We also assume that certain identifiability conditions, which include a persistency of excitation condition, hold. For a fixed integer n, we propose an adaptive update law (a differential equation in time) driven by real-time input-output data for estimating the first n + 1 coefficients in the numerator and the denominator of the transfer function. We show that the estimates for the transfer function coefficients generated by the update law are close to the true values at large times provided n is sufficiently large (the estimates converge to the true values as time and n tend to infinity). The unknown parameter can be reconstructed using the transfer function coefficient estimates obtained with n large and the algebraic expressions relating the transfer function coefficients to the unknown parameter. We also provide a numerical scheme for verifying the identifiability conditions and for choosing n sufficiently large so that the value of the reconstructed parameter is close to the true value. The class of systems to which our approach is applicable includes many partial differential equations with constant/spatially-varying coefficients and distributed/boundary input and output. We illustrate the efficacy of our approach using three examples: a delay system with four unknown scalars, a 1D heat equation with two unknown scalars and a 1D wave equation with an unknown linearly-varying (in space) coefficient.

Optimised correction polynomial functions for the Flux Reconstruction method in time-harmonic electromagnetism

The Flux Reconstruction (FR) method is classically used in the Computational Fluid Dynamics field. However, its use for the simulation of electromagnetic wave propagation is not as developed yet. Following on from the development of a priori error estimates for the 1D wave equations, we introduce optimisation problems to allow an adaptation of the FR correction polynomial functions to the discretisation parameters. Showing notable accuracy gains in 1D, especially in the preasymptotic regime, we generalise this procedure to the 3D Maxwell’s equations, leading to similar interesting possibilities to reduce the computational cost for a given accuracy.

Study of Support Motion of a Finite Bar with a Boundary Damper and a Spring Using Analytical Approaches and FEM

In this paper, we solve the vibration problem of a finite bar with a viscously damped boundary in conjunction with a spring on the same side subject to the support motion on the other side. To avoid the computation of complex modes and the difficulty of orthogonal conditions for the partial differential equation (PDE) of a continuous system, two alternatives are employed. One is the analytical derivation by using the diamond rule of the method of characteristics. The advantage is that this method can yield an analytical solution. However, the disadvantage is that the method can be only applied to some simple problems such as a linear system and the one-dimensional (1D) wave equation. The other numerical method, finite element method (FEM), is incorporated into a general-purpose program to solve this problem. The advantage of the FEM is that this methodology can be applied to solve various problems such as different PDEs. Therefore, the solution obtained by using the FEM is compared with that of the analytical solution. Two special cases, only a spring and a damper alone, are also considered by using the FEM. Interestingly, the same silent area is captured in the displacement profile by using the FEM for the three cases. We also find that the displacement profiles have slope discontinuity which only occurs on the characteristic line. After the wave front arrives at the boundary, various responses appear due to different boundaries, spring, damper and both. It is found that the displacement amplitude of the general case is smaller than the other two special cases. This result matches the application of engineering practice.

Dispersive and Strichartz estimates for 3D wave equation with a Laguerre potential

Abstract Dispersive and Strichartz estimates are obtained for solutions to the wave equation with a Laguerre potential in spatial dimension three. To obtain the desired dispersive estimate, based on the spectral properties of the Schrödinger operator involved, we subsequently prove the dispersive estimate for the corresponding Schrödinger semigroup, obtain a Gaussian-type upper bound, establish Bernstein-type inequalities, and finally pass to the Müller–Seeger’s subordination formula. The desired Strichartz estimates follow by the established dispersive estimate and the standard argument of Keel–Tao.

Electromagnetic waves propagation in thin heterogenous coaxial cables. Comparison between 3D and 1D models

<abstract><p>This work deals with wave propagation into a coaxial cable, which can be modelled by the 3D Maxwell equations or 1D simplified models. The usual model, called the telegrapher's model, is a 1D wave equation of the electrical voltage and current. We derived a more accurate model from the Maxwell equations that takes into account dispersive effects. These two models aim to be a good approximation of the 3D electromagnetic fields in the case where the thickness of the cable is small. We perform some numerical simulations of the 3D Maxwell equations and of the 1D simplified models in order to validate the usual model and the new one. Moreover, we show that, while the usual telegrapher model is of order one with respect to the thickness of the cable, the dispersive 1D model is of order two.</p></abstract>

Convergence analysis of a Krylov subspace spectral method for the 1D wave equation in an inhomogeneous medium

Convergence analysis of a Krylov subspace spectral method for the 1D wave equation in an inhomogeneous medium

Numerical simulation and GPU computing for the 2D wave equation with variable coefficient

Numerical simulation and GPU computing for the 2D wave equation with variable coefficient

A Hard-Constraint Wide-Body Physics-Informed Neural Network Model for Solving Multiple Cases in Forward Problems for Partial Differential Equations

In the fields of physics and engineering, it is crucial to understand phase transition dynamics. This field involves fundamental partial differential equations (PDEs) such as the Allen–Cahn, Burgers, and two-dimensional (2D) wave equations. In alloys, the evolution of the phase transition interface is described by the Allen–Cahn equation. Vibrational and wave phenomena during phase transitions are modeled using the Burgers and 2D wave equations. The combination of these equations gives comprehensive information about the dynamic behavior during a phase transition. Numerical modeling methods such as finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) are often applied to solve phase transition problems that involve many partial differential equations (PDEs). However, physical problems can lead to computational complexity, increasing computational costs dramatically. Physics-informed neural networks (PINNs), as new neural network algorithms, can integrate physical law constraints with neural network algorithms to solve partial differential equations (PDEs), providing a new way to solve PDEs in addition to the traditional numerical modeling methods. In this paper, a hard-constraint wide-body PINN (HWPINN) model based on PINN is proposed. This model improves the effectiveness of the approximation by adding a wide-body structure to the approximation neural network part of the PINN architecture. A hard constraint is used in the physically driven part instead of the traditional practice of PINN constituting a residual network with boundary or initial conditions. The high accuracy of HWPINN for solving PDEs is verified through numerical experiments. One-dimensional (1D) Allen–Cahn, one-dimensional Burgers, and two-dimensional wave equation cases are set up for numerical experiments. The properties of the HWPINN model are inferred from the experimental data. The solution predicted by the model is compared with the FDM solution for evaluating the experimental error in the numerical experiments. HWPINN shows great potential for solving the PDE forward problem and provides a new approach for solving PDEs.

Data Characterization in Dynamical Inverse Problem for the 1D Wave Equation with Matrix Potential

Data Characterization in Dynamical Inverse Problem for the 1D Wave Equation with Matrix Potential

Encouraging students to understand the 1D wave equation

Encouraging students to understand the 1D wave equation

Physics-constrained neural networks for half-space seismic wave modeling

Forward modeling of seismic waves using physics-informed neural networks (PINNs) has attracted much attention. However, a notable challenge arises when modeling seismic wave propagation in large domains (i.e., a half-space), PINNs may encounter the issue of "soft constraint failure". To address this problem, we propose a novel framework called physics-constrained neural networks (PCNNs) specifically designed for modeling seismic wave propagation in a half-space. The method of images is incorporated to effectively implement the free stress boundary conditions of the Earth's surface, leading to the successful propagation of plane waves and cylindrical waves in a half-space. We analyze the training dynamics of neural networks when solving two-dimensional (2D) wave equations from the neural tangent kernel (NTK) perspective. An adaptive training algorithm is introduced to mitigate the unbalanced gradient flow dynamics of the different components of the loss function of PINNs/PCNNs. Furthermore, to tackle the complex behavior of seismic waves in layered media, a sequential training strategy is considered to enhance network scalability and solution accuracy. The results of numerical experiments demonstrate the accuracy and effectiveness of our approach.

Uniform decay rate estimates for the 2D wave equation posed in an inhomogeneous medium with exponential growth source term, locally distributed nonlinear dissipation, and dynamic Cauchy–Ventcel–type boundary conditions

AbstractWe study the wellposedness and stabilization for a Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a exponential growth source term and a nonlinear damping distributed around a neighborhood ω of the boundary according to the geometric control condition. We, in particular, generalize substantially the work due to Almeida et al. (Commun. Contemp. Math. 23 (2021), no. 03, 1950072), in what concerns an exponential growth for source term instead of a polynomial one. We give a proof based on the truncation of a equivalent problem and passage to the limit in order to obtain in one shot, the energy identity as well as the observability inequality, which are the essential ingredients to obtain uniform decay rates of the energy. We show that the energy of the equivalent problem goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase space. One advantage of our proof is that the decay rate is independent of the nonlinearity.

Manipulating the loss calculation to enhance the training process of physics-informed neural networks to solve the 1D wave equation

Manipulating the loss calculation to enhance the training process of physics-informed neural networks to solve the 1D wave equation

A priori compression of convolutional neural networks for wave simulators

Convolutional neural networks are now seeing widespread use in a variety of fields, including image classification, facial and object recognition, medical imaging analysis, and many more. In addition, there are applications such as physics-informed simulators in which accurate forecasts in real time with a minimal lag are required. The present neural network designs include millions of parameters, which makes it difficult to install such complex models on devices that have limited memory. Compression techniques might be able to resolve these issues by decreasing the size of CNN models that are created by reducing the number of parameters that contribute to the complexity of the models. We propose a compressed tensor format of convolutional layer, a priori, before the training of the neural network. 3-way kernels or 2-way kernels in convolutional layers are replaced by one-way fiters. The overfitting phenomena will be reduced also. The time needed to make predictions or time required for training using the original Convolutional Neural Networks model would be cut significantly if there were fewer parameters to deal with. In this paper we present a method of a priori compressing convolutional neural networks for finite element (FE) predictions of physical data. Afterwards we validate our a priori compressed models on physical data from a FE model solving a 2D wave equation. We show that the proposed convolutional compression technique achieves equivalent performance in the prediction error as classical convolutional layers with fewer trainable parameters(around 20%) and lower memory footprint.

A space-time domain RBF method for 2D wave equations

In the present study, we demonstrate the feasibility to reveal the numerical solution of the multi-dimensional wave equations. A simple semi-analytical meshless method was proposed to obtain the numerical solution of the wave equation with a newly-proposed space-time radial basis function to enhance the numerical stability. The wave equation was discretized into equivalent algebraic equations. By specifying boundary and initial conditions, the wave propagation in a two-dimensional domain can be virtually reconstructed. Our results exhibit that the semi-analytical meshless method is suitable and efficient for solving multi-dimensional wave equations.

Simulations in Science and Engineering for University Education

In this paper, the strategies and tests that guarantee a reliable solution of partial differential equations (PDE) solved with the finite difference method are proposed. The elements to be considered are presented so that the solution of equations in partial derivatives is the closest to reality. The experience of the proponents and previous work play an important role in validating simulations in science and engineering problems. A series of tests are proposed that must be carried out to validate and make the simulation results reliable, due to the fact that the selection of the finite difference scheme, the initial data, the boundary conditions and the restrictions in the time step, the stability and convergence of the solution cannot be guaranteed a priori. There are situations in which the solution is stable but converges to values ??that have no physical meaning. To illustrate the strategies and tests inherent in a simulation, such that the solutions can be trusted, the 3D wave equation (three spatial dimensions and one temporal dimension) is solved numerically, by implementing a code in FORTRAN 90, implementing a scheme finite difference to second order approximation. The stability and convergence of the solution are studied according to the schemes proposed in this research, in such a way that the results are reliable, and guarantee the possible practical implementation of the solutions. Finally, the evolution of the scalar field for different times is shown, validated with the principle of conservation of energy. The strategies implemented in this work to guarantee the reliability of the simulations are also valid when solving EDP problems with neural networks, genetic algorithms and other intelligent computing techniques.

High-Order Spectral Method of Density Estimation for Stochastic Differential Equation Driven by Multivariate Gaussian Random Variables

There are some previous works on designing efficient and high-order numerical methods of density estimation for stochastic partial differential equation (SPDE) driven by multivariate Gaussian random variables. They mostly focus on proposing numerical methods of density estimation for SPDE with independent random variables and rarely research density estimation for SPDE is driven by multivariate Gaussian random variables. In this paper, we propose a high-order algorithm of gPC-based density estimation where SPDE driven by multivariate Gaussian random variables. Our main techniques are (1) we build a new multivariate orthogonal basis by adopting the Gauss–Schmidt orthogonalization; (2) with the newly constructed orthogonal basis in hand, we first assume the unknown function in the SPDE has the stochastic general polynomial chaos (gPC) expansion, second implement the stochastic gPC expansion for the SPDE in the multivariate Gaussian measure space, and third we obtain and numerical calculation deterministic differential equations for the coefficients of the expansion; (3) we used high-order algorithm of gPC-based for density estimation and moment estimation. We apply the newly proposed numerical method to a known random function, stochastic 1D wave equation, and stochastic 2D Schnakenberg model, respectively. All the presented stochastic equations are driven by bivariate Gaussian random variables. The efficiency is compared with the Monte-Carlo method based on the known random function.