Courant-Friedrichs-Lewy Research Articles

An efficient level set model with self-similarity for texture segmentation

Textures widely exist in the natural scenes while traditional level set models generally use only intensity information to construct energy and ignore the inherent texture features. Thus these models have difficulty in segmenting texture images especially when the texture objects have similar intensity to the background. To solve this problem, we propose a new level set model for texture segmentation that considers the impact of local Gaussian distribution fitting (LGDF), local self-similarity (LSS) and a new numerical scheme on the evolving contour. The proposed method first introduces a texture energy term based on the local self-similarity texture descriptor to the LGDF model, and then the evolving contour could effectively snap to the textures boundary. Secondly, a lattice Boltzmann method (LBM) is deployed as a new numerical scheme to solve the level set equation, which can break the restriction of the Courant–Friedrichs–Lewy (CFL) condition that limits the time step of iterations in former numerical schemes. Moreover, GPU acceleration further improves the efficiency of the contour evolution. Experimental results show that our model can effectively handle the segmentation of synthetic and natural texture images with heavy noises, intensity inhomogeneity and messy background. At the same time, the proposed model has a relatively low complexity.

Solution Stabilization and Convergence Acceleration for the Harmonic Balance Equation System

This paper presents an efficient approach for stabilizing solution and accelerating convergence of a harmonic balance equation system for an efficient analysis of turbomachinery unsteady flows due to flutter and blade row interaction. The proposed approach combines the Runge–Kutta method with the lower upper symmetric Gauss Seidel (LU-SGS) method and the block Jacobi method. The LU-SGS method, different from its original application as an implicit time marching scheme, is used as an implicit residual smoother with under-relaxation, allowing big Courant–Friedrichs–Lewy (CFL) numbers (in the order of hundreds), leading to significant convergence speedup. The block Jacobi method is introduced to implicitly integrate the time spectral source term of a harmonic balance equation system, in order to reduce the complexity of the direct implicit time integration by the LU-SGS method. The implicit treatment of the time spectral source term thus greatly augments the stability region of a harmonic balance equation system in the case of grid-reduced frequency well above ten. Validation of the harmonic balance flow solver was carried out using linear cascade test data. Flutter analysis of a transonic rotor and blade row interaction analyses for a transonic compressor stage were presented to demonstrate the stabilization and acceleration effect by the combination of the LU-SGS and the block Jacobi methods. The influence of the number of Jacobi iterations on solution stabilization is also investigated, showing that two Jacobi iterations are sufficient for stability purpose, which is much more efficient than existing methods of its kind in the open literature.

The effects of plastic waves on the numerical convergence of the viscous–plastic and elastic–viscous–plastic sea-ice models

The plastic wave speed is derived from the linearized 1-D version of the widely used viscous–plastic (VP) and elastic–viscous–plastic (EVP) sea-ice models. Courant–Friedrichs–Lewy (CFL) conditions are derived using the propagation speed of the wave. 1-D numerical experiments of the VP, EVP and EVP⁎ models successfully recreate a reference solution when the CFL conditions are satisfied, in agreement with the theory presented. The IMplicit–EXplicit (IMEX) method is shown to effectively alleviate the plastic wave CFL constraint on the timestep in the implicitly solved VP model in both 1-D and 2-D. In 2-D, the EVP and EVP⁎ models show first order error in the simulated velocity field when the plastic wave is not resolved. EVP simulations are performed with various advective timestep, number of subcycles, and elastic-wave damping timescales. It is found that increasing the number of subcycles beyond that needed to resolve the elastic wave does not improve the quality of the solution. It is found that reducing the elastic wave damping timescale reduces the spatial extent of first order errors cause by the unresolved plastic wave. Reducing the advective timestep so that the plastic wave is resolved also reduces the velocity error in terms of magnitude and spatial extent. However, the parameter set required for convergence to within the error bars of satellite (RGPS) deformation fields is impractical for use in climate model simulations. The behavior of the EVP⁎ method is analogous to that of the EVP method except that it is not possible to reduce the damping timescale with α=β.

Error analysis of method of lines (MOL) via generalized interpolating moving least squares (GIMLS) approximation

This paper gives an error analysis for the method of lines (MOL) using generalized interpolating moving least squares (GIMLS) approximation. In this study, error bound for the time-dependent linear and nonlinear second-order differential equations in d-dimension will be obtained, when the GIMLS method is used for approximating the spatial variables. Also, the well-known Courant–Friedrichs–Lewy (CFL) condition will be derived in both cases (linear and nonlinear equations). Finally, numerical examples will be reported to confirm the ability of the proposed technique.

Accurate numerical simulation of moisture front in porous material

When comparing measurements to numerical simulations of moisture transfer through porous materials a rush of the experimental moisture front is commonly observed in several works shown in the literature, with transient models that consider only the diffusion process. Thus, to overcome the discrepancies between the experimental and the numerical results, this paper proposes to include the moisture advection transfer in the governing equation. To solve the advection-diffusion or the so-called convection differential equation, it is first proposed two efficient numerical schemes whose efficiencies are investigated for both linear and nonlinear cases. The first scheme, Scharfetter–Gummel, presents a Courant-Friedrichs-Lewy (CFL) condition but it is more accurate and faster than the second one, the well-known Crank–Nicolson approach. Furthermore, the Scharfetter–Gummel scheme has the advantages of being well-balanced and asymptotically preserved. Then, to conclude, results of the convective moisture transfer problem obtained by means of the Scharfetter–Gummel numerical scheme are compared to experimental data from the literature. The inclusion of an advective term in the model may clearly lead to better results than purely diffusive models.

A matrix-exponential decomposition based time-domain method for calculating the defect states of scalar waves in two-dimensional periodic structures

A time-domain method for calculating the defect states of scalar waves in two-dimensional (2D) periodic structures is proposed. In the time-stepping process of the proposed method, the column vector containing the spatially sampled field values is updated by multiplying it with an iteration matrix, which is written in a matrix-exponential form. The matrix-exponential is first computed by using the Suzuki's decomposition based technique of the fourth order, in which the Floquet–Bloch boundary conditions are incorporated. The obtained iteration matrix is then squared to enlarge the time-step that can be used in the time-stepping process (namely, the squaring technique), and the small nonzero elements in the iteration matrix is finally pruned to improve the sparse structure of the matrix (namely, the pruning technique). The numerical examples of the super-cell calculations for 2D defect-containing phononic crystal structures show that, the fourth order decomposition based technique for the matrix-exponential computation is much more efficient than the frequently used precise integration technique (PIT) if the PIT is of an order greater than 2. Although it is not unconditionally stable, the proposed time-domain method is particularly efficient for the super-cell calculations of the defect states in a 2D periodic structure containing a defect with a wave speed much higher than those of the background materials. For this kind of defect-containing structures, the time-stepping process can run stably for a sufficiently large number of the time-steps with a time-step much larger than the Courant–Friedrichs–Lewy (CFL) upper limit, and consequently the overall efficiency of the proposed time-domain method can be significantly higher than that of the conventional finite-difference time-domain (FDTD) method. Some physical interpretations on the properties of the band structures and the defect states of the calculated periodic structures are also presented.

Relaxing the CFL Condition for the Wave Equation on Adaptive Meshes

The Courant–Friedrichs–Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. This paper shows how a simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities.

Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling

The mass-lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points, known as cubature points. To date, such points are only known analytically in tensor domains, such as quadrilateral or hexahedral elements. Thus, the diagonal-mass-matrix spectral element method (SEM) in non-tensor domains always relies on numerically computed interpolation points or quadrature points. However, only the cubature points for degrees 1 to 6 are known, which is the reason that we have developed a p-norm-based optimization algorithm to obtain higher-order cubature points. In this way, we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9. The dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration. Simultaneously, the Lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties. As a result, such points provide both good interpolation properties and integration accuracy. The Courant–Friedrichs–Lewy (CFL) numbers are tabulated for the conventional Fekete-based triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubature-based TSEM (OTSEM). A complementary study demonstrates the spectral convergence of the OTSEM. A numerical example conducted on a half-space model demonstrates that the OTSEM improves the accuracy by approximately one order of magnitude compared to the conventional Fekete-based TSEM. In particular, the accuracy of the 7th-order OTSEM is even higher than that of the 14th-order Fekete-based TSEM. Furthermore, the OTSEM produces a result that can compete in accuracy with the quadrilateral SEM (QSEM). The high accuracy of the OTSEM is also tested with a non-flat topography model. In terms of computational efficiency, the OTSEM is more efficient than the Fekete-based TSEM, although it is slightly costlier than the QSEM when a comparable numerical accuracy is required.

Semi-Lagrangian off-lattice Boltzmann method for weakly compressible flows.

The lattice Boltzmann method is a simulation technique in computational fluid dynamics. In its standard formulation, it is restricted to regular computation grids, second-order spatial accuracy, and a unity Courant-Friedrichs-Lewy (CFL) number. This paper advances the standard lattice Boltzmann method by introducing a semi-Lagrangian streaming step. The proposed method allows significantly larger time steps, unstructured grids, and higher-order accurate representations of the solution to be used. The appealing properties of the approach are demonstrated in simulations of a two-dimensional Taylor-Green vortex, doubly periodic shear layers, and a three-dimensional Taylor-Green vortex.

Adaptive implicit finite element methods for multicomponent compressible flow in heterogeneous and fractured porous media

AbstractThis work presents adaptive implicit first‐order and second‐order discontinuous Galerkin (DG) methods for the transport of multicomponent compressible fluids in heterogeneous and fractured porous media, discretized by triangular, quadrilateral, and hexahedral grids. The adaptive implicit method (AIM) combines the advantages of purely explicit or implicit methods (in time). In grid cells with high fluxes or low pore volumes, the transport update is done implicitly to alleviate the Courant‐Friedrichs‐Lewy (CFL) time step constraints of the conditionally stable explicit approach. Grid cells with a large CFL condition are updated explicitly. Combined, this allows higher efficiency than explicit methods, but it reduces the “penalty” of implicit methods, which exhibit high numerical dispersion and are more computationally and storage expensive per time step. The advantages of AIM are modest for uniform grids and rock properties. However, in heterogeneous or fractured reservoirs explicit methods may become impractical, while a fully implicit approach introduces unnecessary numerical dispersion and is overkill for low‐permeability layers and matrix blocks. In such applications, AIM is shown to be significantly more efficient and accurate. The division between explicit and implicit grid cells is made adaptively in space and time. This allows for a high level of explicitness and can also adapt to high fluxes caused by, e.g., viscous and gravitational flow instabilities. Numerical examples demonstrate the powerful features of AIM to model, e.g., solute transport, carbon sequestration in saline aquifers, and miscible gas injection in fractured oil and gas reservoirs.

Local Time-Step TLM Unstructured Block Meshing for Electromagnetic and Bio-Thermal Applications

Many practical problems are of multiscale nature. This issue appears when relatively small regions inside the computational domain have a large impact on the overall behavior of the structure. This requires using small mesh size in these regions of interest for adequately representing their geometrical details. Using uniform fine grid everywhere is computationally very exhaustive. The use of irregular structured grids may lead to some memory cost reduction as compared to uniform fine mesh. However, it imposes a smaller time step and yields higher dispersion as cell aspect ratio deviates from unity. The alternate approach is to use block meshing which allows fine discretization in regions with fine details and larger mesh size in quasi homogenous regions. The main disadvantage of this technique is that a global time step corresponding to the Courant–Friedrichs–Lewy (CFL) limit in the smallest cells should be used to avoid long-term instability. In this paper, we present a time-domain transmission-line matrix (TLM) algorithm for interfacing block regions that allows the use of local time steps. Hence, cubic cells can be optimally used in all regions with substantial coarseness error reduction. Applied to both electromagnetic and heat transfer TLM models, simulation experiments are presented to show the behavior and performances of the algorithm.

Performance evaluation of the general characteristics based off-lattice Boltzmann scheme and DUGKS for low speed continuum flows

The general characteristics based off-lattice Boltzmann scheme proposed by Bardow et al. [1] (hereafter Bardow's scheme) and the discrete unified gas kinetic scheme (DUGKS) [2] are two methods that successfully overcome the time step restriction by the collision time, which is commonly seen in many other kinetic schemes. In this work, we first perform a theoretical analysis of the two schemes in the finite volume framework by comparing their numerical flux evaluations. It is found that the effect of collision term is considered in the evaluation of the cell interface distribution function in both schemes, which explains why they can overcome the time step restriction and can give accurate results even as the time step is much larger than the collision time. The difference between the two schemes lies in the treatment of the integral of the collision term when evaluating the cell interface distribution function, in which Bardow's scheme uses the rectangular rule while DUGKS uses the trapezoidal rule. The performance of the two schemes, i.e., accuracy, stability, and efficiency are then compared by simulating several two dimensional flows, including the unsteady Taylor–Green vortex flow, the steady lid-driven cavity flow, and the laminar boundary layer problem. It is observed that, DUGKS can give more accurate results than Bardow's scheme with a same mesh size. Furthermore, the numerical stability of Bardow's scheme decreases as the Courant–Friedrichs–Lewy (CFL) number approaches to 1, while the stability of DUGKS is not affected by the CFL number apparently as long as CFL<1. It is also observed that DUGKS is twice as expensive as the Bardow's scheme with the same mesh size.

Large Time Step Roe scheme for a common 1D two-fluid model

We present the Large Time Step (LTS) extension of the Roe scheme and apply it to a standard two-fluid model. Herein, LTS denotes a class of explicit methods that are not limited by the CFL (Courant–Friedrichs–Lewy) condition, allowing us to use very large time steps compared to standard explicit methods. The LTS method was originally developed in the nineteen eighties (LeVeque, 1985), where the Godunov scheme was extended to the LTS Godunov scheme. In the present work, the relaxation of the CFL condition is achieved by increasing the domain of dependence. This might lead to difficulties when it comes to boundary and source terms treatment. We address and discuss these difficulties and propose different ways to treat them. For a shock tube test case, where there are neither source terms nor difficulties associated with the boundaries, the method increases both accuracy and efficiency. For a water faucet test case that includes a source term, the method increases the efficiency, while the accuracy strongly depends on the appropriate treatment of boundary conditions and source terms.

One-dimensional Lagrangian implicit hydrodynamic algorithm for Inertial Confinement Fusion applications

A new one-dimensional hydrodynamic algorithm, specifically developed for Inertial Confinement Fusion (ICF) applications, is presented. The scheme uses a fully conservative Lagrangian formulation in planar, cylindrical, and spherically symmetric geometries, and supports arbitrary equations of state with separate ion and electron components. Fluid equations are discretized on a staggered grid and stabilized by means of an artificial viscosity formulation. The space discretized equations are advanced in time using an implicit algorithm. The method includes several numerical parameters that can be adjusted locally. In regions with low Courant–Friedrichs–Lewy (CFL) number, where stability is not an issue, they can be adjusted to optimize the accuracy. In typical problems, the truncation error can be reduced by a factor between 2 to 10 in comparison with conventional explicit algorithms. On the other hand, in regions with high CFL numbers, the parameters can be set to guarantee unconditional stability. The method can be integrated into complex ICF codes. This is demonstrated through several examples covering a wide range of situations: from thermonuclear ignition physics, where alpha particles are managed as an additional species, to low intensity laser–matter interaction, where liquid–vapor phase transitions occur.

An implicit solver for 1D arterial network models.

In this study, the 1D blood flow equations are solved using a newly proposed enhanced trapezoidal rule method (ETM), which is an extension to the simplified trapezoidal rule method. At vessel junctions, the conservation of mass and conservation of total pressure are held as system constraints using Lagrange multipliers that can be physically interpreted as external flow rates. The ETM scheme is compared with published arterial network benchmark problems and a dam break problem. Strengths of the ETM scheme include being simple to implement, intuitive connection to lumped parameter models, and no restrictive stability criteria such as the Courant-Friedrichs-Lewy (CFL) number. The ETM scheme does not require the use of characteristics at vessel junctions, or for inlet and outlet boundary conditions. The ETM forms an implicit system of equations, which requires only one global solve per time step for pressure, followed by flow rate update on the elemental system of equations; thus, no iterations are required per time step. Consistent results are found for all benchmark cases, and for a 56-vessel arterial network problem, it gives very satisfactory solutions at a spatial and time discretization that results in a maximum CFL of 3, taking 4.44 seconds per cardiac cycle. By increasing the time step and element size to produce a maximum CFL number of 15, the method takes only 0.39second per cardiac cycle with only a small compromise on accuracy.

Time-Accurate Flow Simulations Using a Finite-Volume Based Lattice Boltzmann Flow Solver with Dual Time Stepping Scheme

In this study, the Lattice Boltzmann Method (LBM) is implemented through a finite-volume approach to perform 2D, incompressible, and time-accurate fluid flow analyses on structured grids. Compared to the standard LBM (the so-called stream and collide scheme), the finite-volume approach followed in this study necessitates more computational effort, but the major limitations of the former on grid uniformity and Courant–Friedrichs–Lewy (CFL) number that is to be one are removed. Even though these improvements pave the way for the possibility of solving more practical fluid flow problems with the LBM, time-accurate simulations are still restricted due to the stability criteria dictated by high-aspect ratio grid cells that are usually required for adequate resolution of boundary layers and the stiffness due to the nature of the equation that are being solved. To overcome this limitation, a Dual Time Stepping (DTS) scheme, which iterates the solution in pseudo time using an Implicit-Explicit (IMEX) Runge–Kutta method while advancing the solution in physical time with an explicit scheme (backward difference formula), is developed and implemented. The accuracy of the resulting flow solver is evaluated using benchmark flow problems and overall second-order accuracy is demonstrated.

Application of the characteristics-based sectional method to spatially varying aerosol formation and transport

The characteristics-based ssolution. It is easy to verify thatectional method (CBSM) offers an Eulerian description of an internally mixed aerosol. It was shown to be robust and capable of exact preservation of lower order moments, allowing for highly skewed sectional droplet size distributions. In this paper we apply CBSM to a spatially varying flow, by incorporating the fractional step method. In this way an accurate time integration of the spatial terms in the transport equations for the velocity, mass fractions and sectional droplet concentrations is achieved. Integrating CBSM into the compressible PISO (Pressure-Implicit with Splitting of Operators) algorithm allows for phase change and corresponding changes in pressure. We apply CBSM to a lid-driven cavity flow. First, the steady state isothermal flow solution is validated against published data. Next, by releasing a saturated vapor into the cavity while cooling the walls, we simulate the formation of aerosol. The accuracy of the solution is studied, as well as the performance of the CBSM scheme in the spatially varying context. The solution of the velocity is shown to be accurate, even at CFL (Courant–Friedrichs–Lewy) numbers of unity. The feasibility of the developed method is demonstrated in a 3D complex geometry studying the aerosol generation via nucleation of hot vapors cooled by a dilution stream of cold air in a double-mixing tee system. The sectional approach delivers detailed information about the aerosol formation and size distribution of the droplets in the domain.

Flow models and numerical schemes for single/two-phase transient flow in one dimension

In the two-phase flow field, a traditional mathematical model for simulating the transition of severe slugging flow presents a challenge when liquid slugs completely block pipelines. Accordingly, an advanced and practical slug model that is derived from a mixture model associated with a slip closure is essential to solving the problem in cooperation with the two-fluid model. The model can offset numerical instability that arises from the discontinuous function of the friction factor across the transition from one flow pattern to the other. Two numerical schemes, the non-iterative and the iterative, are developed, and the proposed schemes can stably predict the transient problems under the Courant–Friedrichs–Lewy (CFL) condition for semi-implicit/implicit schemes. In the present work, pressure transients produced by a complex phenomenon, named water hammer effect, are captured using the single-phase flow model in one-dimension to verify the applicability of the numerical schemes and the friction factor model. At last, the analysis of the two-phase transient flow in a pipeline-riser system indicates that the significant advantage of the present schemes is the robustness that the numerical prediction of the severe slugging behaviour is accurate and stable.

Relaxing the CFL condition for the wave equation on adaptive meshes

AbstractThe Courant‐Friedrichs‐Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re‐entrant corners. A simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On the properties of energy stable flux reconstruction schemes for implicit large eddy simulation

We begin by investigating the stability, order of accuracy, and dispersion and dissipation characteristics of the extended range of energy stable flux reconstruction (E-ESFR) schemes in the context of implicit large eddy simulation (ILES). We proceed to demonstrate that subsets of the E-ESFR schemes are more stable than collocation nodal discontinuous Galerkin methods recovered with the flux reconstruction approach (FRDG) for marginally-resolved ILES simulations of the Taylor–Green vortex. These schemes are shown to have reduced dissipation and dispersion errors relative to FRDG schemes of the same polynomial degree and, simultaneously, have increased Courant–Friedrichs–Lewy (CFL) limits. Finally, we simulate turbulent flow over an SD7003 aerofoil using two of the most stable E-ESFR schemes identified by the aforementioned Taylor–Green vortex experiments. Results demonstrate that subsets of E-ESFR schemes appear more stable than the commonly used FRDG method, have increased CFL limits, and are suitable for ILES of complex turbulent flows on unstructured grids.