Local Analytic Solutions Research Articles

A bathtub model with nonlinear velocity–density relation

The bathtub model is an applicable and effective tool for simulating the characteristics of traffic dynamics in downtown areas during morning rush hour, especially the characteristics of hypercongested traffic. In order to reduce the complexity of the research, previous studies have adopted a linear velocity–density relationship. This has limitations in analyzing the cost and duration of hypercongestion in traffic dynamics. In this paper, we develop a bathtub model with one kind of nonlinear velocity–density Relation, i.e., Pipes–Munjal Relation, based on continuous scheduling preferences and analyze its solutions in both no-toll user equilibrium and social optimum states. Due to the intractability in solving this bathtub model, there exists local analytical solutions, but for a complete solution of the model, only numerical solutions are available. The results indicate that as the value of n increases, the model can simulate traffic dynamics with higher cost and longer duration of hypercongestion under equilibrium, and can simulate traffic states with lower system cost under social optimum. We also explore the results of implementing optimal time-varying toll to realize social optimum. This study expands the relevant theories of bathtub models and provides a new perspective for traffic congestion.

Comparison of Hydrodynamic Processes Modelling Results in Shallow Water Bodies Based on 3D Model and 2D Model Averaged by Depth

Introduction. Two-dimensional hydrodynamic models have proven their ability to adequately describe the processes of runoff and transportation in rivers, lakes, estuaries, deltas and seas. Practice shows that even where significant three-dimensional effects are expected, for example, with wind flows, a two-dimensional approach can work effectively. However, in some cases, the two-dimensional model does not accurately reflect the actual flow structures. For example, in shallow waters with complex bathymetry, heterogeneous terrain and dynamics can lead to a non-uniform velocity profile. The aim of the study is to develop a basis for determining in which cases a two-dimensional model averaged in depth is sufficient for modelling hydrodynamic processes in shallow waters like the Azov Sea, and in which cases it is advisable to use a three-dimensional model to obtain accurate results.Materials and Methods. Local analytical solutions have been obtained for the propagation of the predominant singular progressive wave in a shallow, well-mixed reservoir. Advective terms and Coriolis terms are neglected, the vortex viscosity is assumed to be constant, and the lower friction term is linearized. Special attention is paid to the latter, since the characteristics of the models significantly depend on the method of determining the coefficients of lower friction. The analytical method developed in the study shows that certain combinations of higher flow velocities (u ≈˃ 1 m/s) and water depths (d ˃ 50 m) can cause significant differences between the results of the depth-averaged model and the model containing vertical information.Results. The results obtained are verified by numerical simulation of stationary and non-stationary periodic flows in a schematized rectangular basin. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modelling averaged in depth. Both simulations show good compliance with analytical solutions.Discussion and Conclusions. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear effects — nonlinearities caused by higher-order terms in the equations of motion, i.e. terms of advective acceleration and friction, and nonlinear effects caused by geometric nonlinearities, this is due, for example, to different water depths and reservoir widths, which will be important when modelling a real sea.

Piece-wise analytic solution with double shocks for 3-dimensional isentropic irrotational flow

We study the generalized Riemann problem for the 3-dimensional irrotational isentropic flow with analytic data. The existence of local analytic solutions containing two shock fronts is established under minimal compatibility conditions.

A modified Lyapunov method and its applications to ODE

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard‐Lindelöf method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in physics, in which we show that our method possesses an excellent precision even with low iterations.

Shallow Fully Connected Neural Network Training by Forcing Linearization into Valid Region and Balancing Training Rates

A new supervisory training rule for a shallow fully connected neural network (SFCNN) is proposed in this present study. The proposed training rule is developed based on local linearization and analytical optimal solutions for linearized SFCNN. The cause of nonlinearity in neural network training is analyzed, and it is removed by local linearization. The optimal solution for the linearized SFCNN, which minimizes the cost function for the training, is analytically derived. Additionally, the training efficiency and model accuracy of the trained SFCNN are improved by keeping estimates within a valid range of the linearization. The superiority of the proposed approach is demonstrated by applying the proposed training rule to the modeling of a typical nonlinear pH process, Boston housing prices dataset, and automobile mileage per gallon dataset. The proposed training rule shows the smallest modeling error and the smallest iteration number required for convergence compared with several previous approaches from the literature for the case study.

Local and global analyticity for the μ-Novikov equation

ABSTRACT Considered herein is the Cauchy problem of the μ-Novikov equation in the periodic setting. We first prove the existence and uniqueness of local analytic solution of this problem by using the autonomous Ovsyannikov theorem. Then based on the local well-posedness and the blow-up scenario of this problem, the existence of the global solution is established. Finally, we obtain the global-in-time analyticity of the solution of this problem.

Nonlinear mixed-dimension model for embedded tubular networks with application to root water uptake

We present a numerical scheme for the solution of nonlinear mixed-dimensional PDEs describing coupled processes in embedded tubular network system in exchange with a bulk domain. Such problems arise in various biological and technical applications such as in the modeling of root-water uptake, heat exchangers, or geothermal wells. The nonlinearity appears in form of solution-dependent parameters such as pressure-dependent permeability or temperature-dependent thermal conductivity. We derive and analyze a numerical scheme based on distributing the bulk-network coupling source term by a smoothing kernel with local support. By the use of local analytical solutions, interface unknowns and fluxes at the bulk-network interface can be accurately reconstructed from coarsely resolved numerical solutions in the bulk domain. Numerical examples give confidence in the robustness of the method and show the results in comparison to previously published methods. The new method outperforms these existing methods in accuracy and efficiency. In a root water uptake scenario, we accurately estimate the transpiration rate using only a few thousand 3D mesh cells and a structured cube grid whereas other state-of-the-art numerical schemes require millions of cells and local grid refinement to reach comparable accuracy.

Nonlocal wrinkling instabilities in bilayered systems using peridynamics

Wrinkling instabilities occur when a stiff thin film bonded to an elastic substrate undergoes compression. Regardless of the nature of compression, this phenomenon has been extensively studied through local models based on classical continuum mechanics. However, the experimental behavior is not yet fully understood and the influence of nonlocal effects remains largely unexplored. The objective of this paper is to fill this gap from a computational perspective by investigating nonlocal wrinkling instabilities in a bilayered system. Peridynamics (PD), a nonlocal continuum formulation, serves as a tool to model nonlocal material behavior. This manuscript presents a methodology to precisely predict the critical conditions by employing an eigenvalue analysis. Our results approach the local solution when the nonlocality parameter, the horizon size, approaches zero. An experimentally observed influence of the boundaries on the wave pattern is reproduced with PD simulations which suggests nonlocal material behavior as a physical origin. The results suggest that the level of nonlocality of a material model has quantitative influence on the main wrinkling characteristics, while most trends qualitatively coincide with predictions from the local analytical solution. However, a relation between the film thickness and the critical compression is revealed that is not existent in the local theory. Moreover, an approach to determine the peridynamic material parameters across a material interface is established by introducing an interface weighting factor. This paper, for the first time, shows that adding a nonlocal perspective to the analysis of bilayer wrinkling by using PD can significantly advance our understanding of the phenomenon.

An efficient analytical global–local (AGL) analysis of the Lamb wave scattering problem for detecting a horizontal crack in a stiffened plate

This paper addresses a forward problem using an analytical global–local (AGL) method based on the physics of the Lamb wave propagation in the presence of a crack in the complex structure which would be beneficial for the inverse problem of damage detection. Discontinuity in a structure acts as a damage source and interacts with a Lamb wave. The global analytical solution determines the wave generation by a transmitter, wave propagation through the structure, and detection by a receiver. The local analytical solution determines the scattering coefficients for the Lamb waves at the discontinuity location. These frequency-dependent scatter coefficients are calculated considering transmission, reflection, and mode conversion. An analytical method called “complex mode expansion with vector projection (CMEP)” is used to calculate the scattering coefficients of Lamb wave modes from geometric discontinuities. The scattered wave field from a discontinuity is expanded in terms of complex Lamb wave modes with unknown scatter coefficients. These unknown coefficients are obtained from the boundary conditions using a vector projection utilizing the power expression. Two test cases are considered in this paper: (a) a plate with a pristine stiffener and (b) a plate with a cracked stiffener. Complex-valued scattering coefficients are calculated from 50 to 350 kHz for S0 incident waves. Scatter coefficients are compared for both cases to identify the suitable frequency range to excite a Lamb wave to detect the crack. The frequency-dependent complex-valued scattering coefficients are then inserted into the global analytical model. Therefore, in combination AGL method provides the exact analytical Lamb wave solution for the simulation of Lamb wave propagation and interaction with a discontinuity. By comparing the waveforms for both pristine stiffener and cracked stiffener, the crack can be detected. An FEM transient analysis was also performed to calculate the scattered wave signals. FEM results agree well with the AGL predicted results. An experiment was also performed to validate the AGL result. The obtained experimental results match well with the CMEP analytical predictions. The present AGL method is a highly computationally efficient simulation approach which allows performing virtual experiments for structural health monitoring applications.

A unified space–time fully-transversed general nodal expansion method for multidimensional and multigroup steady and transient nuclear reactor neutronics problems

A new unified space–time fully-transversed general nodal expansion method (STFT_GNEM) is developed to solve the multidimensional and multigroup steady and transient neutron diffusion problems by making full use of the transverse-integration technique and the respective advantages of the two popular nodal methods, the analytical nodal method (ANM) and the nodal expansion method (NEM). Specifically, the developed STFT_GNEM can easily obtain the ANM’s local analytical solutions of all the transverse-integrated equations in the spatial coordinate directions based on the basis function expansion framework of the traditional NEM. The temporal operators are appropriately discretized using the transverse-integration procedure in the same way as the spatial operators and then the transverse-integrated equations in the temporal direction are also analytically solved so that both the steady and transient nuclear reactor neutronics problems can be easily integrated into a unified nodal discretization formulation. Besides, the STFT_GNEM code can be easily developed from the existing NEM code. In this paper, the TWIGL reactor kinetics problem, the LRA super prompt-critical transient problem and the three dimensional transient LMW problem are analyzed to test the numerical properties of STFT_GNEM. Numerical results show that the STFT_GNEM code can track the reference solutions very well even on the coarse meshes and for large time-step sizes, which show the effectiveness and the potential of STFT_GNEM for steady and transient nuclear reactor neutronics problems.

General nodal expansion method for multi-dimensional neutronics/thermal-hydraulic coupled problems in pebble-bed core systems

Motivated by the high efficiency and accuracy of nodal methods on the coarse meshes and the desire to solve the large-scale coupled problems in pebble-bed high temperature gas cooled reactor (HTR), a general nodal expansion method (GNEM) is successfully developed and extended to solve multi-dimensional neutronics/thermal-hydraulic(N/TH) coupled problems in pebble-bed reactor core systems. The developed GNEM combines the respective advantages of traditional nodal expansion methods (NEM) and modified nodal integral methods (MNIM). Specifically, the local analytical solutions of transverse integral equations in MNIM are obtained with the basis function expansion framework and nodal balance equation in NEM. The unified discrete formulation is also developed so that it’s convenient for GNEM to solve neutron diffusion equations, convention diffusion equations, pressure poisson equations, heat transfer equations and porous media flow models in the pebble bed core systems. Then all the governing equations of the N/TH coupled problems can be discretized by the GNEM on the coarse meshes so that the computational scale of coupled problems is greatly reduced for the desired precision. To achieve higher accuracy for the coupled problems on the coarse meshes, the high-order information is successfully transferred between the coupled terms by using GNEM. Finally, the coupled GNEM code is formally developed for N/TH coupled problems in pebble-bed reactor core systems. Numerical results show that the coupled GNEM code can track the reference solutions very well on the coarse meshes and show the high-accuracy advantage and the potential of GNEM for complicated multi-dimensional N/TH coupled problems. Further research is needed to add the coupled GNEM to the HTR analysis code-TINTE in order to simultaneously and effectively solve multi-dimensional coupled problems for realistic HTR reactor systems.

The SPACE 1.0 model: a Landlab component for 2-D calculation of sediment transport, bedrock erosion, and landscape evolution

Abstract. Models of landscape evolution by river erosion are often either transport-limited (sediment is always available but may or may not be transportable) or detachment-limited (sediment must be detached from the bed but is then always transportable). While several models incorporate elements of, or transition between, transport-limited and detachment-limited behavior, most require that either sediment or bedrock, but not both, are eroded at any given time. Modeling landscape evolution over large spatial and temporal scales requires a model that can (1) transition freely between transport-limited and detachment-limited behavior, (2) simultaneously treat sediment transport and bedrock erosion, and (3) run in 2-D over large grids and be coupled with other surface process models. We present SPACE (stream power with alluvium conservation and entrainment) 1.0, a new model for simultaneous evolution of an alluvium layer and a bedrock bed based on conservation of sediment mass both on the bed and in the water column. The model treats sediment transport and bedrock erosion simultaneously, embracing the reality that many rivers (even those commonly defined as bedrock rivers) flow over a partially alluviated bed. SPACE improves on previous models of bedrock–alluvial rivers by explicitly calculating sediment erosion and deposition rather than relying on a flux-divergence (Exner) approach. The SPACE model is a component of the Landlab modeling toolkit, a Python-language library used to create models of Earth surface processes. Landlab allows efficient coupling between the SPACE model and components simulating basin hydrology, hillslope evolution, weathering, lithospheric flexure, and other surface processes. Here, we first derive the governing equations of the SPACE model from existing sediment transport and bedrock erosion formulations and explore the behavior of local analytical solutions for sediment flux and alluvium thickness. We derive steady-state analytical solutions for channel slope, alluvium thickness, and sediment flux, and show that SPACE matches predicted behavior in detachment-limited, transport-limited, and mixed conditions. We provide an example of landscape evolution modeling in which SPACE is coupled with hillslope diffusion, and demonstrate that SPACE provides an effective framework for simultaneously modeling 2-D sediment transport and bedrock erosion.

Reflection and transmission of acoustic energy by an elastic plate with structural discontinuities forced by multiple angle oblique broadband sound waves

An analysis is presented for acoustic reflection and transmission from an infinite fluid-loaded plate with spatially periodic discontinuities. The plate, with similar or dissimilar acoustic fluids on both sides, is excited by an oblique wave incident acoustic field. The fully-coupled structural/acoustic problem is treated by the method of Analytical-Numerical Matching (ANM). The ANM framework separates the problem into global numerical and local analytical solutions, and handles rapid spatial variation around the structural discontinuities in closed form, improving the numerical accuracy and convergence rate. The ANM approach includes a novel way to handle difficulties associated with coincidence frequencies. The periodic spatial discontinuities, modeled by various boundary conditions, create deviations from specular directivity with multiple reflection and transmission angles, the effect being most pronounced at structural resonances. The periodic discontinuities redirect part of the structural energy in...

Local Analytic Solutions of an Iterative Functional Differential Equation with Deviating Arguments Depending on the State Derivative

Local Analytic Solutions of an Iterative Functional Differential Equation with Deviating Arguments Depending on the State Derivative

Prediction of reflection and transmission by an elastic barrier with periodic structural discontinuities forced by oblique acoustic waves

An analysis method is developed for acoustic reflection and transmission from an infinite fluid-loaded flexible barrier with spatially periodic discontinuities. The barrier has acoustic fluids on both sides, which can be dissimilar. The structure is excited by an oblique-wave incident acoustic field. The fully-coupled structural/acoustic problem is treated by the method of Analytical-Numerical Matching (ANM). The ANM framework separates the problem into a global numerical solution and local analytical solutions. ANM handles rapid spatial around the structural discontinuities, improving the accuracy and convergence rate of reflected and transmitted pressures. Furthermore, the ANM approach offers a way to handle the mathematical difficulties associated with coincidence frequencies. The periodic spatial discontinuities create variations from simple specular-like directivity with multiple reflection and transmission angles, the effect being most pronounced at structural resonances. The periodic discontinuities can be thought of as redirecting a portion of the structural energy into resonant substructures having wavenumbers different from the oblique wave forcing, reradiating reflection and transmission fields with different directivity patterns. Discrete frequency and broadband results are presented. The goal is to develop efficient first-principles methods for structural-acoustic reflection and transmission between coupled acoustic spaces and into surrounding media.

A Stable Analytical Framework for Isometric Shape-from-Template by Surface Integration.

Shape-from-Template (SfT) reconstructs the shape of a deforming surface from a single image, a 3D template and a deformation prior. For isometric deformations, this is a well-posed problem. However, previous methods which require no initialization break down when the perspective effects are small, which happens when the object is small or viewed from larger distances. That is, they do not handle all projection geometries. We propose stable SfT methods that accurately reconstruct the 3D shape for all projection geometries. We follow the existing approach of using first-order differential constraints and obtain local analytical solutions for depth and the first-order quantities: the depth-gradient or the surface normal. Previous methods use the depth solution directly to obtain the 3D shape. We prove that the depth solution is unstable when the projection geometry tends to affine, while the solution for the first-order quantities remain stable for all projection geometries. We therefore propose to solve SfT by first estimating the first-order quantities (either depth-gradient or surface normal) and integrating them to obtain shape. We validate our approach with extensive synthetic and real-world experiments and obtain significantly more accurate results compared to previous initialization-free methods. Our approach does not require any optimization, which makes it very fast.

Prediction of broadband high-frequency acoustic reflection and transmission from elastic plates with structural discontinuities

A model for the acoustic reflection and transmission for flexible boundaries with structural discontinuities is developed. The system studied is an infinite, fluid loaded plate with periodically fixed constraints that is driven by an incident acoustic field. To treat the local discontinuities of the plate, the coupled structural/acoustic problem is treated by applying the method of Analytical-Numerical Matching (ANM). The ANM framework, which separates the problem into a global numerical solution and a local analytical solution, offers an approach to handle mathematical difficulties associated with the coincidence frequencies of the system. Accurately treating the rapid spatial variation around the discontinuities improves the overall accuracy and convergence of the computational analysis to predict the resulting pressure levels. Results for an infinite plate with discontinuities are compared to a baffled plate with a fluid loading correction introduced to the structural wavenumber. Simple closed-form ban...

Normal forms of functional-differential equations of Beardon type and Briot–Bouquet type equations in the complex domain

We study local analytic solutions of the functional-differential equation of the form \({h(\psi(z)) = b(z) h(z) h^\prime(z) + d(z)h(z)^{2}}\) which are called Beardon type functional-differential equations. All functions involved are supposed to be holomorphic in a neighbourhood of zero. Special cases are the equations f(kz) = kf(z) f′(z) where k is a complex number, \({k \neq 0}\), and \({f(\varphi(z)) = a(z) f(z) f'(z)}\) with given \({\varphi}\) and a. The class of these equations is invariant under transformations \({h \to \alpha h, \alpha(z) \neq 0}\) for all z in a neighbourhood of zero, of the unknown function and \({z \to T(z)}\) of the argument z. In particular, we are interested to know under which conditions a Beardon type functional-differential equation can be transformed to the simplified (normal form) \({h(kz) = k h(z) h'(z) + c(z) h(z)^2}\) where \({k \in \mathbb {C} \backslash\left\{0\right\}}\). We solve this normal form by another transfomation to a so-called Briot–Bouquet type functional-differential equation.

Chain integral solutions to tautological systems

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system τ, arising from the universal family of Calabi–Yau hypersurfaces Y_a in a G-variety X of dimension n. First, we construct a natural topological correspondence between relative cycles in H_n (X−Y_a, ∪ D−Y_a) bounded by the union of G-invariant divisors ∪D in X to the solution sheaf of τ, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in Y_a, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of H_n (X−Y_a,∪ D−Y_a). In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann–Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.

A new run-up algorithm based on local high-order analytic expansions

The practically important problem of the wave run-up is studied in this article in the framework of Nonlinear Shallow Water Equations (NSWE). The main novelty consists in the usage of high order local asymptotic analytical solutions in the vicinity of the shoreline. Namely, we use the analytical techniques introduced by S. Kovalevskaya and the analogy with the compressible gas dynamics (i.e. gas outflow problem into the vacuum). Our run-up algorithm covers all the possible cases of the wave slope on the shoreline and it incorporates the new analytical information in order to determine the shoreline motion to higher accuracy. The application of this algorithm is illustrated in several important practical examples. Finally, the simulation results are compared with the well-known analytical and experimental predictions.