Implicit Splitting Research Articles

Numerical prediction of hole profile in laser drilling process and experimental validation

In the present paper, a simple numerical method is proposed to predict the hole contour in a workpiece due to the action of a multi-pulsed laser. A 3D transient heat conduction equation has been used to predict the temperature at various points in the workpiece subjected to multi-pulsed (Gaussian beam (TEM00) mode) laser beam. Subsequently, the hole contour is predicted from the melt isotherm contour in the workpiece. A finite difference implicit splitting scheme has been chosen to solve the above 3D transient heat conduction equation because of several advantages of this techniques over other techniques. The predicted hole contour has been verified with the experimentally obtained hole contour under similar condition of laser and process parameters. It is observed that the numerically predicted hole contour is matching fairly well with the experimental hole contour. The second objective of this paper is to exploit this numerical model to obtain an optimum combination of laser and process parameters for getting a good quality hole. Hence, the proposed numerical model can be used to set optimum process parameters for laser-drilling operation in industrial applications.

O dinamici miješanja slatke i morske vode u morskim sustavima

Interaction between freshwater and seawater can significantly influence marine life and create important productive and dynamic natural habitats, such as estuaries. We study such processes and for this purpose we develop a new mathematical model with the system of coupled nonlinear conservation equations that govern the interaction of the two incompressible fluids with different properties. The system is solved by using implicit operator splitting and mixed finite elements methods. Finally, the numerical scheme has been used to provide simulations of two real world scenarios.

A Depth-averaged Two Dimensional Shallow Water Model to Simulate Flow-rigid Vegetation Interactions

This paper presents a well-balanced depth-averaged 2D shallow water model for simulating flow-rigid vegetation interactions. The rigid vegetation is modeled as vertical cylinders. A formula of drag force induced by the rigid vegetation is included in the momentum equations as a sink term. Since the bed friction and drag force terms have similar expressions, they are incorporated to be discretized using a splitting implicit scheme which can avoid an exaggerated force when the water depth becomes weak. Limiting value of the incorporated resistance term is derived to ensure stability. The proposed scheme is solved in a finite volume Godunov-type framework based on rectangular mesh with the HLLC approximate Riemann solver to discretize the convection part of equations, while a finite difference method is used to discretize the remaining terms. The MUSCL method is employed to achieve second-order accuracy in space and the second-order accuracy in time is obtained by applying the Runge-Kutta method. The model is tested against measured laboratory experimental data of the interaction of solitary waves with emergent, rigid vegetation. The computed results show good agreement with the experimental data.

Three-dimensional simulations of dilute and concentrated suspensions using smoothed particle hydrodynamics

A three-dimensional model for a suspension of rigid spherical particles in a Newtonian fluid is presented. The solvent is modeled with smoothed particle hydrodynamics method, which takes into account exactly the long-range multi-body hydrodynamic interactions between suspended spheres. Short-range lubrication forces which are necessary to simulate concentrated suspensions, are introduced pair-wisely based on the analytical solution of Stokes equations for approaching/departing objects. Given that lubrication is singular at vanishing solid particle separations, an implicit splitting integration scheme is used to obtain accurate results and at the same time to avoid prohibitively small simulation time steps. Hydrodynamic interactions between solid particles, at both long-range and short-range limits, are verified against theory in the case of two approaching spheres in a quiescent medium and under bulk shear flow, where good agreements are obtained. Finally, numerical results for the suspension viscosity of a many-particle system are shown and compared with analytical solutions available in the dilute and semi-dilute case as well as with previous numerical results obtained in the concentrated limit.

Explicit schemes for parabolic and hyperbolic equations

Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on a time step. More promising explicit schemes are associated with explicit–implicit splitting of the problem operator (Saul’yev asymmetric schemes, explicit alternating direction (ADE) schemes, group explicit method). These schemes belong to the class of unconditionally stable schemes, but they demonstrate bad approximation properties. These explicit schemes are treated as schemes of the alternating triangle method and can be considered as factorized schemes where the problem operator is splitted into the sum of two operators that are adjoint to each other. Here we propose a multilevel modification of the alternating triangle method, which demonstrates better properties in terms of accuracy. We also consider explicit schemes of the alternating triangle method for the numerical solution of boundary value problems for hyperbolic equations of second order. The study is based on the general theory of stability (well-posedness) for operator-difference schemes.

Reprint of Domain decomposition multigrid methods for nonlinear reaction–diffusion problems

In this work, we propose efficient discretizations for nonlinear evolutionary reaction–diffusion problems on general two-dimensional domains. The spatial domain is discretized through an unstructured coarse triangulation, which is subsequently refined via regular triangular grids. Following the method of lines approach, we first consider a finite element spatial discretization, and then use a linearly implicit splitting time integrator related to a suitable decomposition of the triangulation nodes. Such a procedure provides a linear system per internal stage. The equations corresponding to those nodes lying strictly inside the elements of the coarse triangulation can be decoupled and solved in parallel using geometric multigrid techniques. The method is unconditionally stable and computationally efficient, since it avoids the need for Schwarz-type iteration procedures. In addition, it is formulated for triangular elements, thus yielding much flexibility in the discretization of complex geometries. To illustrate its practical utility, the algorithm is shown to reproduce the pattern-forming dynamics of the Schnakenberg model.

Simplified, Analytically Solvable Model Problem for Non-Analog Monte Carlo Methods

In Monte Carlo particle transport, it is important to change the variance of calculations of relatively rare events with a technique known as non-analog Monte Carlo. In order to reduce the variance and the computation time, biasing techniques are introduced to accelerate the calculation convergence without changing the outcome. However these variance reduction techniques are often complex and can only be solved by computer codes. In this study, a simplified, analytically solvable model problem is introduced for non-analog Monte Carlo methods. A sample problem for neutrons passing through a thick shield is simulated. The drawback of this simulation is the expensive computation time and large variance for analog Monte Carlo methods. So, biasing techniques like implicit capture and splitting are introduced and the problem is solved analytically.

Domain decomposition multigrid methods for nonlinear reaction–diffusion problems

In this work, we propose efficient discretizations for nonlinear evolutionary reaction–diffusion problems on general two-dimensional domains. The spatial domain is discretized through an unstructured coarse triangulation, which is subsequently refined via regular triangular grids. Following the method of lines approach, we first consider a finite element spatial discretization, and then use a linearly implicit splitting time integrator related to a suitable decomposition of the triangulation nodes. Such a procedure provides a linear system per internal stage. The equations corresponding to those nodes lying strictly inside the elements of the coarse triangulation can be decoupled and solved in parallel using geometric multigrid techniques. The method is unconditionally stable and computationally efficient, since it avoids the need for Schwarz-type iteration procedures. In addition, it is formulated for triangular elements, thus yielding much flexibility in the discretization of complex geometries. To illustrate its practical utility, the algorithm is shown to reproduce the pattern-forming dynamics of the Schnakenberg model.

Influence of fluid properties on bubble formation, detachment, rising and collapse; Investigation using volume of fluid method

Numerical simulations have been carried out to investigate the formation and motion of single bubble in liquids using volume-of-fluid (VOF) method using the software platform of FLUENT 6.3. Transient conservation mass and momentum equations with considering the effects of surface tension and gravitational force were solved by the pressure implicit splitting operator (PISO) algorithm to simulate the behavior of gas-liquid interface movements in the VOF method. The simulation results of bubble formation and characteristics were in reasonable agreement with experimental observations and available literature results. Effects of fluid physical properties, operation conditions such as orifice diameter on bubble behavior, detachment time, bubble formation frequency and bubble diameter were numerically studied. The simulations showed that bubble size and bubble detachment times are linear functions of surface tension and decrease exponentially with the increase in liquid density. In contrast, only a small influence of the fluid viscosity on bubble size and detachment time was observed. Bubble collapse at a free surface simulation with VOF method was also investigated.

Flow Behavior of Viscoelastic Polymer Solution in Porous Media

Polymer solution flowing in porous media is difficult to characterize and simulate due to the effect caused by the rheological behavior and physicochemical reaction. In this study, a mathematical model of viscoelastic polymer flooding for enhancing oil recovery is presented. The model combines a viscoelastic constitutive model, together with a modified permeability model and a new relative permeability model. The effect of the rheology of polymer solution in porous media, the parameters of pore throat, the residual resistance factor, and the residual oil saturation reduction related to polymer's viscoelasticity have all been taken into account in this new model. We use the implicit operator splitting method to handle the concentration equations, the conjugate gradient method to solve the implicit pressure equations, and the third-order total variation diminishing method to handle the upwind scheme for explicit coefficients. The model is verified with the experimental data. In addition, the numerical model is used to analyze the displacement characteristics of polymer solution flow in porous media influenced by the rheological behavior and porous characters. Also, a typical case of polymer flooding simulated by this new model is demonstrated.

Tau leaping of stiff stochastic chemical systems via local central limit approximation

Stiffness manifests in stochastic dynamic systems in a more complex manner than in deterministic systems; it is not only important for a time-stepping method to remain stable but it is also important for the method to capture the asymptotic variances accurately. In the context of stochastic chemical systems, time stepping methods are known as tau leaping. Well known existing tau leaping methods have shortcomings in this regard. The implicit tau method is far more stable than the trapezoidal tau method but underestimates the asymptotic variance. On the other hand, the trapezoidal tau method which estimates the asymptotic variance exactly for linear systems suffers from the fact that the transients of the method do not decay fast enough in the context of very stiff systems. We propose a tau leaping method that possesses the same stability properties as the implicit method while it also captures the asymptotic variance with reasonable accuracy at least for the test system S1↔S2. The proposed method uses a central limit approximation (CLA) locally over the tau leaping interval and is referred to as the LCLA-τ. The CLA predicts the mean and covariance as solutions of certain differential equations (ODEs) and for efficiency we solve these using a single time step of a suitable low order method. We perform a mean/covariance stability analysis of various possible low order schemes to determine the best scheme. Numerical experiments presented show that LCLA-τ performs favorably for stiff systems and that the LCLA-τ is also able to capture bimodal distributions unlike the CLA itself. The proposed LCLA-τ method uses a split implicit step to compute the mean update. We also prove that any tau leaping method employing a split implicit step converges in the fluid limit to the implicit Euler method as applied to the fluid limit differential equation.

A 2D well-balanced shallow flow model for unstructured grids with novel slope source term treatment

Within the framework of the Godunov-type cell-centered finite volume (CCFV) scheme, this paper proposes a 2D well-balanced shallow water model for unstructured grids. In this model, the face-based van Albada limiting scheme is employed in conjunction with a directional correction to reconstruct second order spatial values at the midpoint of the considered face. The Harten, Lax and van Leer approximate Riemann solver with the Contact wave restored (HLLC) is applied to compute the fluxes of mass and momentum, while the splitting implicit method is utilized to solve the friction source terms. The novel aspects of the model include the new limited directional correction with which the new local extrema caused by the unlimited correction are prevented efficiently, the simplified non-negative water depth reconstruction used to get rid of numerical instabilities and in turn to preserve mass conservation at wet–dry interfaces and the novel slope source term treatment which suits complex unstructured grids well by transforming the slope source of a cell into fluxes at its faces. This model is able to preserve the C-property and mass conservation, to achieve good convergence to steady state, to capture discontinuous flows and to handle complex flows involving wetting and drying over uneven beds on unstructured grids with poor connectivity in an accurate, efficient and robust way. These capabilities are verified against analytical solutions, numerical results of alternative models and experimental and field data.

Implicit finite-difference simulations of seismic wave propagation

We propose a new finite-difference modeling method, implicit both in space and in time, for the scalar wave equation. We use a three-level implicit splitting time integration method for the temporal derivative and implicit finite-difference operators of arbitrary order for the spatial derivatives. Both the implicit splitting time integration method and the implicit spatial finite-difference operators require solving systems of linear equations. We show that it is possible to merge these two sets of linear systems, one from implicit temporal discretizations and the other from implicit spatial discretizations, to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm. We give the complete derivations of the implicit splitting time integration method and the implicit spatial finite-difference operators, and present the resulting discretized formulas for the scalar wave equation. We conduct a thorough numerical analysis on grid dispersions of this new implicit modeling method. We show that implicit spatial finite-difference operators greatly improve the accuracy of the implicit splitting time integration simulation results with only a slight increase in computational time, compared with explicit spatial finite-difference operators. We further verify this conclusion by both 2D and 3D numerical examples.

Regular and irregular wave impacts on floating body

Fully nonlinear wave–body interactions for a stationary floating structure under regular and irregular waves for different water depths, wave heights and periods are studied in a 2-D numerical wave tank. The tank model is based on Reynolds-averaged Navier–Stokes equations and renormalization group k–ε model. The equations are discretized based on the finite volume method. The pressure implicit splitting of operators scheme is employed to treat the pressure–velocity coupling and a compressive interface capturing scheme is used to capture the free surface on mashes of arbitrary topology. The calculated results for regular wave simulation, irregular wave propagation and wave impacts on floating body are compared with the theoretical/experimental data and the numerical results agree well with analytical/experimental solutions. The mean and maximum wave impacts, including rotational moment, on body are obtained. The effects of water depth, wave height and period on forces and moment have been investigated and the calculated results for irregular waves are compared with those induced by regular waves.

Two-dimensional volume of fluid simulation studies on single bubble formation and dynamics in bubble columns

In this paper, the volume of fluid (VOF) model in conjunction with continuum surface force (CSF) model was used to numerically investigate the single bubble formation and dynamics in the bubble columns on the software platform of Fluent 6.3. A set of transient conservation equations of mass and momentum taking surface tension and gravitational force effects into consideration were solved by pressure implicit splitting operator (PISO) algorithm and a piecewise linear interface calculation (PLIC) was applied to characterize the behavior of gas–liquid interface movement in the VOF method. The simulation results of bubble formation and dynamics compare well with available literature results. The effects of physical properties including surface tension, liquid viscosity and density, gas or liquid operation conditions and orifice size on the single bubble generation, detachment, rising and coaxial bubble coalescence were systematically analyzed, and the effect of superficial liquid velocity on single bubble behavior was especially discussed. It is found that non-zero superficial liquid velocity enhances the bubble detachment, decreases the bubble size, and delays the coaxial bubble coalescence obviously. Increasing superficial liquid velocity largely raises the velocity of the leading bubble and enlarging orifice gas velocity mainly accelerates the second bubble of two coalescence bubbles.

Block alternating splitting implicit iteration methods for saddle‐point problems from time‐harmonic eddy current models

SUMMARYFor the saddle‐point problems arising from the finite element discretizations of the hybrid formulations of the time‐harmonic eddy current problems, we establish a class of block alternating splitting implicit iteration methods and demonstrate its unconditional convergence. Experimental results are given to show the feasibility and effectiveness of this class of iterative methods when they are employed as preconditioners for Krylov subspace methods such as GMRES and BiCGSTAB. Copyright © 2011 John Wiley & Sons, Ltd.

Operator splitting implicit integration factor methods for stiff reaction–diffusion–advection systems

For reaction–diffusion–advection equations, the stiffness from the reaction and diffusion terms often requires very restricted time step size, while the nonlinear advection term may lead to a sharp gradient in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. For reaction–diffusion systems with both stiff reaction and diffusion terms, implicit integration factor (IIF) method and its higher dimensional analog compact IIF (cIIF) serve as an efficient class of time-stepping methods, and their second order version is linearly unconditionally stable. For nonlinear hyperbolic equations, weighted essentially non-oscillatory (WENO) methods are a class of schemes with a uniformly high order of accuracy in smooth regions of the solution, which can also resolve the sharp gradient in an accurate and essentially non-oscillatory fashion. In this paper, we couple IIF/cIIF with WENO methods using the operator splitting approach to solve reaction–diffusion–advection equations. In particular, we apply the IIF/cIIF method to the stiff reaction and diffusion terms and the WENO method to the advection term in two different splitting sequences. Calculation of local truncation error and direct numerical simulations for both splitting approaches show the second order accuracy of the splitting method, and linear stability analysis and direct comparison with other approaches reveals excellent efficiency and stability properties. Applications of the splitting approach to two biological systems demonstrate that the overall method is accurate and efficient, and the splitting sequence consisting of two reaction–diffusion steps is more desirable than the one consisting of two advection steps, because CWC exhibits better accuracy and stability.

Locally Limited and Fully Conserved RKDG2 Shallow Water Solutions with Wetting and Drying

This work extends a well-balanced second-order Runge-Kutta discontinuous Galerkin (RKDG2) scheme to provide conservative simulations for shallow flows involving wetting and drying over irregular topographies with friction effects. For this purpose, a wetting and drying technique designed originally for a finite volume (FV) scheme is improved and implemented, which includes the discretization of friction source terms via a splitting implicit integration approach. Another focus of this work is to design a fully conserved RKDG2 scheme to provide conservative solutions for both mass and momentum through a local slope limiting process. Several steady and transient benchmark tests with/without friction effects are simulated to validate the new solver and demonstrate the effects of different slope limiting processes, i.e. globally and locally slope limiting processes.

Fully Coupled Transient Heat Transfer and Melt Filling Simulations in Rapid Heat Cycle Molding with Steam Heating

Rapid heat cycle molding (RHCM) is a novel injection molding technology, in which injection mold is rapidly heated to a high temperature, usually higher than the glass transition temperature of the polymer material, before melt-injection and rapidly cooled down to solidify the shaped polymer melt in mold cavity for ejection. Since the elevated mold temperature can eliminate the unwanted premature melt freezing during filling stage, the melt flow resistance is greatly reduced and the filling ability of the polymer melt is also significantly improved. As a result, plastic parts with excellent surface appearance can be obtained. In this study, a three-dimensional numerical model coupled with heat transfer analysis and melt-filling processes for RHCM with steam heating was established. The thermal response analysis for the heating stage of RHCM process was performed by solving heat conduction equations. The heat transfer analysis results right after the mold cavity surface is heated up to the required temperature are taken as initial temperature conditions of the mold cavity for the following melt filling simulation. The pressure implicit splitting of operations solution algorithm was used to solve the pressure-velocity coupled Navier-Stokes equation for melt filling process. The moving interface between melt and air was captured by using the volume of fluid method. The energy equations for melt filling process were solved in a coupled manner for the cavity and mold domain at the matrix level. The proposed fully-coupled numerical model was applied in simulation of the molding processes, including a two-dimensional rectangular cavity with different heating times and a three-dimensional large scale LCD panel with a stem-heated stationary mold. The results show that the fully coupled numerical method provides reliable temperature and flow field predictions with the thermal response analysis and melt flow estimation.

Wave-body interactions for a surface-piercing body in water of finite depth

Nonlinear wave-body interactions for a stationary surface-piercing body in water of finite depth with flat and sloping bottoms are simulated in a two-dimensional numerical wave tank, which is constructed mainly based on the spatially averaged Navier-Stokes equations with the k – ɛ model for simulating the turbulence. The equations are discretized based on the finite volume method and the scheme of the pressure implicit splitting of operators is employed to solve the Navier-Stokes equations. By using the force time histories, the mean and higher-harmonic force components are calculated. The computational results are shown to be in good agreement with experimental and numerical results of other researchers. Then, the horizontal force, the vertical force and the moment on the surface-piercing body under nonlinear regular waves with flat and sloping bottoms are obtained. The results indicate that the bottom topographies have a significant influence on the wave loads on the surface-piercing body.