Scalar Transport Problems Research Articles

Parallel ensemble Kalman method with total variation regularization for large-scale field inversion

Field inversion is often encountered in data-driven computational modeling to infer latent spatial–varying parameters from available observations. The ensemble Kalman method is emerging as a useful tool for solving field inversion problems due to its derivative-free merits. However, the method is computationally prohibitive for large-scale field inversion with high-dimensional observation data, which necessitates developing a practical efficient implementation strategy. In this work, we propose a parallel implementation of the ensemble Kalman method with total variation regularization for large-scale field inversion problems. It is achieved by partitioning the computational domain into non-overlapping subdomains and performing local ensemble Kalman updates at each subdomain parallelly. In doing so, the computational complexity of the ensemble-based inversion method is significantly reduced to the level of local subdomains. Further, the total variation regularization is employed to smoothen the physical field over the entire domain, which can reduce the inference discrepancy caused by missing covariances near subdomain interfaces. The capability of the proposed method is demonstrated in three field inversion problems with increasing complexity, i.e., the diffusion problem, the scalar transport problem and the Reynolds averaged Navier-Stokes closure problem. The numerical results show that the proposed method can significantly improve computational efficiency with satisfactory inference accuracy.

An efficient HLL-based scheme for capturing contact-discontinuity in scalar transport by shallow water flow

An efficient numerical scheme is developed for coupled hydrodynamic and scalar transport systems to guarantee the conservation and the positivity-preserving properties for water depth and scalar concentration. A second-order well-balanced positivity-preserving central-upwind scheme based on the finite volume method is adopted to discretize both the Saint-Venant system and the advective fluxes in the scalar transport system. In particular, an anti-diffusion modification is augmented in an ad hoc manner to the Harten–Lax–van Leer approximate Riemann solver for the scalar transport system, with the aim of significantly reducing the numerical diffusion near contact discontinuities. The proposed scheme is validated through seven numerical experiments, wherein the advection and diffusion processes of scalar transport are considered either separately or in combination. Accuracy is measured based on the difference between the numerical results and analytical solutions, and the performance of the proposed scheme is assessed by comparing it with that of the existing scheme. Convergence analysis confirms that the proposed scheme is accurate to the second order. The stationary steady state with wet-dry fronts is simulated, which verifies that the proposed scheme can preserve the exact C-property. The results for the pure diffusion case reveal that the original scheme has limitations in precisely predicting scalar diffusion when the diffusion coefficient is small. By contrast, the proposed scheme accurately approximates scalar diffusion, even at low diffusivity. The results under flow conditions with various Froude and Péclet numbers confirm that the proposed scheme is more accurate than the existing scheme in solving scalar transport problems over a wide range of shallow water flows.

An enhanced non-oscillatory BFECC algorithm for finite element solution of advective transport problems

In this paper, the so-called “back and forth error compensation correction (BFECC)” methodology is utilized to improve the solvers developed for the advection equation. Strict obedience to the so-called “discrete maximum principle” is enforced by incorporating a gradient-based limiter into the BFECC algorithm. The accuracy of the BFECC algorithm in capturing the steep-fronts in hyperbolic scalar-transport problems is improved by introducing a controlled anti-diffusivity. This is achieved at the cost of performing an additional backward sub-solution-step and modifying the formulation of the error compensation accordingly. The performance of the proposed methodology is assessed by solving a series of benchmarks utilizing different combinations of the BFECC algorithms and the underlying numerical schemes. Results are presented for both the structured and unstructured meshes.

An GPU-accelerated particle tracking method for Eulerian–Lagrangian simulations using hardware ray tracing cores

To address the high computational cost of particle tracking for realistic Eulerian–Lagrangian simulations, a novel efficient and robust particle tracking method (RT method) for unstructured meshes is presented. The method, for the first time, leverages both hardware ray tracing (RT) cores and GPU parallel computing technology to accelerate Eulerian–Lagrangian simulations. The method includes a hardware-accelerated hosting cell locator using bounding volume hierarchy tree (BVH) and a robust treatment of particle-wall interaction (multiple specular reflection) using an improved neighbor searching approach. The method is implemented in a GPU-accelerated open-source code, which is verified against a reference neighbor-searching particle-tracking method (NS method) and experimental observations. To evaluate the performance of our method, several numerical simulations of fluid-driven scalar transport problem are solved. Using a verification case, we show that the particle distribution simulated by our code is in a good agreement with an experimental observation. Tracking failures and stuck particles are not observed in any simulations. Benchmark results indicate that our RT method leads to a roughly 1.8−2.0× performance improvement compared to the reference NS method for large-scale simulations (millions of mesh cells and particles).

Boundary variation diminished conservative semi-Lagrangian method for both compressible and incompressible flows

Motivated by the enlightenment that diminishing the jump at the cell boundary can effectively reduce numerical dissipation near the critical region, and a novel constrained interpolation profile conservative semi-Lagrangian method is proposed based on a newly designed boundary variation diminishing algorithm. First, a constrained interpolation profile conservative semi-Lagrangian scheme with the piecewise tangent of hyperbola for interface capturing scheme is proposed as one candidate to represent jump-like discontinuities. Second, the constrained interpolation profile conservative semi-Lagrangian scheme with a fourth-order weighted essentially non-oscillatory limiter is used as another candidate to keep the high-order and non-oscillatory reconstruction for smooth solutions. The selection criterion of these two candidates is designed by minimizing the total variations of the first derivative at cell boundaries. A unified pressure-based projection formulation with a fractional step procedure is implemented with the proposed scheme to simulate both compressible and incompressible flows. A variety of numerical tests are studied, including linear and nonlinear scalar wave transport problems and compressible and incompressible flow problems. Results show that the proposed method can effectively eliminate numerical oscillation and diffusion, suggesting it has great potential to be applied to various types of engineering problems including both compressible and incompressible flows.

Pressure scrambling effects and the quantification of turbulent scalar flux model uncertainties

Closure models for the turbulent scalar flux are an important source of uncertainty in Reynolds-averaged-Navier-Stokes (RANS) simulations of scalar transport. This paper presents an approach to quantify this uncertainty in simulations of complex engineering flows. The approach addresses the uncertainty in modeling the pressure scrambling (PS) effect, which is the primary mechanism balancing the productions in scalar flux dynamics. Inspired by the two classical phenomenological theories of return-to-isotropy (RI) and isotropization-of-production (IP), we assume that the most likely directions of the PS term are around a fan-shaped region bounded by the RI and IP directions. Subsequently, we propose a strategy that requires two additional simulations, defining perturbations of the PS directions towards the RI and IP limits. The approach is applied to simulations of forced heat convection in a complex pin-fin array configuration, and shows favorable monotonic properties and bounding behaviors for various quantities of interest (QoIs) relevant to heat transfer. To conclude, the results are analyzed from the perspective of transverse scalar transport in a shear flow; the analysis indicates that the proposed approach is likely to exhibit monotonic behaviors in a wide range of scalar transport problems.

Dissipation-based continuation method for multiphase flow in heterogeneous porous media

In reservoir simulation, solution of the coupled systems of nonlinear algebraic equations that are associated with fully-implicit (backward Euler) discretization is challenging. Having a robust and efficient nonlinear solver is necessary in order for reservoir simulation to serve as the primary tool for managing the recovery processes of large-scale reservoirs. However, there are several outstanding challenges that are intimately connected to the highly nonlinear nature of the problem. Given a set of sources and sinks, the variation in the total velocity can span many orders of magnitude due to extreme contrasts in the permeability field in large-scale subsurface porous formations. Moreover, multiple and complex saturation fronts must be properly resolved throughout the three-dimensional reservoir model of interest. Add to that numerical simulation studies entail making field-scale predictions over many decades, and the challenge of developing robust and efficient nonlinear solvers across a very wide parameter space becomes clear. Here, we develop a continuation method based on the use of a dissipation operator. We focus on nonlinear two-phase flow and transport in heterogeneous formations in the presence of viscous, gravitational, and capillary forces. The homotopy is constructed by adding numerical dissipation to the coupled discrete conservation equations. A continuation parameter is introduced to control the amount of dissipation. Numerical evidence of multi-dimensional models and detailed analysis of single-cell problems are used to explain how the dissipation operator improves the nonlinear convergence of the coupled system of equations. An adaptive strategy to determine the dissipation coefficient is proposed. The dissipation level is computed locally for each cell interface. We demonstrate the efficiency of the dissipation-based continuation (DBC) nonlinear solver using several examples, including 1D scalar transport and 2D heterogeneous problems with fully-coupled flow and transport. The DBC solver has better convergence properties compared with the standard damped-Newton solvers used in reservoir simulation.

Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space

SummaryIn a previous paper, the authors presented an elemental enriched space to be used in a finite‐element framework (EFEM) capable of reproducing kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the interelement boundaries. In this previous publication, only scalar transport problems were solved (thermal problems). In the present work, these ideas are generalized to vectorial unknowns, in particular, the incompressible Navier‐Stokes equations for multifluid flows presenting internal moving interfaces. The advantage of the EFEM compared with global enrichment is the significant reduction in computing time when the internal interface is moving. In the EFEM, the matrix to be solved at each time step has not only the same amount of degrees of freedom (DOFs) but also the same connectivity between the DOFs. This frozen matrix graph enormously improves the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that it allows a linear variation of the jump, thus improving the convergence rate, compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finite‐element code is extremely easy with the version presented here because the new shape functions are based on the usual finite‐element method shape functions for triangles or tetrahedrals, and once the internal DOFs are statically condensed, the resulting elements have exactly the same number of unknowns as the nonenriched finite elements.

Variational multiscale modeling with discontinuous subscales: analysis and application to scalar transport

We examine a variational multiscale method in which the unresolved fine-scales are approximated element-wise using a discontinuous Galerkin method. We establish stability and convergence results for the methodology as applied to the scalar transport problem, and we prove that the method exhibits optimal convergence rates in the SUPG-norm and is robust with respect to the Peclet number if the discontinuous subscale approximation space is sufficiently rich. We apply the method to isogeometric NURBS discretizations of steady and unsteady transport problems, and the corresponding numerical results demonstrate that the method is stable and accurate in the advective limit even when low-order discontinuous subscale approximations are employed.

A fully conservative high‐order upwind multi‐moment method using moments in both upwind and downwind cells

SummaryWe propose a fully conservative high‐order upwind multi‐moment method for the conservation equation. The proposed method is based on a third‐order polynomial interpolation function and semi‐Lagrangian formulation and is a variant of the constrained interpolation profile conservative semi‐Lagrangian scheme with third‐order polynomial function method. The third‐order interpolation function is constructed based on three constraints in the upwind cell (two boundary values and a cell average) and a constraint in the downwind cell (a cell center value). The proposed method shows fourth‐order accuracy in a benchmark problem (sine wave propagation). We also propose a less oscillatory formulation of the proposed method. The less oscillatory formulation can minimize numerical oscillations. These methods were validated through scalar transport problems, and compressible flow problems (shock tube and 2D explosion problems). Copyright © 2016 John Wiley & Sons, Ltd.

On a consistent high-order finite difference scheme with kinetic energy conservation for simulating turbulent reacting flows

The main objective of this study is to present an accurate and consistent numerical framework for turbulent reacting flows based on a high-order finite difference (HOFD) scheme. It was shown previously by Desjardins et al. (2008) [4] that a centered finite difference scheme discretely conserving the kinetic energy and an upwind-biased scheme for the scalar transport can be combined into a useful scheme for turbulent reacting flows. With a high-order spatial accuracy, however, an inconsistency among discretization schemes for different conservation laws is identified, which can disturb a scalar field spuriously under non-uniform density distribution. Various theoretical and numerical analyses are performed on the sources of the unphysical error. From this, the derivative of the mass-conserving velocity and the local Péclet number are identified as the primary factors affecting the error. As a solution, an HOFD stencil for the mass conservation is reformulated into a flux-based form that can be used consistently with an upwind-biased scheme for the scalar transport. The effectiveness of the proposed formulation is verified using two-dimensional laminar flows such as a scalar transport problem and a laminar premixed flame, where unphysical oscillations in the scalar fields are removed. The applicability of the proposed scheme is demonstrated in an LES of a turbulent stratified premixed flame.

Localized Boundary-Domain Integro-partial Differential Formulations for Transient Scalar Transport Problems

Integro-partial differential equations, though very challenging can still be solved using conventional numerical techniques. Although these problems have been severally solved, it is still noticeable how little has been said about them in boundary element method (BEM) literature. A major reason for this is that for those problems where an encounter with the problem domain becomes a necessity, BEM’s inadequacies become highly apparent. Moreover for such situations, the fundamental solution is either not available in a cheaply computable form or a considerable numerical effort is required to handle this numerical challenge. Sometimes, the fundamental solutions may exist in a form that is highly non-local and lead to system of equations with a fully populated matrix that is cumbersome to handle numerically, especially for field problems. In the work reported herein, we adopt a fundamental solution of an auxiliary form of a governing partial differential equation coupled with the Green’s identity to discretize and localize an integro-partial differential transport equation by conversion into a boundary-domain form which is amenable to a hybrid boundary integral numerical formulation. It is observed that this numerical formulation is straightforward and yields accurate results when compared with those found in literature.

A second order residual based predictor–corrector approach for time dependent pollutant transport

We present a second order residual distribution scheme for scalar transport problems in shallow water flows. The scheme, suitable for the unsteady cases, is obtained adapting to the shallow water context the explicit Runge-Kutta schemes for scalar equations 1. The resulting scheme is decoupled from the hydrodynamics yet the continuity equation has to be considered in order to respect some important numerical properties at discrete level. Beyond the classical characteristics of the residual formulation presented in 1,2, we introduce the possibility to iterate the corrector step in order to improve the accuracy of the scheme. Another novelty is that the scheme is based on a precise monotonicity condition which guarantees the respect of the maximum principle. We thus end up with a scheme which is mass conservative, second order accurate and monotone. These properties are checked in the numerical tests, where the proposed approach is also compared to some finite volume schemes on unstructured grids. The results obtained show the interest in adopting the predictor-corrector scheme for pollutant transport applications, where conservation of the mass, monotonicity and accuracy are the most relevant concerns.

A micromechanics-based nonlocal constitutive equation incorporating three-point statistics for random linear elastic composite materials

A variational formulation employing the minimum potential and complementary energy principles is used to derive a micromechanics-based nonlocal constitutive equation for random linear elastic composite materials, relating ensemble averages of stress and strain in the most general situation when mean fields vary spatially. All information contained in the energy principles is retained; we employ stress polarization trial fields utilizing one-point statistics so that the resulting nonlocal constitutive equation incorporates up through three-point statistics. The variational structure is developed first for arbitrary heterogeneous linear elastic materials, then for randomly inhomogeneous materials, then for general n-phase composite materials, and finally for two-phase composite materials, in which case explicit variational upper and lower bounds on the nonlocal effective modulus tensor operator are derived. For statistically uniform infinite-body composites, these bounds are determined even more explicitly in Fourier transform space. We evaluate these in detail in an example case: longitudinal shear of an aligned fiber or void composite. We determine the full permissible ranges of the terms involving two- and three-point statistics in these bounds, and thereby exhibit explicit results that encompass arbitrary isotropic in-plane phase distributions; we also develop a nonlocal “Milton parameter”, the variation of whose eigenvalues throughout the interval [0, 1] describes the full permissible range of the three-point term. Example plots of the new bounds show them to provide substantial improvement over the (two-point) Hashin–Shtrikman bounds on the nonlocal operator tensor, for all permissible values of the two- and three-point parameters. We next discuss further applications of the general nonlocal operator bounds: to any three-dimensional scalar transport problem e.g. conductivity, for which explicit results are given encompassing the full permissible ranges of the two- and three-point statistics terms for arbitrary three-dimensional isotropic phase distributions; and to general three-dimensional composites, where explicit results require future research. Finally, we show how the work just summarized, treating elastostatics, can be generalized to elastodynamics, first in general, then explicitly for the longitudinal shear example.

A conservative finite volume scheme with time-accurate local time stepping for scalar transport on unstructured grids

In this article we propose a new conservative high resolution TVD (total variation diminishing) finite volume scheme with time-accurate local time stepping (LTS) on unstructured grids for the solution of scalar transport problems, which are typical in the context of water quality simulations. To keep the presentation of the new method as simple as possible, the algorithm is only derived in two space dimensions and for purely convective transport problems, hence neglecting diffusion and reaction terms. The new numerical method for the solution of the scalar transport is directly coupled to the hydrodynamic model of Casulli and Walters (2000) that provides the dynamics of the free surface and the velocity vector field based on a semi-implicit discretization of the shallow water equations. Wetting and drying is handled rigorously by the nonlinear algorithm proposed by Casulli (2009). The new time-accurate LTS algorithm allows a different time step size for each element of the unstructured grid, based on an element-local Courant-Friedrichs-Lewy (CFL) stability condition. The proposed method does not need any synchronization between different time steps of different elements and is by construction locally and globally conservative. The LTS scheme is based on a piecewise linear polynomial reconstruction in space–time using the MUSCL–Hancock method, to obtain second order of accuracy in both space and time.The new algorithm is first validated on some classical test cases for pure advection problems, for which exact solutions are known. In all cases we obtain a very good level of accuracy, showing also numerical convergence results; we furthermore confirm mass conservation up to machine precision and observe an improved computational efficiency compared to a standard second order TVD scheme for scalar transport with global time stepping (GTS). Then, the new LTS method is applied to some more complex problems, where the new scalar transport scheme has also been coupled to a semi-implicit model for the simulation of the free surface hydrodynamics, including nonlinear wetting and drying. The last application shown in this paper is carried out on a real geometry, for which we have taken available DTM data of the lake Guaiba in Brazil. Comparisons have been made in all cases with a second order TVD scheme based on GTS. For the new LTS algorithm we report a significant reduction in computational effort, with a savings of CPU time of the order of up to 95%.

Level set topology optimization of scalar transport problems

This paper studies level set topology optimization of scalar transport problems, modeled by an advection-diffusion equation. Examples of such problems include the transport of energy or mass in a f...

Erratum to: Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

Erratum to: Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

High resolution methods for scalar transport problems in compliant systems of arteries

In this paper a new semi-implicit high resolution scheme for the simulation of advection–diffusion problems in compliant arterial systems is proposed. Such transport problems are not only of great importance for the modeling of drug delivery processes, but also for the simulation of continuous processes occurring in the human body such as the exchange of oxygen, carbon dioxide, nutrients and toxics. Assuming cylindrical geometry and axially symmetric blood flow, a finite volume scheme for scalar transport on unstructured staggered grids is derived. It is shown how both mass conservation and maximum principle can be assured by the present method. Since the discrete maximum principle imposes a CFL type restriction on the time step, the scalar transport equation is solved using a consistent local time-stepping approach in order to not affect the unconditional stability of the underlying semi-implicit scheme for the hydrodynamics. It is a key feature of the present approach that the radial profiles of axial velocity and scalar concentration are computed directly from first principles and that no heuristic model for the velocity profile is needed as in classical one-dimensional approaches, which are still frequently used for the simulation of artery trees. The knowledge of radial velocity and concentration gradients is fundamental for the exchange processes happening across the vessel walls. The accuracy of the proposed approach is validated on one- and two-dimensional test problems with exact solution. An example for scalar transport in a model artery tree with 55 branches rounds off the numerical test problems discussed in this paper.

Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887---888, 2001; J Comput Phys 219:513---531, 2006; Comput Appl Math 218:79---87, 2008; J Comput Phys 228:2517---2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145---192, 2012; J Comput Appl Math 236:2317---2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915---4946, 2004; Int J Numer Methods Fluids 42:265---295, 2003; Kuzmin and Moller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525---558, 2002; J Comput Phys 198:131---158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming $$P_1$$ and $$Q_1$$ finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix---Raviart element (Crouzeix and Raviart in RAIRO R3 7:33---76, 1973) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher---Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97---111, 1992). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection---diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix---Raviart element. Good resolution of smooth and discontinuous profiles is attested to $$Q_1^\mathrm{nc}$$ approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix---vector multiplications is addressed.

Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

The present article proposes a new hybrid Eulerian–Lagrangian numerical method, based on a volume particle meshing of the Eulerian grid, for solving transport equations. The approach, called Volume Of Fluid Sub-Mesh method (VOF-SM), has the advantage of being able to deal with interface tracking as well as advection–diffusion transport equations of scalar quantities. The Eulerian evolutions of a scalar field could be obtained on any orthogonal curvilinear grid thanks to the Lagrangian advection and a redistribution of particles on the Eulerian grid. The Eulerian concentrations result from the projection of the volume and scalar informations handled by the particles. The particle velocities are interpolated from the Eulerian velocity field. The VOF-SM method is validated on several scalar interface tracking and transport problems and is compared to existing schemes within the literature. It is finally coupled to a Navier–Stokes solver and applied to the simulation of two free-surface flows, i.e. the two-dimensional buckling of a viscous jet during the filling of a square mold and the three-dimensional dam-break flow in a tank.