Predictor-corrector Procedure Research Articles

Numerical simulation of coupled Klein–Gordon–Schrödinger equations: RBF partition of unity

The coupled Klein–Gordon–Schrödinger equations have significant implications in quantum field theory, particle physics, cosmology, and nonlinear dynamics. In this study, we propose an efficient method for numerically simulating this system. The proposed approach involves employing the radial basis function partition of unity for spatial discretization. This method utilizes scaled Lagrange basis functions with polyharmonic spline kernels, taking advantage of the scalability property of the polyharmonic kernel to ensure stability in the approximation process. The resulting spatially discretized system yields a nonlinear time-dependent set of differential equations. To solve this system, we combine an implicit central finite difference scheme with a predictor–corrector procedure to overcome the nonlinearity. In the numerical results section, we assess the efficiency, accuracy, and versatility of the proposed method by conducting several simulations in large domains over extended periods. We present and compare these results with those obtained using alternative methods, demonstrating the effectiveness of the proposed approach in accurately solving the coupled Klein–Gordon–Schrödinger equations.

A non-hydrostatic model for wave evolution on a submerged trapezoidal breakwater

A depth-averaged non-hydrostatic model is formulated to investigate wave evolution on a water channel with a submerged trapezoidal breakwater. This model is an extension of nonlinear shallow water equations that includes hydrodynamic pressure and vertical velocity. In the momentum equation, a diffusion term is also considered to represent the turbulence effects in the system. The equations are solved numerically using a combination of a staggered finite volume method and predictor–corrector procedure. Comparisons are made against three independent laboratory experiments of wave propagation over submerged breakwaters. The level of agreement is higher than with Boussinesq-type and RANS models. The numerical scheme is also used to study the effect of the height, length, and diffusion coefficient of the breakwater on wave propagation. We have found that those characteristics affect the wave similarly by smoothing the wave shape and significantly reducing the transmitted wave amplitude.

An adaptive wavelet method for nonlinear partial differential equations with applications to dynamic damage modeling

Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet-based technique to solve a coupled system of nonlinear partial differential equations (PDEs) while resolving features on a wide range of spatial and temporal scales. The algorithm exploits the multiresolution nature of wavelet basis functions to solve initial-boundary value problems on finite domains with a sparse multiresolution spatial discretization. By leveraging wavelet theory and embedding a predictor-corrector procedure within the time advancement loop, we dynamically adapt the computational grid and maintain accuracy of the solutions of the PDEs as they evolve. Consequently, our method provides high fidelity simulations with significant data compression. We present verification of the algorithm and demonstrate its capabilities by modeling high-strain rate damage nucleation and propagation in nonlinear solids using a novel Eulerian-Lagrangian continuum framework.

Continuous-time predator-prey-like systems used to investigate the question: Can human consciousness help eliminate temporary mosquito breeding sites from around human habitats?

Mosquito breeding-sites are an essential resource for mosquito growth, yet an understanding of their existence and creation, in the case of temporary habitats, have not been mathematically studied. Hence, we use a system of two first order nonlinear ordinary differential equations to model and theoretically investigate the dynamics of temporary mosquito breeding-sites formation in a uniform environment. The model captures the dynamic interplay between community action, climatic factors and availability of temporary mosquito breeding-sites by interpreting the possible pathways and environmental processes leading to temporary breeding-sites formation. Model’s analysis show it is possible to eliminate temporary breeding-sites from around a close vicinity to human habitats by increasing the level of human consciousness, with human consciousness measured via human response to community action. Different feedback response functions are used to excite breeding-site removal and community action. When the response functionals are both constants, an indicator function is identified, whose size can indicate whether in the long-run, community action can lead to the removal and elimination of temporary breeding-sites near human habitats. Using a predictor–corrector procedure that fits real climatic data to a continuous periodic function, we demonstrate how climatic variables can be included in the model and how the time variation of temperature and precipitation in a given area can be constructed just by appropriately choosing the parameters of a sinusoidal function and then correcting the output using nonlinear least squares analysis. Numerical simulation results are used to complement our analytical results showing the different dynamical behaviours that can be obtained. When breeding-site densities are high, human consciousness starts rising and its effect can lead to the reduction of breeding-site density, measured, and interpreted via community action. In some scenarios, the response can lead to complete removal of the breeding sites, while in others it may enter a limit cycle with different transient dynamics determined by the temperature parameters, waning rate of consciousness and breeding site impact on igniting consciousness. The novelty in this manuscript lies in the idea in which the focus is on modelling temporary mosquito breeding-sites, a process that is natural, yet neglected. Additionally, the incorporation of human consciousness highlights the need for a better understanding of human factors in neglected population dynamics studies and to attempt to correlate consciousness to adherence and its influence on disease control. Our mathematical model to address the problem of mosquito breeding-site formation, an important problem that has minimally been addressed, serves as a foundational model that can be extended to incorporate stochasticity in breeding site formation.

High order asymptotic preserving discontinuous Galerkin methods for gray radiative transfer equations

In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs). Inspired by the works in [6,55], we propose to penalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term. A hyperbolic, namely Δt=O(h) in the transport regime where Δt and h are the time step and mesh size respectively, and unconditional stability instead of parabolic time step restriction Δt=O(h2) in the diffusive regime are obtained, which are also free from the photon mean free path. We further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations within each outer iterative loop. For the resulting scheme, only an implicit system for the macroscopic variable needs to be solved, while the microscopic variable can be updated explicitly. Besides, the nonlinear implicit system is decoupled to a linear positive definite system followed by nonlinear algebraic equations due to the Picard iteration. Namely, high dimension, implicit treatment and nonlinearity are all decoupled. Our scheme is shown to be AP and asymptotically accurate (AA). Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is high order accurate, effective and efficient.

Reactive power optimization under interval uncertainty of renewable power generation based on a security limits method

With the integration of high penetration renewable power generation, the voltage security of power systems is threatened. The reactive power optimization including interval uncertainty (RPOIU) model is applied for developing a voltage control strategy to ensure that the state variables of a power grid reside within their safe operating limits under uncertainty of renewable power generation and load demand input data. However, the RPOIU model is not solved properly in terms of solution efficiency and optimization effect at present. This paper proposes a novel method for solving the RPOIU model, which is denoted as the security limits method (SLM). The proposed SLM first computes the security limits of the state variables of the RPOIU model using an interval power flow method, i.e., the optimizing-scenarios method, and then transforms the RPOIU model into a deterministic model whose constraints conform to the established security limits. The deterministic model is then efficiently solved using the interior point method. We verify that the voltage control strategy obtained by solving the deterministic model is able to ensure that the values of the state control variables satisfy the constraints of the original RPOIU model, but with more conservative security limits. This issue is addressed by modifying the security limits of the SLM through defining an interval ratio, and applying an iterative predictor-corrector procedure. The result obtained by the proposed SLM and modified SLM based on numerical simulations are compared with those obtained by a previously proposed method for effectively solving the RPOIU model, and analysis of the results demonstrates the advantages, effectiveness, and good applicability of the proposed SLM.

Mapping exomoon trajectories around Earth-like exoplanets

ABSTRACT We consider a system in which both the parent star and the Earth-like exoplanet move on circular orbits. Using numerical methods, such as the orbit classification technique, we study all types of trajectories of possible exomoons around the exoplanet. In particular, we scan the phase space around the exoplanet and we distinguish between bounded, collisional, and escaping trajectories, considering both retrograde and prograde types of motion. In the case of bounded regular motion, we also use the grid method and a standard predictor-corrector procedure for revealing the corresponding network of symmetric periodic solutions, while we also compute their linear stability.

A MULTIRESOLUTION ADAPTIVE WAVELET METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to solve partial differential equations (PDEs) with features evolving on a wide range of spatial and temporal scales. To meet these challenges, we present a multiresolution wavelet algorithm to solve PDEs with significant data compression and explicit error control. We discretize in space by projecting fields and spatial derivative operators onto wavelet basis functions. We provide error estimates for the wavelet representation of fields and their derivatives. Then, our estimates are used to construct a sparse multiresolution discretization which guarantees the prescribed accuracy. Additionally, we embed a predictor-corrector procedure within the temporal integration to dynamically adapt the computational grid and maintain the accuracy of the solution of the PDE as it evolves. We present examples to highlight the accuracy and adaptivity of our approach.

A 3D OpenFOAM based finite volume solver for incompressible Oldroyd-B model with infinity relaxation time

In this paper, we develop a Finite Volume solver for a 3D incompressible Oldroyd-B model with infinity relaxation time. The Finite Volume solver is implemented by using a leading open-source computational mechanics software OpenFOAM. We have imposed the divergence free condition as a constraint on the momentum equation to derive a pressure equation and a predictor-corrector procedure is applied when solving the velocity field. Both stability analysis and numerical experiments are given to show the robustness and accuracy of our algorithm. Two concrete examples on a cubical domain and a dumbbell are computed and illustrated.

Application of two-dimensional hydrodynamics numerical modelling in mangrove planting of coastal areas

A two-dimensional numerical model is presented for application in mangrove planting of coastal areas. The model is discretized with finite volume method based on an unstructured mesh and employs the Roe-MUSCL scheme and the predictor-corrector procedure for time stepping. The model is validated primarily by the observed river runoff data at the upstream of the Jiaojiang River and tidal level along the open boundary obtained with the TPXO7.2 global tidal model. It is then applied to study tidal hydrodynamics in the Dachuan River of the Baisha Bay within the Jiaojiang River estuary during the spring, middle and neap tidal cycles in July and August 2016. The model results show well agreements between the predicted and observed tidal level and tidal current process. Finally, referred to existing research on critical tidal level and duration of Kandelia candel planting in coastal areas of the southeast China, a proper planting condition of K. candel seedlings are determined, after adopting the mean tidal level and duration calculated from the model results.

Численное моделирование теплопередачи и переноса металла при электронно-лучевом аддитивном формообразовании с подачей присадочной проволоки

The development of a mathematical model provides an analysis of heat transfer and metal flow during wire-based electron-beam additive manufacturing described. The procedure for solving the heat equation for the metal in the solid phase and the Navier-Stokes equations in the liquid phase, based on the use of the finite-difference method and the predictor-corrector procedure is described. An algorithm for numerical approximation of the motion of the free surface of the melt, using the concept of the volume of fluid (VOF) is describer. The original method for calculating the effect of surface tension forces, based on the numerical calculation of the surface curvature index is proposed. The results of simulating the melting of a wire element localized above a substrate made of 316L steel are described

On the shape optimisation of the force networks of masonry structures

A novel lumped stresses network approach for the analysis of the mechanical behaviour of masonry structures is presented. It is formulated through a network of lumped stresses, making use of polyhedral stress functions and a variational approximation of the continuous equilibrium problem. The proposed method allows for modelling masonry structures as no-tension elements and gives, in the case of curved masonry members, the optimised surface through a predictor-corrector procedure and the stress function describing the membrane stress. The given approach offers a useful tool for predicting the crack pattern of unreinforced masonry structures and the associated stress fields. It is validated against some benchmark case studies dealing with a hemispherical dome, a groin vault, a cloister vault and a masonry beam.

Static and dynamic nonlinear response of masonry walls

A nonlocal damage-plastic model is proposed to investigate the mechanical response of masonry elements, under static and dynamic actions. The adopted constitutive relationship is able to capture degrading mechanisms due to propagation of microcracks and accumulation of irreversible strains. Moreover, the stiffness recovery, due to re-closure of tensile cracks when material undergoes compression strains, is taken into account to properly simulate the masonry cyclic response. The proposed relationship is implemented in a 2Dplane stress finite element and a predictor–corrector procedure is developed to solve the nonlinear evolution problem of damage and plastic variables.The model validation is carried out by comparing numerical and experimental results on masonry panels under cyclic quasi-static conditions. Then, to analyze the effect of the evolution of degrading mechanisms on masonry dynamic response, the frequency response curves of a slender wall are evaluated by highlighting the influence of the proposed masonry constitutive relationship with respect to other widely studied models characterized by nonlinear invariant restoring forces.

Three-Layer Non-hydrostatic Staggered Scheme for Free Surface Flow

AbstractIn this paper, a finite difference algorithm using a three-layer approximation for the vertical flow region to solve the 2D Euler equations is considered. In this algorithm, the pressure is split into hydrostatic and hydrodynamic parts, and the predictor-corrector procedure is applied. In the predictor step, the momentum hydrostatic model is formulated. In the corrector step, the hydrodynamic pressure is accommodated after solving the Laplace equation using the Successive Over Relaxation (SOR) iteration method. The resulting algorithm is first tested to simulate a standing wave over an intermediate constant depth. Dispersion relation of the scheme is derived, and it is shown to agree with the analytical dispersion relation for kd < π with 94% accuracy. The second test case is a solitary wave simulation. Our computed solitary wave propagates with constant velocity, undisturbed in shape, and confirm the analytical solitary wave. Finally, the scheme is tested to simulate the appearance of the undular bore. The result shows a good agreement with the result from the finite volume scheme for the Boussinesq-type model by Soares-Frazão and Guinot (2008).

An enriched damage-frictional cohesive-zone model incorporating stress multi-axiality

The paper presents an interphase cohesive zone model (CZM) incorporating stress multi-axiality devised to capture, by simplified micro-modeling, the influence of the in-plane strain and stress state in the mechanical response of the CZM. Moreover, the model is able to account for the Poisson-related effect in the interphase, which can play an important role in the modeling of heterogeneous masonry elements. From the constitutive point of view, the proposed formulation couples damage and friction by addressing a smooth transition from a quasi-brittle response to a residual frictional behavior described by a Coulomb law with unilateral contact. As in-plane stresses are accounted for, damage activation and evolution are governed by a Drucker–Prager law with linear softening. A predictor-corrector procedure based on a backward Euler scheme is detailed for integrating the nonlinear evolutive problem together with the related tangent operator which consistently linearizes the algorithmic strategy. This framework is embedded into a kinematically-enriched finite element interphase formulation incorporating stress multi-axiality. The modeling features of the resulting numerical tool are tested both at the local level, for the typical interphase point, and in meso-structural tests consisting of brick-mortar triplets, investigating the capability of the proposed model and numerical procedure to simulate the brick-mortar decohesion mechanism during confined slip tests.

Dynamic Voltage Stability Assessment in Large Power Systems With Topology Control Actions

This paper proposes a tractable and scalable algorithm to identify and analyze bifurcation points of a large-scale power system model, which are directly related to dynamic voltage instability problems. Different types of bifurcations are analyzed, including: saddle-node (fold), Hopf, singularity-induced and limit-induced. An algorithm that combines optimization and predictor-corrector procedure is proposed for equilibrium tracing. The algorithm is based on calculation of only critical (rightmost and closest-to-zero) eigenvalues. The proposed algorithm is extended to the case of dynamic voltage stability assessment for power systems with optimized topology (simultaneously subjected to the topology control changes and generation re-dispatch). The proposed approach is illustrated on two (medium- and large-scale real-world) test power systems.

Integration algorithm for covariance nonstationary dynamic analysis using equivalent stochastic linearization

Deterministic mechanical systems subject to stochastic dynamic actions, such as wind or earthquakes, have to be properly evaluated using a stochastic approach. For nonlinear mechanical systems it is necessary to approximate solutions using mathematical tools, as the stochastic equivalent linearization. It is a simple approach from the theoretical point of view, but needs numerical techniques whose computational complexity increases in case of nonstationary excitations. In this paper a procedure to solve covariance analysis of stochastic linearized systems in the case of nonstationary excitation is proposed. The nonstationary Lyapunov differential matrix covariance equation for the linearized system is solved using a numerical algorithm which updates linearized system coefficient matrix at each step. The technique used is a predictor–corrector procedure based on backward Euler method. Accuracy and computational costs are analysed showing the efficiency of the proposed procedure.

A finite volume method for numerical simulation of shallow water models with porosity

In the current work, we present a new finite volume method for numerical simulation of one and two dimensions of shallow water equations with porosity. The introduction of a porosity into shallow water equations leads to modified expressions for the fluxes and source terms. The proposed method consists of two stages, which can be viewed as a predictor–corrector procedure. The first stage (predictor) of the scheme depends on a local parameter allowing to control diffusion, which modulate by using the limiters theory. The second stage (corrector) recovers the conservation equation. Numerical results are presented for shallow water equations with porosity. It is found that the proposed finite volume method offers a robust and accurate approach for solving shallow water equations with source term and porosity.

Viscoelastic computations for reverse roll coating with dynamic wetting lines and the Phan-Thien-Tanner models

The computational modelling of reverse roll coating with dynamic wetting line has been analysed for various non-Newtonian viscoelastic materials appealing to the Phan-Thien-Tanner (PTT) network class of models suitable for typical polymer solutions, with properties of shear thinning and strain hardening/softening. The numerical technique utilizes a hybrid finite element-sub-cell finite volume algorithm with a dynamic free-surface location, drawing upon a fractional-staged predictor-corrector semi-implicit time-stepping procedure of an incremental pressure-correction form. The numerical solution is investigated following a systematic study which allows for parametric variation in elasticity (We-variation), extensional hardening-softening (e), and solvent fraction (β). Under incompressible flow conditions, linear PTT (LPTT) and exponential PTT (EPTT) models were used to solve the paint strip coatings, under reverse roll-coating configuration. This involves two-dimensional planar reverse roll-coating domains, considering a range of Weissenberg numbers (We) up to critical levels, addressing velocity fields and vortex development, pressure and lift profiles, shear rate, and stress fields. Various differences are observed when comparing solutions for these constitutive models. Concerning the effects of elasticity, increase in We stimulates vortex structures, which are visible at both the downstream meniscus and upstream narrowest nip region, whilst decreasing the peak pressure and lift values at the nip constriction. At low values (e > 0.5, β = 0.1) of extensional viscosity, the LPTT flow fields were much easier to extract, attaining critical We levels up to unity, in contrast to critical We levels of 0.4 for EPTT solutions. This finding is reversed at higher extensional viscosity levels (e < 0.5). This trend reveals qualitative agreement with theoretical studies. Noting flow behaviour under EPTT solution, increasing the peak level of strain hardening/softening is found to stimulate vortex activity around the nip region, with a corresponding increase in peak pressure and lift values.

A 2D Cosserat finite element based on a damage-plastic model for brittle materials

A 2D Cosserat model with a damage-plastic isotropic constitutive law for brittle-like materials is presented. Different damaging mechanisms in tension and compression are considered. The plastic flow is described by introducing a suitable yield function and evolution laws in terms of effective stresses. Small strain and displacement assumptions hold. A 4-node finite element is formulated with three degrees-of-freedom for each node and a predictor–corrector procedure is used to solve the damage-plastic evolutionary problem. The lateral response of walls under monotonic and cyclic horizontal actions is analyzed and a satisfactory description of the global response curves and damaging mechanisms is obtained.