Fractional Step Algorithm Research Articles

Two-phase two-layer SNS-PFEM for hydromechanical geotechnical large deformation problems

A two-phase one-layer stabilised node-based strain-smoothing Particle Finite Element Method (SNS-PFEM) was previously proposed, but it lacked the capability to simulate water-soil coupling. To address this limitation, this study presents an improved approach called the two-phase two-layer SNS-PFEM, specifically designed to handle such problems. In this method, two overlapping mesh layers are employed to independently represent the two phases. Within this framework, the water phase is characterised as an incompressible Newtonian fluid solved using the PFEM-T3 method, while the soil phase is described using an elastoplastic constitutive model, effectively addressed through a modified SNS-PFEM. To obtain the solution for the proposed two-phase two-layer approach, a fractional step algorithm is utilised, allowing efficient and accurate computations. To validate the proposed method, it is applied to various soil-water coupling problems, encompassing both quasi-static and large deformation scenarios. The first set of simulations are (1) simulation of 1D Terzaghi's consolidation test and (2) simulation of fully submerged soil subjected to gravitational load. Next, the proposed method is applied to simulate cases involving large deformation problems: (1) flow passing through a porous dam, (2) a submerged granular landslide and its induced waves and (3) a submarine progressive landslide. The results of all simulations demonstrate the robustness and versatility of the proposed method in effectively analysing large deformation multi-phase problems in the field of geotechnical engineering. This advancement is a significant step forward in accurately simulating complex interactions between water and soil, opening up new possibilities for research and analysis in geotechnical applications.

A staggered projection scheme for viscoelastic flows

We develop a numerical scheme for the flow of viscoelastic fluids, including the Oldroyd-B and FENE-CR constitutive models. The space discretization is staggered, using either the Marker-And-Cell (MAC) scheme for structured nonuniform grids, or the Rannacher and Turek (RT) nonconforming low-order finite element approximation for general quadrangular or hexahedral meshes. The time discretization uses a fractional-step algorithm where the solution of the Navier–Stokes equations is first obtained by a projection method and then the transport-reaction equation for the conformation tensor is solved by a finite volume scheme. In order to obtain consistency, the space discretization of the divergence of the elastic part of the stress tensor in the momentum balance equation is derived using a weak form of the MAC scheme. For stability and accuracy purposes, the solution of the transport-reaction equation for the conformation tensor is split into pure convection steps, with a change of variable to the log-conformation tensor, and a reaction step, which consists in solving one ODE per cell via an Euler scheme with local sub-cycling. Numerical computations for the flow in the lid-driven cavity at Weissenberg numbers above one and the flow around a confined cylinder confirm the efficiency of the scheme.

Multiscale, multiphysics modeling of saturated granular materials in large deformation

We propose a multiscale, multiphysics approach by coupling two-phase material point method (MPM) with discrete element method (DEM) (MPM-DEM) to simulate the hydro-mechanical coupling responses of saturated granular media from small strain en route to large deformation under either quasi-static or dynamic loading conditions. The multiscale scheme is featured by (a) using a two-phase MPM in conjunction with the u−v−p formulation to solve the solid–fluid interactions in the macroscopic domain of a boundary value problem of saturated porous media, and (b) employing a DEM assembly comprised of arbitrarily shaped particles to provide path-dependent effective constitutive responses for each material point under complex loading conditions. A semi-implicit integration scheme based on the fractional step algorithm is further implemented in the proposed multiscale approach to improve its overall efficiency. The proposed approach is validated by the one-dimensional consolidation test before being further used to simulate more challenging problems, including the cyclic shaking test, the column collapse, and the wave propagation in anisotropic saturated porous media. We demonstrate that the proposed MPM-DEM approach is powerful and versatile in capturing the complicated static and dynamic multiphysics interactions exhibited in saturated granular media that could be of practical importance in various engineering settings. We further establish connections between these macroscopic observations with their underlying microstructural mechanisms to offer multiscale insights into the complicated dynamic responses of saturated sand.

Numerical simulation of two-dimensional biomagnetic shear flow

The current work focuses on numerical analysis of two-dimensional simple shear flow of biomagnetic fluid flowing in the vicinity of non-uniform magnetic field. Accordingly, the numerical model is constructed with the help of semi-implicit fractional step algorithm utilizing a finite volume method to solve the corresponding continuity and Navier-Stokes equations. Numerical simulations are mainly focussed on changing Reynolds number (Re), magnetic number (Mn) and Stuart number(N). The results reveal that multiple recirculation zones emerge as a result of the combined influence of flow and magnetic field. Out of the three parameters considered, the formation of vortices is not affected by variation of Stuart number compared to Reynolds and magnetic numbers.

An immersed boundary projection method for solving the fluid-rigid body interaction problems

We develop an immersed boundary projection method for solving the Naiver-Stokes equations and Newton-Euler equations to simulate the fluid-rigid body interactions in two and three dimensions. A novel fractional step algorithm is introduced for which fast solvers can be applied by exploiting the algebraic structure of the underlying schemes. The Navier-Stokes equations are decoupled while the Newton-Euler equations are solved simultaneously with a constraint equation of the immersed boundary force density. In contrast to previous works, the present method preserves both the fluid incompressibility and the kinematic constraint of the rigid body dynamics at a discrete level simultaneously while maintaining numerical stability. We demonstrate the numerical results of the present method involving spherical and spheroidal rigid bodies with a moderate range of density ratios, which are congruent with the results in the literature.

Moving immersed boundary method for fluid–solid interaction

A strongly coupled algorithm is presented for the incompressible fluid–rigid body interaction using the moving immersed boundary method. The pressure and the boundary force are treated as Lagrange multipliers to enforce the incompressibility and no-slip wall constraints. To compute the two unknowns from the velocity field, we adopt the fractional step algorithm and successively apply the two constraints. A Poisson equation and a moving force equation are derived for the pressure and the boundary force, respectively. As both coefficient matrices are formulated to be symmetric and positive-definite, the resulting linear systems are solved efficiently with the conjugate gradient solver. The strongly coupled nonlinear fluid–solid system is achieved by a fixed-point iteration. To improve the computational efficiency, we only iterate the moving force equation with the rigid body motion equation, and the time-consuming pressure Poisson equation is solved once the sub-iteration is finished. The proposed method is validated with various benchmark tests, and the results compare well with the literature.

Dynamic heat transfer in a tall annulus: Effects of the porous matrix for low Darcy numbers

This work focuses on the transient natural convection in vertical annuli partially filled with a porous medium, characterized by a low value of Darcy number. The generalized porous medium model has been implemented and solved by using a validated and stable version of a fully explicit fractional step algorithm. Sensitivity analyses are performed here by changing both the geometrical features of the cavity and the porous domain, both the properties of the porous medium. The results are compared with those obtained for the same cavity in absence of the porous insert, in order to investigate how it influences the transient evolution of the Nusselt number.

Arbitrary-rate relaxation techniques for the numerical modeling of compressible two-phase flows with heat and mass transfer

We describe compressible two-phase flows by a single-velocity six-equation flow model, which is composed of the phasic mass and total energy equations, one volume fraction equation, and the mixture momentum equation. The model contains relaxation source terms accounting for volume, heat and mass transfer. The equations are numerically solved via a fractional step algorithm, where we alternate between the solution of the homogeneous hyperbolic portion of the system via a HLLC-type wave propagation scheme, and the solution of a sequence of three systems of ordinary differential equations for the relaxation source terms driving the flow toward mechanical, thermal and chemical equilibrium. In the literature often numerical relaxation procedures are based on simplifying assumptions, namely simple equations of state, such as the stiffened gas one, and instantaneous relaxation processes. These simplifications of the flow physics might be inadequate for the description of the thermodynamical processes involved in various flow problems. In the present work we introduce new numerical relaxation techniques with two significant properties: the capability to describe heat and mass transfer processes of arbitrary relaxation time, and the applicability to a general equation of state. We show the effectiveness of the proposed methods by presenting several numerical experiments.

A semi-implicit fractional step algorithm on staggered meshes for simulating a compressible two-layer mixed-flow model

We present here a numerical scheme based on the classical SIMPLE algorithm for the time-stepping and on a MAC discretization on staggered meshes for the estimation of the spatial derivatives. This scheme is applied here to a two-layer two-fluid model that has been proposed in previous works of the literature in order to perform computations of gas–liquid flows in pipes with gravity effects. This model accounts for pressure relaxation and velocity relaxation between the two phases through source terms. Thanks to new forms of these source terms, their discretization can be easily introduced in the different steps of the SIMPLE algorithm. The whole algorithm then allows to couple convective terms and source terms without suffering from the CFL limitation of the explicit Godunov-type schemes. A detailed description of the scheme is proposed and some numerical tests have been performed in order to assess the behavior of the approximated solutions when pressure–velocity relaxation is taken into account.

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.

Dynamic analysis of large deformation problems in saturated porous media by smoothed particle finite element method

This study presents a new formulation of smoothed particle finite element method (SPFEM) for dynamic problems in two-phase saturated porous media. A node integration method is used for both solid and water phases, which helps stabilize the coupled formulation when low-order triangle elements are used. Two different time integration schemes (i.e. explicit velocity Verlet and semi-implicit fractional step methods) are used to solve the coupled SPFEM formulations. In particular, a novel weakly-compressible fractional step algorithm is proposed, the key virtue of which is its capability to simulate high-frequency dynamic wave propagation in the porous media. The proposed coupled SPFEM is validated by comparing the numerical results with the analytical solution of two poroelastic examples, focusing on 1D wave propagation and 1D consolidation problems. Three more numerical tests are presented to further validate the proposed coupled SPFEM in simulating 2D wave propagation, self-weights slumping block and seepage flow induced progressive failure of embankment, and a very good agreement with the numerical results reported in the literature is obtained.

An efficient 3D non-hydrostatic model for predicting nonlinear wave interactions with fixed floating structures

This paper presents a three-dimensional (3D) non-hydrostatic model for the prediction of the interaction between nonlinear waves and fixed floating structures. The model solves the incompressible Euler equation by use of a semi-implicit, fractional step algorithm. The water surface elevation is treated as a single-valued function of horizontal position. In order to deal with floating structures, a new numerical algorithm is proposed which combines the immersed boundary method and the global continuity equation in the pressurized region (flow region under the structure). This new algorithm holds the symmetry of the Poisson equation and therefore results in an efficient model. The developed model is validated with the data of two test cases involving 3D nonlinear wave interactions with a floating structure. The model results are compared with experimental data or results of other models. The proposed model exhibits generally good agreement with experimental data and/or other model results, demonstrating its accuracy in resolving 3D nonlinear wave interaction with floating structures.

A Cell-Centered Semi-Lagrangian Finite Volume Method for Solving Two-Dimensional Coupled Burgers’ Equations

A cell-centered finite volume semi-Lagrangian method is presented for the numerical solution of two-dimensional coupled Burgers’ problems on unstructured triangular meshes. The method combines a modified method of characteristics for the time integration and a cell-centered finite volume for the space discretization. The new method belongs to fractional-step algorithms for which the convection and the viscous parts in the coupled Burgers’ problems are treated separately. The crucial step of interpolation in the convection step is performed using two local procedures accounting for the element where the departure point is located. The resulting semidiscretized system is then solved using a third-order explicit Runge-Kutta scheme. In contrast to the Eulerian-based methods, we apply the new method for each time step along the characteristic curves instead of the time direction. The performance of the current method is verified using different examples for coupled Burgers’ problems with known analytical solutions. We also apply the method for simulation of an example of coupled Burgers’ flows in a complex geometry. In these test problems, the new cell-centered finite volume semi-Lagrangian method demonstrates its ability to accurately resolve the two-dimensional coupled Burgers’ problems.

Propagation of Solitary Wave Around Conical Island in Level-Set Finite Element Framework

Lee, H., 2021. Propagation of solitary wave around conical island in level-set finite element framework. In: Lee, J.L.; Suh, K.-S.; Lee, B.; Shin, S., and Lee, J. (eds.), Crisis and Integrated Management for Coastal and Marine Safety. Journal of Coastal Research, Special Issue No. 114, pp. 16–20. Coconut Creek (Florida), ISSN 0749-0208. Though the solitary wave is a single wave, it consists of a complex spectrum of frequencies, which enables in-depth analysis and reliable generation both in the laboratory and in the numerical model. For these reasons, solitary waves are known to be a good candidate for the description of waves including tsunamis since they effectively model the important effects of the long wave on the coasts very well. In addition, it is supposed to propagate over constant depth without appreciable changes, allowing for consistent referencing of its offshore or incident wave height. In this study, the level-set scheme, which is combined with the incompressible Navier-Stokes solver based on the fractional step algorithm in the framework of the finite element method, was applied to the modeling of wave propagation and runup on a circular conical island. Unstructured hexahedral meshes were generated for this purpose and MPI (Message Passing Interface) based parallel algorithms were developed to accelerate the computation. The physical behavior of waves are discussed in detail and the numerical results (e.g., runup heights) are compared with the experimental data. The good agreements were observed.

An Experimental Investigation of Viscoelastic Flow in a Contraction Channel.

In order to assess the predictive capability of the S–MDCPP model, which may describe the viscoelastic behavior of the low-density polyethylene melts, a planar contraction flow benchmark problem is calculated in this investigation. A pressure-stabilized iterative fractional step algorithm based on the finite increment calculus (FIC) method is adopted to overcome oscillations of the pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique in combination with the streamline upwind Petrov-Galerkin (SUPG) method are employed to calculate the viscoelastic flow. The equal low-order finite elements interpolation approximations for velocity-pressure-stress variables can be applied to calculate the viscoelastic contraction flows for LDPE melts. The predicted velocities agree well with the experimental results of particle imagine velocity (PIV) method, and the pattern of principal stress difference calculated by the S-MDCPP model has good agreement with the results measured by the flow induced birefringence (FIB) device. Numerical and experimental results show that the S-MDCPP model is capable of accurately capturing the rheological behaviors of branched polymers in complex flow.

Non-hydrostatic model for internal wave generations and propagations using immersed boundary method

A non-hydrostatic model is developed to predict internal wave generations and propagations. The model employs a semi-implicit, fractional step algorithm to solve the governing equations based on the Cartesian grid system. Immersed boundary method is incorporated to deal with the complex geometry of uneven bottoms. The combination of the z-level vertical coordinate and the immersed boundary method enables the model to treat uneven bottoms and meanwhile to avoid numerical errors in the calculation of the baroclinic pressure force. To consider the effect of stratification, a variant of the Smagorinsky sub-grid scale model is employed to calculate the eddy viscosity and diffusivity. The capability of the model in resolving internal wave generation by tidal forcing over an idealized Gaussian-shaped topography is firstly validated by comparing the developed model with a terrain-following non-hydrostatic model. Then, two laboratory-scale test cases involving internal solitary wave propagations over variable topographies are considered. It can be found that the model captures the main characteristics of waveform evolution including internal hydraulic jump, wave breaking and waveform inversion. Thus, the proposed model provide an alternative to simulate internal wave generations and propagations.

Algorithms for Constrained Best-fit Alignment

Manufacturing complex structures as planes requires the assembly of several pieces. The first step in the process is to align the pieces. This article is concerned with some mathematical and computational aspects of new algorithms devoted to the alignment of the pieces. We describe the properties of suitable algorithms to handle a non standard constrained optimization problem that occurs in the assembly process of a manufactured product. Then we present two kinds of algorithms: the first based on a fractional step algorithm and the second on a local search algorithm. We assess them on real cases and compare their results with an evolutionary algorithm for difficult non-linear or non-convex optimization problems in continuous domain.

An adoption of the Spalart–Allmaras turbulence model for two- and three-dimensional free surface environmental flows

In this paper, the Spalart–Allmaras turbulence model, very popular for compressible aerospace applications, is adapted for incompressible free surface environmental flows. The Spalart–Allmaras model is a Reynolds-averaged Navier–Stokes model which solves for one transport equation of a viscosity-like variable. Besides its efficiency and reasonable computational cost, the model can take into account all sources of turbulence production and destruction. The Spalart–Allmaras model was implemented in two- and three-dimensional cases within a massively parallel software. For two-dimensional applications, the Spalart–Allmaras model was used together with the shallow water equations. Although dealing with turbulence with depth-averaged shallow water equations may be perplexing, horizontal turbulent structures could be adequately retrieved. For three-dimensional applications, the Spalart–Allmaras model was introduced with the incompressible Navier–Stokes equations. Numerical resolution is based on a fractional step algorithm that, unlike in a majority of published studies, considers a real free surface, a σ-transform moving mesh and a non-hydrostatic pressure distribution. Numerical resolution uses a mixed finite volume-finite element algorithm. The advection part could be handled, among other options, using a characteristics-based scheme. The Spalart–Allmaras model was validated against experimental laboratory observations and compared to other turbulence models. Results confirmed the potential of the adapted SA turbulence model, as both averaged and fluctuating fields (i.e. kinetic energy and velocity) showed quality comparable to, or even better, than those obtained by the well-known k-ϵ turbulence model.

An efficient selective cell-based smoothed finite element approach to fluid-structure interaction

This paper describes an efficient and simple selective cell-based smoothed finite element method (CS-FEM) for partitioned fluid–structure interaction. Depending on a fractional-step fluid solver, a selective smoothed integration scheme is proposed for the Navier–Stokes equations in stationary and deforming domains. A simple hourglass stabilization is then introduced into the under-integrated smoothed Galerkin weak form of the fractional-step algorithm. As a result, the computational efficiency is considerably boosted in comparison with existing CS-FEM formulation. Meanwhile, the CS-FEM is applied to spatially discretize the elastodynamics equations of nonlinear solids as usual. After discussing the mesh moving strategy, the gradient smoothing is performed in each individual interface element to evaluate the fluid forces acting on oscillating rigid and flexible bodies. The block Gauss–Seidel procedure is employed to couple all interacting fields under the arbitrary Lagrangian–Eulerian description. Several numerical examples are presented to demonstrate the desirable efficiency and accuracy of the proposed methodology.

An immersed boundary projection method for simulating the inextensible vesicle dynamics

We develop an immersed boundary projection method (IBPM) based on an unconditionally energy stable scheme to simulate the vesicle dynamics in a viscous fluid. Utilizing the block LU decomposition of the algebraic system, a novel fractional step algorithm is introduced by decoupling all solution variables, including the fluid velocity, fluid pressure, and the elastic tension. In contrast to previous works, the present method preserves both the fluid incompressibility and the interface inextensibility at a discrete level simultaneously. In conjunction with an implicit discretization of the bending force, the present method alleviates the time-step restriction, so the numerical stability is assured by non-increasing total discrete energy during the simulation. The numerical algorithm takes a linearithmic complexity by using preconditioned GMRES and FFT-based solvers. The grid convergence studies confirm the solution variables exhibit first-order convergence rate in L2-norm. We demonstrate the numerical results of the vesicle dynamics in a quiescent fluid, Poiseuille flow, and shear flow, which are congruent with the results in the literature.