Operator Splitting Scheme Research Articles

Detecting Water Bodies on RADARSAT Imagery

This paper presents a novel geodesic active contour (GAC) model based on an edge detector for rapid detection of water bodies from spaceborne synthetic aperture radar (SAR) imagery with high speckle noise. The original edge indicator function based on gradients is replaced by an edge indicator function based on the ratio of exponentially weighted averages (ROEWA) operator. Thus, the capability of edge detection and the accuracy of locating edges are greatly improved, which makes the model more appropriate for SAR images. In addition, an enhancing term is added to the original model's energy function in order to boost the strength for the contour's evolution. An unconditionally stable additive operator splitting (AOS) scheme and a fast algorithm for re-initialization of the level set function are adopted, which not only enhances the model's stability, but also speeds up the model's convergence remarkably. The experimental results on simulated and real RADARSAT-1/-2 images show its efficiency and accuracy.

On the Acceleration of AOS Schemes for Nonlinear Diffusion Filtering

Based on the renowned additive operator splitting (AOS) schemes that ensure equal treatment of all coordinate axis, accelerated AOS schemes for solving regularized Perona-Malik (P-M) equation are presented. These ameliorated AOS schemes are stable unconditionally, consistent with nonlinear parabolic equations under certain circumstance and demonstrate a remarkably economical CPU time. The paper is intended to discuss the properties of the modified AOS schemes in several aspects. Finally, experiments show that as well as the edges and detailed information of images are preserved, the proposed AOS schemes are at least twice more efficient than the conventional AOS schemes.

A third accurate operator splitting method

Operator splitting is a widely used procedure in the numerical solution of initial and boundary value problems of partial differential equations. From the theoretical point of view, the success of splitting is primarily determined by the splitting error. In this paper, a third-order accurate operator splitting scheme is introduced. Stability condition and its local splitting error are given. Finally, consistency analysis is presented.

A two-stage method for oil slick segmentation

In this paper we propose a two-stage algorithm for oil slick segmentation in synthetic aperture radar (SAR) images. In the first stage, we propose a new variational model to reduce speckles in non-textured SAR images. Applications to simulated and real SAR images show that the method is well balanced in the quality of the conventional criteria. Then, in the second stage, we use the fast Chan–Vese (CV) model and the level set method to segment the oil slick in the de-speckled SAR image. The additive operator splitting (AOS) scheme is used in the numerical implementation to improve computational efficiency. Experimental results show that our two-stage algorithm is effective for oil slick segmentation in SAR images.

Three dimensional fluid–structure interaction under pulsatile flow by using distributed Lagrange multiplier method

In this article we discuss an application of a Lagrange multiplier based fictitious domain method for the simulation of moving leaflets in an unsteady flow generated by pressure gradients during the systolic phase of the cardiac cycle in three dimensional geometry. The mathematical model includes the Navier–Stokes equations coupled with the Euler–Newton equations describing fluid–structure interaction for the generalized Neumann boundary conditions on upstream and downstream boundaries. The solution method includes the finite element method combined with an operator-splitting scheme, where the fictitious formulation allows the flow calculations to be in a fixed rectangular parallelepiped to predict the dynamic structural positions and flow field during the valve opening and closing phases.

Orientation-Matching Minimization for Image Denoising and Inpainting

In this paper, we propose an orientation-matching functional minimization for image denoising and image inpainting. Following the two-step TV-Stokes algorithm (Rahman et al. in Scale space and variational methods in computer vision, pp. 473---482, Springer, Heidelberg, 2007; Tai et al. in Image processing based on partial differential equations, pp. 3---22, Springer, Heidelberg, 2006; Bertalmio et al. in Proc. conf. comp. vision pattern rec., pp. 355---362, 2001), a regularized tangential vector field with zero divergence condition is first obtained. Then a novel approach to reconstruct the image is proposed. Instead of finding an image that fits the regularized normal direction from the first step, we propose to minimize an orientation matching cost measuring the alignment between the image gradient and the regularized normal direction. This functional yields a new nonlinear partial differential equation (PDE) for reconstructing denoised and inpainted images. The equation has an adaptive diffusivity depending on the orientation of the regularized normal vector field, providing reconstructed images which have sharp edges and smooth regions. The additive operator splitting (AOS) scheme is used for discretizing Euler-Lagrange equations. We present the results of various numerical experiments that illustrate the improvements obtained with the new functional.

A phase field model of electromechanical fracture

Structural reliability analyses of piezoelectric solids need the modeling of failure under coupled electromechanical actions. However, the numerical simulation of failure due to fracture based on sharp crack discontinuities may suffer in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. In this work, we develop a framework of diffusive fracture in piezoelectric solids. We start our investigation with the definition of a crack surface functional of the phase field that Γ -converges for vanishing length-scale parameter to a sharp crack topology. This functional provides the basis for the definition of suitable dissipation functions which govern the evolution of the crack phase field. Based on experimental results available in the literature, we suggest a non-associative dissipative framework where the fracture phase field is driven by the mechanical part of the coupled electromechanical driving force. This accounts for a hierarchical view that considers (i) the decrease of stiffness due to mechanical rupture as the primary action that is followed by (ii) the decrease of electric permittivity due to the generated free space. The proposed definition of mechanical and electrical parts of the fracture driving force follows in a natural format from a kinematic assumption, that decomposes the total strains and the total electric field into energy–enthalpy-producing parts and fracture parts, respectively. Such an approach allows the insertion of well-known anisotropic piezoelectric storage functions without change. We end up with a three-field-problem that couples the displacement with the electric potential and the fracture phase field. The latter is governed by a micro-balance equation, which appears in a very transparent form in terms of a history field containing a maximum fracture source obtained in the time history of the electromechanical process. This representation allows the construction of a very robust algorithmic treatment based on a operator split scheme, which successively updates in a typical time step the history field, the crack phase field and finally the two piezoelectric fields. The proposed model is considered to be the canonically simple scheme for the simulation of diffusive electromechanical crack propagation in solids. We demonstrate its modeling capacity by means of representative numerical examples.

Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

The numerical analysis of a dynamic constrained optimization problem is presented. It consists of a global minimization problem that is coupled with a system of ordinary differential equations. The activation and the deactivation of inequality constraints induce discontinuity points in the time evolution. A numerical method based on an operator splitting scheme and a fixed point algorithm is advocated. The ordinary differential equations are approximated by the Crank-Nicolson scheme, while a primal-dual interior-point method with warm-starts is used to solve the minimization problem. The computation of the discontinuity points is based on geometric arguments, extrapolation polynomials and sensitivity analysis. Second order convergence of the method is proved when an inequality constraint is activated. Numerical results for atmospheric particles confirm the theoretical investigations.

A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. Following our recent work [C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically-consistent phase field models of fracture: Variational principles and multi-field fe implementations, International Journal for Numerical Methods in Engineering DOI:10.1002/nme.2861] on phase-field-type fracture, we propose in this paper a new variational framework for rate-independent diffusive fracture that bases on the introduction of a local history field. It contains a maximum reference energy obtained in the deformation history, which may be considered as a measure for the maximum tensile strain obtained in history. It is shown that this local variable drives the evolution of the crack phase field. The introduction of the history field provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a new algorithmic treatment of diffusive fracture. Here, we propose an extremely robust operator split scheme that successively updates in a typical time step the history field, the crack phase field and finally the displacement field. A regularization based on a viscous crack resistance that even enhances the robustness of the algorithm may easily be added. The proposed algorithm is considered to be the canonically simple scheme for the treatment of diffusive fracture in elastic solids. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples.

Moving objects segmentation based on piecewise constant Mumford-Shah model solving by additive operator splitting

We propose a moving objects segmentation method for color image sequences based on the piecewise constant Mumford-Shah model (also known as the C-V model) solving by the semi-implicit additive operator splitting (AOS) scheme, which is unconditionally stable, fast, and easy to implement. The method first uses the Gaussian mixture model for background modeling and then subtracts the background to obtain the moving regions that are the handling objects of our method. As a result of the introduction of the AOS scheme, we could use a rather large time step and still maintain the stability of the evolution process. Additionally, the method can easily be parallelized because the AOS scheme decomposes the equations into a sequence of one-dimensional (1-D) systems. The experimental results demonstrate that under real moving objects video tests, the AOS scheme accelerates the evolution of the curve and significantly reduces the number of iterations, and also demonstrates the validity of our method.

Anisotropic Diffusion for Hyperspectral Imagery Enhancement

Among all enhancement techniques being developed over the past two decades, anisotropic diffusion has received much attention and experienced significant developments, with promising results and applications in various specific domains. The elegant property of the technique is that it can enhance images by reducing undesirable intensity variability within the objects in the image, while improving SNR and enhancing the contrast of the edges in scalar and, more recently, vector-valued images, such as color, multispectral, and hyperspectral imagery. In this paper, we present an alternative hyperspectral anisotropic diffusion scheme that takes into account the recent advances and the specificities of hyperspectral remote sensing. In addition, the proposed anisotropic diffusion algorithm can improve the classification accuracy of hyperspectral imagery by reducing the spatial and spectral variability of the image, while preserving the edges of objects. It is also revealed that the additive operator splitting scheme of our method can increase computer efficiency. Qualitative experiments, based on a real hyperspectral remote sensing image, show significant improvements in visual effects when using our method. Quantitative analyses, based on classification accuracies, confirm the superiority and validity of the proposed diffusion algorithm.

An operator splitting scheme for coupling macroscopic transport and grain growth in a two-phase multiscale solidification model: Part II – Application of the model

In this paper, the application of the two-phase solidification model with a splitting scheme for coupling macroscopic transport and grain growth, described in part I of this work, is presented with simulations of solidification of a binary alloy in a rectangular cavity. The splitting method is first verified by comparing the predictions with a classical directly coupled method. The verification is performed for a critical limiting case, as identified in part I, with infinitely fast microscopic diffusion (lever rule) and a fixed solid in the mushy zone. A further test is performed to show the capability of our model, where macroscopic transport and diffusion-controlled grain growth kinetics are coupled by the splitting scheme, in predicting recalescence. Thereafter, the complete model, as presented in part I, is applied to simulate solidification in its full complexity, considering the grain motion and grain growth kinetics. With the help of the model predictions the influence of the motion of globular solid grains on the transport phenomena during solidification and formation of macrosegregation is discussed.

GeoSysBRNS—A flexible multidimensional reactive transport model for simulating biogeochemical subsurface processes

The description of reactive transport processes in subsurface environments requires a sound understanding of both the biogeochemical complexity of the system and the spatially resolved transport of reactive species. However, most existing reactive transport models, for example in the field of contaminant hydrology, are specialized either in the simulation of the reactive or of the flow and transport processes. In this paper, we present and test the coupling of two highly flexible codes for the simulation of reactive transport processes in the subsurface: the Biogeochemical Reaction Network Simulator (BRNS), which contains a solver for kinetically and thermodynamically constrained biogeochemical reactions, and GeoSys/RockFlow, a multidimensional finite element subsurface flow and transport simulator. The new model, named GeoSysBRNS, maintains the full flexibility of the original models. The coupling is handled using an operator splitting scheme, which allows the reactive solver to be compiled into a problem specific library that is accessed by the transport simulator at runtime. The accuracy of the code coupling within GeoSysBRNS is demonstrated using two benchmark problems from the literature: a laboratory experiment on organic carbon degradation in a sand column via multiple microbial degradation pathways, and a dispersive mixing controlled bioreactive transport problem in aquifers, assuming three different reaction kinetics.

On the preconditioned conjugate gradient solution of a Stokes problem with Robin-type boundary conditions

We discuss the solution by a preconditioned conjugate gradient algorithm of a Stokes problem with Robin-type boundary conditions. This Stokes problem is encountered when applying an appropriate operator-splitting scheme to the time-discretization of a system modeling the interaction of an incompressible viscous fluid with a deformable thin elastic structure. The main contribution of this Note is the identification of a preconditioner operating in the pressure space that reduces substantially the number of iterations when compared to a conjugate gradient algorithm equipped with the canonical scalar product of L 2 . The results of numerical experiments show the validity of our approach. To cite this article: R. Glowinski, G. Guidoboni, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

An operator splitting scheme for coupling macroscopic transport and grain growth in a two-phase multiscale solidification model: Part I – Model and solution scheme

We present a two-phase multiscale solidification model, which accounts for coupled heat, mass, solute and momentum transfer and notably describes the motion of globular equiaxed grains. On the microscopic scale the model takes into account nucleation, finite diffusion in both solid and liquid phases, and resolves the diffusion-controlled phase change by a solute balance at the solid–liquid interface coupled with a thermodynamic equilibrium condition for the interface. The main idea of the paper is the solution algorithm of the multiscale model. We propose and present in detail an operator splitting method that resolves the global conservation equations in two stages. In the first solution stage we resolve and integrate globally only the convection terms. In the second solution stage we resolve and integrate locally the contributions of the grain growth kinetics, i.e. the phase change and the microscopic diffusion. We analyze this splitting method based on a scaling analysis that shows that the justification of the method stems from the separation of time scales of convection, diffusion and phase change. We discuss this scale separation in detail and define a test case that is presented and analyzed in a companion paper.

The method of fundamental solutions for solving incompressible Navier–Stokes problems

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) is proposed to solve the primitive variables formulation of the Navier–Stokes equations. The MFS is a meshless method since it is free from the mesh generation and numerical integration. We will transform the Navier–Stokes equations into simple advection–diffusion and Poisson differential operators via the operator-splitting scheme or the so-called projection method, instead of directly using the more complicated fundamental solutions (Stokeslets) of the unsteady Stokes equations. The resultant velocity advection–diffusion equations and the pressure Poisson equation are then calculated by using the MFS together with the Eulerian–Lagrangian method (ELM) and the method of particular solutions (MPS). The proposed meshless numerical scheme is a first attempt to apply the MFS for solving the Navier–Stokes equations in the moderate-Reynolds-number flow regimes. The lid-driven cavity flows at the Reynolds numbers up to 3200 for two-dimensional (2D) and 1000 for three-dimensional (3D) are chosen to validate the present algorithm. Through further simulating the flows in the 2D circular cavity with an eccentric rotating cylinder and in the 3D cube with a fixed sphere inside, we are able to demonstrate the advantages and flexibility of the proposed meshless method in the irregular geometry and multi-dimensional flows, even though very coarse node points are used in this study as compared with other mesh-dependent numerical schemes.

Vortex formation in a cavity with oscillating walls

The vortex formation in a two-dimensional Cartesian cavity, which their vertical walls move simultaneously with an oscillatory velocity and the horizontal walls are fixed pistons, is studied numerically. The governing equations were solved with a finite element method combined with an operator splitting scheme. We analyzed the behavior of vortical structures occurring inside a cavity with an aspect ratio of height-to-width of 1.5 for three different displacement amplitudes of the vertical oscillatory walls (amplitude/width=Y=0.2, 0.4, and 0.8) and Reynolds numbers based on the cavity width of 50, 500, and 1000. Two vortex formation mechanisms are identified: (a) the shear, oscillatory motion of the moving boundaries coupled with the fixed walls that provide a translational symmetry-breaking effect and (b) the sharp changes in the flow motion when the flow meets the corners of the cavity. The vortex cores were identified using the Jeong–Hussain criterion and it is found that the area occupied by the cores decreases as the Reynolds number increases and increases as Y increases. All flows studied are cyclic symmetric and for low Y and Re values they are also symmetric with respect to the vertical axis dividing the cavity in two sides. In asymmetric flows, the unbalance between the vortices on each side of the midvertical line generates a vortex that occupies the central part of the cavity. The breakdown of the axial symmetry was studied for a fixed value of the oscillation amplitude, Y=0.8, taking Re as the bifurcation parameter. The results indicate that the symmetry is broken through a supercritical pitchfork bifurcation.

Error analysis of stabilized semi-implicit method of Allen-Cahn equation

We consider in this paper the stabilized semi-implicit (in time) scheme and the splitting scheme for the Allen-Cahn equation $\phi_t-\Delta\phi+$e$^-2f(\phi)=0$ arising from phase transitions in material science. For the stabilized first-order scheme, we show that it is unconditionally stable and the error bound depends on e-1 in some lower polynomial order using the spectrum estimate of [2, 10, 11]. In addition, the first- and second-order operator splitting schemes are proposed and the accuracy are tested and compared with the semi-implicit schemes numerically.

Hybrid Simulation with Improved Operator-Splitting Integration Using Experimental Tangent Stiffness Matrix Estimation

Improved numerical integration procedures are essential for the extension of hybrid numerical and experimental simulation to large and complex structural systems. While implicit integration algorithms are widely used in pure numerical simulations for their superior stability and accuracy, their direct application to hybrid simulation has been partially limited by difficulties in estimating the tangent stiffness matrix of multi-degree-of-freedom experimental substructures. Current applications of hybrid simulation using integrators with improved stability have mostly resorted to methods that are noniterative or utilize the initial stiffness matrix for iterative corrections. To improve the accuracy of integration procedures for hybrid simulation, a new method for online estimation of experimental tangent stiffness is proposed. The stiffness estimation procedure is tailored for fast online applications by transforming the measurements into a coordinate system, which reduces the number of unknown stiffness coefficients that need to be updated during the simulation. The updated experimental stiffness matrix is used in a modified operator-splitting integration scheme to improve the accuracy of hybrid simulations with highly nonlinear experimental substructures. The application and effectiveness of the proposed approach is demonstrated through hybrid simulations with multi-degree-of-freedom experimental substructures.

3D hybrid Cartesian/immersed-boundary finite-element analysis of heat and flow patterns in a two-roll mill

A fully three-dimensional time-dependent Navier–Stokes model with forced convection is developed to numerically investigate the heat and flow patterns of the two-roll mill system with two inner rotating cylinders. Such direct numerical simulations are usually limited by the difficulties from huge computational cost and complex boundary treatment. For a fast numerical process, we can use the operator-splitting scheme with the BTD term to advance the solution in temporal evolution. To implement the calculation over a Cartesian grid, the hybrid Cartesian/immersed-boundary finite-element method is employed for spatial discretization. In the authors’ previous study [D.L. Young, C.L. Chiu, C.M. Fan, A hybrid Cartesian/immersed-boundary finite-element method for simulating heat and flow patterns in a two-roll mill, Numer. Heat Transfer B 51 (3) (2007) 251–274], we have developed a simplified 2D numerical model to analyze the heat and flow patterns on the cross section of two-roll-mill flow under the assumption of infinite length in the third (vertical) direction. However, the 2D solutions could not completely represent the realistic physical phenomena unless a 3D algorithm is developed. In this study we then paid the particular attention to develop a 3D model to investigate the vertical heat and flow behaviors, including 3D features of the vortex structure, periodic oscillation and chaotic instabilities. It is found that the proposed 3D model is able to cover the 2D features if the assumptions of 2D conditions are fulfilled.