Staggered Grid Discretization Research Articles

Dual-porosity model for centrifugal filtration and washing dynamics of absorbent materials

Filtration and washing are widely employed in various industrials, including chemical, pharmaceutical, and environmental protection. Clarifying the dynamics of these processes is essential for quality control and parameters optimization. Traditional filtration theories mainly focus on non-absorbent materials and often consider washing separately from filtration, lacking in continuous process modeling. Consequently, the dynamic modeling theory for continuous filtration and washing processes of materials with bound water remains incomplete. This study integrates multiphase fluid dynamics theory with traditional modeling methods to establish a dynamic model for centrifugal filtration and alcohol washing of absorbent materials like nitrocellulose(NC) fibers using a dual-porosity approach. An implicit numerical solution algorithm and a dynamic simulation system for centrifugal filtration and washing processes were developed using staggered grid discretization and the damped Newton iteration method. Numerical simulations were conducted on three different process schemes for NC fiber slurry: single dewatering, one-time washing, and two-time washing. The effects of process parameters such as pre-dewatering time, alcohol volume addition, and first dewatering time on the filtration and washing processes were analyzed. Experimental validation with cotton fiber slurry filtration and alcohol washing confirmed the accuracy of the dynamic model and numerical algorithm. This research provides a reliable theoretical and numerical foundation for the dynamic simulation and mechanism study of filtration and washing processes for absorbent materials like NC fibers.

Decompression and Fracturing Caused by Magmatically Induced Thermal Stresses

AbstractStudies of host rock deformation around magmatic intrusions usually focus on the development of stresses directly related to the intrusion process. This is done either by considering an inflating region that represents the intruding body, or by considering multiphase deformation. Thermal processes, especially volume changes caused by thermal expansion are typically ignored. We show that thermal stresses around upper crustal magma bodies are likely to be significant and sufficient to create an extensive fracture network around the magma body by brittle yielding. At the same time, cooling induces decompression within the intrusion, which can promote the appearance of a volatile phase. Volatile phases and the development of a fracture network around the inclusion may thus be the processes that control magmatic‐hydrothermal alteration around intrusions. This suggests that thermal stresses likely play an important role in the development of magmatic systems. To quantify the magnitude of thermal stresses around cooling intrusions, we present a fully compressible 2D visco‐elasto‐plastic thermo‐mechanical numerical model. We utilize a finite difference staggered grid discretization and a graphics processing unit based pseudo‐transient solver. First, we present purely thermo‐elastic solutions, then we include the effects of viscous relaxation and plastic yielding. The dominant deformation mechanism in our models is determined in a self‐consistent manner, by taking into account stress, pressure, and temperature conditions. Using experimentally determined flow laws, the resulting thermal stresses can be comparable to or even exceed the confining pressure. This suggests that thermal stresses alone could result in the development of a fracture network around magmatic bodies.

Voxel‐based finite elements with hourglass control in fast Fourier transform‐based computational homogenization

AbstractThe power of fast Fourier transform (FFT)‐based methods in computational micromechanics critically depends on a seamless integration of discretization scheme and solution method. In contrast to solution methods, where options are available that are fast, robust and memory‐efficient at the same time, choosing the underlying discretization scheme still requires the user to make compromises. Discretizations with trigonometric polynomials suffer from spurious oscillations in the solution fields and lead to ill‐conditioned systems for complex porous materials, but come with rather accurate effective properties for finitely contrasted materials. The staggered grid discretization, a finite‐volume scheme, is devoid of bulk artifacts in the solution fields and works robustly for porous materials, but does not handle anisotropic materials in a natural way. Fully integrated finite‐element discretizations share the advantages of the staggered grid, but involve a higher memory footprint, require a higher computational effort due to the increased number of integration points and typically overestimate the effective properties. Most widely used is the rotated staggered grid discretization, which may also be viewed as an underintegrated trilinear finite element discretization, which does not impose restrictions on the constitutive law, has fewer artifacts than Fourier‐type discretizations and leads to rather accurate effective properties. However, this discretization comes with two downsides. For a start, checkerboard artifacts are still present. Second, convergence problems occur for complex porous microstructures. The work at hand introduces FFT‐based solution techniques for underintegrated trilinear finite elements with hourglass control. The latter approach permits to suppress local hourglass modes, which stabilizes the convergence behavior of the solvers for porous materials and removes the checkerboards from the local solution field. Moreover, the hourglass‐control parameter can be adjusted to “soften” the material response compared to fully integrated elements, using only a single integration point for nonlinear analyses at the same time. To be effective, the introduced technology requires a displacement‐based implementation. The article exposes an efficient way for doing so, providing minimal interfaces to the most commonly used solution techniques and the appropriate convergence criterion.

FFT‐based homogenization with mixed uniform boundary conditions

AbstractThe fast Fourier transform (FFT) based homogenization method of Moulinec and Suquet has been established as a fast, accurate, and robust tool for periodic homogenization in solid mechanics. In a finite element context, Pahr and Zysset have introduced nonperiodic boundary conditions (PMUBC) for homogenization problems. We show how to implement PMUBC efficiently in an FFT‐based code using discrete sine and cosine transforms. Compared with the domain mirroring approach, we reduce the runtime by a factor of 2 to 3, and the memory requirements by a factor of 8. We show that the use of periodic boundary conditions for nonperiodic geometries yields vastly different results than with PMUBC. Furthermore, we examine the influence of the discretization method by comparing the staggered grid discretization with a finite element discretization.

3-D DC Resistivity Forward Modeling Using the Multi-resolution Grid

We implemented a novel multi-resolution grid approach to direct current resistivity (DCR) modeling in 3-D. The multi-resolution grid was initially developed to solve the electromagnetic forward problem and helped to improve the modeling efficiency. In the DCR forward problem, the distribution of the electric potentials in the subsurface is estimated. We consider finite-difference staggered grid discretization, which requires fine grid resolution to accurately model electric potentials around the current electrodes and complex model geometries near the surface. Since the potential variations attenuate with depth, the grid resolution can be decreased correspondingly. The conventional staggered grid fixes the horizontal grid resolution that extends to all layers. This leads to over-discretization and therefore unnecessary high computational costs (time and memory). The non-conformal multi-resolution grid allows the refinement or roughening for the grid’s horizontal resolution with depth, resulting in a substantial reduction of the degrees of freedom, and subsequently, computational requirements. In our implementation, the coefficient matrix maintains its symmetry, which is beneficial for using the iterative solvers and solving the adjoint problem in inversion. Through comparison with the staggered grid, we have found that the multi-resolution grid can significantly improve the modeling efficiency without compromising the accuracy. Therefore, the multi-resolution grid allows modeling with finer horizontal resolutions at lower computational costs, which is essential for accurate representation of the complex structures. Consequently, the inversion based on our modeling approach will be more efficient and accurate.

Total roto-translational variation

We consider curvature depending variational models for image regularization, such as Euler's elastica. These models are known to provide strong priors for the continuity of edges and hence have important applications in shape-and image processing. We consider a lifted convex representation of these models in the roto-translation space: In this space, curvature depending variational energies are represented by means of a convex functional defined on divergence free vector fields. The line energies are then easily extended to any scalar function. It yields a natural generalization of the total variation to the roto-translation space. As our main result, we show that the proposed convex representation is tight for characteristic functions of smooth shapes. We also discuss cases where this representation fails. For numerical solution, we propose a staggered grid discretization based on an averaged Raviart-Thomas finite elements approximation. This discretization is consistent, up to minor details, with the underlying continuous model. The resulting non-smooth convex optimization problem is solved using a first-order primal-dual algorithm. We illustrate the results of our numerical algorithm on various problems from shape-and image processing.

1-D transient numerical modeling of counter-current two-phase stratified flow inside a medium temperature solar linear collector

Two-phase gravity-driven and gravity-stratified flow regime inside a pipe, which is present in many engineering applications, is an attractive option for solar cooling/heating/power production using Rankine cycles, absorption cycles or any other thermodynamic application by means of vapor as working fluid. This paper offers a numerical model of this flow configuration that copes with transient phenomena, like unsteadiness of solar radiation, among others. The mathematical model consists of 1-D balance equations for mass and momentum for both fluids and energy for both fluids and the wall of the pipe that absorbs the solar radiation. The model is characterized by the fact that the area (or height) of the liquid layer is treated as a dependent variable forming part of the solution. The numerical method consists in a finite volume staggered grid discretization of the governing equations. Mass flow and liquid area are calculated with a semi-implicit pressure based method and the transient terms are treated with the explicit first stage singly implicit Runge-Kutta (ESDIRK) method. The calculation of the mass transfer rate from liquid to vapor is calculated iteratively by a guess-and-correct mass transfer algorithm, specially developed for stratified flows. The results show the applicability and benefits of this model for the not so well known counter-current stratified two-phase with evaporation/boiling. Additionally, the performance of the mass transfer algorithm is discussed showing that it is monotonic decreasing and linearly convergent.

Numerical solution of one-dimensional transient, two-phase flows with temporal fully implicit high order schemes: Subcooled boiling in pipes

An extension for 1-D transient two-phase flows of the SIMPLE-ESDIRK method, initially developed for incompressible viscous flows by Ijaz is presented. This extension is motivated by the high temporal order of accuracy demanded to cope with fast phase change events. This methodology is suitable for boiling heat exchangers, solar thermal receivers, etc. The methodology of the solution consist in a finite volume staggered grid discretization of the governing equations in which the transient terms are treated with the explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) method. It is suitable for stiff differential equations, present in instant boiling or condensation processes. It is combined with the semi-implicit pressure linked equations algorithm (SIMPLE) for the calculation of the pressure field. The case of study consists of the numerical reproduction of the Bartolomei upward boiling pipe flow experiment. The steady-state validation of the numerical algorithm is made against these experimental results and well known numerical results for that experiment. In addition, a detailed study reveals the benefits over the first order Euler Backward method when applying 3rd and 4th order schemes, making emphasis in the behaviour when the system is subjected to periodic square wave wall heat function disturbances, concluding that the use of the ESDIRK method in two-phase calculations presents remarkable accuracy and computational advantages.

A finite difference solver for incompressible Navier–Stokes flows in complex domains

Modern CFD applications require the treatment of general complex domains to accurately model the emerging flow patterns. In the present work, a new low order finite difference scheme is employed and tested for the numerical solution of the incompressible Navier–Stokes equations in a complex domain described in curvilinear coordinates. A staggered grid discretization is used on both the physical and computational domains. A subgrid based computation of the Jacobian and the metric coefficients of the transformation is used. The incompressibility condition, properly transformed in curvilinear coordinates, is enforced by an iterative procedure employing either a modified local pressure correction technique or the globally defined numerical solution of a general elliptic BVP. Results obtained by the proposed overall solution technique, exhibit very good agreement with other experimental and numerical calculations for a variety of domains and grid configurations. The overall numerical solver effectively treats the general complex domains.

Computational homogenization of elasticity on a staggered grid

SummaryIn this article, we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers. Our method shares some properties with the FFT‐based homogenization technique of Moulinec and Suquet, which has received widespread attention recently because of its robustness and computational speed. These similarities include the use of FFT and the resulting performing solvers. The staggered grid discretization, however, offers three crucial improvements. Firstly, solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by Moulinec–Suquet's discretization. Secondly, the iteration numbers of our solvers are bounded independently of the grid size and the contrast. In particular, our solvers converge for three‐dimensional porous structures, which cannot be handled by Moulinec–Suquet's method. Thirdly, the finite difference discretization allows for algorithmic variants with lower memory consumption. More precisely, it is possible to reduce the memory consumption of the Moulinec–Suquet algorithms by 50%. We underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale. Copyright © 2015 John Wiley & Sons, Ltd.

Propagation characteristics of Rayleigh wave under gravity based on FDTD method

R wave is short for Rayleigh wave. R wave,on which the effect of gravity is generally inevitable,has received considerably extensive attention and been widely used in many fields,such as geological exploration,nondestructive test,etc. The effects of gravity on R wave velocity for elastic parameters were further analyzed based on available R wave velocity function,and the applicable range of the approximate one was given as well. Wave propagation through quasi-half-space isotropic linear elastic medium with gravity,under the excitation of differentiated Gaussian pulse( DGP) was studied by finite difference time domain( FDTD) method with staggered grid discretization,and extended boundary conditions were introduced.Velocity values obtained were closer to the theoretical ones. Meanwhile,the influence of gravity on time domain,frequency domain response was discussed. By analyzing the theoretical and numerical results,it is necessary to take the effect of gravity into account in simulation for more accurate prediction.

Analysis of some projection method based preconditioners for models of incompressible flow

In this paper, several projection method based preconditioners for various incompressible flow models are studied. In the derivations of these projection method based preconditioners, we use three different types of the approximations of the inverse of the Schur complement, i.e., the exact inverse, the Cahouet–Chabard type approximation and the BFBt type approximation. We illuminate the connections and the distinctions between these projection method based preconditioners and those related preconditioners. For the preconditioners using the Cahouet–Chabard type approximation, we show that the eigenvalues of the preconditioned systems have uniform bounds independent of the parameters and most of them are equal to 1. The analysis is based on a detailed discussion of the commutator difference operator. Moreover, these results demonstrate the stability of the staggered grid discretization and reveal the effects of the boundary treatment. To further illustrate the effectiveness of these projection method based preconditioners, numerical experiments are given to compare their performances with those of the related preconditioners. Generalizations of the projection method based preconditioners to other saddle point problems are also discussed.

Optimal Transport with Proximal Splitting

This article reviews the use of first order convex optimization schemes to solve the discretized dynamic optimal transport problem, initially proposed by Benamou and Brenier. We develop a staggered grid discretization that is well adapted to the computation of the $L^2$ optimal transport geodesic between distributions defined on a uniform spatial grid. We show how proximal splitting schemes can be used to solve the resulting large scale convex optimization problem. A specific instantiation of this method on a centered grid corresponds to the initial algorithm developed by Benamou and Brenier. We also show how more general cost functions can be taken into account and how to extend the method to perform optimal transport on a Riemannian manifold.

Domain robust preconditioning for a staggered grid discretization of the Stokes equations

To the best of our knowledge, most numerical methods for solving discretizations of the Stokes equations depend on the Schur complement and thus the convergence rate depends on the aspect ratio of the domain. We present a preconditioner for a staggered grid discretization of the Stokes equations which is proven to be independent of the aspect ratio. Finally numerical experiments confirm the robustness of the method.

An adaptive staggered grid finite difference method for modeling geodynamic Stokes flows with strongly variable viscosity

Here we describe a new staggered grid formulation for discretizing incompressible Stokes flow which has been specifically designed for use on adaptive quadtree‐type meshes. The key to our new adaptive staggered grid (ASG) stencil is in the form of the stress‐conservative finite difference constraints which are enforced at the “hanging” velocity nodes between resolution transitions within the mesh. The new ASG discretization maintains a compact stencil, thus preserving the sparsity within the matrix which both minimizes the computational cost and enables the discrete system to be efficiently solved via sparse direct factorizations or iterative methods. We demonstrate numerically that the ASG stencil (1) is stable and does not produce spurious pressure oscillations across regions of grid refinement, which intersect discontinuous viscosity structures, and (2) possesses the same order of accuracy as the classical nonadaptive staggered grid discretization. Several pragmatic error indicators that are used to drive adaptivity are introduced in order to demonstrate the superior performance of the ASG stencil over traditional nonadaptive grid approaches. Furthermore, to demonstrate the potential of this new methodology, we present geodynamic examples of both lithospheric and planetary scales models.

A tensor artificial viscosity using a finite element approach

We derive a tensor artificial viscosity suitable for use in a 2D or 3D unstructured arbitrary Lagrangian–Eulerian (ALE) hydrodynamics code. This work is similar in nature to that of Campbell and Shashkov [1]; however, our approach is based on a finite element discretization that is fundamentally different from the mimetic finite difference framework. The finite element point of view leads to novel insights as well as improved numerical results. We begin with a generalized tensor version of the Von Neumann–Richtmyer artificial viscosity, then convert it to a variational formulation and apply a Galerkin discretization process using high order Gaussian quadrature to obtain a generalized nodal force term and corresponding zonal heating (or shock entropy) term. This technique is modular and is therefore suitable for coupling to a traditional staggered grid discretization of the momentum and energy conservation laws; however, we motivate the use of such finite element approaches for discretizing each term in the Euler equations. We review the key properties that any artificial viscosity must possess and use these to formulate specific constraints on the total artificial viscosity force term as well as the artificial viscosity coefficient. We also show, that under certain simplifying assumptions, the two-dimensional scheme from [1] can be viewed as an under-integrated version of our finite element method. This equivalence holds on general distorted quadrilateral grids. Finally, we present computational results on some standard shock hydro test problems, as well as some more challenging problems, indicating the advantages of the new approach with respect to symmetry preservation for shock wave propagation over general grids.

A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g. the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics 207 (2005) 457–492.]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier–Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton–Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.

Simulating vortex motion in superconducting films with the time-dependent Ginzburg - Landau equations

The time-dependent Ginzburg - Landau equations (TDGL) model a thin-film superconductor of finite size placed under a magnetic field. For numerical computation, we use a staggered grid discretization, a technique well known in numerical fluid mechanics. Some properties of the solutions are established. An efficient explicit - implicit method based on the forward Euler method is developed. In our simulations, we impose natural boundary conditions at the edge of the superconductor. With suitable choices of parameters (corresponding to physical superconductors of type II) and the strength of the external magnetic field, the steady-state solutions exhibit vortices. When a variable strength magnetic field, simulating a transient current, is introduced, we observe motion of the vortices in a periodic pattern.