Implicit Fractional Step Research Articles

Computational multiphase periporomechanics for unguided cracking in unsaturated porous media

AbstractIn this article, we formulate and implement a computational multiphase periporomechanics paradigm for unguided fracturing in unsaturated porous media assuming passive pore air pressure. The same governing equation for the solid phase applies on and off cracks. Crack formation in this framework is autonomous, requiring no prior estimates of crack topology. As a new contribution, an energy‐based criterion for arbitrary crack formation is formulated using the peridynamic effective force state for unsaturated porous media. Unsaturated fluid flow in the fracture space is modeled in a simplified way in line with the nonlocal formulation of unsaturated fluid flow in the bulk. The formulated unsaturated fracturing periporomechanics is numerically implemented through an implicit fractional step algorithm in time and a two‐phase mixed meshless method in space. The two‐stage operator split converts the coupled periporomechanics problem into an undrained deformation and fracture problem and an unsaturated fluid flow in the deformed skeleton configuration. Numerical simulations of in‐plane open and shear cracking are conducted to validate the accuracy and robustness of the fracturing unsaturated periporomechanics model. Then numerical examples of wing cracking and nonplanar cracking in unsaturated soil specimens are presented to demonstrate the efficacy of the proposed multiphase periporomechanics paradigm for unguided cracking in unsaturated porous media.

Influence of the turbulence modeling on the simulation of unsteady cavitating flows

The present work is focused on the analysis of cloud cavitation, i.e. high-speed unsteady liquid /vapor flows characterized by periodical large-scale shedding. The study relies on the numerical modeling of cavitating flows with a homogeneous approach coupled with the Reynolds-Averaged Navier-Stokes (RANS) framework. A new algorithm based on an implicit fractional step method is used to solve the time-dependent mass, momentum and transport equation for the void fraction. It is combined with an original treatment of the cavitation source term, which enables to maintain the void fraction within its physical range [0 1] throughout the calculation. No artificial numerical limitation is applied, so the overall mass conservation of the method is not deteriorated. The influence of the turbulence modeling on the results is investigated by applying a k-ε RNG and a k-ω SST model with and without the modification of the turbulent viscosity proposed by Reboud et al.[1,2]. Cavitation on a 2D Venturi type section and a NACA hydrofoil are both considered. The present work confirms the capability of the modified k-ε RNG and k-ω SST models to reproduce the main features of cloud cavitation. The role of the transport of principal turbulent shear stress in the mechanism of flow unsteadiness is discussed, as well as the validity of the Bradshaw's assumption for adverse pressure gradient. The benefit and the limitations of the modification of the turbulent viscosity are shown: while it enables to reproduce the main features of the flow behavior, significant discrepancies in the local shear stress are still found.

Numerical investigation of viscous effects on the nonlinear Burgers equation

This research article presents the numerical solution of the viscous Burgers equation. The diagonally implicit fractional step $\theta$ (DIFST) scheme is used for the time discretization and the space derivative is discretized by the conforming finite element method with quadrilateral mesh. The viscosity effect on the shock wave is calculated with an estimation of the $L_2$ error. For comparison of different time discretization schemes, three test problems are computed. The stability and accuracy of the schemes are given by estimating the $L_2$ error norm. Numerical simulation for one- and two-dimensional problems are given and illustrated graphically. The effect of the viscosity parameter on the nonlinearity of the Burger equation is computed. The stability of the schemes for different time steps with CPU time is also given.

A massively-parallel, unstructured overset method to simulate moving bodies in turbulent flows

An unstructured overset method capable of performing direct numerical simulation (DNS) and large eddy simulation (LES) of many (O(105)) moving bodies, utilizing many computational cores (O(105)), in turbulent, incompressible fluid flow is presented. Unstructured meshes are attached to bodies and placed within a fixed background domain. Body meshes are allowed to arbitrarily overlap and move throughout the domain. Within each mesh a high resolution, unstructured, non-dissipative finite volume method is used to solve for the flow field. Boundary conditions for each mesh are provided by interpolation from flow solutions on overlapping meshes. When many unstructured meshes of different resolution overlap, care is required in the connection between the different flow solutions. An interpolant is created which seeks to preserve volume conservation of flow quantities between meshes regardless of overlapping mesh differences. An implicit fractional step method is used for time advancement, requiring the calculation of a predicted fluid velocity and corrector pressure field. For the predictor step, the resulting interpolation is directly introduced into the implicit equations for the predicted flow field. For the corrected pressure field, the continuity between meshes is weakly enforced using a penalty formulation. The pressure formulation is symmetric, positive-definite and non-singular resulting in a formulation which is readily solvable using traditional iterative matrix inversion techniques. An Arbitrary Euler-Lagrangian (ALE) method coupled to a 6 degrees of freedom rigid body equation system (6-DOF) is used for body motion. For rotation, a quaternion representation is used to solve Euler's equations of rigid body motion. A linear spring damper model, which uses geometry information readily available from the overset assembly process, is used for collisions. Validation of the method for canonical flow fields is presented including assessment of order of accuracy and kinetic energy conservation properties. Particle-resolved direct numerical simulation (PR-DNS) of single particles in various flow fields are presented for validation. PR-DNS results of 50,000 spherical particles freely moving within turbulent channel flow are shown as a demonstration of the method at full scale. LES results of a marine propeller under crashback conditions are shown to demonstrate the ability to simulate highly unsteady turbulent flows over complex, moving geometries.

An adaptive time‐stepping scheme for the numerical simulation of Cahn‐Hilliard equation with variable mobility

AbstractIn spinodal decomposition phenomena, the initial perturbations evolve at a faster time scale while there is a slower growth at a later time. Therefore, using uniform small time‐steps for tracking the fast dynamics is computationally expensive. On the other hand, uniform large time‐steps may overlook the rapid changes. In this article, we propose an adaptive time‐stepping scheme for faster time scale simulations of spinodal decomposition by solving the Cahn‐Hilliard equation numerically. We consider the double well potential having a polynomial of 6th‐order along with 4th‐order polynomial for the variable mobility. A diagonally implicit fractional step θ‐scheme(DIFSTS) for temporal discretization while conforming finite element method for spatial discretization is used. Accuracy and efficiency of the method are given and simulations of 2D spinodal decomposition are illustrated graphically.

A mathematical model of tumor hypoxia targeting in cancer treatment and its numerical simulation

Solid tumor includes the areas where oxygen concentration is very low called hypoxia, often in the surrounding areas of necrosis. Hypoxic cells in these areas are resistant to chemotherapy and radiation therapy. The presence of hypoxia and necrosis enables tumor-selective treatment, including hypoxia-activated prodrugs, tumor hypoxia-specific gene therapy and tumor-targeting bacterial therapy. This article deals with the mathematical formulation of tumor hypoxia-targeting by introducing a decay parameter of oxygen in the model given by Kolobov et al. (2009) and Avila et al. (2013). The well-posedness of the governing partial differential equations and numerical simulation are provided. For the purpose of numerical simulations, the conforming Q1 finite element method for space discretization and second-order diagonally implicit fractional step θ-scheme for temporal discretization are used. The effect of oxygen on hypoxia, necrotic region decay and the maximum age of growing tumor cells are computed and illustrated graphically. It is observed that the distribution of nutrients in tissues have substantial effect on tumor growth rate and structure.

Three dimensional heat transfer from a square cylinder at low Reynolds numbers

Three-dimensional unsteady flow and heat transfer from an isothermal square cylinder subjected to crossflow of air is numerically investigated. The governing continuity, momentum and energy equations are solved using implicit fractional step method. The first order spatial derivatives are discretized using a third-order upwind scheme while the second order derivatives from the viscous terms are discretized by central-difference formula. The momentum equations are solved separately using Crank-Nicholson time-stepping method. The Poisson type equations are solved using Jacobi method with Chebyshev acceleration procedure.The heat transfer characteristics of a square cylinder subjected cross flow of air and the three-dimensionality of the flow and its effects are assessed. Three dimensional instantaneous isotemperature surfaces and in the wake region and local Nusselt number variations on the cylinder surfaces along z-axis were provided. The time histories of the Nusselt numbers at the cylinder surfaces are obtained. Also instantaneous and mean deviation of the local Nusselt numbers on the cylinder surfaces both in spanwise direction and x-y plane are provided for Re = 185 and 250 which are representative of typical A-mode and B-mode vortex structures in spanwise direction respectively.

Numerical Model for Laminar Flow in Agitated Vessel by tow Blades Impeller

A large number of chemicals, biochemical or petrochemical industry operations are performed in stirred tanks or in mechanically agitated vessels. The optimum operating mode of these equipments requires a detailed knowledge of the hydrodynamic behavior induced by the agitator. In this piece of work the characterization of the laminar viscous fluid flow fields in a cylindrical stirred tank is agitated by inclined blades anchor agitator studied. The computational fluid dynamic (CFD) model based on an implicit fractional step scheme and control volume method was developed for the spatial discretization of the Navier-Stokes, formulated in Cartesian coordinates primitives variables (u, v, p, T) on unstructured triangular mesh. Some simulations of the flow around an anchor with straights blades allowed validating the used method. We have analyzed the influence of the tilt blades degree on the hydrodynamic flow behaviors, such as the stream function, the velocity field, the velocities components, and the power consumption. The comparison between some of the obtained results with literature data, have showed a satisfactory agreement.

An efficient time-stepping scheme for numerical simulation of dendritic crystal growth

In this article, we present an adaptive time-stepping technique for numerical simulation of dendritic crystal growth model. The diagonally implicit fractional step -scheme for time discretisation a...

Theory and application of fractional step characteristic finite difference method in numerical simulation of second order enhanced oil production

A kind of second-order implicit fractional step characteristic finite difference method is presented in this paper for the numerically simulation coupled system of enhanced (chemical) oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in l2 norm is derived. This method has been applied successfully to the numerical simulation of enhanced oil production in actual oilfields, and the simulation results are quite interesting and satisfactory.

Theory and application of numerical simulation method of capillary force enhanced oil production

A kind of second-order implicit upwind fractional step finite difference methods are presented for the numerical simulation of coupled systems for enhanced (chemical) oil production with capillary force in the porous media. Some techniques, e.g., the calculus of variations, the energy analysis method, the commutativity of the products of difference operators, the decomposition of high-order difference operators, and the theory of a priori estimate, are introduced. An optimal order error estimate in the l2 norm is derived. The method is successfully used in the numerical simulation of the enhanced oil production in actual oilfields. The simulation results are satisfactory and interesting.

Numerical Investigation of Vortex Shedding Modes of Flow past Two Circular Cylinders in Side-by-Side Using Immersed Boundary Method

The vortex shedding modes of flow past two circular cylinders in side-by-side arrangement are investigated numerically in this paper. The simulations are carried out using a ghost cell immersed boundary method which imposes the boundary condition through reconstruction of the local velocity field near the immersed boundary. The two-dimensional unsteady incompressible Navier-Stokes equations are solved using an implicit fractional step method based on cell-center, collocated arrangement of the primary variables. Vorticity contours of the flow around the cylinders and force time histories are presented. Anti-phase and in-phase vortex shedding modes were found to exist in the flow simulation. These results of simulations were in agreement with phenomena observed in experiment and numerical results of previous researchers.

Ghost Cell Immersed Boundary Method for Simulating Flow over Two Circular Cylinders in Tandem Arrangement

In this work, a ghost cell immersed boundary method is applied to the numerical simulation of a uniform flows over a circular cylinder and two circular cylinders in tandem arrangement. The Navier-Stokes equations are solved using an implicit fractional step method employed on collocated arrangement variables. Immersed boundary method permit the use of structured Cartesian meshes to simulate flows involving complex boundaries. The shedding of vortices and flow interference between two circular cylinders in tandem arrangement are investigated numerically. The calculations are validated against the experimental and numerical results obtained by other researchers to prove the accuracy and effectiveness.

Domain decomposition operator splitting for mimetic finite difference discretizations of non-stationary problems

In this paper, we propose reliable and efficient numerical methods for solving semilinear, time-dependent partial differential equations of reaction–diffusion type. The original problem is first integrated in time by using a linearly implicit fractional step Runge–Kutta method. This method takes advantage of a suitable partitioning of the diffusion operator based on domain decomposition techniques. The resulting semidiscrete problem is fully discretized by means of a mimetic finite difference method on quadrilateral meshes. Due to the previous splitting, the totally discrete scheme can be reduced to a set of uncoupled linear systems which can be solved in parallel. The overall algorithm is unconditionally stable and second-order convergent in both time and space. These properties are confirmed by numerical experiments.

Convergence of fractional step mimetic finite difference discretizations for semilinear parabolic problems

This paper deals with the numerical solution of semilinear parabolic problems by means of efficient parallel algorithms. We first consider a mimetic finite difference method for the spatial semidiscretization. The connection of this method with an appropriate mixed finite element technique is the key to prove the convergence of the semidiscrete scheme. Next, we propose and analyze the use of a linearly implicit fractional step Runge–Kutta method as time integrator. The choice of suitable operator splittings related to an adequate decomposition of the spatial domain makes it possible to obtain totally discrete schemes that can be easily parallelized. A numerical test is shown to illustrate the theoretical results.

Simulation of spilling breaking waves using a two phase flow CFD model

A two phase flow CFD model has been developed for 2D spilling breaking wave simulations. A mass conservative level set method similar to Olsson and Kreiss [Olsson E, Kreiss G. A conservative level set method for two phase flow. J Comput Phys 2005;210(1):225–46] is implemented for capturing the air–water interface. The solver is discretised using a finite volume method based on a curvilinear coordinate system. A fully implicit fractional step method is used to advance simulations in time. The solver has been tested and validated by repeating benchmark results of dam breaking simulation and travelling solitary wave simulation. Finally, we employ this solver to simulate spilling breaking waves in the surf zone. Our results show that surface elevations, the location of the breaking point and undertow profiles can generally be well captured. We have also found that temporal and spatial schemes may have significant impacts on computational results.

Locally linearized fractional step methods for nonlinear parabolic problems

This work deals with the efficient numerical solution of a class of nonlinear time-dependent reaction–diffusion equations. Via the method of lines approach, we first perform the spatial discretization of the original problem by applying a mimetic finite difference scheme. The system of ordinary differential equations arising from that process is then integrated in time with a linearly implicit fractional step method. For that purpose, we locally decompose the discrete nonlinear diffusion operator using suitable Taylor expansions and a domain decomposition splitting technique. The totally discrete scheme considers implicit time integrations for the linear terms while explicitly handling the nonlinear ones. As a result, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled parallelizable linear subsystems. The convergence of the proposed methods is illustrated by numerical experiments.

J0601-3-3 フラクショナルステップ法による多孔質内流動解析スキームの開発(PEFC内の凝縮水挙動)

Solid matrix drag term usually results in a significant pressure drop across the porous medium. The general momentum conservation equation reduces to the extended Darcy's law for the flow in porous media. So Darcy resistance force is very important for the numerical simulation inside the porous media. A new implicit fractional step algorithm is used for the numerical treatment of Darcy drag term and its effect on the residual error of continuity is studied in the present investigation. Faster convergence and reduced mass conservation error is found by using this algorithm.

Effects of an adverse pressure gradient on a turbulent boundary layer

Direct numerical simulations were performed to investigate the effects of an adverse pressure gradient (APG) on a turbulent boundary layer. A fully implicit fractional step method was employed to simulate the flows. Equilibrium APG flows were established using a power-law free-stream distribution, U ∞ ∼ x m . The streamwise length from the inlet was made sufficiently long that the change in free-stream velocity associated with the APG did not influence the main stream. The flow with zero pressure gradient (ZPG) was also simulated and compared with the APG flows. The spatially developing characteristics of the turbulence stresses in the non-equilibrium APG flows were carefully examined. The instantaneous flow fields and vorticity fluctuations were analyzed to characterize the response of the outer turbulence to an APG. The present numerical results showed that the mean flows are greatly affected by an APG, and the coherent structures in the outer layer of the APG flows were more activated than those in the ZPG flow which may be attributed to increased turbulence intensities, shear stresses and pressure fluctuations in the APG systems. Examination of the Reynolds stress budget revealed that the energy redistribution was enhanced in the outer layer of the APG flows compared to the ZPG flow.

A new class of second order linearly implicit fractional step methods

A new family of linearly implicit fractional step methods is proposed and analysed in this paper. The combination of one of these time integrators with a suitable spatial discretization permits a very efficient numerical solution of semilinear parabolic problems. The main quality of this new family of methods, compared to other existing time integrators of this type, is that they are stable when the spatial differential operator is decomposed in a number m of “simpler” operators which do not necessarily commute. We prove that these methods satisfy this general stability result as well as they are second order consistent. Both consistency and stability are proven for an operator splitting in an arbitrary number m of terms ( m ⩾ 2 ). Finally, a numerical experiment illustrates these theoretical results in the last section of the paper.