非线性无力场的Low-Lou解法探讨

The equations of motion for a hydrodynamic solid are studied for a pressure-density relation appropriate to pulse propagation in solid and compactible media. After collapse of the compactible material, the pressure-density relation is assumed to have the form P = A[(ρ/ρ0)γ−1]. A semianalytical method of solution is used on the resulting one-dimensional set of nonlinear partial differential equations. By using characteristic variables, the nonlinear differential equations are transformed to a set of linear partial differential equations whose solution is then obtained numerically. Applications are made to aluminum, for which the constants A and γ are known, and to compactible materials using an assumed value for γ. This investigation serves to define the various types of phenomena that can occur in a semi-infinite compactible material that is shocked by a rectangular pressure pulse. In particular, the analysis shows that the shock-wave penetration depth at the end of the first shock-rarefaction interaction has a minimum when regarded as a function of the material porosity.

A number of stability results for general systems of linear and nonlinear partial differential equations that describe distributed parameter networks and systems are obtained by means of Liapunov's direct method. Conditions for the asymptotic stability of the null solution of linear time-invariant distributed parameter systems are given. This result is then extended to a class of distributed parameter systems consisting of a linear system and a zero-memory nonlinear elment described by a set of Lurie-like partial differential equations. Bounded input-bounded output stability for the above two classes of systems is established by extension of a result of Goldwyn, Chao, and Chang for lumped parameter systems. Finally, it is shown how Malkin's theorem on the stability under persistent disturbances can be extended to provide a solution to the dynamic range problem for systems described by a general set of nonlinear partial differential equations.

This work explores the evolution of pore structure by considering the coupling effect of ionic transport and chemical reaction, especially salt precipitation. The reactive transport system is based on two sets of equations: a set of partial differential equations based on mass conservation for the ionic transport, and a set of nonlinear algebraic equations based on the law of mass action for the chemical processes. By adopting this model, transport properties such as ionic concentration and mass of chemical components are implemented. The porosity and the water saturation degree are also considered and examined in our work. Besides, we propose a probabilistic-based porous network representing the evolution of microstructure induced by precipitation/dissolution. The probabilistic porous network is the interconnection between the microstructure and the macro properties. With a given water saturation degree and porosity calculated by the reactive transport model, we could establish the water-dependent evolution of pore structure in cement-based materials. Such a numerical model could be used to interpret the mechanism of the local precipitation/dissolution process in pore scales, which cannot be implemented by experimental measurements.

Over the last couple of decades the demand for high precision and enhanced performance of physical systems has been steadily increasing. This demand often results in miniaturization and complex design, thus increasing the need for complex nonlinear control methods. Some of the state of the art nonlinear methods are stymied by the requirement of full state information, model and parameter uncertainties, mathematical complexity, etc. For many scenarios it is nearly impossible to consider all the uncertainties during the design of a feedback controller. Additionally, while designing a modelbased nonlinear control there is no standard mechanism to incorporate performance measures. Some of the mentioned issues can be addressed by using online learning. Animals and humans have the ability to share, explore, act or respond, memorize the outcome and repeat the task to achieve a better outcome when they encounter the same or a similar scenario. This is called learning from interaction. One instance of this approach is reinforcement learning (RL). However, RL methods are hindered by the curse of dimensionality, non-interpretability and non-monotonic convergence of the learning algorithms. This can be attributed to the intrinsic characteristics of RL, as it is a modelfree approach and hence no standard mechanism exists to incorporate a priori model information. In this thesis, learning methods are proposed which explicitly use the available system knowledge. This can be seen as a new class of approaches that bridge model-based and model-free methods. These methods can address some of the hurdles mentioned earlier. For example, i) a prior system information can speed up the learning, ii) new control objectives can be achieved which otherwise would be extremely difficult to attain using only model-based methods, iii) physical meaning can be attributed to the learned controller. The developed approach is as follows: themodel of the given physical system is represented in the port-Hamiltonian (PH) form. For the system dynamics in PH form a passivity-based control (PBC) law is formulated, which often requires the solution to a set of partial differential equations (PDEs). Instead of finding an analytical solution, the PBC control law is parameterized using an unknown parameter vector. Then, by using a variation of the standard actor-critic learning algorithm, the unknown parameters can be learned online. Using the principles of stochastic approximation theory, a proof of convergence for the developed method is shown. The proposedmethods are evaluated for the stabilization and regulation ofmechanical and electro-mechanical systems. The simulation and experimental results show comparable learning curves. In the final part of the thesis a novel integral reinforcement learning approach is developed to solve for the optimal output tracking control problem for a set of linear heterogeneous multi-agent systems. Unlike existing methods, this approach does not need to solve either the output regulator equation or requires a p-copy of the leader’s dynamics in the agent’s control law. A detailed numerical evaluation has been conducted to show the feasibility of the developed method.

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas expanding through a two-dimensional nozzle and into a surrounding low-density environment. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, the variable hard sphere model is used as a molecular model and the no time counter method is employed as a collision sampling technique. The results of both the FDDO and the DSMC methods show good agreement. The FDDO method requires less computational effort than the DSMC method by factors of 10 to 40 in CPU time, depending on the degree of rarefaction.

The study of natural convection from a heat source located at the bottom of partially top vented enclosure is presented. Air cooling for electronic component situated at the bottom of a top vented enclosure is studied theoretically. The flow is assumed to be laminar, steady and of constant physical properties. The process is described by continuity, momentum and energy partial differential equations, which are be expressed in Cartesian coordinates system. Due to the nature of the studied problem and with proper transformation of the problem dependent and independent variables, these governing equations are transformed to a set of dimensionless partial differential equations. This set of differential equations is transformed to set of difference equations by the implantation of finite difference technique. Accordingly, one can obtain the solutions for this problem, which are obtained by application of the well known Gauss Siedel iteration method. A computer program is developed to solve the present proposed mathematical model. According to this solution, the values of Nusselt number, for different values of Rayleigh number are obtained. Comparisons between the present obtained results and those results obtained in previous theoretical results are performed for parametric variations of the vent opening size and shape. Also a correlation for Nusselt number as a function of Rayleigh number and equivalent diameter of the vent is proposed.

Purpose – The purpose of this paper is to focus on the numerical modelling of transient natural convection flow of an incompressible viscous nanofluid past an impulsively started semi-infinite vertical plate with variable surface temperature. Design/methodology/approach – The problem is governed by the coupled non-linear partial differential equations with appropriate boundary conditions. A robust, well-tested, Crank-Nicolson type of implicit finite-difference method, which is unconditionally stable and convergent, is used to solve the governing non-linear set of partial differential equations. Findings – The local and average values of the skin-friction coefficient (viscous drag) and the average Nusselt number (the rate of heat transfer) decreased, while the local Nusselt number increased for all nanofluids, namely, aluminium oxide-water, copper-water, titanium oxide-water and silver-water with an increase in the temperature exponent m. Selecting aluminium oxide as the dispersing nanoparticles leads to the maximum average Nusselt number (the rate of heat transfer), while choosing silver as the dispersing nanoparticles leads to the minimum local Nusselt number compared to the other nanofluids for all values of the temperature exponent m. Also, choosing silver as the dispersing nanoparticles leads to the minimum skin-friction coefficient (viscous drag), while selecting aluminium oxide as the dispersing nanoparticles leads to the maximum skin-friction coefficient (viscous drag) for all values of the temperature exponent m. Research limitations/implications – The Brinkman model for dynamic viscosity and Maxwell-Garnett model for thermal conductivity are employed. The governing boundary layer equations are written according to The Tiwari-Das nanofluid model. A range of nanofluids containing nanoparticles of aluminium oxide, copper, titanium oxide and silver with nanoparticle volume fraction range less than or equal to 0.04 are considered. Practical implications – The present simulations are relevant to nanomaterials thermal flow processing in the chemical engineering and metallurgy industries. This study also provides an important benchmark for further simulations of nanofluid dynamic transport phenomena of relevance to materials processing, with alternative computational algorithms (e.g. finite element methods). Originality/value – This paper is relatively original and illustrates the influence of variable surface temperature on transient natural convection flow of a viscous incompressible nanofluid and heat transfer from an impulsively started semi-infinite vertical plate.

This article presents a mathematical model for characterizing the dynamic behavior of spherical shock waves in a self-gravitating, radiating ideal gas with the presence of an azimuthal magnetic field, emphasizing adiabatic conditions. The analysis assumes that the environment just ahead of the shock wave is stationary, and it accounts for variations in density, magnetic field, and fluid velocity within the disturbed medium just behind the shock front. Furthermore, the impact of thermal radiation within the context of an optically thin limit is incorporated into the energy equation of the governing system. Employing the Lie invariance method, the set of partial differential equations governing the flow within this medium is transformed into a system of nonlinear ordinary differential equations through the use of similarity variables. Four distinct cases of similarity solutions are derived by selecting different values for the arbitrary constants associated with the generators. Among these four cases, only two yield similarity solutions, one assuming a power-law shock path and the other an exponential-law shock path. In the case of a power-law shock path, the resulting set of nonlinear ordinary differential equations is numerically solved using the 4th-order Runge-Kutta method in MATLAB software. The article thoroughly discusses the influence of various parameters, including γ (adiabatic index of the gas), Ma−2 (Alfvén–Mach number), ϕ (ambient density exponent), and G0 (gravitational parameter), on the flow properties. The findings are presented graphically to provide a comprehensive understanding of the effects of these parameters.

Dynamic simulation of reversible solid-fluid reactions in nonisothermal porous spheres with Stefan-Maxwell diffusion

This research aims to study the 3D magnetohydrodynamics stagnation-point flow (SPF) over a horizontal plane surface (HPS) carrying water-based graphene oxide (GO) nanoparticles caused by an irregular heat source/sink used in heat transfer procedures. In addition, a Tiwari–Das model is used to inspect the dynamics of fluid flow behavior and heat transmission features of the nanoparticles with experiencing the impacts of thermal radiation. The acquired nonlinear set of partial differential equations (PDEs) is transfigured to a system of ordinary differential equations (ODEs) using similarity transformations. The accumulative dimensionless ODEs are then further tackled in MATLAB using the bvp4c solver. Tables and figures are prepared for the execution of several relevant constraints along with nodal/saddle indicative parameter, internal heat source/sink parameter, radiation parameter and nanoparticles volume fraction which divulges and clarify more accurately the posited quantitative data and graphical findings. Also, the velocity profile decelerated in the axial and transverse coordinate axes for a higher value of the nanoparticle volume fraction but the dimensionless temperature distribution is augmented. Additionally, thermal boundary layer thickness and profile of temperature enriches with higher impressions of radiation constraint. However, the internal heat sink factor declines the profiles of temperature while escalating with the superior value of the internal heat source parameter.

The goal of diffusion-weighted magnetic resonance imaging (DWI) is to infer the structural connectivity of an individual subject’s brain in vivo. To statistically study the variability and differences between normal and abnormal brain connectomes, a mathematical model of the neural connections is required. In this paper, we represent the brain connectome as a Riemannian manifold, which allows us to model neural connections as geodesics. This leads to the challenging problem of estimating a Riemannian metric that is compatible with the DWI data, i.e., a metric such that the geodesic curves represent individual fiber tracts of the connectomics. We reduce this problem to that of solving a highly nonlinear set of partial differential equations (PDEs) and study the applicability of convolutional encoder-decoder neural networks (CEDNNs) for solving this geometrically motivated PDE. Our method achieves excellent performance in the alignment of geodesics with white matter pathways and tackles a long-standing issue in previous geodesic tractography methods: the inability to recover crossing fibers with high fidelity. Code is available at https://github.com/aarentai/Metric-Cnn-3D-IPMI .

Introduces a method to locate the leakage position in a pipeline estimating flows deviations with minimal order nonlinear observers when only measurements at the extremes of the duct are available, and calculating the leakage position using a set of nonlinear equations at the leak point. Taking into a account that the fluid model in the duct is given by a set of partial differential equations, a finite dimension nonlinear model of minimal order for FDI is obtained assuming that flows can be measured at the end points of the line, pressions are the control variables, and require the uniform observability condition for each of the outputs. The system properties of uniform observability and affinity in the control allows us to improve the residuals generator response previously reported in the literature by linear observers. Moreover, the residuals generator design problem is simplified, since two independent simple nonlinear estimators for flows deviations which are sensitive to any leak can be designed using either fixed or varying gain observers. For the leakage position estimation a set of three nonlinear equations must be on-line solved in which the unknowns are leak rate, its position and pressure at the leak position. The performance of the approach is demonstrated using simulated and experimental data taken from a water pilot pipeline.

Analysis of boundary layer MHD Darcy-Forchheimer radiative nanofluid flow with soret and dufour effects by means of marangoni convection

Resin transfer moulding: mathematical modelling and numerical simulations

A set of partial differential equations governing orthotropic shallow shells with arbitrary quadratic midsurface is reduced into an eight order partial differential equation with one function as unknown. By use of the method of plane wave decomposition this partial differential equation is transformed into an ordinary differential equation. The fundamental solutions for orthotropic shallow shells are presented in the form of a definite integral. The calculation of the fundamental solutions is discussed in detail.