Operator Splitting Scheme Research Articles

求解Allen-Cahn方程的两种高效数值格式

基于算子分裂思想，本文提出了求解Allen-Cahn方程的两种高效算子分裂格式。首先将此方程分裂为线性项与非线性项两个部分：对线性部分，通过二阶中心差分法与四阶紧致差分法分别进行数值计算，并利用傅里叶方法给出了稳定性的分析；对非线性部分进行精确求解。最后，通过数值算例比较两种算子分裂格式的数值解差异以及运行效率，而且验证了两种格式满足能量递减规律。 Based on the idea of operator splitting, this paper proposes two efficient operator splitting schemes for the Allen-Cahn equation. The original equation is divided into linear and nonlinear parts. The linear part is approximated numerically by two different schemes: the second-order center difference scheme and the fourth-order compact difference scheme. The stability analysis of both schemes is discussed according to a Fourier stability analysis. The nonlinear part is solved accurately. Numerical comparisons are carried out to verify the accuracy and efficiency of the proposed methods, and to verify the given schemes satisfied the law of decreasing energy.

On the Alternate Direction Implicit (ADI) Method for Solving Heat Transfer in Composite Stamping

Thermostamping of thermoplastic matrix composites is a process where a preheated blank is rapidly shaped in a cold matching mould. Predictive modelling of the main physical phenomena occurring in this process requires an accurate prediction of the temperature field. In this paper, a numerical method is proposed to simulate this heat transfer. The initial three-dimensional heat equation is handled using an additive decomposition, a thin shell assumption, and an operator splitting strategy. An adapted resolution algorithm is then presented. It results in an alternate direction implicit decomposition: the problem is solved successively as a 2D surface problem and several one-dimensional through thickness problems. The strategy was fully validated versus a 3D calculation on a simple test case and the proposed strategy is shown to enable a tremendous calculation speed up. The limits of applicability of this method are investigated with two parametric studies, one on the thickness to width ratio and the other one on the effect of curvature. These conditions are usually fulfilled in industrial cases. Finally, even though the method was developed under linear assumption (constant material properties), the strategy validity is extended to multiply, temperature dependant (nonlinear) case using an industrial test case. Because of the standard methods involved, the proposed ADI method can readily be implemented in existing software.

A structure preserving scheme for the Kolmogorov–Fokker–Planck equation

In this paper we introduce a numerical scheme which preserves the behavior of solutions to the Kolmogorov Equation as time tends to infinity. The method presented is based on a self-similar change of variables technique to transform the Kolmogorov Equation into a new form, such that the problem of designing structure preserving schemes, for the original equation, amounts to building a standard scheme for the transformed equation. This transformation also has the added benefit of allowing for an exact operator splitting scheme, whereas in the original form a standard operator splitting was only second-order. Finally, we verify the preservation of long time behavior through numerical simulations.

Computing numerical solutions of the pseudo-parabolic Buckley–Leverett equation with dynamic capillary pressure

We present numerical approaches for solving a pseudo-parabolic partial differential equation, which models incompressible two phase flow in porous media taking into account dynamic effects in the capillary pressure. First, we briefly discuss two numerical schemes based on the operator splitting technique. Our numerical experiments show that the standard splitting, widely used to solve parabolic problems, may fail when applied to pseudo-parabolic models. As an illustration, we give an example for this case. So we present an operator splitting scheme based on a dispersive-like character that obtains correct numerical solutions. Then, we discuss an unsplit efficient numerical modelling, locally conservative by construction. This framework is based on a fully coupled space–time mixed hybrid finite element/volume discretization approach in order to account for the delicate local nonlinear balance between the numerical approximations of the hyperbolic flux and the pseudo-parabolic term, but linked to a natural dispersive-like character of the full pseudo-parabolic equation. We compare our numerical results with approximate solutions constructed with methods recently introduced in the specialized literature, in order to establish that we are computing the expected qualitative behaviour of the solutions.

Deep face liveness detection based on nonlinear diffusion using convolution neural network

A face-spoofing attack occurs when an imposter manipulates a face recognition and verification system to gain access as a legitimate user by presenting a 2D printed image or recorded video to the face sensor. This paper presents an efficient and non-intrusive method to counter face-spoofing attacks that uses a single image to detect spoofing attacks. We apply a nonlinear diffusion based on an additive operator splitting scheme. Additionally, we propose a specialized deep convolution neural network that can extract the discriminative and high-level features of the input diffused image to differentiate between a fake face and a real face. Our proposed method is both efficient and convenient compared with the previously implemented state-of-the-art methods described in the literature review. We achieved the highest reported accuracy of 99% on the widely used NUAA dataset. In addition, we tested our method on the Replay Attack dataset which consists of 1200 short videos of both real access and spoofing attacks. An extensive experimental analysis was conducted that demonstrated better results when compared to previous static algorithms results. However, this result can be improved by applying a sparse autoencoder learning algorithm to obtain a more distinguishable diffused image.

Hybrid level set method based on image diffusion

In this paper, a new hybrid diffusion-based level set method is proposed to efficiently address the complex image segmentation problem. Different from the traditional methods, the proposed method is performed on image diffusion space rather than intensity space. Firstly, the nonlinear diffusion based on total variation flow and additive operator splitting scheme is performed on the original intensity image to obtain the diffused image. Then, the local diffusion energy term is constructed by performing homomorphic unsharp masking operation on diffused image so as to implement a local piecewise constant search. To avoid trapping into local minimum produced by local energy, the global diffusion energy term is formed by approximating diffused image in a global piecewise constant way. Besides, the regularization energy term is included to have penalization effect on evolving contour length and maintenance of level set function being signed distance function. By minimizing the overall energy functional which is a linear combination of local energy, global energy and regularization energy, the evolving contour can be driven to approach the object boundary. The experiments on different characteristics of complex images have shown that the proposed method can achieve satisfying segmentation performance accompanied with some good properties, i.e. the robustness to initial parameter and contour setting, noise insensitivity, quick and stable convergence.

On the merits of extrapolation-based stiff ODE solvers for combustion CFD

In applications including compression-ignition engines, there is a need to accommodate more realistic chemistry in computational fluid dynamics (CFD) simulations. Here, we consider approaches where a chemical mechanism is implemented in an application CFD code, an operator-splitting strategy is used to isolate the chemical source terms, and a stiff ordinary differential equation (ODE) solver is used to compute the changes in composition due to chemical reactions for each computational element. Chemical source terms often dominate the computational effort, and reducing the high computational cost associated with realistic chemistry has been the subject of extensive research. This includes work on improved stiff ODE solvers, which in most cases has centered on backward differentiation formula (BDF) methods. Here a different class of solvers is considered, based on extrapolation methods. Key elements of stiff ODE solvers are reviewed briefly, focusing on differences between BDF methods and extrapolation methods. Issues related to using a stiff ODE solver with operator splitting in a CFD code (where the solver is repeatedly stopped and restarted) are emphasized. Homogeneous-reactor results are presented first. There the relationship between user-specified error tolerances and solution accuracy is explored, tradeoffs between accuracy and CPU time are shown, and close-to-linear increase in CPU time with increasing chemical mechanism size is demonstrated. Engine results are presented next, including both homogeneous-charge compression-ignition engines, and direct-injection (nonhomogeneous) compression-ignition engines. There some results are presented where the stiff ODE solver is combined with a dynamic adaptive chemistry scheme. In all cases, it is found that the extrapolation solver offers significant advantages in accuracy and computational efficiency compared to the BDF solver. While the results presented are for one BDF solver (CVODE) and one extrapolation solver (SEULEX), it is anticipated that the insight into how stiff ODE solvers behave in combustion CFD will be broadly applicable to other solvers in these general classes.

Analysis of the operator splitting scheme for the Allen–Cahn equation

ABSTRACTThis paper presents two numerical schemes for solving the Allen–Cahn equation representing a model for antiphase domain coarsening in a binary mixture. Based on the operator splitting method, the Allen–Cahn equation was divided into a linear and a nonlinear sub equation: the linear sub equation was solved by a Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size; the nonlinear sub equation was then solved analytically due to the availability of a closed-form solution. The stability and error analysis of the numerical solution are presented in detail. Numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the schemes are unconditionally stable and second-order accurate in time.

Additive domain decomposition operator splittings—convergence analyses in a dissipative framework

We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.

Semi-implicit iterative methods for low Mach number turbulent reacting flows: Operator splitting versus approximate factorization

Two formally second-order accurate, semi-implicit, iterative methods for the solution of scalar transport–reaction equations are developed for Direct Numerical Simulation (DNS) of low Mach number turbulent reacting flows. The first is a monolithic scheme based on a linearly implicit midpoint method utilizing an approximately factorized exact Jacobian of the transport and reaction operators. The second is an operator splitting scheme based on the Strang splitting approach. The accuracy properties of these schemes, as well as their stability, cost, and the effect of chemical mechanism size on relative performance, are assessed in two one-dimensional test configurations comprising an unsteady premixed flame and an unsteady nonpremixed ignition, which have substantially different Damköhler numbers and relative stiffness of transport to chemistry. All schemes demonstrate their formal order of accuracy in the fully-coupled convergence tests. Compared to a (non-)factorized scheme with a diagonal approximation to the chemical Jacobian, the monolithic, factorized scheme using the exact chemical Jacobian is shown to be both more stable and more economical. This is due to an improved convergence rate of the iterative procedure, and the difference between the two schemes in convergence rate grows as the time step increases. The stability properties of the Strang splitting scheme are demonstrated to outpace those of Lie splitting and monolithic schemes in simulations at high Damköhler number; however, in this regime, the monolithic scheme using the approximately factorized exact Jacobian is found to be the most economical at practical CFL numbers. The performance of the schemes is further evaluated in a simulation of a three-dimensional, spatially evolving, turbulent nonpremixed planar jet flame.

Nonlinear Advection–Diffusion–Reaction Phenomena Involved in the Evolution and Pumping of Oil in Open Sea: Modeling, Numerical Simulation and Validation Considering the Prestige and Oleg Naydenov Oil Spill Cases

The main goal of this article is to improve upon a previous model used to simulate the evolution of oil spots in the open sea and the effect of a skimmer ship pumping oil out from the spots. The concentration of the pollutant is subject to the effects of wind and sea currents, diffusion, and the pumping action of a skimmer (i.e., cleaning) ship that follows a pre-assigned trajectory. This implies that the mathematical model is of the advection---diffusion---reaction type. A drawback of our previous model was that diffusion was propagating with infinite velocity; in this article, we use an improved model relying on a nonlinear diffusion term, implying that diffusion propagates with finite velocity. To reduce numerical diffusion when approximating the advection part of the model, we consider second order discretization schemes with nonlinear flux limiters. We consider also absorbing boundary conditions to insure accurate results near the boundary. To reduce CPU time we use an operator-splitting scheme for the time discretization. Finally, we also introduce the modeling of coastlines and dynamic sources of pollutant. The novel approach we advocate in this article is validated by comparing our numerical results with real life measurements from the Oleg Naydenov and the Prestige oil spills, which took place in Spain in 2015 and 2002, respectively.

A sparse stiff chemistry solver based on dynamic adaptive integration for efficient combustion simulations

A sparse stiff chemistry solver based on dynamic adaptive hybrid integration (AHI-S) is developed and demonstrated for efficient combustion simulations. In a previous study, a dynamic adaptive method for hybrid integration (AHI) was developed to speed up the time integration of chemically reacting flows with detailed chemistry. The AHI method solves the fast subcomponent of chemistry implicitly and the slow subcomponent of chemistry and transport explicitly, and it was shown that AHI is more accurate and efficient than the operator-splitting schemes when there are significant radical sources from the transport term. In the present study, the AHI method is first improved to minimize the number of nontrivial entries in the Jacobian. Sparse matrix techniques are further integrated into AHI to achieve high computational efficiency. The performance of the new AHI-S solver is investigated in constant-pressure auto-ignition systems using different mechanisms that consist of 9–2878 species. It is shown that the computational cost of the AHI-S solver is overall linearly proportional to the mechanism size and is comparable to that of evaluating reaction rates using CHEMKIN-II subroutines. The AHI-S solver achieves speed-up factors ranging from approximately 10, for the 9-species hydrogen mechanism, to approximately 3000, for the 2878-species biodiesel mechanism, compared with the fully implicit VODE solver with Jacobian evaluated through numerical perturbations and factorized with dense matrix operations. It is further found that for mechanisms with less than approximately 100 species, the time saving of AHI-S is primarily attributed to the reduced size of the implicit core of the governing equations, while for mechanisms with more than 100 species, the computational cost of VODE is dominated by the dense LU factorization, such that the time saving of AHI-S is mostly attributed to the sparse LU factorization. The AHI-S solver is then applied to unsteady perfectly stirred reactors involving extinction and re-ignition. Speed-up factors from 50 to 30,000 are achieved compared with the Strang splitting scheme with the chemistry substeps implicitly integrated with VODE, while speed-up factors of 10–100 are achieved compared with the Strang splitting scheme implemented with the sparse stiff LSODES solver. In the end, the performance of AHI-S is investigated in one-dimensional (1-D) unsteady freely propagating laminar premixed flames for a methane/air mixture, for which the time step size in AHI-S is limited by the fastest transport process. A speed-up factor of approximately 200 is achieved compared with the Strang splitting scheme for fixed time step sizes between 10−8s and 10−6s.

Multi-GPU unsteady 2D flow simulation coupled with a state-to-state chemical kinetics

In this work we are presenting a GPU version of a CFD code for high enthalpy reacting flow, using the state-to-state approach. In supersonic and hypersonic flows, thermal and chemical non-equilibrium is one of the fundamental aspects that must be taken into account for the accurate characterization of the plasma and state-to-state kinetics is the most accurate approach used for this kind of problems. This model consists in writing a continuity equation for the population of each vibrational level of the molecules in the mixture, determining at the same time the species densities and the distribution of the population in internal levels. An explicit scheme is employed here to integrate the governing equations, so as to exploit the GPU structure and obtain an efficient algorithm. The best performances are obtained for reacting flows in state-to-state approach, reaching speedups of the order of 100, thanks to the use of an operator splitting scheme for the kinetics equations.

Adaptive time stepping in pseudo-dynamic testing of earthquake driven structures

The problem of assessing errors in implementing time-marching algorithms in the context of pseudo-dynamic seismic testing of structures is considered. These errors occur in implementing the numerical and experimental steps of the test procedure. The study investigates how a linearized variational equation can be augmented with the governing equation of motion to track the effect of the errors, and, accordingly, adjust the step size of integration adaptively to keep a global error norm within specified limits. The governing augmented equations are integrated using an explicit operator splitting scheme. Additional efforts, in terms of evaluation of the tangent stiffness matrix, are shown to become necessary while modelling the errors. Illustrative examples include numerical studies on a set of nonlinear systems and an experimental study on a geometrically nonlinear two-storied building frame. The experimental results from pseudo-dynamic test are shown to compare reasonably well with pertinent results from an effective force test.

Homogenization in micro-magneto-mechanics

Ferromagnetic materials are characterized by a heterogeneous micro-structure that can be altered by external magnetic and mechanical stimuli. The understanding and the description of the micro-structure evolution is of particular importance for the design and the analysis of smart materials with magneto-mechanical coupling. The macroscopic response of the material results from complex magneto-mechanical interactions occurring on smaller length scales, which are driven by magnetization reorientation and associated magnetic domain wall motions. The aim of this work is to directly base the description of the macroscopic magneto-mechanical material behavior on the micro-magnetic domain evolution. This will be realized by the incorporation of a ferromagnetic phase-field formulation into a macroscopic Boltzmann continuum by the use of computational homogenization. The transition conditions between the two scales are obtained via rigorous exploitation of rate-type and incremental variational principles, which incorporate an extended version of the classical Hill---Mandel macro-homogeneity condition covering the phase field on the micro-scale. An efficient two-scale computational scenario is developed based on an operator splitting scheme that includes a predictor for the magnetization on the micro-scale. Two- and three-dimensional numerical simulations demonstrate the performance of the method. They investigate micro-magnetic domain evolution driven by macroscopic fields as well as the associated overall hysteretic response of ferromagnetic solids.

Simulation of non-linear acoustic field and thermal pattern of phased-array high-intensity focused ultrasound (HIFU)

Purpose: HIFU becomes an effective and non-invasive modality of solid tumour/cancer ablation. Simulation of the non-linear acoustic wave propagation using a phased-array transducer in multiple layered media using different focusing strategies and the consequent lesion formation are essential in HIFU planning in order to enhance the efficacy and efficiency of treatment.Materials and methods: An angular spectrum approach with marching fractional steps was applied in the wave propagation from phased-array HIFU transducer, and diffraction, attenuation, and non-linearity effects were accounted for by a second-order operator splitting scheme. The simulated distributions of the first three harmonics along and transverse to the transducer axis were compared to the hydrophone measurements. The bioheat equation was used to simulate the subsequent temperature elevation using the deposited acoustic energy, and lesion formation was determined by the thermal dose.Results: Better agreement was found between the measured harmonics distribution and simulation using the proposed algorithm than the Khokhlov–Zabozotskaya–Kuznetsov equation. Variable focusing of the phased-array transducer (geometric focusing, transverse shifting and the generation of multiple foci) can be simulated successfully. The shifting and splitting of focus was found to result in significantly less temperature elevation at the focus and the subsequently, the smaller lesion size, but the larger grating lobe grating lobe in the pre-focal region.Conclusions: The proposed algorithm could simulate the non-linear wave propagation from the source with arbitrary shape and distribution of excitation through multiple tissue layers in high computation accuracy. The performance of phased-array HIFU can be optimised in the treatment planning.

Using the PPML approach for constructing a low-dissipation, operator-splitting scheme for numerical simulations of hydrodynamic flows

An approach for constructing a low-dissipation numerical method is described. The method is based on a combination of the operator-splitting method, Godunov method, and piecewise-parabolic method on the local stencil. Numerical method was tested on a standard suite of hydrodynamic test problems. In addition, the performance of the method is demonstrated on a global test problem showing the development of a spiral structure in a gravitationally unstable gaseous galactic disk.

Improvement of accuracy of multi-scale models of Li-ion batteries by applying operator splitting techniques

In this work operator splitting techniques have been applied successfully to improve the accuracy of multi-scale Lithium-ion (Li-ion) battery models. A slightly simplified Li-ion battery model is derived, which can be solved on one time scale and multiple time scales. Different operator splitting schemes combined with different approximations are compared with the non-splitted reference solution in terms of stability, accuracy and processor cost. It is shown, that the reverse Strang–Marchuk splitting combined with the implicit scheme to solve the diffusion operator and Newton method to approximate the non-linear source term can improve the accuracy of the commonly applied vertical (sequential) multi-scale models by almost 3 times without considerably increasing the processor cost.

COMPARISON OF IMPLICIT SCHEMES TO SOLVE EQUATIONS OF RADIATION HYDRODYNAMICS WITH A FLUX-LIMITED DIFFUSION APPROXIMATION: NEWTON–RAPHSON, OPERATOR SPLITTING, AND LINEARIZATION

ABSTRACT Radiation is an important process of energy transport, a force, and a basis for synthetic observations, so radiation hydrodynamics (RHD) calculations have occupied an important place in astrophysics. However, although the progress in computational technology is remarkable, their high numerical cost is still a persistent problem. In this work, we compare the following schemes used to solve the nonlinear simultaneous equations of an RHD algorithm with the flux-limited diffusion approximation: the Newton–Raphson (NR) method, operator splitting, and linearization (LIN), from the perspective of the computational cost involved. For operator splitting, in addition to the traditional simple operator splitting (SOS) scheme, we examined the scheme developed by Douglas & Rachford (DROS). We solve three test problems (the thermal relaxation mode, the relaxation and the propagation of linear waves, and radiating shock) using these schemes and then compare their dependence on the time step size. As a result, we find the conditions of the time step size necessary for adopting each scheme. The LIN scheme is superior to other schemes if the ratio of radiation pressure to gas pressure is sufficiently low. On the other hand, DROS can be the most efficient scheme if the ratio is high. Although the NR scheme can be adopted independently of the regime, especially in a problem that involves optically thin regions, the convergence tends to be worse. In all cases, SOS is not practical.

A parallel alternating direction method with application to compound l1-regularized imaging inverse problems

We derive a parallel alternating direction method of multipliers (PADMM) and apply it to compound l1-regularized imaging inverse problems. The proposed method is capable of locating the saddle point of large-scale convex minimization problems with the sum of several nonsmooth but proximable terms. Using an operator splitting strategy, the objective is decomposed into subproblems that are conveniently, individually and simultaneously solved. With the assistance of the Moreau decomposition, our method excludes auxiliary variables that exist in the ADMM and possesses a compacter structure. Thus, the proposed method is preferable in distributed computation. The convergence proof and convergence rate analysis are presented. Application to both image restoration and image compressed sensing demonstrates the effectiveness and efficiency of the proposed method.