Helmholtz decomposition with a scalar Poisson equation in elastic anisotropic media

P-and S-wave separation plays an important role in elastic reverse-time migration.It can reduce the artifacts caused by crosstalk between different modes and improve image quality.In addition,P-and Swave separation can also be used to better understand and distinguish wave types in complex media.At present,the methods for separating wave modes in anisotropic media mainly include spatial nonstationary filtering,low-rank approximation,and vector Poisson equation.Most of these methods require multiple Fourier transforms or the calculation of large matrices,which require high computational costs for problems with large scale.In this paper,an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain.For 2D problems,the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation.Therefore,compared with existing methods based on pseudoHelmholtz decomposition operators,this method can significantly reduce the computational cost.Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly.

Implementation of the Integrated Green’s Function Method for 3D Poisson’s Equation in a Large Aspect Ratio Computational Domain

The solution of Poisson’s Equation plays an important role in many areas, including modeling high-intensity and high-brightness beams in particle accelerators. For the computational domain with a large aspect ratio, the integrated Green’s function method has been adopted to solve the 3D Poisson equation subject to open boundary conditions. In this paper, we report on the efficient implementation of this method, which can save more than a factor of 50 computing time compared with the direct brute force implementation and its improvement under certain extreme conditions.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

Derivation of a Revised Tsiolkovsky Rocket Equation That Predicts Combustion Oscillations

Our study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be derived using a 1D elastic collision model of the momentum exchange between the differential propellant mass element (dm) and the rocket final mass (m1), in which dm initially travels forward to collide with m1 and rebounds to exit through the exhaust nozzle with a velocity that is known as the effective exhaust velocity ve. We observe that such a model does not explain how dm was able to acquire its initial forward velocity without the support of a reactive mass traveling in the opposite direction. We show instead that the initial kinetic energy of dm is generated from dm itself by a process of self-combustion and expansion. In our ideal rocket with a single particle dm confined inside a hollow tube with one closed end, we show that the process of self-combustion and expansion of dm will result in a pair of differential particles each with a mass dm/2, and each traveling away from one another along the tube axis, from the center of combustion. These two identical particles represent the active and reactive sub-components of dm, co-generated in compliance with Newton’s third law of equal action and reaction. Building on this model, we derive a linear momentum ODE of the system, the solution of which yields what we call the Revised Tsiolkovsky Rocket Equation (RTRE). We show that RTRE has a mathematical form that is similar to TRE, with the exception of the effective exhaust velocity (ve) term. The ve term in TRE is replaced in RTRE by the average of two distinct exhaust velocities that we refer to as fast-jet, vx1, and slow-jet, vx2. These two velocities correspond, respectively, to the velocities of the detonation pressure wave that is vectored directly towards the exhaust nozzle, and the retonation wave that is initially vectored in the direction of rocket propagation, but subsequently becomes reflected from the thrust surface of the combustion chamber to exit through the exhaust nozzle with a time lag behind the detonation wave. The detonation-retonation phenomenon is supported by experimental evidence in the published literature. Finally, we use a convolution model to simulate the composite exhaust pressure wave, highlighting the frequency spectrum of the pressure perturbations that are generated by the mutual interference between the fast-jet and slow-jet components. Our analysis offers insights into the origin of combustion oscillations in rocket engines, with possible extensions beyond rocket engineering into other fields of combustion engineering.

High-Order Spatial FDTD Solver of Maxwell’s Equations for Terahertz Radiation Production

We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma. We examined the performance of the applied scheme, in this context, we implemented the developed model to study selected phenomena in terahertz radiation production, such as the excitation energy and conversion efficiency of the produced THz radiation, in addition to the influence of the pulse chirping on properties of the produced radiation. The obtained numerical results have clarified that the applied HO-FDTD scheme is precisely accurate to solve Maxwell’s equations and sufficiently valid to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma.

A gated recurrent unit model to predict Poisson’s ratio using deep learning

Static Poisson’s ratio(vs)is crucial for determining geomechanical properties in petroleum applications,namely sand production.Some models have been used to predict vs;however,the published models were limited to specific data ranges with an average absolute percentage relative error(AAPRE)of more than 10%.The published gated recurrent unit(GRU)models do not consider trend analysis to show physical behaviors.In this study,we aim to develop a GRU model using trend analysis and three inputs for predicting n s based on a broad range of data,n s(value of 0.1627-0.4492),bulk formation density(RHOB)(0.315-2.994 g/mL),compressional time(DTc)(44.43-186.9 μs/ft),and shear time(DTs)(72.9-341.2μ s/ft).The GRU model was evaluated using different approaches,including statistical error an-alyses.The GRU model showed the proper trends,and the model data ranges were wider than previous ones.The GRU model has the largest correlation coefficient(R)of 0.967 and the lowest AAPRE,average percent relative error(APRE),root mean square error(RMSE),and standard deviation(SD)of 3.228%,1.054%,4.389,and 0.013,respectively,compared to other models.The GRU model has a high accuracy for the different datasets:training,validation,testing,and the whole datasets with R and AAPRE values were 0.981 and 2.601%,0.966 and 3.274%,0.967 and 3.228%,and 0.977 and 2.861%,respectively.The group error analyses of all inputs show that the GRU model has less than 5% AAPRE for all input ranges,which is superior to other models that have different AAPRE values of more than 10% at various ranges of inputs.

一种三维Poisson-Nernst-Planck方程的虚单元计算

利用虚单元方法在多面体网格上求解一种三维稳态Poisson-Nernst-Planck(PNP)方程,并给出PNP方程的虚单元离散形式,推导电势方程及离子浓度方程的刚度矩阵与荷载向量的矩阵表达式.数值实验结果表明,在3种不同的多面体网格下实现了PNP方程的虚单元计算,数值解在L^(2)和H^(1)范数下均达到最优阶.

线性Poisson-Boltzmann方程的虚单元L^(2)误差估计

针对一类线性Poisson-Boltzmann方程的虚单元L^(2)误差估计进行分析.首先引入正则化的线性Poisson-Boltzmann方程,将原问题转化为非奇性Poisson-Boltzmann方程.然后给出L^(2)范数的误差估计.最后在四边形和五边形混合多边形网格上进行数值实验,数值结果验证了理论分析的正确性.

Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions

We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this work.In particular,to better capture the sharpness of the solution,we propose the adaptive sampling methods(ASMs)based on the residual and the gradient of the solution.We first present a residual only-based ASM denoted by ASMⅠ.In this approach,we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains,then we add new residual points in the sub-domain which has the largest mean absolute value of the residual,and those points which have the largest absolute values of the residual in this sub-domain as new residual points.We further develop a second type of ASM(denoted by ASMⅡ)based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the solution.The procedure of ASMⅡis almost the same as that of ASMⅠ,and we add new residual points which have not only large residuals but also large gradients.To demonstrate the effectiveness of the present methods,we use both ASMⅠand ASMⅡto solve a number of PDEs,including the Burger equation,the compressible Euler equation,the Poisson equation over an Lshape domain as well as the high-dimensional Poisson equation.It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASMⅠor ASMⅡ,and both methods deliver much more accurate solutions than the original PINNs with the same number of residual points.Moreover,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠalgorithm.This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution.Furthermore,we also employ the similar adaptive sampling technique for the data points of boundary conditions(BCs)if the sharpness of the solution is near the boundary.The result of the L-shape Poisson problem indicates that the present method can significantly improve the efficiency,stability,and accuracy.

Topology Optimization of Metamaterial Microstructures for Negative Poisson’s Ratio under Large Deformation Using a Gradient-Free Method

Negative Poisson’s ratio(NPR)metamaterials are attractive for their unique mechanical behaviors and potential applications in deformation control and energy absorption.However,when subjected to significant stretching,NPR metamaterials designed under small strain assumption may experience a rapid degradation in NPR performance.To address this issue,this study aims to design metamaterials maintaining a targeted NPR under large deformation by taking advantage of the geometry nonlinearity mechanism.A representative periodic unit cell is modeled considering geometry nonlinearity,and its topology is designed using a gradient-free method.The unit cell microstructural topologies are described with the material-field series-expansion(MFSE)method.The MFSE method assumes spatial correlation of the material distribution,which greatly reduces the number of required design variables.To conveniently design metamaterials with desired NPR under large deformation,we propose a two-stage gradient-free metamaterial topology optimization method,which fully takes advantage of the dimension reduction benefits of the MFSE method and the Kriging surrogate model technique.Initially,we use homogenization to find a preliminary NPR design under a small deformation assumption.In the second stage,we begin with this preliminary design and minimize deviations in NPR from a targeted value under large deformation.Using this strategy and solution technique,we successfully obtain a group of NPR metamaterials that can sustain different desired NPRs in the range of[−0.8,−0.1]under uniaxial stretching up to 20% strain.Furthermore,typical microstructure designs are fabricated and tested through experiments.The experimental results show good consistency with our numerical results,demonstrating the effectiveness of the present gradientfree NPR metamaterial design strategy.

一类非线性Poisson-Nernst-Planck方程的边平均有限元计算

针对一类非线性Poisson-Nernst-Planck数学模型,为提高数值求解效率并保证数值求解稳定性,推导了边平均有限元离散格式,并给出了数值求解的耦合迭代算法。在对网格进行一些适当假设的情况下,边平均有限元离散格式的刚度矩阵是一个M-阵,数值求解比标准有限元方法更稳定。数值结果表明,边平均有限元方法的L^(2)模误差收敛阶达到最优阶,且在自由度相同情况下,边平均有限元方法所用CPU时间大约是标准有限元方法的1/3。

Poisson-Nernst-Planck方程组大解的整体存在性的一个新证明方法

目的研究来源于半导体理论中刻画带电粒子漂移和扩散现象的Poisson-Nernst-Planck方程组,建立该方程组大解的整体存在性。方法利用加权法、扰动法和反证法进行研究。结果通过引入新的权函数建立了扰动方程在Chemin-Lerner混合时空空间中的非线性估计,然后利用反证法建立了Poisson-Nernst-Planck方程组大解的整体存在性。结论2类带电粒子密度函数之差在适定性理论中起着更重要的作用。

一类含时Poisson-Nernst-Planck方程的虚单元计算

针对一类含时Poisson-Nernst-Planck(PNP)方程,为避免在解决实际问题时有限元法中的网格适应性问题,构造了L^(2)投影算子与Gummel迭代相结合的虚单元算法。该算法允许以更简单的方式设计和分析新的格式,可以灵活处理各种网格,对于多边形或多面体单元甚至非凸单元组成的网格剖分都可以很好地处理,使得虚单元法可以适应于任意多边形网格,大大降低了网格的生成难度。给出了虚单元算法在三角形网格、四边形网格、非凸网格下的数值算例。数值实验结果表明,在这3种多边形网格上,L^(2)和H^(1)模的收敛阶分别为二阶和一阶,均达到了最优阶。

基于重心插值配点法求解变系数广义Poisson方程

提出了一种求解二维变系数广义Poisson方程的重心插值配点法,重心插值配点法是一种真正的无网格配点方法,其在计算域内不用划分网格,得出的数值解在一定误差范围内可以无限接近精确解,同时具有很好的数值稳定性.本文中的数值算例均采用重心有理插值配点法和重心Lagrange插值配点法来计算,然后通过与解析解的比较,可以得出该方法具有快速、准确和易于数值计算的优点,是一种较好的数值计算方法.

带有投资收益率和双保费复合Poisson-Geometric风险模型的研究

研究了保费收入为线性增长和随机保费的风险模型,且随机保费的保单数服从复合Poisson过程,理赔次数服从复合Poisson-Geometric过程.应用全概率公式和积分变换公式,推导了该模型Gerber-Shiu折现罚金函数满足的更新方程,并当随机保费、理赔过程均服从特定指数分布时,得到了该模型破产概率的解析解,最后通过数值模拟对理论进行了分析验证.

A LARGE DEVIATION PRINCIPLE FOR THE STOCHASTIC GENERALIZED GINZBURG-LANDAU EQUATION DRIVEN BY JUMP NOISE

In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise.Here,we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle,which was initially proposed by Matoussi,Sabbagh and Zhang(2021).

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Tiêu Equation Experimentation of Understanding by Energy Transfer Quantum Mechanics

Background: The Tiêu equation has a ground roots approach to the process of Quantum Biology and goes deeper through the incorporation of Quantum Mechanics. The process can be measured in plant, animal, and human usage through a variety of experimental or testing forms. Animal studies were conducted for which, in the first day of the study all the animals consistently gained dramatic weight, even as a toxic substance was introduced as described in the introduction of the paper to harm animal subjects which induced weight loss through toxicity. Tests can be made by incorporating blood report results. Human patients were also observed to show improvement to their health as administration of the substance was introduced to the biological mechanism and plants were initially exposed to the substance to observe results. This is consistent with the Tiêu equation which provides that wave function is created as the introduction of the substance to the biological mechanism which supports Quantum Mechanics. The Tiêu equation demonstrates that Quantum Mechanics moves a particle by temperature producing energy thru the blood-brain barrier for example. Methods: The methods for the Tiêu equation incorporate animal studies to include the substance administered through laboratory standards using Good Laboratory Practices under Title 40 C.F.R. § 158. Human patients were treated with the substance by medical professionals who are experts in their field and have knowledge to the response of patients. Plant applications were acquired for observation and guidance of ongoing experiments of animals’ representative for the biologics mechanism. Results: The animal studies along with patient blood testing results have been an impressive line that has followed the Tiêu equation to consistently show improvement in the introduction of the innovation to biologic mechanisms. The mechanism responds to the substance by producing energy to the mechanism with efficient effect. For plant observations, plant organisms responded, and were seen as showing improvement thru visual observation.

Solution of Laguerre’s Differential Equations via Modified Adomian Decomposition Method

This paper presents a technique for obtaining an exact solution for the well-known Laguerre’s differential equations that arise in the modeling of several phenomena in quantum mechanics and engineering. We utilize an efficient procedure based on the modified Adomian decomposition method to obtain closed-form solutions of the Laguerre’s and the associated Laguerre’s differential equations. The proposed technique makes sense as the attitudes of the acquired solutions towards the neighboring singular points are correctly taken care of.