Predictor-corrector Algorithm Research Articles

Multi-Process Integrated Planning and Guidance for Return of Reusable Aircraft Under Time Constraint

As the launch frequency of reusable launch vehicle (RLV) increases, the requirement of landing site management is increasing. Especially, the return flight time of RLV should be constrained to improve the utilization of landing site. In this paper, a multi-process integrated (MPI) planning and guidance algorithm for deorbit phase and reentry phase is proposed to meet the mission requirement of RLV. Firstly, the states of RLV at reentry point and deorbit point are given by MPI planning algorithm through inverse calculation. To be specific, by taking the terminal state as the initial value, the reentry point is obtained by the backward calculation based on a cosine-quadratic bank angle profile. According to the relationship among the reentry point, the deorbit time and the deorbit range, the parameters of deorbit phase can be calculated by efficient interpolation based on deorbit ability analysis. The bank angle profile is adjusted to make estimated total flight range approximate the predefined total range by particle swarm optimization algorithm. After that, guidance algorithm considering the terminal position and time constraints of the whole flight phase is given, and the trajectory optimization problem related to brake angle and start-up time is transformed into a series of convex optimization problems in deorbit phase, which can be solved by primal-dual interior-point method with better computational performance and satisfactory accuracy. In the reentry phase, the bank angle profile is designed and updated with improved predictor-corrector algorithm to correct the error during the flight. Finally, the feasibility and robustness of the propose algorithm are verified by multi-mission planning guidance simulation and Monte Carlo simulation.

High-order discretization of a stable time-domain integral equation for 3D acoustic scattering

We develop a high-order, explicit method for acoustic scattering in three space dimensions based on a combined-field time-domain integral equation. The spatial discretization, of Nyström type, uses Gaussian quadrature on panels combined with a special treatment of the weakly singular kernels arising in near-neighbor interactions. In time, a new class of convolution splines is used in a predictor-corrector algorithm. Experiments on a torus and a perturbed torus are used to explore the stability and accuracy of the proposed scheme. This involved around one thousand solver runs, at up to 8th order and up to around 20,000 spatial unknowns, demonstrating 5–9 digits of accuracy. In addition we show that parameters in the combined field formulation, chosen on the basis of analysis for the sphere and other convex scatterers, work well in these cases.

Improved adaptive variance reduction algorithm based on RMC code for deep penetration problems

An adaptive variance reduction algorithm based on RMC code was developed for deep penetration problems. The adaptive variance reduction algorithm uses the conservation of penetration rate and a predictor–corrector algorithm to quickly determine exponential importance parameters or equal-gradient importance parameters with fast iterative convergence. The adaptive variance reduction algorithm greatly accelerates one-dimensional deep penetration problems. However, the previous algorithm does not have capacity of energy bias nor does it support calculations on three-dimensional models. The adaptive variance reduction algorithm is improved to allow for energy bias and for three-dimensional models. The improved algorithm was applied to the HBR2 benchmark with the average Figure of Merit (FOM) of RMC code improved 243 fold. Therefore, this improved adaptive variance reduction algorithm can efficiently deal with complex engineering shielding problems.

A Predictor–Corrector Algorithm for Fast Polynomial Chaos-Based Uncertainty Quantification of Multi-Walled Carbon Nanotube Interconnects

In this article, a predictor–corrector algorithm for the fast polynomial chaos (PC)-based uncertainty quantification (UQ) of multi-walled carbon nanotube (MWCNT) interconnect networks is presented. The proposed algorithm intelligently combines the numerical efficiency of the approximate equivalent single conductor (ESC) model of the MWCNT interconnect network with the rigor and accuracy of a multi-conductor circuit (MCC) model. Consequently, this algorithm significantly accelerates the generation of the PC surrogate models (or metamodels) of the network responses for minimal loss in accuracy. These metamodels can be probed efficiently and repeatedly to quantify the impact of manufacturing and fabrication process uncertainty on the MWCNT network responses.

Phase-field-based lattice Boltzmann model for liquid-gas-solid flow.

Based on phase-field theory, we develop a lattice Boltzmann (LB) model for liquid-gas-solid flow from multiphase and particle dynamics algorithms. A modified bounce-back method is developed for the velocity-based LB approach. A curved boundary treatment with second-order accuracy based on velocity interpolation is developed. We propose a predictor-corrector scheme algorithm for specifying the three-phase contact angle on curved boundaries within the framework of structured Cartesian grids. In order to make the algorithm more stable, we combine the implicit particle velocity update scheme and the Galilean invariant momentum exchange method. The proposed method is validated through several single- and multicomponent fluid test cases. It was found the surface tension force associated with the interface acting on the solid structures can be captured. We simulate the sinking of a circular cylinder due to gravity, the numerical results agree well with the experimental data. Finally, we apply the method to the self-assembly process of multiple floating cylinders on water surface.

Predictor–Corrector Nodal Integral Method for simulation of high Reynolds number fluid flow using larger time steps in Burgers ’ equation

Nodal Integral Methods (NIM), although frequently used for solving neutron transport equations, have not found wider acceptance for solving fluid flow problems. One of the pivotal reasons behind this is the lack of efficient non-linear solvers for the system of coupled algebraic equations obtained by these methods, thus making such methods prohibitively expensive for highly nonlinear (higher Reynolds number) fluid flow problems. Successive improvements in earlier attempts resulted in an improved version called Modified NIM (MNIM) scheme; following which, a further modified version appeared (called as M2NIM), that used the concept of the delayed coefficients. Upon comparing the solutions obtained using MNIM and M2NIM schemes, respectively, it was observed that although the modified scheme with delayed coefficients (M2NIM) has significantly faster convergence, it is less accurate. The convergence and accuracy are very likely to suffer to a significant extent in the high Reynolds number fluid flows, especially when a large time-step is to be considered. In order to resolve these difficulties, a new type of physics-based predictor–corrector algorithm is proposed in the present study. In the proposed computational algorithm, the linearized M2NIM scheme is used to predict the solution, whereas the MNIM scheme is utilized to improve the predictive guess. The novelty of such a hybrid numerical algorithm comes from the fact that it combines the advantage of the faster convergence of M2NIM with the accuracy of the MNIM scheme. The algorithm also benefits from the unique inclusion of Jacobian-free Newton-Krylov (JFNK) method, which helps in getting rid of the formation of large Jacobian matrices, thereby reducing unnecessary computational overhead. The proposed methodology has been applied to solve a non-linear convection–diffusion problem, represented by the Burgers’ equation. The computational results for bothone-dimensional and two-dimensional Burgers’ equation are presented to demonstrate the effectiveness of the developed novel algorithm, as well as, the advantage it offers over the existing numerical methods.

Phase-Field Material Point Method for dynamic brittle fracture with isotropic and anisotropic surface energy

A novel phase field material point method is introduced for robust simulation of dynamic fracture in elastic media considering the most general case of anisotropic surface energy. Anisotropy is explicitly introduced through a properly defined crack density functional. The particular case of impact driven fracture is treated by employing a discrete field approach within the material point method setting. In this, the equations of motion and phase field governing equations are solved independently for each discrete field using a predictor–corrector algorithm. Contact at the interface is resolved through frictional contact conditions. The proposed method is verified using analytical predictions. The influence of surface energy anisotropy and loading conditions on the resulting crack paths is assessed through a set of benchmark problems. Comparisons are made with the standard Phase Field Finite Element Method and experimental observations.

Dual Prediction–Correction Methods for Linearly Constrained Time-Varying Convex Programs

Devising efficient algorithms to solve continuously-varying strongly convex optimization programs is key in many applications, from control systems to signal processing and machine learning. In this context, solving means to find and track the optimizer trajectory of the continuously-varying convex optimization program. Recently, a novel prediction–correction methodology has been put forward to set up iterative algorithms that sample the continuously-varying optimization program at discrete time steps and perform a limited amount of computations to correct their approximate optimizer with the new sampled problem and predict how the optimizer will change at the next time step. Prediction–correction algorithms have been shown to outperform more classical strategies, i.e., correction-only methods. Typically, prediction–correction methods have asymptotical tracking errors of the order of $h^2$ , where $h$ is the sampling period, whereas classical strategies have order of $h$ . Up to now, prediction–correction algorithms have been developed in the primal space, both for unconstrained and simply constrained convex programs. In this paper, we show how to tackle linearly constrained continuously-varying problem by prediction–correction in the dual space and we prove similar asymptotical error bounds as their primal versions.

Numerical solution of a modified anomalous diffusion equation with nonlinear source term through meshless singular boundary method

In this study, singular boundary method is employed for solving a modified anomalous diffusion process in two dimensional space with nonlinear source term with initial and Dirichlet-type boundary conditions. The process is modeled as a two dimensional nonlinear time-fractional sub-diffusion equation in sense of Riemann-Liouville fractional derivatives. A splitting scheme is applied to split the solution of the inhomogeneous governing equation into homogeneous solution and particular solution. We present the numerical operation for calculating the particular solution and homogeneous solution. For gaining approximation particular solution and homogeneous solution we employed MPS method and SBM method, respectively. We use θ-weighted finite difference method as time discretization for time derivatives. We employ Predictor-Corrector Algorithm for the nonlinear source term. A comparison check between SBM and other methods is given to show the accuracy of SBM applying on the equation. Consequently, some numerical examples with different domains is tested and compared with the exact analytical solutions to display the validity and accuracy of the numerical method and compared with other methods.

The analysis of periodic orbits generated by Lagrangian solutions of the restricted three-body problem with non-spherical primaries

The present paper deals with the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies. We have illustrated the periodic orbits for different values of μ, h, σ1 and σ2 (h is energy constant, μ is mass ratio of the two primaries, σ1 and σ2 are oblateness factors). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by (Karimov and Sokolsky, 1989). We have applied the predictor-corrector algorithm to construct the periodic orbits in an attempt to unveil the effect of oblateness of the primaries by taking the fixed values of parameters μ, h, σ1 and σ2.

New hybrid cycle jump approach for predicting low-cycle fatigue behavior by a micromechanical model with the damage induced anisotropy concept

In this work, a new hybrid cycle jump approach was proposed to predict the low-cycle fatigue (LCF) response using a well-established micromechanical model. The damage activation/deactivation phenomenon was formulated at the macroscale. Using the concept of rate independent plasticity with the small strains assumption, the damage deactivation effect on the LCF response and its lifetime was accurately described by the model under simple and complex cyclic loading paths. Three algorithms namely, Taylor 1st and 2nd order and predictor-corrector were utilized for estimating the model response after each cycle jump. An optimum precision factor (ρ) value was defined and then tested via a suitable compromise between computation times and accuracy. In order to numerically determine the cyclic jump length (ΔN), a new hybrid approach was developed based on the overall von Mises stress and the overall damage together with monitoring the accumulated slip. Then, a comparative study revealed that the predictor-corrector algorithm was opted due to its robustness compared to the other algorithms. The numerical study was carried out using two different cyclic loading configurations. One was under simple tension-compression and the other under complex tension-torsion with 90° out-of-phase angle. The model response was described under two distinct states: (i) cycle-by-cycle and (ii) cycle jump concept. For each numerical state, the overall and local responses were recorded. With the new hybrid cycle jump approach, the model response highlights that the greater, the number of cycles, the greater, the gain in computation time but the less, the accuracy. For example, in TC for Nf = 10,000 cycles, a recorded gain in computation time was attained 77% (against 95% by means of cycles number) with a relative error of 4%; whereas in TT90 of Nf = 2124, the computation gain became 43.4% (and 73.4% with respect to number of jumped cycles) with a relative error of 3%.

Asymptotics of a catenoid liquid bridge between two spherical particles with different radii and contact angles

A liquid bridge between two neighboring particles is commonly observed in nature and various industrial processes. An accurate prediction of the profile of a liquid bridge is significantly important in particulate flows, while it is an analytically challenging task as well. In this paper, we develop an asymptotic solution for a catenoid liquid bridge profile, which is the minimal surface ensuring the minimum total surface energy. Our asymptotic solution is based on a rapid convergent predictor-corrector algorithm that considers different factors including boundary conditions, volume conservation, and geometrical relations while providing the relationship between the liquid bridge profile, bridge radius, half-filling angles, and creeping distances. Therefore, this asymptotic solution of the catenoid of the liquid bridge is applicable to general scenarios of any two neighboring particles of either equal or different sizes having identical or different contact angles. In order to validate the proposed asymptotic solution, we performed comprehensive experiments where the observed and predicted liquid bridge profiles and the resultant capillary forces from both the approaches are found closely matching. Moreover, we also investigate and report the influence of the radii ratio, contact angles, particle distances, and the liquid bridge volumes on its profiles.

Numerical considerations for the simulation of proppant transport through fractures

Injection of a proppant slurry (fluid mixed with granular material) into a reservoir to keep fractures open is a common procedure in hydraulic fracturing treatments. This article presents the numerical methodology for simulation of proppant transport through a hydraulic fracture using the Finite Volume Method. Proppant models commonly used in the hydraulic fracture literature solve the linearized advection equation; this work presents solution methods for the nonlinear form of the proppant flux equation.The complexities of solving the nonlinear hyperbolic advection equation that governs proppant transport are tackled, particularly handling shock waves that are generated due to the nonlinear flux function and the spatially-varying width and pressure gradient along the fracture. A critical time step is derived for the proppant transport problem solved using an explicit solution strategy. A predictor-corrector algorithm is developed to constrain the proppant from exceeding the physically admissible range. The model is able to capture the mechanism of proppant bridging which occurs in sections of narrow fracture width, tip screen-out which occurs where fracture becomes saturated with proppant, and flushing of proppant into new fracture segments. The results are verified by comparing with characteristic solutions, and the model is used to simulate proppant transport through a KGD fracture.

Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics

Lagrangian particle methods based on detailed atomic and molecular models are powerful computational tools for studying the dynamics of microscale and nanoscale systems. However, the maximum time step is limited by the smallest oscillation period of the fastest atomic motion, rendering long-time simulations very expensive. To resolve this bottleneck, we propose a supervised parallel-in-time algorithm for stochastic dynamics (SPASD) to accelerate long-time Lagrangian particle simulations. Our method is inspired by bottom-up coarse-graining projections that yield mean-field hydrodynamic behavior in the continuum limit. Here as an example, we use the dissipative particle dynamics (DPD) as the Lagrangian particle simulator that is supervised by its macroscopic counterpart, i.e., the Navier-Stokes simulator. The low-dimensional macroscopic system (here, the Navier-Stokes solver) serves as a predictor to supervise the high-dimensional Lagrangian simulator, in a predictor-corrector type algorithm. The results of the Lagrangian simulation then correct the mean-field prediction and provide the proper microscopic details (e.g., consistent fluctuations, correlations, etc.). The unique feature that sets SPASD apart from other multiscale methods is the use of a low-fidelity macroscopic model as a predictor. The macro-model can be approximate and even inconsistent with the microscale description, but SPASD anticipates the deviation and corrects it internally to recover the true dynamics. We first present the algorithm and analyze its theoretical speedup, and subsequently we present the accuracy and convergence of the algorithm for the time-dependent plane Poiseuille flow, demonstrating that SPASD converges exponentially fast over iterations, irrespective of the accuracy of the predictor. Moreover, the fluctuating characteristics of the stochastic dynamics are identical to the unsupervised (serial in time) DPD simulation. We also compare the performance of SPASD to the conventional spatial decomposition method, which is one of the most parallel-efficient methods for particle simulations. We find that the parallel efficiency of SPASD and the conventional spatial decomposition method are similar for a small number of computing cores, but for a large number of cores the performance of SPASD is superior. Furthermore, SPASD can be used in conjunction with spatial decomposition for enhanced performance. Lastly, we simulate a two-dimensional cavity flow that requires more iterations to converge compared to the Poiseuille flow, and we observe that SPASD converges to the correct solution. Although a DPD solver is used to demonstrate the results, SPASD is a general framework and can be readily applied to other Lagrangian approaches including molecular dynamics and Langevin dynamics.

Predictor-corrector inexact smoothing algorithm for symmetric cone complementarity problems with Cartesian P0-property

In this paper we consider the symmetric cone complementarity problem with Cartesian P0-property (denoted by P0-SCCP) which includes the well-known monotone symmetric cone complementarity problem. We propose a predictor-corrector inexact smoothing algorithm for solving the P0-SCCP and prove that the method is globally and locally quadratically convergent under suitable assumptions. Especially, we prove that our algorithm can generate a bounded iteration sequence when the solution set of the P0-SCCP is nonempty and bounded, or the solution set of the monotone SCCP is nonempty. Moreover, the proposed algorithm solves the linear systems in both predictor step and corrector step only approximately by using an inexact Newton method. Hence when one solves large-scale SCCPs, our algorithm can save much computation work compared to existing smoothing-type algorithms. Numerical results confirm these good theoretical properties.

Fluctuating hydrodynamics of electrolytes at electroneutral scales

At mesoscopic scales electrolyte solutions are modeled by the fluctuating generalized Poisson-Nernst-Planck (PNP) equations [J.-P. P\'eraud et al., Phys. Rev. F, 1(7):074103, 2016]. However, at length and time scales larger than the Debye scales, electrolytes are effectively electroneutral, and the charged-fluid PNP equations become too stiff to solve numerically. Here we formulate the isothermal incompressible equations of fluctuating hydrodynamics for reactive multispecies mixtures involving charged species in the electroneutral limit, and design a numerical algorithm to solve these equations. Our model does not assume a dilute electrolyte solution but rather treats all species on an equal footing, accounting for cross-diffusion and non-ideality using Maxwell-Stefan theory. By enforcing local electroneutrality as a constraint, we obtain an elliptic equation for the electric potential that replaces the Poisson equation in the fluctuating PNP equations. We develop a second-order midpoint predictor-corrector algorithm to solve either the charged-fluid or electroneutral equations with only a change of the elliptic solver. We use the electroneutral algorithm to study a gravitational fingering instability, triggered by thermal fluctuations, at an interface where an acid and base react to neutralize each other. Our results demonstrate that, because the four ions diffuse with very different coefficients, one must treat each ion as an individual species, and cannot treat the acid, base, and salt as neutral species. This emphasizes the differences between electrodiffusion and classical Fickian diffusion, even at electroneutral scales.

Supersonic Dual Bell Axisymmetric Minimum Length Nozzle Conception for High Propulsion Thrust

The aim of this work is to develop a new numerical computing program to design a new form of a dual bell axisymmetric supersonic minimum length nozzle contour, adapted to two different altitudes, usually at sea level and space, giving a supersonic uniform and parallel flow to the exit section. The first bell has a uniform and parallel sonic inlet and a low-altitude adaptation, with a reduced supersonic Mach number, while the second bell has a supersonic inlet and uniform and parallel flow and high altitude adaptation with high supersonic Mach number. The two bells are attached to an inflection point at the exit of the first bell. The passage from low altitude to high altitude is done without any mechanical activation. The purpose of this type of nozzle is the possibility of supersonic flight in two different regimes adapted into two different altitudes, under the assumption of a calorically and thermally perfect gas. This type of nozzle has an inflection point at the attachment point of the second bell to the first bell. The design is done using the method of characteristics. Solving the equations is done numerically by the predictor corrector algorithm. The validation of the results is controlled by the convergence of the critical sections ratio, calculated numerically, to that given by the theory. In this case, all the design parameters automatically converge towards the desired solution.

Performance improvement of supersonic nozzles design using a high-temperature model

The aim of this paper is to discuss the development of new contours of axisymmetric supersonic nozzles giving a uniform and parallel flow at the exit section, to improve the aerodynamic performances compared to the minimum length nozzle, by increasing the exit Mach number and the thrust coefficient, and by reduction of the nozzle's mass, while holding the same throat section between the two nozzles. The new nozzle is named the best performance nozzle. Its form contains a cylindrical central body and an external wall for the flow redress. The study is done at high temperature, lower than the dissociation threshold of the molecules. The variation of the specific heats with the temperature is considered. The design is made by the method of characteristics. The predictor-corrector algorithm is used to make the numerical resolution of the obtained nonlinear algebraic equations. The validation of results is made by the convergence of the numerical critical sections ratio with that given by the theory. The comparison of the results is made with the minimum length nozzle since it is currently used in the aerospace propulsion. The design depends on M E, T0, y body, y*, and the mesh generation. The application is done with air. A computational fluid dynamics verification for the under nozzle expressed regime has shown that a flow separation with the wall is observed because of the side-loads, which are reduced for this new nozzle compared to the minimum length nozzle.

PULSEDYN—A dynamical simulation tool for studying strongly nonlinear chains

We introduce PULSEDYN, a particle dynamics program written in C++, to solve the equations of motion of many-body nonlinear systems in one dimension. PULSEDYN contains a suite of commonly used potentials (Fermi–Pasta–Ulam–Tsingou, Toda, Morse and Lennard-Jones) and solvers (velocity-Verlet and Gear 5th order predictor–corrector) for particle dynamics. PULSEDYN is designed to perform scientifically accurate simulations using these calculations. For accessing the built-in features of PULSEDYN, no knowledge of programming is expected, apart from the creation of a parameter file using predefined commands for running the executable. Therefore, simulations for research projects can be performed with minimal code writing using PULSEDYN. PULSEDYN is distributed under the GNU GPL v3 license and the source code and the executable are available on Github. The writing style emphasizes organization and legibility. Therefore, we anticipate that it would serve as a good template for users who may wish to adapt the code to their specific needs. In this manuscript, we first discuss PULSEDYN and its features in detail. Then, we show results reproduced from literature using PULSEDYN for the following cases (i) Soliton propagation and collisions in the (integrable) Toda lattice (ii) Recurrence phenomena, decay of localized excitations and solitary wave collision in the Fermi–Pasta–Ulam–Tsingou lattice and (iii) Solitary wave propagation in the Morse and Lennard-Jones lattices. Finally, we present a new result on a problem of fundamental historical importance using PULSEDYN. We show by means of explicit specific heat calculations, that in the limit of strong nonlinearity, the quartic Fermi–Pasta–Ulam–Tsingou system approaches equipartition at late times. Program summaryProgram Title: PULSEDYNProgram Files doi:http://dx.doi.org/10.17632/fc8483t7jz.1Licensing provisions: GNU General Public License 3 (GPL)Programming Language: C++Supplementary Material: user manual, documentationNature of problem: We solve a Newtonian particle dynamics problem in one dimension for a variety of commonly used nonlinear manybody systems. The goal is that the user should be able to run scientifically accurate simulations with minimal effort. Benchmarking should also be easy to perform by different users running simulations across different platforms.Solution method: The code uses a parameter file interface with a set of commands for a range of pre-built functionalities. By providing the code and the needed parameter file to the user we hope to make scientifically accurate simulations easier to perform. We provide the Gear 5th order predictor–corrector algorithm as well as the velocity-Verlet algorithm and the following potentials — the Toda potential, Morse Potential, Lennard-Jones Potential and a polynomial potential up to 4th order. To make benchmarking consistent we release the software as open source and free to distribute.

Infrared small dim target detection based on region proposal

This paper presents a fast detection method for infrared small dim target based on region proposal. Detection is divided into two parts: region proposal determination and target precise position extraction. First, potential target positions are extracted by a novel direction detection method based on stationary wavelet transform (SWT). Region proposal can be achieved by self-cross validation of the sequence of potential target positions. Based on the region proposal, a predictor-corrector algorithm is applied to obtain target precise position of current frame. Background prediction is acquired by a unified multi-scale temporal profile (UMTP) model with adaptive fusion and corrected by the 3σ criterion. Results show that the proposed method can preserve almost exactly the background while eliminating target pixels and has the ability to cope with multiple targets simultaneously. Based on the region proposal, real-time processing can be achieved.