Complex Initial Conditions Research Articles

An Orbital-attitude Coupled Control Framework for a Full-scale Flexible Electric Solar Wind Sail Spacecraft in Orbital Transformation Missions

In this work, a novel orbital-attitude coupled control framework is developed for the electric solar wind sail (E-sail) spacecraft, aiming to handle the orbital transformation and attitude maneuver simultaneously. In the framework, the desired attitude parameters and the slow-varying voltage component are provided by the orbital control strategy, while the current attitude parameters and the fast-varying voltage component are manipulated by the attitude control strategy to approach their desired values. The orbital-attitude coupled characteristics of the E-sail spacecraft, including the flexibility-induced coupling effect, are fully described by the referenced nodal coordinate formulation. Considering the input saturation conditions, the governing equation for the orbital control strategy is then derived, in which the in-plane and out-of-plane displacement and velocity errors are prescribed as the state variables to be eliminated. An integral sliding mode control (ISMC) scheme is proposed to improve the robustness against the unmeasurable disturbance term. A model predictive control (MPC) scheme is introduced to enhance the convergence efficiency, where a quadratic optimization is performed to plan the desired attitude parameters and voltage components within the prediction horizon. To evaluate the control performance in the orbital transformation and attitude maneuver missions on the displaced non-Keplerian orbit, a series of scenarios with complex initial conditions are simulated under different control schemes, including the ISMC-MPC compound scheme. The results show that the control strategy designed under the rigid-body assumptions may not be feasible for the flexible E-sail spacecraft, while the investigated control strategy realizes the accurate and efficient convergence of the orbital and attitude variables on both the rigid and flexible E-sail spacecraft with the tether deformation stabilized.

Solving Viscous Burgers’ Equation: Hybrid Approach Combining Boundary Layer Theory and Physics-Informed Neural Networks

In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers’ equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations.

Analysis of a First-Order Delay Model under a History Function with Discontinuity

This paper analyzes the first-order delay equation y′(t)=αy(t)+βy(t−τ) subject to a history function in addition to an initial condition that assumes discontinuity at t=0. The method of steps is successfully applied to derive the exact solution in an explicit form. In addition, a unified formula is provided to describe the solution in any finite sub-interval of the problem’s domain. The characteristics and properties of the solution are theoretically investigated and then confirmed through several plots. The behavior of the solution and its derivative are examined and interpreted. The results show that the method of steps is an effective method of solution to treat the current delay model. The present successful analysis can be used to investigate other delay models with complex initial conditions. Furthermore, the present approach can be generalized to include the inhomogeneous version of the current model without using numerical methods.

Sequencing Initial Conditions in Physics-Informed Neural Networks

The scientific machine learning (SciML) field has introduced a new class of models called physics-informed neural networks (PINNs). These models incorporate domain-specific knowledge as soft constraints on a loss function and use machine learning techniques to train the model. Although PINN models have shown promising results for simple problems, they are prone to failure when moderate level of complexities are added to the problems. We demonstrate that the existing baseline models, in particular PINN and evolutionary sampling (Evo), are unable to capture the solution to differential equations with convection, reaction, and diffusion operators when the imposed initial condition is non-trivial. We then propose a promising solution to address these types of failure modes. This approach involves coupling Curriculum learning with the baseline models, where the network first trains on PDEs with simple initial conditions and is progressively exposed to more complex initial conditions. Our results show that we can reduce the error by 1 – 2 orders of magnitude with our proposed method compared to regular PINN and Evo.

A multiphase model developed for mesoscopic heat and mass transfer in thawing frozen soil based on lattice Boltzmann method

Subgrade engineering in cold region collapsed and settled under the influence of global warming, this was mainly due to the thawing frozen soil. The heat and mass transfers in thawing frozen soils are generally described with continuity equations on the macroscopic scale. However, the processes of ice melting and water migration in soil were rarely described on the mesoscopic scale. In this study, a multiphase pseudo-potential model with enthalpy-based module was developed based on lattice Boltzmann method to predict the distributions of temperature and water content in the thawing process of frozen soil. The force source terms in pseudo-potential model were given as adhesive force, cohesive force, temperature drive force and bulk force to describe the interaction between multiple components in thawing process, and the enthalpy-based module was employed to realize the process of phase transition in soil. The complex initial conditions of thawing process were measured and then set in simulation for the first time. The results showed that as compared with the measured soil temperatures and water contents at different top soil temperature (5 °C and 10 °C) and times (0, 12, and 36 h), the relative errors of model were 0.01 % ∼ 0.72 % and 0.10 % ∼ 19.31 % for the soil temperatures and water contents, respectively. Further, the developed pseudo-potential model can also be used to detail the mesoscopic heat and mass transfers in thawing soils.

Morphological evolution via surface diffusion learned by convolutional, recurrent neural networks: Extrapolation and prediction uncertainty

We use a convolutional, recurrent neural network approach to learn morphological evolution driven by surface diffusion. To this aim we first produce a training set using phase field simulations. Intentionally, we insert in such a set only relatively simple, isolated shapes. After proper data augmentation, training and validation, the model is shown to correctly predict also the evolution of previously unobserved morphologies and to have learned the correct scaling of the evolution time with size. Importantly, we quantify prediction uncertainties based on a bootstrap-aggregation procedure. The latter proved to be fundamental in pointing out high uncertainties when applying the model to more complex initial conditions (e.g., leading to the splitting of high aspect-ratio individual structures). The automatic smart augmentation of the training set and design of a hybrid simulation method are discussed.

Generalized solutions of the Riccati equation

In this paper, we consider a nonlinear differential equation of the first order of the Riccati hierarchy. The concept of a generalized solution for such an equation cannot be introduced within the framework of the classical theory of generalized functions because the product of generalized functions is not defined. To introduce the concept of a generalized solution, two approaches are considered. In the first approach, approximation by solutions of the Cauchy problem with complex initial conditions is used, and generalized solutions are defined as limits of approximating families in the sense of convergence in D′(¡). It is shown that there are two generalized solutions of the Cauchy problem. The type of solution depends on whether the poles of the approximating solution are located in the upper or lower half-plane. The second approach uses approximation with a system of equations. It is shown that there are many approximating systems, meanwhile, generalized solutions of the Cauchy problem depend on the choice of the approximating system.

Un análisis comparativo de los métodos: miméticos, diferencias finitas y elementos finitos para problemas estacionarios 1-dimensional

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

Irregular Dust Features around Intermediate-mass Young Stars with GPI: Signs of Youth or Misaligned Disks?

Abstract We are undertaking a large survey of over 30 disks using the Gemini Planet Imager (GPI) to see whether the observed dust structures match spectral energy distribution predictions and have any correlation with stellar properties. GPI can observe near-infrared light scattered from dust in circumstellar environments using high-resolution Polarimetric Differential Imaging with coronagraphy and adaptive optics. The data have been taken in the J and H bands over two years, with inner working angles of 0.″08 and 0.″11, respectively. Ahead of the release of the complete survey results, here we present five objects with extended and irregular dust structures within 2″ of the central star. These objects are FU Ori, MWC 789, HD 45677, Hen 3-365, and HD 139614. The observed structures are consistent with each object being a pre-main-sequence star with protoplanetary dust. The five objects’ circumstellar environments could result from extreme youth and complex initial conditions, from asymmetric scattering patterns due to shadows cast by misaligned disks, or in some cases from interactions with companions. We see complex U ϕ structures in most objects that could indicate multiple scattering or result from the illumination of companions. Specific key findings include the first high-contrast observation of MWC 789 revealing a newly discovered companion candidate and arc, and two faint companion candidates around Hen 3-365. These two objects should be observed further to confirm whether the companion candidates are comoving. Further observations and modeling are required to determine the causes of the structures.

The Geodynamic World Builder: a solution for complex initial conditions in numerical modeling

Abstract. The Geodynamic World Builder is an open-source code library intended to set up initial conditions for computational geodynamic models in both Cartesian and spherical geometries. The inputs for the JavaScript Object Notation (JSON)-style parameter file are not mathematical but rather a structured nested list describing tectonic features, e.g., a continental, an oceanic or a subducting plate. Each of these tectonic features can be assigned a specific temperature profile (e.g., plate model) or composition label (e.g., uniform). For each point in space, the Geodynamic World Builder can return the composition and/or temperature. It is written in C++ but can be used in almost any language through its C and Fortran wrappers. Various examples of 2-D and 3-D subduction settings are presented. The Geodynamic World Builder comes with an extensive online user manual.

Computation fluid dynamics model of first-flush runoff through a hydrodynamic separator

Abstract Urban rainwater harvesting systems are widely used to solve urban water shortage. However, the initial rainwater collected from urban roads contains numerous particles that severely limit its reuse. This study designed a hydrodynamic separator on a separated sewer systems (between the rainwater collecting main pipe and intercepting main canal) to remove particulate pollutants, avoid rainwater reflux, and realize staggered regulation during flood peak. In this work, a 3D model of the hydrodynamic separator was constructed using SolidWorks software, and state-of-the-art FLUENT computational fluid dynamics was applied in fine-scale simulation of the hydrodynamic separator under complex initial conditions. Numerical tests were performed to validate the capability of the hydrodynamic separator in removing particulate pollutants under different initial conditions. The results of numerical tests showed that particle trajectories are relatively regular spirals in the hydrodynamic separator, whereas large-diameter particles feature high removal rates. Particle removal rate increased with increasing diameter-depth ratio and height difference between the overflow and inlet pipes. Considering the particle removal rate, the optimal order of angles between the overflow and inlet pipes is 90° > 150° > 60° > 120°. The parameters of the hydrodynamic separator in removing initial rainfall particulate pollutants were obtained, and a pilot-scale experiment was conducted, providing theoretical importance in reducing energy consumption of rainwater harvesting pretreatment systems.

Improvement on an analytical finite line source model considering complex initial and boundary conditions: Part 1, model development and validation

For a ground coupled heat pump system, the variations of ground surface temperature and undisturbed subsurface temperature have pronounced effects on the heat transfer between the ground heat exchanger and surrounding soil. However, how to quantify these effects analytically has long been a tricky issue. In this paper, finite line source model under more complex initial and boundary conditions (FLSCC) is presented and solved based on an existing model, and transient solutions of sub-problems are given, to investigate the effects of ground surface temperature and subsurface temperature variations. To examine the performance of this model, mean borehole surface temperature simulated by FLSCC was compared to that from five conventional models in a case study. Results show that long term behavior of FLSCC is the most close to the reference duct ground heat storage model (DST). The relative mean deviation between the two models at each simulation time step is less than 4.02% from 10 h to 100 years, and the mean value throughout the simulated 100 years is only 1.05%. Also, its short term behavior is close to a validated numerical mode. In addition, the new model is also validated with experimental data from two thermal response tests. Errors between tested and simulated mean fluid temperature calculated by FLSCC are smaller than that of conventional models. Conclusively, the improved model is suitable for dealing with simulation projects at any locations and from any starting point in a year, and also deal with thermal response test data interpretation. It is an effective supplement to finite line source model, and is helpful for fast and reliable practical engineering design.

Fickian and non-Fickian diffusion of cosmic rays

Abstract Fluid approximations to cosmic ray (CR) transport are often preferred to kinetic descriptions in studies of the dynamics of the interstellar medium (ISM) of galaxies, because they allow simpler analytical and numerical treatments. Magnetohydrodynamic simulations of the ISM usually incorporate CR dynamics as an advection–diffusion equation for CR energy density, with anisotropic, magnetic-field-aligned diffusion with the diffusive flux assumed to obey Fick’s law. We compare test particle and fluid simulations of CRs in a random magnetic field. We demonstrate that a non-Fickian prescription of CR diffusion, which corresponds to the telegraph equation for the CR energy density, can be easily calibrated to match the test particle simulations with great accuracy. In particular, we consider a random magnetic field in the fluid simulation that has a lower spatial resolution than that used in the particle simulation to demonstrate that an appropriate choice of the diffusion tensor can account effectively for the unresolved (subgrid) scales of the magnetic field. We show that the characteristic time that appears in the telegraph equation can be physically interpreted as the time required for the particles to reach a diffusive regime and we stress that the Fickian description of the CR fluid is unable to describe complex boundary or initial conditions for the CR energy flux.

Mathematical Modeling of Release Kinetics from Supramolecular Drug Delivery Systems.

Embedding of active substances in supramolecular systems has as the main goal to ensure the controlled release of the active ingredients. Whatever the final architecture or entrapment mechanism, modeling of release is challenging due to the moving boundary conditions and complex initial conditions. Despite huge diversity of formulations, diffusion phenomena are involved in practically all release processes. The approach in this paper starts, therefore, from mathematical methods for solving the diffusion equation in initial and boundary conditions, which are further connected with phenomenological conditions, simplified and idealized in order to lead to problems which can be analytically solved. Consequently, the release models are classified starting from the geometry of diffusion domain, initial conditions, and conditions on frontiers. Taking into account that practically all solutions of the models use the separation of variables method and integral transformation method, two specific applications of these methods are included. This paper suggests that “good modeling practice” of release kinetics consists essentially of identifying the most appropriate mathematical conditions corresponding to implied physicochemical phenomena. However, in most of the cases, models can be written but analytical solutions for these models cannot be obtained. Consequently, empiric models remain the first choice, and they receive an important place in the review.

Application of the homotopy analysis method to multispecies reactive transport equations with general initial conditions

Multispecies solute transport through soils is characterized by the dynamic processes of convection, diffusion, decay and a series of chemical reactions, posing difficulties when seeking to derive analytical solutions related to subsurface hydrology. At present, most analytical solutions for multiple-species contaminant transport problems are based on the Laplace transform or Fourier transform technique, a decomposition strategy such as ingenious transform format and matrix diagonalization procedure, classical integral methods, the generalized integral transform technique, etc. These methods have been limited to some cases with simple initial conditions, identical retardation coefficients for all species, and complicated mathematical procedures, which restrict their use for cases under complex initial conditions. To cope with these limitations, the homotopy analysis method (HAM) is implemented to solve multispecies reactive transport models with more general, smooth initial conditions. Applying an auxiliary linear and nonlinear operator, a zero-order deformation equation is derived to obtain an approximate solution with high accuracy (different cases with various initial and boundary conditions). The semi-analytical solutions are compared with exact analytical solutions from the literature; good agreement between the resulting simulation and analytical solutions has been achieved. This illustrates that HAM is one of the most effective methods for solving the multispecies reactive transport models with more general and smooth initial conditions.

An efficient ray-tracing method and its application to Gaussian beam migration in complex multilayered anisotropic media

We have developed a fast ray-tracing method for multiple layered inhomogeneous anisotropic media, based on the generalized Snell’s law. Realistic geologic structures continuously varying with embedded discontinuities are parameterized by adopting cubic B-splines with nonuniformly spaced nodes. Because the anisotropic characteristic is often closely related to the interface configuration, this model parameterization scheme containing the natural inclination of the corresponding layer is particularly suitable for tilted transverse isotropic models whose symmetry axis is generally perpendicular to the direction of the layers. With this model parameterization, the first- and second-order spatial derivatives of the velocity within the interfaces can be effectively obtained, which facilitates the amplitude computation in dynamic ray tracing. By using complex initial conditions for the dynamic ray system and taking the multipath effect into consideration, our method is applicable to Gaussian beam migration. Numerical experiments of our method have been used to verify its effectiveness, practicability, and efficiency in memory storage and computation.

Ultrafast evolution of electric fields from high-intensity laser-matter interactions

The interaction of high-power ultra-short lasers with materials offers fascinating wealth of transient phenomena which are in the core of novel scientific research. Deciphering its evolution is a complicated task that strongly depends on the details of the early phase of the interaction, which acts as complex initial conditions. The entire process, moreover, is difficult to probe since it develops close to target on the sub-picosecond timescale and ends after some picoseconds. Here we present experimental results related to the fields and charges generated by the interaction of an ultra-short high-intensity laser with metallic targets. The temporal evolution of the interaction is probed with a novel femtosecond resolution diagnostics that enables the differentiation of the contribution by the high-energy forerunner electrons and the radiated electromagnetic pulses generated by the currents of the remaining charges on the target surface. Our results provide a snapshot of huge pulses, up to 0.6 teravolt per meter, emitted with multi-megaelectronvolt electron bunches with sub-picosecond duration and are able to explore the processes involved in laser-matter interactions at the femtosecond timescale.

A simple analytical infiltration model for short-duration rainfall

Many infiltration models have been proposed to simulate infiltration process. Different initial soil conditions and non-uniform initial water content can lead to infiltration simulation errors, especially for short-duration rainfall (SHR). Few infiltration models are specifically derived to eliminate the errors caused by the complex initial soil conditions. We present a simple analytical infiltration model for SHR infiltration simulation, i.e., Short-duration Infiltration Process model (SHIP model). The infiltration simulated by 5 models (i.e., SHIP (high) model, SHIP (middle) model, SHIP (low) model, Philip model and Parlange model) were compared based on numerical experiments and soil column experiments. In numerical experiments, SHIP (middle) and Parlange models had robust solutions for SHR infiltration simulation of 12 typical soils under different initial soil conditions. The absolute values of percent bias were less than 12% and the values of Nash and Sutcliffe efficiency were greater than 0.83. Additionally, in soil column experiments, infiltration rate fluctuated in a range because of non-uniform initial water content. SHIP (high) and SHIP (low) models can simulate an infiltration range, which successfully covered the fluctuation range of the observed infiltration rate. According to the robustness of solutions and the coverage of fluctuation range of infiltration rate, SHIP model can be integrated into hydrologic models to simulate SHR infiltration process and benefit the flood forecast.

Uncertainties in vertical groundwater fluxes from 1‐D steady state heat transport analyses caused by heterogeneity, multidimensional flow, and climate change

AbstractSteady state 1‐D analytical solutions to estimate groundwater fluxes from temperature profiles are an attractive option because they are simple to apply, with no complex boundary or initial conditions. Steady state solutions have been applied to estimate both aquifer scale fluxes as well as to estimate groundwater discharge to streams. This study explores the sources of uncertainty in flux estimates from regional scale aquifers caused by sensor precision, aquifer heterogeneity, multidimensional flow and variations in surface temperature due to climate change. Synthetic temperature profiles were generated using 2‐D groundwater flow and heat transport models with homogeneous and heterogeneous hydraulic and thermal properties. Temperature profiles were analyzed assuming temperature can be determined with a precision between 0.1°C and 0.001°C. Analysis of synthetic temperature profiles show that the Bredehoeft and Papadopulos (1965) method can provide good estimates of the mean vertical Darcy flux over the length of the temperature profile. Reliable flux estimates were obtained when the ratio of vertical to horizontal flux was as low as 0.1, and in heterogeneous media, providing that temperature at the upper boundary was constant in time. However, temporal increases in surface temperature led to over‐estimation of fluxes. Overestimates increased with time since the onset of, and with the rate of surface warming. Overall, the Bredehoeft and Papadopulos (1965) method may be more robust for the conditions with constant temperature distributions than previously thought, but that transient methods that account for surface warming should be used to determine fluxes in shallow aquifers.

Dynamics of the equation in the complex plane

We investigate the dynamics of the second-order rational difference equation in the title with complex parameters and arbitrary complex initial conditions. Very little is known about rational difference equation in the complex domain. The solutions of such equation exhibit many rich and complicated asymptotic characters. Analysis of the local stability of the equilibria and periodicity of solutions has been carried out. We further exhibit several interesting characteristics of the solutions of this equation, using computations, which do not arise when we consider the same equation with positive real parameters and initial conditions. We pose several open problems and conjectures of paramount importance regarding boundedness and global asymptotic convergence. It is our belief that the analysis done here is generic in nature and can be helpful in the study of other difference equations in the complex domain.