Control Volume Scheme Research Articles

Analysis of non-Fickian diffusion problems in a composite medium

The one-dimensional non-Fickian diffusion problems in a two-layered composite medium for finite and semi-infinite geometry are analyzed by using a hybrid application of the Laplace transform technique and control-volume method in conjunction with the hyperbolic shape functions, where the effect of the potential field is taken into account. The Laplace transform method used to remove the time-dependent terms in the governing differential equation and boundary conditions, and then the transformed equations are discretized by the control volume scheme. To evidence the accuracy of the present numerical method, a comparison between the present numerical results and analytical solution is made for the constant potential gradient. Results show that the present numerical results are accurate for various values of the potential gradient, relaxation time ratio, and diffusion coefficient ratio. It can be found that these values play an important role in the present problem. An interesting finding is that when the mass wave encounters an interface of the dissimilar materials, a portion of the wave is reflected and the rest is transmitted. The speed of propagation can change owing to the penetration of the mass wave into the region of the different material. The wave nature is significant only for short times and quickly dissipates with time.

THREE-DIMENSIONAL SIMULATION OF FLIP CHIP ENCAPSULATION PROCESS

T Flip chip package is the most important technology in IC package for the necessary of scale, velocity and cost by the development of semiconductor technology and the innovation of computer product. It has the advantage of low cost, low interface and small volume in IC package. This paper indicates that the analysis for package of the solder ball and chip. A finite element simulation of moving boundaries in a three-dimensional inertia-free, incompressible flow is presented. A control volume scheme with a fixed finite element mesh is employed to predict fluid front advancement. The epoxy is used for the package material. The injection situation uses for one line injection, L line injection, U line injection and central point injection location. The injection process uses for different parameters (mold temperature, injection temperature, injection pressure, injection time). The results show that the mold temperature is the most factor for processing parameters. It indicates that the L line injection is the best injection situation on flip chip package.

Simulation of Liquid Composite Molding Based on Control-Volume Finite Element Method

The control-volume finite element method (CVFEM) is a widely used numerical tool for modeling the polymer molding process because it is more user friendly and more robust than the conventional finite element or finite difference methods. This work compares the accuracy and stability of five upwind schemes of CVFEM for simulating the non-isothermal flow in the reactive liquid composite molding (LCM) processes. The results show that the mass weighted skew upwind (MAW) and the one-point upwind schemes are the most robust for simulating non-isothermal reactive flow in complicated flow domains and mesh shape. The streamline upwind control-volume (SUCV) scheme provides better numerical accuracy for simulating isothermal reactive flow. However, it tends to give unrealistic solutions at nodes close to the inlet region when simulating non-isothermal reactive flow in complicated geometry with obtuse triangular mesh. The two-point upwind scheme is inferior to MAW, the one-point upwind, and the SUCV schemes. The exponential and linear shape functions are not suitable for simulating the reactive LCM processes.

High‐order filtering for control volume flow simulation

AbstractA general methodology is presented in order to obtain a hierarchy of high‐order filter functions, starting from the standard top‐hat filter, naturally linked to control volumes flow simulations. The goal is to have a new filtered variable better represented in its high resolved wavenumber components by using a suitable deconvolution. The proposed formulation is applied to the integral momentum equation, that is the evolution equation for the top‐hat filtered variable, by performing a spatial reconstruction based on the approximate inversion of the averaging operator. A theoretical analysis for the Burgers' model equation is presented, demonstrating that the local de‐averaging is an effective tool to obtain a higher‐order accuracy. It is also shown that the subgrid‐scale term, to be modeled in the deconvolved balance equation, has a smaller absolute importance in the resolved wavenumber range for increasing deconvolution order. A numerical analysis of the procedure is presented, based on high‐order upwind and central fluxes reconstruction, leading to congruent control volume schemes. Finally, the features of the present high‐order conservative formulation are tested in the numerical simulation of a sample turbulent flow: the flow behind a backward‐facing step. Copyright © 2001 John Wiley & Sons, Ltd.

NUMERICAL ANALYSIS OF NON-FICKIAN DIFFUSION PROBLEMS IN A POTENTIAL FIELD

The primary difficulty encountered in the numerical solution of non-Fickian diffusion problems is numerical oscillations in the vicinity of sharp discontinuities. The present study applies a hybrid numerical scheme of the Laplace transform technique and the controlvolume method in conjunction with the hyperbolic shape functions to investigate the one-dimensional non-Fickian diffusion problems in the presence of a potential field for finite or semi-infinite geometry. The Laplace transform method used to remove the time-dependent terms in the governing differential equation and boundary conditions, and then the transformed equations are discretized by the control-volume scheme. To show the accuracy of the present numerical method, a comparison of the mass concentration distribution between the present numerical results and the analytic solutions is made. Results show that the present numerical results agree well with the analytic solutions and do not exhibit numerical oscillations in the vicinity of the jump discontinuity for various potential values. The potential gradient dV/dx has a great effect on the mass concentration distribution. The strength of the jump discontinuity is reduced as the value of the dimensionless potential gradient is increased.

Three‐dimensional simulation of microchip encapsulation process

AbstractA finite element simulation of moving boundaries in a three‐dimensional inertiafree, incompressible flow is presented. A control volume scheme with a fixed finite element mesh is employed to predict fluid front advancement. Fluid front advancement and pressure variation in a flow domain similar to the mold cavity used for microchip encapsulation are predicted. The predicted fluid front advancement and pressure variation are in good agreement with the corresponding experimental results. As the difference in the thicknesses of mold cavities above and below the microchip is changed, the weld line location and pressure variation during mold filling are found to change significantly.

Accuracy in the context of a control‐volume model

A shallow‐water control‐volume model is used to demonstrate features affecting the accuracy of numerical ocean‐atmosphere calculations. This model provides a context for illustrating the importance of consistent use of fields defined at points and fields averaged over cell volumes, cell faces and other surfaces. The accurate transformation between various averaged fields is demonstrated. Application of semi‐Lagrangian and COSMIC advection within N‐cycle time stepping is explored and compared with a control volume scheme that is time step limited. A fourth‐order accurate implicit solution for the potential update is demonstrated using two applications of a second‐order full multigrid solver with a defect correction. The defect correction does not make a significant correction to the potential update for our control‐volume formulation. It follows, therefore, that a single application of the second‐order solver effectively gives fourth‐order accuracy. On the other hand, potential gradients do need to be calculated using fourth‐order differencing or accuracy will suffer. Furthermore, calculating potential gradients at second order within an otherwise fourth‐order accurate spatial differencing scheme can result in unphysical eddies developing.

Inviscid and viscous simulations of spoiler performance

AbstractInviscid and viscous computational models have been used to predict two-dimensional steady separated flow around an aerofoil with a spoiler. First, an attempt is made to construct an inviscid panel method in which piece-wise linear vortex panels are placed on the aerofoil and spoiler. The separated region is modelled with potential flow analysis by using free vortex sheets placed on the separation streamline from the spoiler tip and aerofoil trailing-edge. The shapes of the vortex sheets require an iteration to be established. The calculation is based on an assumption of the length of the free vortex sheets. Second, a viscous procedure based on a finite control volume scheme is used to solve the Navier-Stokes equations for turbulent flow by making used of the k-ε model. An algebraic pressure correction is incorporated in the calculation. The predictions from the two models are in reasonable agreement with experimental measurements.

The control volume finite element numerical solution technique applied to creep in softwoods

This paper presents the novel application of the control volume finite element method (CVFE), otherwise known as the vertex-centered control volume scheme, to creep problems. The discretisation procedure is valid for both structured and unstructured meshes thus enabling complex geometries to be modelled. Analytical solutions to engineering problems involving creep often do not exist, therefore, a comparison with the well established numerical solution technique, the finite element method, has been included to provide a benchmark solution.

Solution of the Rarefied Gas Lubrication Equation Using an Additive Correction Based Multigrid Control Volume Method

The ever shortening product cycle for magnetic recording disk drives demands a fast and accurate numerical prediction of the slider’s flying characteristics during the design stage. A computationally efficient multigrid control volume method is developed in this paper for the solution of the very high bearing number and shaped rail air bearing problems. The control volume schemes for discretizing the Reynolds lubrication equation are based on various convection-diffusion formulations, while the solution of the resulting discretization equations is accelerated by an additive correction based multigrid method. A comparison study using the 50 percent tripad and Headway AAB sliders demonstrates a significant improvement in the solver’s performance as compared to the single-grid method.

A staggered control volume scheme for unstructured triangular grids

SUMMARY The purpose of this work is to introduce and validate a new staggered control volume method for the simulation of 2D=axisymmetric incompressible flows. The present study introduces a numerical procedure for solving the Navier‐Stokes equations using the primitive variable formulation. The proposed method is an extension of the staggered grid methodology to unstructured triangular meshes for a control volume approach which features ease of handling of irregularly shaped domains. Two alternative elements are studied: transported scalars are stored either at the sides of an element or at its vertices, while the pressure is always stored at the centre of an element. Two interpolation functions were investigated for the integration of the momentum equations: a skewed mass-weighted upwind function and a flow-oriented exponential shape function. The momentum equations are solved over the covolume of a side or of a vertex and the pressure‐velocity coupling makes use of a localized linear reconstruction of the discontinuous pressure field surrounding an element in order to obtain the pressure gradient terms. The pressure equation is obtained through a discretization of the continuity equation which uses the triangular element itself as the control volume. The method is applied to the simulation of the following test cases: backward-facing step flow, flow over a two-dimensional obstacle and flow in a pipe with sudden contraction of cross-sectional area. All numerical investigations are compared with experimental data from the literature. A grid convergence and error analysis study is also carried out for flow in a driven cavity. Results compared favourably with experimental data and so the new control volume scheme is deemed well suited for the prediction of incompressible flows in complex geometries. # 1997 John Wiley & Sons, Ltd.

A staggered control volume scheme for unstructured triangular grids

A staggered control volume scheme for unstructured triangular grids

MODELLING OF RELEASE OF GAS FROM HIGH-PRESSURE PIPELINES

SUMMARY The problem investigated is the break of a high-pressure pipeline carrying natural single-phase gas which may condensate (retrograde) when the pressure drops. Single-phase non-ideal gas is assumed using a generalized equation of state. Taking advantage of the choked massflow condition, the break is split into a pipe flow problem and a dispersion flow problem, both solved using a finite difference control volume scheme. The transient flow field from the pipeline break location is expanded analytically, using an approximation of the governing equations, until ambient pressure is reached and matched to the corresponding gas dispersion flow field using as subgrid model a jet box with a time-varying equivalent nozzle area as an internal boundary of the dispersion domain. The turbulence models used for the pipe and dispersion flow fields are an empirical model of Reichard and the k-* model for buoyant flow respectively. The pipe flow simulations indicate that the flow from the pipeline might include dispersed condensate which will affect quantitatively the mass flow rate from the pipeline and qualitatively the gas dispersion if the condensate rains out. The transient dispersion simulation shows that an entrainment flow field develops and mixes supersaturated gas with ambient warmer air to an unsaturated mixture. Because of the inertia of the ambient air, it takes time to develop the entrainment flow field. As a consequence of this and the decay of the mass Aow with time, the lower flammability limit of the gasair mixture reaches its most remote downstream position relatively early in the simulation (about 15 s) and withdraws closer to the break location.

SIMPLIFIED CONTROL-VOLUME FINITE-ELEMENT METHOD

Localized vector algebra treatment of nonorthogonality is applied to two-dimensional quadrilateral control volumes using Cartesian base vectors in a primitive variable formulation of the Navier-Stokes equations for steady incompressible laminar flow. With optional grid-aligned, locally analytic interpolation, a simplified control-volume finite-element scheme is presented. Discretization of source terms, determination of interface convection-diffusion fluxes, pressure correction factors, and geometric quantities are described briefly. Results of three test cases provide useful initial insights into the performance of the method. The conclusion is reached that a simple finite-volume-based approach to nonorthogonality has been achieved.

Effects of Pull Speed on Die Wall Temperatures for Flat Composites of Various Sizes

Properties of pultruded composites depend strongly on processing variables such as pull speed. Die wall temperatures may change depending on the size of the composite and the pull speed due to heat absorption or heat generation by the composite. It is the objective of this study to emphasize the importance of predicting and understanding the impact of die wall temperatures on centerline temperatures and degree of cure for composites of various thicknesses pultruded at different pull speeds. In order to accomplish this goal, a transient, three-dimensional numerical thermochemical heat transfer model for the heating section of the pultruder was developed. The governing energy and species equations for the composite were solved using a finite difference control volume scheme. The kinetic parameters for Shell EPON 9420 epoxy resin, determined using a heat flux type differential scanning calorimeter (DSC), were employed in this study. Previous researchers have tended to ignore the variations in die wall temperature distribution or have relied on experimentally obtained die wall temperature data to develop models and make predictions. This research overcomes those restrictions by completely predicting the temperature profiles in the entire die, thereby providing the pultrusion engineer with the tools to design a heating section of a pultrusion machine. An operational envelope has also been developed to establish guidelines for maximum pull speeds that can be used to obtain a specified degree of cure in the composite.

A control volume approach for integral viscoelastic models and its application to contraction flow of polymer melts

A numerical scheme based on the control volume discretization is developed for integral-type viscoelastic models. The strain history tracking can be done more efficiently using finite volumes with staggered grids than using finite elements. The Semi-Implicit Method for Pressure Linked Equations (SIMPLE) with alternating direction implicit (ADI) iterations is employed to solve the momentum equations on staggered grids, with the non-Newtonian stress contribution integrated upstream along stream-lines and treated as a source term in each control volume. The new method gave better computing stability and solution smoothness than its finite element counterpart. The new scheme is used to simulate the 5.75:1 abrupt circular contraction flow of a low-density polyethylene (LDPE) melt with KBKZ model, for which experimental data on recirculation vortex growth with elasticity are available in the literature. A comparison of the numerical results with experimental data on the opening angle of vortex shows good agreement and some improvement over previous finite element computations. A mesh refinement study with three meshes demonstrated good mesh convergence. The present control volume scheme is found to be more than twice as fast as its finite element counterpart in dealing with integral viscoelastic models, at least in simple geometries.

A Mathematical Model for Natural Convection Moisture Migration in Stored Grain

Thermal gradients that develop due to the seasonal variations in ambient temperatures cause natural convection air-flow and migration of moisture within bulk grain. In this article, a mathematical model of moisture migration is developed by reflecting the change in air moisture on the grain moisture through the sorption isotherm. Thus, the effect of temperature and moisture gradients are clearly separated in the conservation of moisture equation. The differential equations for energy are developed by using moisture dependent thermal properties. Natural convection Darcy flows are computed by solving an equation for the pressure, which represents the continuity equation. The resulting unsteady, nonlinear, coupled differential equations are numerically solved using a control-volume scheme. Natural convection moisture migration is also experimentally simulated under laboratory conditions. The validity of the numerical model is evaluated by comparing the experimental results with the predicted results.

Generation of oriented three‐dimensional Delaunay grids suitable for the control volume integration method

AbstractA fully automatic algorithm for three‐dimensional mesh generation is presented. The algorithm preserves the robustness and time efficiency of the finite octree technique; replacing octrees by more general 2–4–8‐ trees, it is capable of generating oriented meshes. In a post‐processing step, meshes are transformed in order to satisfy the Delaunay criterion, and so that non‐overlapping, closed control volumes can be defined for each mesh point through edge bisectors. The method is shown to be appropriate for modelling semiconductor devices, where the control volume scheme is the method of choice due to the peculiarities of the partial differential equations involved, and where locally refined and oriented meshes are essential to describe the relevant internal physical quantities adequately while keeping the number of mesh points practical.

Mathematical Model for Moisture Diffusion in Stored Grain Due to Temperature Gradients

Seasonal variations in ambient temperatures can cause safe moisture contents of stored grain to become unsafe due to migration or redistribution of moisture within the storage. Such migration of moisture generally occurs due to thermal gradients in the stored grain and depends on various factors related to the grain quality, bin size, and local climatic conditions. In this article, a mathematical model for the simultaneous heat and moisture transfer is developed by incorporating a sorption isotherm relationship to predict moisture migration in stored grain due to diffusion. The coupled, nonlinear transport equations are solved numerically by using a control-volume scheme. The predicted results show good agreement with the available experimental data in the literature. This model is also applied to predict moisture migration under conditions approximating winter and summer storage periods. The predictions of moisture and temperature distribution are consistent with field observations.

Numerical analysis for hyperbolic heat conduction

A new numerical simulation of the hyperbolic heat conduction problem is investigated. The primary difficulty encountered in the numerical solution of such a problem is numerical oscillations in the vicinity of sharp discontinuities. In this work, it is shown that the hybrid technique based on the Laplace transform and control volume methods can successfully be applied to suppress these oscillations. The Laplace transform method is used to remove the time-dependent terms, and then the transformed equations are discretized by the control volume scheme. Various comparative examples involving a nonlinear problem with surface radiation and the hyperbolic heat conduction in a composite region are illustrated to verify the accuracy of the present method. Due to the application of the Laplace transform method, the present technique does not need to consider the effects of the Courant number on the numerical results.