Predictor-corrector Steps Research Articles

Phase-field formulated meshless simulation of Rayleigh-Taylor instability problem

The interface between two immiscible Newtonian liquids with different densities and the same viscosity, influenced by gravity, is based on the Phase-Field Method (PFM) formulation. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in the inertial and buoyancy terms (non-Boussinesq formulation). The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in two-dimensional Cartesian coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. Implicit time discretisation is performed for the NS and CH equations in the momentum predictor and Phase-Field (PF) variable corrector steps of PISO, while the momentum corrector steps solve the NS equation explicitly. This solution procedure has improved stability compared to Chorin’s pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the author’s previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris.

An Improved Data Augmentation Scheme for Model Predictive Control Policy Approximation

This paper considers the problem of data generation for MPC policy approximation. Learning an approximate MPC policy from expert demonstrations requires a large data set consisting of optimal state-action pairs, sampled across the feasible state space. Yet, the key challenge of efficiently generating the training samples has not been studied widely. Recently, a sensitivity-based data augmentation framework for MPC policy approximation was proposed, where the parametric sensitivities are exploited to cheaply generate several additional samples from a single offline MPC computation. The error due to augmenting the training data set with inexact samples was shown to increase with the size of the neighborhood around each sample used for data augmentation. Building upon this work, this letter paper presents an improved data augmentation scheme based on predictor-corrector steps that enforces a user-defined level of accuracy, and shows that the error bound of the augmented samples are independent of the size of the neighborhood used for data augmentation.

A Simplified Linearized Lattice Boltzmann Method for Acoustic Propagation Simulation.

A simplified linearized lattice Boltzmann method (SLLBM) suitable for the simulation of acoustic waves propagation in fluids was proposed herein. Through Chapman-Enskog expansion analysis, the linearized lattice Boltzmann equation (LLBE) was first recovered to linearized macroscopic equations. Then, using the fractional-step calculation technique, the solution of these linearized equations was divided into two steps: a predictor step and corrector step. Next, the evolution of the perturbation distribution function was transformed into the evolution of the perturbation equilibrium distribution function using second-order interpolation approximation of the latter at other positions and times to represent the nonequilibrium part of the former; additionally, the calculation formulas of SLLBM were deduced. SLLBM inherits the advantages of the linearized lattice Boltzmann method (LLBM), calculating acoustic disturbance and the mean flow separately so that macroscopic variables of the mean flow do not affect the calculation of acoustic disturbance. At the same time, it has other advantages: the calculation process is simpler, and the cost of computing memory is reduced. In addition, to simulate the acoustic scattering problem caused by the acoustic waves encountering objects, the immersed boundary method (IBM) and SLLBM were further combined so that the method can simulate the influence of complex geometries. Several cases were used to validate the feasibility of SLLBM for simulation of acoustic wave propagation under the mean flow.

A fractional-step lattice Boltzmann method for multiphase flows with complex interfacial behavior and large density contrast

In the present study, a robust fractional-step lattice Boltzmann (FSLB) method is proposed to simulate the mass transfer phenomenon in incompressible multiphase flows with complex interfacial behavior and large density contrast. The previous simplified lattice Boltzmann method recovers the continuity equation in first-order accuracy and reconstructs the corrector step by directly applying the complex central difference scheme on the macroscopic variables. However, the present FSLB method employs the Chapman-Enskog expansion analysis to reconstruct the convection and diffusion terms of the macroscopic governing equations, and uses the equilibrium and non-equilibrium distribution functions to establish the predictor-corrector step. The intermediate variables are predicted by the equilibrium distribution functions without the consideration of the source terms, and then the physical variables are corrected by the non-equilibrium distribution functions and the source terms without the evolution of the distribution functions. The reconstructed governing equations in both the predictor and corrector steps can be recovered into fully second-order accuracy through the C-E expansion analysis and the Taylor series expansion analysis. The present FSLB method inherits the excellent performance of kinetic theory from the conventional LB method and the good numerical stability from the matured fractional-step method, which is validated by several benchmark problems, such as Laplace law, bubble merging, interfacial deformation under a magnetic field, Rayleigh-Taylor instability at Reynolds number of 3000, Kelvin-Helmholtz instability at Reynolds number of 5000, and bubble rising at density ratio of 1000. A good agreement between the present numerical results with the published numerical data verifies the capability and reliability of the present FSLB method to handle the multiphase problems with complex interfacial behavior and large density contrast.

Efficient adaptive exponential time integrators for nonlinear Schrödinger equations with nonlocal potential

The performance of exponential-based numerical integrators for the time propagation of the equations associated with the multiconfiguration time-dependent Hartree–Fock (MCTDHF) method for the approximation of the multi-particle Schrödinger equation in one space dimension is assessed. Among the most popular integrators such as Runge–Kutta methods, time-splitting, exponential integrators and Lawson methods, exponential Lawson multistep methods with one predictor–corrector step provide the best stability and accuracy at the least effort. This assessment is based on the observation that the evaluation of the nonlocal terms associated with the potential is the computationally most demanding part of such a calculation in our setting. In addition, the predictor step provides an estimator for the local time-stepping error, thus allowing for adaptive time-stepping which reflects the smoothness of the solution and enables to reliably control the accuracy of a computation in a robust way, without the need to guess an optimal stepsize a priori. One-dimensional model examples are studied to compare different time integrators and demonstrate the successful application of our adaptive methods.

A homotopy continuation inversion of geoelectrical sounding data

In nonlinear inversion of geophysical data, improper initial approximation of the model parameters usually leads to local convergence of the normal Newton iteration methods, despite enforcing constraints on the physical properties. To mitigate this problem, we present a globally convergent Homotopy continuation algorithm to solve the nonlinear least squares problem through a path-tracking strategy in model space. The proposed scheme is based on introducing a new functional to replace the quadratic Tikhonov-Phillips functional. The algorithm implementation includes a sequence of predictor-corrector steps to find the best direction of the solution. The predictor calculates an approximate solution of the corresponding new function in the Homotopy in consequence of using a new value of the continuation parameter at each step of the algorithm. The predicted approximate solution is then corrected by applying the corrector step (e.g., Gauss-Newton method). The global convergence of the Homotopy algorithm is compared with a conventional iterative method through the synthetic and real 1-D resistivity data sets. Furthermore, a bootstrap-based uncertainty analysis is provided to quantify the error in the inverted models derived from the case study. The results of blocky and smooth inversion demonstrate that the presented optimization method outperforms the standard algorithm in the sense of stability, rate of convergence, and the recovered models.

MOC-Z Models for Transient Gaseous Cavitation in Pipe Flow

AbstractThe paper presents models based on the method of characteristics (MOC) for transient gaseous cavitation in pipe flow. Two predictor-corrector steps are considered, in which only the term pe...

On the optimal heat source location of partially heated energy storage process using the newly developed simplified enthalpy based lattice Boltzmann method

The present study aims to fill the knowledge gap in the partially heated energy storage process by finding the optimal heat source location. A new simplified enthalpy based lattice Boltzmann method is proposed for the present work. The new model follows the predictor–corrector step, while the traditional model follows the collision-streaming step. Compared to the traditional model, this new model has the same accuracy, but it has advantages on virtual memory saving and physical boundary treatment due to its unique structure. Besides, in the partially heated energy storage process, the heat source location can pose significant impact on the charging efficiency and the energy storage rate. The optimal heat source location is where the maximum charging efficiency and energy storage rate are achieved. Results indicate that, a small region with the length of 0.1 l can be deemed as the optimal region for the heat source location, for that the charging time and the energy storage rate at various locations inside that region are almost the same. The optimal region decreases with the increase of Rayleigh and Prandtl numbers. In addition, the optimal regions for some specific heat source lengths and Rayleigh numbers are obtained. Based on these results, a map is provided to indicate the variations of the optimal region. According to this map, the optimal heat source locations for more varied conditions can be estimated through the heat source length and the Rayleigh number, which would be quite useful and significant for the thermal design in various engineering applications.

An Efficient Two-Layer Non-Hydrostatic Model for Investigating Wave Run-Up Phenomena

In this paper, we study the maximum run-up of solitary waves on a sloping beach and over a reef through a non-hydrostatic model. We do a modification on the non-hydrostatic model derived by Stelling and Zijlema. The model is approximated by resolving the vertical fluid depth into two-layer system. In contrast to the two-layer model proposed by Stelling, here, we have a block of a tridiagonal matrix for the hydrodynamic pressure. The equations are then solved by applying a staggered finite volume method with predictor-corrector step. For validation, several test cases are presented. The first test is simulating the propagation of solitary waves over a flat bottom. Good results in amplitude and shape preservation are obtained. Furthermore, run-up simulations are conducted for solitary waves climbing up a sloping beach, following the experimental set-up by Synolakis. In this case, two simulations are performed with solitary waves of small and large amplitude. Again, good agreements are obtained, especially for the prediction of run-up height. Moreover, we validate our numerical scheme for wave run-up simulation over a reef, and the result confirms the experimental data.

A step size control scheme based on polynomial regression for continuation power flow calculation

Continuation power flow is an effective approach to analyse static voltage stability by accurately calculating the maximum loading point. To improve the efficiency of continuation power flow model by decreasing its calculation burden, a new step size control scheme for predictor-corrector steps is proposed in this study. Firstly, the predictor tangent vector is obtained according to Jacobin matrix. Secondly, the optimal parameter with the largest sensitivity to load power change is identified by recognizing biggest component in tangent predictor vector. Thirdly, polynomial regression between the optimal parameter and the power increment parameter is analysed. Finally, the improved self-adaption step size control scheme is established in terms of the obtained polynomial regression model. Studies based 14-bus test system are used to demonstrate the efficacy of the proposed scheme. For this study, specific point-to-point maximum loading is calculated.

QpSWIFT: A Real-Time Sparse Quadratic Program Solver for Robotic Applications

In this letter, we present qpSWIFT, a real-time quadratic program (QP) solver. Motivated by the need for a robust embedded QP solver in robotic applications, qpSWIFT employs standard primal-dual interior-point method, along with Mehrotra predictor–corrector steps and Nesterov–Todd scaling. The sparse structure of the resulting Karush–Kuhn–Tucker linear system in the QP formulation is exploited, and sparse direct methods are utilized to solve the linear system of equations. To further accelerate the factorization process, we only modify the corresponding rows of the matrix factors that change during iterations and cache the nonzero Cholesky pattern. qpSWIFT is library free, written in ANSI-C and its performance is benchmarked through standard problems that could be cast as QP. Numerical results show that qpSWIFT outperforms state-of-the-art solvers for small scale problems. To evaluate the performance of the solver, a real-time implementation of the solver in the model predictive control framework through experiments on a quadrupedal robot are presented.

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.

Simplified multiphase lattice Boltzmann method for simulating multiphase flows with large density ratios and complex interfaces

In this paper, a simplified multiphase lattice Boltzmann method (SMLBM) is developed for modeling incompressible multiphase flows with large density ratios and complex interfaces. SMLBM is derived from reconstructing solutions to macroscopic equations recovered from the multiphase lattice Boltzmann model through Chapman-Enskog expansion analysis and resolved in a predictor-corrector step. The Cahn-Hilliard equation is utilized as the interface tracking algorithm, the solution of which is also reconstructed within the lattice Boltzmann framework. The resultant formulation of SMLBM reflects a direct update of macroscopic variables instead of distribution functions, which could remarkably save virtual memory and facilitate implementation of physical boundary conditions. Numerical tests also indicate that our scheme could effectively simulate challenging cases with large density ratios (up to 1000) and complex interfaces, and at high Reynolds numbers.

SALMON: Scalable Ab-initio Light–Matter simulator for Optics and Nanoscience

SALMON (Scalable Ab-initio Light–Mattersimulator for Optics and Nanoscience, http://salmon-tddft.jp) is a software package for the simulation of electron dynamics and optical properties of molecules, nanostructures, and crystalline solids based on first-principles time-dependent density functional theory. The core part of the software is the real-time, real-space calculation of the electron dynamics induced in molecules and solids by an external electric field solving the time-dependent Kohn–Sham equation. Using a weak instantaneous perturbing field, linear response properties such as polarizabilities and photoabsorptions in isolated systems and dielectric functions in periodic systems are determined. Using an optical laser pulse, the ultrafast electronic response that may be highly nonlinear in the field strength is investigated in time domain. The propagation of the laser pulse in bulk solids and thin films can also be included in the simulation via coupling the electron dynamics in many microscopic unit cells using Maxwell’s equations describing the time evolution of the electromagnetic fields. The code is efficiently parallelized so that it may describe the electron dynamics in large systems including up to a few thousand atoms. The present paper provides an overview of the capabilities of the software package showing several sample calculations. Program summaryProgram Title: SALMON: Scalable Ab-initio Light–Matter simulator for Optics and NanoscienceProgram Files doi:http://dx.doi.org/10.17632/8pm5znxtsb.1Licensing provisions: Apache-2.0Programming language: Fortran 2003Nature of problem: Electron dynamics in molecules, nanostructures, and crystalline solids induced by an external electric field is calculated based on first-principles time-dependent density functional theory. Using a weak impulsive field, linear optical properties such as polarizabilities, photoabsorptions, and dielectric functions are extracted. Using an optical laser pulse, the ultrafast electronic response that may be highly nonlinear with respect to the exciting field strength is described as well. The propagation of the laser pulse in bulk solids and thin films is considered by coupling the electron dynamics in many microscopic unit cells using Maxwell’s equations describing the time evolution of the electromagnetic field.Solution method: Electron dynamics is calculated by solving the time-dependent Kohn–Sham equation in real time and real space. For this, the electronic orbitals are discretized on a uniform Cartesian grid in three dimensions. Norm-conserving pseudopotentials are used to account for the interactions between the valence electrons and the ionic cores. Grid spacings in real space and time, typically 0.02 nm and 1 as respectively, determine the spatial and temporal resolutions of the simulation results. In most calculations, the ground state is first calculated by solving the static Kohn–Sham equation, in order to prepare the initial conditions. The orbitals are evolved in time with an explicit integration algorithm such as a truncated Taylor expansion of the evolution operator, together with a predictor–corrector step when necessary. For the propagation of the laser pulse in a bulk solid, Maxwell’s equations are solved using a finite-difference scheme. By this, the electric field of the laser pulse and the electron dynamics in many microscopic unit cells of the crystalline solid are coupled in a multiscale framework.

A splitting method for free surface flows over partially submerged obstacles

Abstract The paper proposes a stable time-splitting method for the numerical simulation of free-surface viscous flows. The key features of the method are a semi-Lagrangian scheme for the level-set function transport improved with MacCormack predictor–corrector step with limiting strategy and an adaptive volume-correction procedure. The spatial discretization is done by a hybrid finite volume/finite difference method on dynamically adaptive hexahedral meshes. Numerical verification is done by comparing full-scale 3D numerical simulations of the sloshing tank and the coastal wave run-up with other numerical and experimental results known from the literature.

A Dispersive Numerical Model for the Formation of Undular Bores Generated by Tsunami Wave Fission

AbstractA two-layer non-hydrostatic numerical model is proposed to simulate the formation of undular bores by tsunami wave fission. These phenomena could not be produced by a hydrostatic model. Here, we derived a modified Shallow Water Equations with involving hydrodynamic pressure using two layer approach. Staggered finite volume method with predictor corrector step is applied to solve the equation numerically. Numerical dispersion relation is derived from our model to confirm the exact linear dispersion relation for dispersive waves. To illustrate the performance of our non-hydrostatic scheme in case of linear wave dispersion and non-linearity, four test cases of free surface flows are provided. The first test case is standing wave in a closed basin, which test the ability of the numerical scheme in simulating dispersive wave motion with the correct frequency. The second test case is the solitary wave propagation as the examination of owing balance between dispersion and nonlinearity. Regular wave propagation over a submerged bar test by Beji is simulated to show that our non-hydrostatic scheme described well the shoaling process as well as the linear dispersion compared with the experimental data. The last test case is the undular bore propagation.

Strongly coupled partitioned six degree-of-freedom rigid body motion solver with Aitken's dynamic under-relaxation

An implicit method of solving the six degree-of-freedom rigid body motion equations based on the second order Adams-Bashforth-Moulten method was utilised as an improvement over the leapfrog scheme by making modifications to the rigid body motion solver libraries directly. The implementation will depend on predictor-corrector steps still residing within the hybrid Pressure Implicit with Splitting of Operators - Semi-Implicit Method for Pressure Linked Equations (PIMPLE) outer corrector loops to ensure strong coupling between fluid and motion. Aitken's under-relaxation is also introduced in this study to optimise the convergence rate and stability of the coupled solver. The resulting coupled solver ran on a free floating object tutorial test case when converged matches the original solver. It further allows a varying 70%–80% reduction in simulation times compared using a fixed under-relaxation to achieve the required stability.

Bed Evolution under Rapidly Varying Flows by a New Method for Wave Speed Estimation

This paper proposes a sediment-transport model based on coupled Saint-Venant and Exner equations. A finite volume method of Godunov type with predictor-corrector steps is used to solve a set of coupled equations. An efficient combination of approximate Riemann solvers is proposed to compute fluxes associated with sediment-laden flow. In addition, a new method is proposed for computing the water depth and velocity values along the shear wave. This method ensures smooth solutions, even for flows with high discontinuities, and on domains with highly distorted grids. The numerical model is tested for channel aggradation on a sloping bottom, dam-break cases at flume-scale and reach-scale with flat bottom configurations and varying downstream water depths. The proposed model is tested for predicting the position of hydraulic jump, wave front propagation, and for predicting magnitude of bed erosion. The comparison between results based on the proposed scheme and analytical, experimental, and published numerical results shows good agreement. Sensitivity analysis shows that the model is computationally efficient and virtually independent of mesh refinement.

A New Total Variation Diminishing Predictor Corrector Approach for Two-Dimensional Shallow Water Flow

This paper presents a new efficient and robust hydrodynamic model for simulation of unsteady shallow water flow. The governing equations of shallow water flows in two dimensional forms are solved using a new total variation diminishing (TVD) MacCormack predictor corrector scheme. In this numerical technique an additional TVD term is added after the traditional predictor corrector steps. The advantage of the present TVD term is that it is very simple and gives accurate results at the same time removing the numerical oscillations. Further, application of semi implicit treatment of the friction slope term helps in flow simulation even with very low water depth. Finally the model is used to analyze a wide variety of hydraulic problems including quiescent water above irregular bed, steady flow over irregular bed, steady flow over irregular bed with a shock, dam break flow over dry bed and dam break flow over wet bed. For each of the cases numerical results are compared with available analytical solution and known experimental data. The agreements between the results are satisfactory.

Underestimation of nuclear fuel burnup – theory, demonstration and solution in numerical models

Monte Carlo methodology provides reference statistical solution of neutron transport criticality problems of nuclear systems. Estimated reaction rates can be applied as an input to Bateman equations that govern isotopic evolution of reactor materials. Because statistical solution of Boltzmann equation is computationally expensive, it is in practice applied to time steps of limited length. In this paper we show that simple staircase step model leads to underprediction of numerical fuel burnup (Fissions per Initial Metal Atom – FIMA). Theoretical considerations indicates that this error is inversely proportional to the length of the time step and origins from the variation of heating per source neutron. The bias can be diminished by application of predictor-corrector step model. A set of burnup simulations with various step length and coupling schemes has been performed. SERPENT code version 1.17 has been applied to the model of a typical fuel assembly from Pressurized Water Reactor. In reference case FIMA reaches 6.24% that is equivalent to about 60 GWD/tHM of industrial burnup. The discrepancies up to 1% have been observed depending on time step model and theoretical predictions are consistent with numerical results. Conclusions presented in this paper are important for research and development concerning nuclear fuel cycle also in the context of Gen4 systems.