Consistent Numerical Scheme Research Articles

An all-Mach consistent numerical scheme for simulation of compressible multi-component fluids including surface tension, cavitation, turbulence modeling and interface sharpening on compact stencils

We present a numerical scheme valid in the range of highly to weakly compressible flows using a single-fluid four equation approach together with multi-component thermodynamic models. The approach can easily be included into existing finite volume methods on compact stencils and enables handling of compressibility of all involved phases including surface tension, cavitation and viscous effects. The mass fraction (indicator function) is sharpened in the two-phase interface region using the algebraic interface sharpening technique Tangent of Hyperbola for INterface Capturing (THINC). The cell face reconstruction procedure for mass fractions switches between an upwind-biased and a THINC-based scheme, along with proper slope limiters and a suitable compression coefficient, respectively. For additional sub-grid turbulence modeling, a fourth order central scheme is included into the switching process, along with a modified discontinuity sensor. The proposed “All-Mach” Riemann solver consistently merges the thermodynamic relationship of the components into the reconstructed thermodynamic variables (like density, internal energy), wherefore we call them All-Mach THINC-based Thermodynamic-Dependent Update (All-Mach THINC-TDU) method. Both, liquid-gas and liquid-vapor interfaces can be sharpened. Surface tension effects are taken into account by using a Continuum Surface Force (CSF) model. In order to reduce spurious oscillations at interfaces we decouple the computation of the interface curvature from the computation of the gradient of the Heaviside function. An explicit, four stage low storage Runge-Kutta method is used for time integration. The proposed methodology is validated against a series of reference cases, such as bubble oscillation/advection/deformation, shock-bubble interaction, a vapor/gas bubble collapse and a multi-component shear flow. The results of a near-critical shock/droplet interaction case are superior to those obtained by WENO3 and OWENO3 schemes and support that the proposed methodology works well with various thermodynamic relations, like the Peng-Robinson equation of state. Finally, the approach is applied to simulate the three-dimensional primary break-up of a turbulent diesel jet in a nitrogen/methane mixture including surface tension effects under typical dual-fuel conditions. The obtained results demonstrate that the methodology enables robust and accurate simulations of compressible multi-phase/multi-component flows on compact computational stencils without excessive spurious oscillations or significant numerical diffusion/dissipation.

Two-scale methods for the normalized infinity Laplacian: rates of convergence

Abstract We propose a monotone and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized infinity Laplacian, which could be related to the family of the so-called two-scale methods. We show that this method is convergent and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right-hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms, are also considered.

Numerical Investigation of Curved Tip Effect on the Performance of 3-D Cavitating Hydrofoils Moving under Free Surface

Curved tip effect on the performance of three-dimensional (3-D) cavitating hydrofoils moving steadily under a free water surface has been investigated numerically by an iterative boundary element method (IBEM) developed before. The IBEM has been modified and extended to solve this problem. The fluid is assumed to be inviscid, incompressible and the flow irrotational. All variables and equations are made non-dimensional to achieve a very quick and consistent numerical scheme. The IBEM is based on the Green’s theorem. The hydrofoil problem and the free surface problem are solved separately with the effects of each other via their potential values in an iterative manner. Both the 3-D hydrofoil surface and the free surface are modelled with constant strength source and constant strength doublet panels. The kinematic boundary condition is applied on the hydrofoil surface while the linearized kinematic and dynamic combined condition is applied on the free water surface. The method was validated extensively before for cavitating hydrofoils but not when combined with tip curvature. It is first applied to a tapered wing and the results are compared with those of experiments. Later, it is applied to a tapered hydrofoil with curved tip upwards or downwards. The effects of curved tip shape on cavitating hydrofoil performance have been investigated. It is found that the curved tip caused an increase in loading and cavitation volume in unbounded flow domain. It is also obtained that curved tip caused a decrease in loading on the hydrofoil for lower chord-based Froude numbers than 0.8.

Stable Computer Method for Solving Initial Value Problems with Engineering Applications

Engineering and applied mathematics disciplines that involve differential equations in general, and initial value problems in particular, include classical mechanics, thermodynamics, electromagnetism, and the general theory of relativity. A reliable, stable, efficient, and consistent numerical scheme is frequently required for modelling and simulation of a wide range of real-world problems using differential equations. In this study, the tangent slope is assumed to be the contra-harmonic mean, in which the arithmetic mean is used as a correction instead of Euler’s method to improve the efficiency of the improved Euler’s technique for solving ordinary differential equations with initial conditions. The stability, consistency, and efficiency of the system were evaluated, and the conclusions were supported by the presentation of numerical test applications in engineering. According to the stability analysis, the proposed method has a wider stability region than other well-known methods that are currently used in the literature for solving initial-value problems. To validate the rate convergence of the numerical technique, a few initial value problems of both scalar and vector valued types were examined. The proposed method, modified Euler explicit method, and other methods known in the literature have all been used to calculate the absolute maximum error, absolute error at the last grid point of the integration interval under consideration, and computational time in seconds to test the performance. The Lorentz system was used as an example to illustrate the validity of the solution provided by the newly developed method. The method is determined to be more reliable than the commonly existing methods with the same order of convergence, as mentioned in the literature for numerical calculations and visualization of the results produced by all the methods discussed, Mat Lab-R2011b has been used.

A consistent scheme of the high-speed source term in probability density function methods for supersonic flows

To improve the particle energy accuracy of the probability density function (PDF) method in supersonic flows, a consistent numerical scheme for the high-speed source term in the particle energy equation is put forward. The proposed scheme is designed and computed based on the characteristic format, which shares the same right and left eigenvectors for the Jacobian of the convection term of the system. Therefore, this scheme is fully consistent with the spatial discretization of the inviscid term in the finite-difference solver of compressible flows. To show the advantages of the proposed scheme, a redundant energy equation is solved along with the Euler equation. The consistent scheme and the quasi scheme are numerically tested and compared in several canonical flows. The results show that the high-speed source term calculated by the consistent scheme generates much less numerical oscillation than the quasi scheme around discontinuities. Due to the accumulation of the numerical errors of the high-speed source term in every single iteration, the energy redundantly computed by the consistent scheme agrees better with the Euler results than the other scheme. Following this new high-speed source computation scheme, the large eddy simulation-PDF method is further developed and tested in a shock tube problem interacting with an isotropic turbulent flow and a supersonic temporally developing mixing layer. The results show that PDF with this consistent high-speed source scheme can improve the energy accuracy as well as turbulent combustion in supersonic flows.

Finite Rate Reaction Mechanism Adapted for Modelling Pseudo-Equilibrium Pyrolysis of Cellulose

This manuscript is related to a formulation for modelling cellulose pyrolysis with a pseudo-equilibrium approach. The objective is to model the kinetics of the cellulose pyrolysis with a semi-global mechanism obtained from the literature in order to obtain the yield and the rate of formation, mainly that of char. The pseudo-equilibrium approach consists of the assumption that the solid phase devolatilisation can be described kinetically—at a finite rate—thus preserving the competitive characteristic between the production of char and tar, while the gas phase can be described directly by means of chemical equilibrium. The aforementioned approach gives a set of ordinary, linear, and nonlinear differential equations that are solved numerically with a consistent numerical scheme (i.e., the Totally Implicit Euler method). Chemical equilibrium was solved using CANTERA coupled with a code written in MATLAB. The results showed that the scheme preserved the tar-gas competitive characteristic for cellulose pyrolysis. The gas phase was defined as a mixture of CO2, CO, H2O, CH4, H2, and N2, showing a similar composition compared to models from the literature. Finally, the extension of the model to biomass in general is straightforward for including hemicellulose and lignin. The formulation is described in detail throughout the document in order to be replicated and evaluated for other biological components.

Exponential Bhatnagar-Gross-Krook integrator for multiscale particle-based kinetic simulations.

Despite the development of an extensive toolbox of multiscale rarefied flow simulators, such simulations remain challenging due to the significant disparity of collisional and macroscopic spatiotemporal scales. Our study offers a novel and consistent numerical scheme for a coupled treatment of particles advection and collision governed by the BGK evolution, honoring positivity of the velocity distribution. Our method shares its framework, in spirit, with the unified gas kinetic class of multiscale schemes. Yet it provides attractive features for particle-based stochastic simulations, readily implementable to existing direct simulation Monte Carlo codes. Two main innovations are integrated in the presented BGK particle method. The first ingredient is a high-order time integration that can be interpreted probabilistically, independent of the time step size. The next one is identifying modified particle distributions that remain invariant under the advection-relaxation evolution. We demonstrate accuracy and performance of the devised scheme for prototypic gas flows over a wide range of rarefaction parameters. Due to the resulting robustness and flexibility of the devised exponential BGK integrator, the scheme paves the way towards more affordable simulations of large-scale and multiscale rarefied gas phenomena.

A multi-objective Bayesian optimization environment for systematic design of numerical schemes for compressible flow

Multi-objective Bayesian optimization (MOBO) is an efficient and robust optimization framework for expensive functions. In this work, we use MOBO to optimize the free parameters of a high-order nonlinear weighted essentially non-oscillatory (WENO) reconstruction scheme to devise a model for implicit large eddy simulations. We concurrently optimize for a low dispersion error and sufficient shock-capturing ability for compressible flows as well as for physically consistent transition occurring in under-resolved flow regions. With our approach, we follow the genealogy of designing implicit sub-grid models. Yet, in contrast to previous works that were limited to incompressible flows, our model is also applicable to compressible flows. Validated results show that the model is able to decrease excessive dissipation in continuous flow regimes, to capture shocks with little dispersive and dissipative errors while achieving a well shaped vortical structures. The proposed framework is general and can be used to design a physically consistent numerical scheme for under-resolved compressible-flow simulations.

A variational asymmetric phase-field model of quasi-brittle fracture: Energetic solutions and their computation.

We derive the variational formulation of a gradient damage model by applying the energetic formulation of rate-independent processes and obtain a regularized formulation of fracture. The model exhibits different behaviour at traction and compression and has a state-dependent dissipation potential which induces a path-independent work. We will show how such formulation provides the natural framework for setting up a consistent numerical scheme with the underlying variational structure and for the derivation of additional necessary conditions of global optimality in the form of a two-sided energetic inequality. These conditions will form our criteria for making a better choice of the starting guess in the application of the alternating minimization scheme to describe crack propagation as quasistatic evolution of global minimizers of the underlying incremental functional. We will apply the procedure for two- and three-dimensional benchmark problems and we will compare the results with the solution of the weak form of the Euler-Lagrange equations. We will observe that by including the two-sided energetic inequality in our solution method, we describe, for some of the benchmark problems, an equilibrium path when damage starts to manifest, which is different from the one obtained by solving simply the stationariety conditions of the underlying functional.

Ergodic SDEs on submanifolds and related numerical sampling schemes

In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measureμon the level set of a smooth functionξ: ℝd→ ℝk, 1 ≤k<d. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff’s ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose invariant measure isμ. Motivated by the previous work of Ciccottiet al.(Commun. Pur. Appl. Math.61(2008) 371–408), as well as the work of Leliévreet al.(Math. Comput.81(2012) 2071–2125), in this paper we construct a family of ergodic diffusion processes on the level set ofξwhose invariant measures coincide with the given one. For the conditional measure, we propose a consistent numerical scheme which samples the conditional measure asymptotically. The numerical scheme doesn’t require computing the second derivatives ofξand the error estimates of its long time sampling efficiency are obtained.

An automatic and parameter-free information-based method for sparse representation in wavelet bases

In this article an information-based method for the selection of expansion coefficients of functions in a Hilbert basis is presented. An information-based measure, namely Entropic NID (ENID), is presented; the optimal separation point between more informative coefficients and less informative ones is selected by evaluating the information contribution of two competing sets of expansion coefficients. A consistent numerical scheme is given to approximate ENID and the numerical error is studied. Numerical simulations are provided to test the behaviour of ENID in different wavelet bases, as well as to perform comparative studies.

Path-dependent American options

In this paper, we investigate a path-dependent American option problem and provide an efficient and implementable numerical scheme for the solution of its associated path-dependent variational inequality. We obtain the viscosity characterization of our value function and suggest a monotone, stable and consistent numerical scheme, the convergence of which is proven thanks to the uniqueness property. We further enrich our study by providing and implementing a numerical algorithm. Some numerical results are also included.

Zika virus: An emerging player in the global scenario

In this article an information-based method for the selection of expansion coefficients of functions in a Hilbert basis is presented. An information-based measure, namely Entropic NID (ENID), is presented; the optimal separation point between more informative coefficients and less informative ones is selected by evaluating the information contribution of two competing sets of expansion coefficients. A consistent numerical scheme is given to approximate ENID and the numerical error is studied. Numerical simulations are provided to test the behaviour of ENID in different wavelet bases, as well as to perform comparative studies.

Consistent numerical schemes for solving nonlinear inverse source problems with the gradient-type algorithms and the Newton-Kantorovich methods

Consistent numerical schemes for solving nonlinear inverse source problems with the gradient-type algorithms and the Newton-Kantorovich methods

Consistent Numerical Schemes for Solving Nonlinear Inverse Source Problems with Gradient-Type Algorithms and Newton–Kantorovich Methods

In this paper, algorithms of solving an inverse source problem for systems of production–destruction equations are considered. Numerical schemes that are consistent to satisfy Lagrange’s identity for solving direct and adjoint problems are constructed. With the help of adjoint equations, a sensitivity operator with a discrete analog is constructed. It links perturbations of the measured values with those of the sought-for model parameters. This operator transforms the inverse problem to a quasilinear system of equations and allows applying Newton–Kantorovich methods to it. A numerical comparison of gradient algorithms based on consistent and inconsistent numerical schemes and a Newton–Kantorovich algorithm applied to solving an inverse source problem for a nonlinear Lorenz model is done.

Novel image inpainting algorithm based on adaptive fourth‐order partial differential equation

Image inpainting is used to restore some damaged or missing information regions of an image based on the surrounding information. Inspired by the fourth-order partial differential equation (PDE), this study presents two adaptive variational functionals for image inpainting, shows a stable variational scheme and investigates existence and uniqueness of solution for the minimising functionals. A consistent numerical discretisation scheme is also constructed for the derived fourth-order PDEs. The proposed models can simultaneously restore the missing, corrupted or undesirable regions and remove noise. Some successful image inpainting experiments and method comparisons are also provided in this study.

A consistent velocity approximation for variable-density flow and transport in porous media

Summary Using a finite-volume nodal-based code, we introduce a new consistent numerical scheme for approximating the seepage velocity in variable-density flow in porous media. It has been known from previous works that the lack of consistency in the numerical approximation for the seepage velocity can severely compromise the accuracy of numerical solutions of salt-transport in porous media. This paper provides a detailed description of the code needed for the new consistent velocity approximation, including numerical approximations and algorithm implementations. The new scheme is tested against two challenging benchmark cases in the field: the hydrostatic test and the salt-pool problem. Results obtained with the newly proposed consistent scheme show the best agreement with the benchmark experiments achieved so far. When an inconsistent approximation is used instead, the accuracy of the numerical results is greatly degraded. The current work confirms that using a consistent velocity approximation is essential for maintaining accuracy in numerical simulations of variable-density flow, especially in cases with high salt mass fraction, when the system of governing equations is strongly coupled.

LCC-Demons: A robust and accurate symmetric diffeomorphic registration algorithm

Non-linear registration is a key instrument for computational anatomy to study the morphology of organs and tissues. However, in order to be an effective instrument for the clinical practice, registration algorithms must be computationally efficient, accurate and most importantly robust to the multiple biases affecting medical images. In this work we propose a fast and robust registration framework based on the log-Demons diffeomorphic registration algorithm. The transformation is parameterized by stationary velocity fields (SVFs), and the similarity metric implements a symmetric local correlation coefficient (LCC). Moreover, we show how the SVF setting provides a stable and consistent numerical scheme for the computation of the Jacobian determinant and the flux of the deformation across the boundaries of a given region. Thus, it provides a robust evaluation of spatial changes. We tested the LCC-Demons in the inter-subject registration setting, by comparing with state-of-the-art registration algorithms on public available datasets, and in the intra-subject longitudinal registration problem, for the statistically powered measurements of the longitudinal atrophy in Alzheimer's disease. Experimental results show that LCC-Demons is a generic, flexible, efficient and robust algorithm for the accurate non-linear registration of images, which can find several applications in the field of medical imaging. Without any additional optimization, it solves equally well intra & inter-subject registration problems, and compares favorably to state-of-the-art methods.

MICROMORPHIC TWO-SCALE MODELLING OF PERIODIC GRID STRUCTURES

The present contribution focuses on the numerical homogenization of periodic grid structures. In order to investigate the micro-to-macroscale transition, a consistent numerical homogenization scheme will be presented, replacing a heterogeneous Cauchy microcontinuum by a homogeneous micromorphic substitute continuum on the macroscale. The extended degrees of freedom, namely, the microdeformation and its gradient, are to be interpreted in terms of geometrical deformation modes and the related loading conditions of the underlying unit cells. Assuming strain energy equivalence of the macro- and the microscale, the effective constitutive properties of a square and a honeycomb grid structure are identified and quantitatively validated in comparison to reference calculations with microscopic resolution.

A consistent numerical scheme for self‐gravitating fluid dynamics

AbstractIn this article, we present a numerical scheme for the 3‐D system of self‐gravitating fluid dynamics in the collisional case as well as in the non‐collisional case. Consistency in the sense of distributions is proved in 1‐D and in absence of pressure. In the other cases consistency is proved under the numerical assumptions of boundedness of the velocity field in the CFL condition and of boundedness of the gradient of the gravitation potential. In 2‐D and 3‐D, concentrations of matter in strings and points can cause a theoretical difficulty in the pressureless case although one observes that the scheme still works. The initial data are L∞ functions in velocity and L1 functions in density. Applications are given to numerical simulations of the role of dark matter and gravitational collapse in cosmology as well as Jeans theory. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013