Order Finite Volume Method Research Articles

Externally guided underfill encapsulation and subsequent filler particle migration during electronics packaging

We present a numerical study of the transient multiphase flow of a liquid encapsulant in a microgap formed between parallel plates with special reference to electronics packaging. The surface tension and polarization forces are assumed to drive the fluid flow. The governing equations are solved using the volume of fluid model within the framework of the finite volume method in order to predict the free surface front of the encapsulant together with the time and spatial evolution of filler particle concentration during the underfill process. The numerical results are found to be in good agreement with available reported experimental data. The influence of the polarization force is found to reduce the encapsulant fill time, increase the flow length and distribute the filler particles to a larger flow length. The new process has practical implications, such as overcoming the partial curing of underfill epoxy before the flow is complete.

Numerical and experimental study of transversal pressure waves at a tube exit

This paper presents investigations on the interaction of a pressure wave with a tube exit. The transversal waves, generated during the reflection of an incident pressure wave at a tube exit, are measured and the results are compared with numerical simulations using a second order finite volume method. The results show clearly a transversal propagation process at the exit plane, the reflected waves becoming plane only at several diameters from the tube end.

DSC solution for flow in a staggered double lid driven cavity

AbstractA benchmark quality solution is presented for flow in a staggered double lid driven cavity obtained by using the wavelet‐based discrete singular convolution (DSC). The proposed wavelet based algorithm combines local methods' flexibility and global methods' accuracy, and hence, is a promising approach for achieving the high accuracy solution of the Navier–Stokes equations. Block structured grids with pseudo‐overlapping subdomains are employed in the present simulation. A third order Runge–Kutta scheme is used for the temporal discretization. Quantitative results are presented, apart from the qualitative fluid flow patterns. The prevalence of rich features of flow morphology, such as two primary vortex patterns, merged single primary vortex patterns, and secondary eddies, makes this problem very attractive and interesting. The problem is quite challenging for the possible existence of numerically induced asymmetric flow patterns and elliptic instability. Important computational issues like consistence, convergence and reliability of the numerical scheme are examined. The DSC algorithm is tested on the single lid driven cavity flow and the Taylor problem with a closed form solution. The double lid driven cavity simulations are cross‐validated with the standard second order finite volume method. Copyright © 2003 John Wiley & Sons, Ltd.

Development and analysis of higher order finite volume methods over rectangles for elliptic equations

Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangular elements; we also find finite volume methods to associate with BDFM2 three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux.

The anti-dissipative, non-monotone behavior of Petrov-Galerkin upwinding

The Petrov–Galerkin method has been developed with the primary goal of damping spurious oscillations near discontinuities in advection dominated flows. For time-dependent problems, the typical Petrov–Galerkin method is based on the minimization of the dispersion error and the simultaneous selective addition of dissipation. This optimal design helps to dampen the oscillations prevalent near discontinuities in standard Bubnov–Galerkin solutions. However, it is demonstrated that when the Courant number is less than 1, the Petrov–Galerkin method actually amplifies undershoots at the base of discontinuities. This is shown in an heuristic manner, and is demonstrated with numerical experiments with the scalar advection and Richards' equations. A discussion of monotonicity preservation as a design criterion, as opposed to phase or amplitude error minimization, is also presented. The Petrov–Galerkin method is further linked to the high-resolution, total variation diminishing (TVD) finite volume method in order to obtain a monotonicity preserving Petrov–Galerkin method.

A Roe Scheme for Ideal MHD Equations on 2D Adaptively Refined Triangular Grids

In this paper we present a second order finite volume method for the resolution of the bidimensional ideal MHD equations on adaptively refined triangular meshes. Our numerical flux function is based on a multidimensional extension of the Roe scheme proposed by Cargo and Gallice for the 1D MHD system. If the mesh is only composed of triangles, our scheme is proved to be weakly consistent with the condition ∇…B=0. This property fails on a cartesian grid. The efficiency of our refinement procedure is shown on 2D MHD shock capturing simulations. Numerical results are compared in case of the interaction of a supersonic plasma with a cylinder on the adapted grid and several non-refined grids. We also present a mass loading simulation which corresponds to a 2D version of the interaction between the solar wind and a comet.

A new theoretically motivated higher order upwind scheme on unstructured grids of simplices

Many approaches exist to define a cell-centered upwind finite volume scheme of higher order on an unstructured grid of simplices. However, real theoretical motivation in the form of a convergence result does not exist for these approaches. Furthermore, some theoretical results of convergence exist for higher order finite volume methods, where no description of the numerical implementation is given to realize the necessary requirements for the convergence theory. Therefore we present in this paper a new limiter function which is motivated by these requirements and ensures a convergent scheme in the theoretical context: The approximated solution converges to the entropy solution in the case of scalar conservation laws in two space dimensions. This new limiter function is combined with a typical class of reconstruction functions very efficiently, which is illustrated by several test examples for scalar conservation laws as well as systems of such laws. In connection with the requirements to be fulfilled, a proof of a maximum principle of the finite volume scheme applied to simplices and dual cells is given. So for the approach of the higher order upwind finite volume scheme on dual cells, as used in several papers, a missing proof is now given. The ideas in this proof are also applied to the discontinuous Galerkin method, so that an existing maximum principle can be improved considerably. The main advantage comes from the fact that no requirements on the discretization of the domain are necessary: no B-triangulations or Delaunay triangulation are needed.

A HYBRID SPECTRAL/FINITE VOLUME METHOD FOR VISCOUS FLOWS

Abstract This paper presents a hybrid spectral/finite volume method for steady-state compressible viscous flows. The method is evaluated for accuracy via test cases for various Mach numbers. The domain is divided into a viscous region and an inviscid region. The viscous region uses the full Navier-Stokes equations, while the inviscid region employs the Euler equations. A high order Chebyshev collocation spectral method is developed for the viscous region to resolve boundary layers. This method avoids the dense grids needed by finite-volume methods to resolve the viscous areas. A low order finite-volume method based on a Lax-Wendroff type scheme is employed for the inviscid region. A special interface formulation is developed for coupling the spectral with the finite-volume method. Comparisons with analytic results as well as convergence histories are presented.

A global integro-differential approach for coupled methods in coupled problems: towards a general package

A package is described which uses several methods to model complex phenomena involving coupled problems. A generator program has been developed for the finite element method, the boundary element method, and the finite volume method in order to describe coupled formulations and to build the object package. Two kinds of coupling techniques have been used, weak and direct ones. >