Piecewise Linear Interface Calculation Research Articles

주조유동 시뮬레이션에서 자유경계면 추적 기법 비교 연구

We applied two numerical schemes to improve accuracy of the solution in the flow simulation of molten metal. One method is Piecewise Linear Interface Calculation (PLIC) method and the other is Donor-Acceptor (D-A) method. In the present work, we have tested simple problems to verify the module of the interface reconstruction algorithms. After validations, accuracy and efficiency of these two methods have compared by simulating various real products. On the numerical simulation of free surface flow, it is possible for PLIC method to track very accurately the interface between phases. PLIC method, however, has the weak point where a lot of computational time hangs, though it shows the more accurate interface reconstruction. Donor-Acceptor method has enough effectiveness in the macro observation of mold filling sequence though it shows the inferior accuracy.

Revisit to the THINC scheme: A simple algebraic VOF algorithm

This short note presents an improved multi-dimensional algebraic VOF method to capture moving interfaces. The interface jump in the THINC (tangent of hyperbola for INterface capturing) scheme is adaptively scaled to a proper thickness according to the interface orientation. The numerical accuracy in computing multi-dimensional moving interfaces is significantly improved. Without any geometrical reconstruction, the proposed method is extremely simple and easy to use, and its numerical accuracy is superior to other existing methods of its kind and comparable to the conventional PLIC (piecewise linear interface calculation) type VOF schemes.

Numerical study for laminar wavy motions of liquid film flow on vertical wall

Film flows are classified into non-wavy laminar, wavy laminar and turbulence along the Reynolds number or the flow stability. Since the wavy motions of the film flows are so intricate and nonlinear, the studies have largely been dependent upon the experimental way. The numerical approaches have been limited on the non-wavy flow regime. To track the free surface position, various numerical techniques such as the VOF (Volume of Fluids), the MAC (Marker and Cell) and the moving grid have been adopted. However those were for a more accurate estimation of the average film thickness and not for capturing the wavy motion. Because the wavy motion highly affects the heat transfer in the film flow, the profound concern for the wavy motion is significant. In this study, the wavy motions of the laminar wavy film flow with the Reynolds number 200–1000 are successfully found by the VOF and PLIC (Piecewise Linear Interface Calculation) method. The numerical results, including the average film thickness, and the wave’s amplitude, frequency and velocity, are compared with the experimental results.

An Improved Volume of Fluid Method for Two-Phase Flow Computations on Collocated Grid System

An improved volume of fluid method called the accurate density and viscosity volume of fluid (ADV-VOF) method is proposed to solve two-phase flow problems. The method has the following features: (1) All operations are performed on a collocated grid system. (2) The piecewise linear interface calculation is used to capture interfaces and perform accurate estimations of cell-edged density and viscosity. (3) The conservative Navier–Stokes equations are solved with the convective term discretized by a second and third order interpolation for convection scheme. (4) A fractional-step method is applied to solve the conservative Navier–Stokes equations, and the BiCGSTAB algorithm is used to solve the algebraic equations by discretizing the pressure-correction equation. The above features guarantee a simple, stable, efficient, and accurate simulation of two-phase flow problems. The effectiveness of the ADV-VOF method is verified by comparing it with the conventional volume of fluid method with rough treatment of cell-edged density and viscosity. It is found that the ADV-VOF method could successfully model the two-phase problems with large density ratio and viscosity ratio between two phases and is better than the conventional volume of fluid method in this respect.

Three-dimensional simulation of underfill process in flip-chip encapsulation

Underfill is an important process in flip-chip encapsulation because of its great impact on the reliability of the electronic packagings. This paper focuses on the study of the fluid flow in capillary-driven underfill encapsulation, and a three-dimensional simulation approach of the underfill process is proposed. Firstly, the driven mechanism of the capillary action is analyzed by first presenting an inter-motivation model, according to which the capillary action is modeled as two different forces, the adhesive tension force close to the wall and the surface tension force on the free surface. Those two forces motivate each other during the fluid flow process. Then, based on this, the fluid flow simulation of the underfill process is established. During the solution, two common Petrov–Galerkin (PG) methods are employed to solve the governing equations. Finally, the piecewise linear interface calculation (PLIC)-flow analysis network (FAN) method is proposed to track and reconstruct the melt-front for each time step. Compared with the experimental data and the predictions of the other numerical methods, the proposed approach shows a good performance in predicting the fluid flow in the capillary-driven underfill process.

Computational fluid dynamics modelling on flow characteristics of two-phase flow in micro-channels

Simulated gas-liquid two-phase horizontal flow characteristics were compared with corresponding experimental data taken from the Triplett. For the simulation of the two-phase flow characteristics, the well-known volume of fluid approach co-operated with the piecewise linear interface calculation interface reconstruction method was carried out. First, a variety of two-phase flow patterns were reproduced. The numerical results give good agreement with the flow regimes expected from the Triplett. Next to that, the two-phase pressure drops in micro-channels were analysed numerically by using the same computational fluid dynamics method. Also, the comparison of pressure drop among the numerical simulations, experimental data and the results calculated by homogeneous equilibrium model was presented. The agreement between numerical results and the existing experimental data was found to be satisfactory. So, it can be deduced that the numerical analysis procedure proposed here can achieve a better prediction for two-phase flow characteristics in micro-channels.

Stabilized filling simulation of microchip encapsulation process

The filling process is a key stage of plastic encapsulation, which mainly affects the reliability of chips. This paper presents a stabilized filling simulation method of microchip encapsulation process. In this method, the non-isothermal and incompressible N–S equations coupled with the Kamal curing equation is introduced in as governing equations, and the Castro–Macosko model is employed as the constitutive model of the epoxy molding compound (EMC). The Galerkin/least-squares (GLS)-stabilized finite element method is used to solve the governing equations. With GLS scheme, numerical surge during the solution is relieved and equal order interpolation for velocity field and pressure field is achieved at the same time. A new kind of Euler’s method named piecewise linear interface calculation (PLIC)–flow analysis network (FAN) is proposed to track and describe the flow fronts for each time step. The experimental and numerical results are found to be in good agreement, which proves the accuracy and stabilization of this method.

A volume of fluid approach for crystal growth simulation

A new approach to simulating the dendritic growth of pure metals, based on a recent volume of fluid (VOF) method with PLIC (piecewise linear interface calculation) reconstruction of the interface, is presented. The energy equation is solved using a diffuse-interface method, which avoids the need to apply the thermal boundary conditions directly at the solid front. The thermal gradients at both sides of the interface, which are needed to obtain the front velocity, are calculated with the aid of a distance function to the reconstructed interface. The advection equation of a discretized solid fraction function is solved using the unsplit VOF advection method proposed by López et al. [J. Comput. Phys. 195 (2004) 718–742] (extended to three dimensions by Hernández et al. [Int. J. Numer. Methods Fluids 58 (2008) 897–921]), and the interface curvature is computed using an improved height function technique, which provides second-order accuracy. The proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions.

Large deformation analysis of adhesive by Eulerian method with new material model

The material model to describe large deformation of a pressure sensitive adhesive (PSA) is presented. A relationship between stress and strain of PSA includes viscoelasticity and rubber-elasticity. Therefore, we propose the material model for describing viscoelasticity and rubber-elasticity, and extend the presented material model to the rate form for three dimensional finite element analysis. After proposing the material model for PSA, we formulate the Eulerian method to simulate large deformation behavior. In the Eulerian calculation, the Piecewise Linear Interface Calculation (PLIC) method for capturing material surface is employed. By using PLIC method, we can impose dynamic and kinematic boundary conditions on captured material surface. The representative two computational examples are calculated to check validity of the present methods.

P.4.02 Effects of varenicline on dopamine D2/3 receptor binding in rats using ex vivo phosphor storage imaging

Calculated gas/vapor–liquid two-phase co-current horizontal flow regimes are compared with experimental data, taken from the Baker chart. For the calculation of the two-phase flow regimes, use is made of the well-known volume of fluid (VOF) multiphase flow model that uses a piecewise linear interface calculation (PLIC) interface reconstruction method in each computational cell, implemented in a computational fluid dynamics (CFD) code.The flow regimes for water–air flow are reproduced. All simulations give good agreement with the flow regimes expected from the Baker chart. Horizontal flow regimes for two-phase liquid–vapor gasoil flow are also reproduced. To the best of our knowledge, this is the first time that the flow regimes for gasoil liquid–vapor flow as predicted by the Baker chart are simulated. Again, a good agreement between the simulated and expected flow regime is obtained. All horizontal flow regimes appearing in the Baker chart can thus be calculated using CFD.

J0102-2-2 生体力学シミュレーションのためのボクセル固体連成解析手法(次世代生命体統合シミュレーション(2))

A solid-solid-fluid interaction scheme is developed for biomechanics simulation. The scheme is based on full Eulerian formulation, which is familiar with medical voxel data. Both solid arid fluid are modeled as an incompressible material and one governing equation treats equations by using the volume fractions of the materials. Both FEM and FDM are used for space discretization of fixed structural mesh. PLIC (Piecewise Linear Interface Calculation) method is used to capture the material interfaces with high accuracy and to impose boundary conditions on the material interfaces.

Modeling the evaporation of a hydrocarbon feedstock in the convection section of a steam cracker

A model has been developed allowing to simulate the flow boiling process of a hydrocarbon feedstock. For the calculation of the different horizontal two-phase flow regimes that evolve during flow boiling, use is made of the Volume Of Fluid (VOF) model that uses a Piecewise Linear Interface Calculation (PLIC) method to reconstruct the interface between both phases in each computational cell. In-house developed codes calculate the mass and energy transfer phenomena occurring during this flow boiling process. As such, an existing CFD code is completed with a newly developed complete evaporation model. The developed model is used to simulate the flow boiling process of a hydrocarbon feedstock in the tubes of a convection section heat exchanger of a steam cracker. The simulation results show a succession of horizontal two-phase flow regimes in agreement with the literature.

A new volume of fluid method in three dimensions—Part II: Piecewise‐planar interface reconstruction with cubic‐Bézier fit

AbstractA new interface reconstruction method in 3D is presented. The method involves a conservative level‐contour reconstruction coupled to a cubic‐Bézier interpolation. The use of the proposed piecewise linear interface calculation (PLIC) reconstruction scheme coupled to a multidimensional time integration provides solutions of second‐order spatial and temporal accuracy. The accuracy and efficiency of the proposed reconstruction algorithm are demonstrated through several tests, whose results are compared with those obtained with other recently proposed methods. An overall improvement in accuracy with respect to other recent methods has been achieved, along with a substantial reduction in the central processing unit time required. Copyright © 2008 John Wiley & Sons, Ltd.

Application of the VOF method based on unstructured quadrilateral mesh

To simulate two-dimensional free-surface flows with complex boundaries directly and accurately, a novel VOF (Volume-of-fluid) method based on unstructured quadrilateral mesh is presented. Without introducing any complicated boundary treatment or artificial diffusion, this method treated curved boundaries directly by utilizing the inherent merit of unstructured mesh in fitting curves. The PLIC (Piecewise Linear Interface Calculation) method was adopted to obtain a second-order accurate linearized reconstruction approximation and the MLER (Modified Lagrangian-Eulerian Re-map) method was introduced to advect fluid volumes on unstructured mesh. Moreover, an analytical relation for the interface’s line constant vs. the volume clipped by the interface was developed so as to improve the method’s efficiency. To validate this method, a comprehensive series of large straining advection tests were performed. Numerical results provide convincing evidences for the method’s high volume conservative accuracy and second-order shape error convergence rate. Also, a dramatic improvement on computational accuracy over its unstructured triangular mesh counterpart is checked.

Numerical simulation of fluid–structure interaction using a combined volume of fluid and immersed boundary method

In this work, a combined immersed boundary (IB) and volume of fluid (VOF) methodology is developed to simulate the interactions of free-surface waves and submerged solid bodies. The IB method is used to account for the no-slip boundary condition at solid interfaces and the VOF method, utilizing a piecewise linear interface calculation, is employed to track free surfaces. The combined model is applied in several case studies, including the propagation of small-amplitude progressive waves over a submerged trapezoidal dike, a solitary wave traveling over a submerged rectangular object, and wave generation induced by a moving bed. Numerical results depicting the free-surface evolutions and velocity fields are in good agreement with either experimental data or numerical results obtained by other researchers. In addition, the simplification of the initial free-surface deformation used in most tsunami earthquake source study is justified by the present model application. The methodology presented in the paper serves as a good tool for solving many practical problems involving free surfaces and complex boundaries.

CFD modeling of all gas–liquid and vapor–liquid flow regimes predicted by the Baker chart

Abstract Calculated gas/vapor–liquid two-phase co-current horizontal flow regimes are compared with experimental data, taken from the Baker chart. For the calculation of the two-phase flow regimes, use is made of the well-known volume of fluid (VOF) multiphase flow model that uses a piecewise linear interface calculation (PLIC) interface reconstruction method in each computational cell, implemented in a computational fluid dynamics (CFD) code. The flow regimes for water–air flow are reproduced. All simulations give good agreement with the flow regimes expected from the Baker chart. Horizontal flow regimes for two-phase liquid–vapor gasoil flow are also reproduced. To the best of our knowledge, this is the first time that the flow regimes for gasoil liquid–vapor flow as predicted by the Baker chart are simulated. Again, a good agreement between the simulated and expected flow regime is obtained. All horizontal flow regimes appearing in the Baker chart can thus be calculated using CFD.

An improved three-dimensional model for interface pressure calculations in free-surface flows

A three-dimensional method for the calculation of interface pressure in the computational modeling of free surfaces and interfaces is developed. The methodology is based on the calculation of the pressure force at the interfacial cell faces and is mainly designed for volume of fluid (VOF) interface capturing approach. The pressure forces at the interfacial cell faces are calculated according to the pressure imposed by each fluid on the portion of the cell face that is occupied by that fluid. Special formulations for the pressure in the interfacial cells are derived for different orientations of an interface. The present method, referred to as pressure calculation based on the interface location (PCIL), is applied to both static and dynamic cases. First, a three-dimensional motionless drop of liquid in an initially stagnant fluid with no gravity force is simulated as the static case and then two different small air bubbles in water are simulated as dynamic cases. A two-fluid, piecewise linear interface calculation VOF method is used for numerical simulation of the interfacial flow. For the static case, both the continuum surface force (CSF) and the continuum surface stress (CSS) methods are used for surface tension calculations. A wide range of Ohnesorge numbers and density and viscosity ratios of the two fluids are tested. It is shown that the presence of spurious currents (artificial velocities present in case of considerable capillary forces) is mainly due to the inaccurate calculation of pressure forces in the interfacial computational cells. The PCIL model reduces the spurious currents up to more than two orders of magnitude for the cases tested. Also for the dynamic bubble rise case, it is shown that using the numerical solver employed here, without PCIL, the magnitude of spurious currents is so high that it is not possible to simulate this type of surface tension dominated flows, while using PCIL, we are able to simulate bubble rise and obtain results in close agreement with the experimental data.

The Simulation of Shape Evolution of Solder Joints during Reflow Process and Its Experimental Validation

A three-dimensional computer aided analysis system has been developed in this study based on fluid dynamics to simulate the shape evolution of solder joints during the reflow process. The solution of velocity and pressure fields is based on SOLA (SOLution Algorithm) scheme, and the method to construct the interface and the transportation of volume fractions of liquid in the cells are coupled with PLIC (Piecewise Linear Interface Calculation) and VOF (Volume of Fluid) technologies. In order to consider the effect of surface tension on a fluid surface, the CSF (Continuum Surface Force) model is employed. The simulated results are compared with experimental measurements and good consistency is observed. Furthermore, the simulated results can reveal the evolution of the molten solder from its initial state to the equilibrium state. It is also capable of analyzing the overflow conditions when the amount of solder deposit is too much to be constrained in the solder pad, which is not achievable by the energy-based technique such as Surface Evolver.

Simulation des écoulements diphasiques dans les stations de pompage

The purpose of the present study is to develop numerical tools for two-phase flow ploblems in pumping stations. Reliable simulation methods of two-phase flow are inevitable in nuclear engineering field because computer programs are important design tools for nuclear reactors which are exposed to severe ambient conditions of high temperature, high pressure and gamma ray irradiation. Mock-up tests of nuclear thermal-hydraulics are very difficult. Hitachi Ltd. has created advanced computer programs for multi-dimensional two-phase flow to analize reactor systems and components. The numerical methods enable accurate and stable interface calculation in spite of large density ratio between the phases, because non-staggered finite volume method, which is suitable for treatment of discontinuity, and minimum numerical diffusion are adopted. The multi-dimensional computer programs employ two types of two-phase flow model. The first one is the extended two-fluid model, which has capability of both interface tracking and ordinary two-fluid model calculations. This model is appropriate for the present purpose because it can treat free surface of channels and entrained bubbly mixture due to breaking waves. The second one is the improved VOF (Volume of Fluid) method in which PLIC (Piecewise Linear Interface Calculation) technique was employed. The interface state calculation is simplified by approximating cubic calculation cells to spherical ones. These computer program were applied to hydraulic studies in Hitachi Industries Co., Ltd., a vender of fluid machinery and pumping systems. It is necessary to evacuate air downstream from siphon outlets as bubble entrainment by water flow in order to operate the plant effectively after pumping station start-up. Experiments were carried out by using a reduced scale test-section. The test section was made of transparent acryl and the cross-section was rectangular. Free surface of water layer on the inclined wall and entrained bubbly mixture into the water pool due to water layer collision with free surface were observed by a video camera. The extended two-fluid model results of transient water level and pressures in the simulated siphon outlets were in good agreement with the experimental data. The predicted pressure histories at the crest and the pump site were in good agreement with the experimental data. The sedimentation in a inlet channel of a pump station was carried out. Three dimensional free surface flow was calculated by the improved VOF method and behavior of suspended sediment in the flow field is calculated. The surface wave velocity in the shallow channel was in good agreement with the theoretical one. Sediment particles had size distribution and drag force and gravity and buoyancy was taken into consideration. Critical tractive force of the sediment on floor was estimated from Schields’ correlation. The predictions of sedimentation area and suspended fraction agreed well with experimental data. These results showed that the present computer programs were able to treat key two-phase flow phenomena in pumping stations.

A PLIC volume tracking method for the simulation of two-fluid flows

Numerical methodologies for computer simulations of two-fluid flows are presented. These methodologies fall into the category of volume tracking methods with piecewise-linear interface calculation (PLIC). The scope of this work is limited to laminar flows of immiscible, non-reacting, incompressible Newtonian fluids, without phase change, in planar two-dimensional geometries. The following novel or enhanced procedures are proposed: a parallelogram scheme for multidimensional advection of the volume-fraction field; a circle-fit technique for the orientation of the interface segments and the calculation of curvature; a novel contact angle treatment; and a staggered formulation for volumetric body forces that can accurately balance pressure forces in the vicinity of the interface. In addition, surface-tension-derived and hydrostatic-derived pressure adjustments are introduced as a means of accurately calculating pressure forces in cells that contain the interface, so as to minimize the non-physical flows that afflict many available volume tracking methods. The proposed method is validated using four test problems that involve simulations of pure advection, a static drop, an oscillating bubble, and a static meniscus. Copyright © 2006 John Wiley & Sons, Ltd.