An enthalpy formulation based on an arbitrarily deforming mesh for solution of the Stefan problem

Traditionally schemes for dealing with the Stefan phase change problem are separated into fixed grif or front tracking (deforming grid) schemes. A standard fixed grid scheme is to use an enthalpy formulation and track the movement of the phase front via a liquid fraction variable. In this paper, an enthalpy formulation is applied on a continuously deforming finite element grid. This approach results in a general numerical scheme that incorporates both front tracking and fixed grid schemes. It is shown how on appropriate setting of the grid velocity a fixed or deforming grid solution can be generated from the general scheme. In addition an approximate front tracking scheme is developed which can produce accurate non-oscillatory predictions at a computational cost close to an efficient fixed grid scheme. The versatility of the general scheme and the approximate front tracking scheme are demonstrated on solution of a number of Stefan problems in both one and two dimensions.     

Simulation of the three-dimensional non-isothermal mold filling process in resin transfer molding

Numerical simulation of resin transfer molding (RTM) is known as a useful method to analyze the process before the mold is actually built. In thick parts, the resin flow is no longer two-dimensional and must be simulated in a fully three-dimensional space. This article presents numerical simulations of three-dimensional non-isothermal mold filling of the RTM process. The control volume/finite element method (CV/FEM) is used in this study. Numerical formulation for resin flow is based on the concept of nodal partial saturation at the flow front. This approach permits to include a transient term in the working equation, removing the need for calculation of time step to track the flow front in conventional scheme. In order to compare the results of the nodal partial saturation concept with the conventional method, a numerical scheme based on the quasi-steady state formulation is also presented. The computer codes developed based on both numerical formulations, allow the prediction of flow front positions; and pressure, temperature and conversion distributions in three-dimensional molds with complicated geometries. The validity of the two schemes is evaluated by comparison with analytical solutions of simple geometries. In all instances excellent agreement is observed. Numerical case studies are provided to demonstrate the effectiveness of the developed computer codes. The results show that the numerical procedure based on the nodal partial saturation concept, developed in this study, provides numerically valid and reasonably accurate predictions.     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part II: The coupled problem of phase-change in porous media

The conceptual framework of a least squares rate variational approach to the formulation of continuously deforming mixed-variable finite element computational scheme for a single evolution equation was presented in Part I.¹ In this paper (Part II), we extend these concepts and present an adaptively deforming mixed variable finite element method for solving general two-dimensional transport problems governed by a system of coupled non-linear partial differential evolution equations. In particular, we consider porous media problems that involve coupled heat and mass transport processes that yield steep continuous moving fronts, and abrupt, discontinuous, moving phase-change interfaces. In this method, the potentials, such as the temperature, pressure and species concentration, and the corresponding fluxes, are permitted to jump in value across the phase-change interfaces. The equations, and the jump conditions, governing the physical phenomena, which were specialized from a general multiphase, multiconstituent mixture theory, provided the basis for the development and implementation of a two-dimensional numerical simulator. This simulator can effectively resolve steep continuous fronts (i.e. shock capturing) without oscillations or numerical dispersion, and can accurately represent and track discontinuous fronts (i.e. shock fitting) through adaptive grid deformation and redistribution. The numerical implementation of this simulator and numerical examples that demonstrate the performance of the computational method are presented in Part III² of this paper.     

注射条件对LCM工艺非饱和流动特性影响

双尺度多孔纤维预制体填充过程中延迟浸润的非饱和流动现象,对基于树脂流过区域为完全饱和区域的充模理论及模拟方法提出了挑战。通过控制体/有限单元（CV/FE）法结合沉浸函数实现了液体模塑成型工艺（LCM）中非饱和填充浸润的数值模拟,并对比了恒压下的实验结果,验证了其可靠性。分析讨论了注射口压力、流量和液体黏度对双尺度多孔纤维织物非饱和填充浸润特性的影响。结果表明:在允许误差内,该数值模拟结果可靠,可用于分析讨论各因素对双尺度多孔织物非饱和流动特性的影响;填充浸润过程中,纤维织物内部非饱和区域长度并非保持不变,而是随着填充浸润的进行经历了4个变化过程;不同注射条件下,压力、流量及黏度对非饱和流动特性影响不同。研究结果对合理控制注射条件及流体特性实现双尺度多孔纤维预制件的完全浸润具有指导意义。      

Application of the level set method to the simulation of resin transfer molding

A new methodology is presented to simulate mold filling in resin transfer molding (RTM) using a combination of the level set and boundary element methods (BEMs). RTM is a composite manufacturing process where a liquid resin is injected in a closed rigid mold containing a dry fibrous reinforcement. Process simulation is motivated by the importance of tracking accurately the motion of the flow front during the mold filling stage. The BEM solves the equation governing the resin flow and the level set method is implemented to track the resin front in the mold. This formulation opens up new opportunities to improve RTM flow simulations and optimize injection molds. The present paper focuses on isothermal resin flow in undeformable porous medium. The implementation of the numerical algorithm is described and several examples of two-dimensional filling with single or multiple injection gates are presented. The robustness of the coupling and the ability to predict accurately the position of the front by this new model are discussed. It is also shown how dry spot formation can be tracked precisely during the simulation and how a generalization of this approach allows predicting resin flow across obstacles.     

An adaptive multiresolution method for parabolic PDEs with time‐step control

We present an efficient adaptive numerical scheme for parabolic partial differential equations based on a finite volume (FV) discretization with explicit time discretization using embedded Runge–Kutta (RK) schemes. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. The costly fluxes are evaluated on the adaptive grid only. Compact RK methods of second and third order are then used to choose automatically the new time step while controlling the approximation error in time. Non‐admissible choices of the time step are avoided by limiting its variation. The implementation of the multiresolution representation uses a dynamic tree data structure, which allows memory compression and CPU time reduction. This new numerical scheme is validated using different classical test problems in one, two and three space dimensions. The gain in memory and CPU time with respect to the FV scheme on a regular grid is reported, which demonstrates the efficiency of the new method. Copyright © 2008 John Wiley & Sons, Ltd.     

Implicit yield function formulation for granular and rock-like materials

The constitutive modelling of granular, porous and quasi-brittle materials is based on yield (or damage) functions, which may exhibit features (for instance, lack of convexity, or branches where the values go to infinity, or ‘false elastic domains’) preventing the use of efficient return-mapping integration schemes. This problem is solved by proposing a general construction strategy to define an implicitly defined convex yield function starting from any convex yield surface. Based on this implicit definition of the yield function, a return-mapping integration scheme is implemented and tested for elastic–plastic (or -damaging) rate equations. The scheme is general and, although it introduces a numerical cost when compared to situations where the scheme is not needed, is demonstrated to perform correctly and accurately.     

An adaptive solution of the 3-D Euler equations on an unstructured grid

Summary A new algorithm to generate the unstructured grid on a curved surface is developed. The advancing front method is used to generate the tetrahedral meshes in the space. An adaptive grid technique is used to enhance the calculation efficiency. The AUSM⁺ (Advection Upstream Splitting Method) scheme which was developed on a structured grid has been extended to be used to the spatial discretization of a cell-centered finite volume formulation on the unstructured grid. A second order spatial accuracy is achieved by applying a novel cell reconstruction procedure which can prevent the solution from exhibiting spurious oscillations without adding a limiter. A 3-D Euler solver for an adaptive tetrahedral grid and numerical results for several cases are presented.     

恒压下树脂在双尺度多孔介质中的流动特性

基于复合材料液态模塑(LCM)工艺过程中存在半饱和区域的实验现象以及对预制体双尺度效应的逐步认识, 一些学者提出用沉浸模型来研究双尺度多孔介质的不饱和流动。通过体积均匀化方法描述了双尺度多孔介质复合材料液态模塑工艺模型的特征, 得到含有沉浸项的双尺度多孔介质的质量守恒方程, 并采用有限元法对方程进行数值求解, 通过具体算例计算了考虑双尺度效应时恒压树脂注射下不同时段的压力分布状态, 得到树脂在填充过程中流动前沿半饱和区域从出现到消失的过程, 采用不同注射压力进行模拟并比较。结果表明, 与单尺度多孔介质模型不同, 双尺度多孔介质模型更能反映实际树脂填充过程中出现的半饱和区域现象。     

On the Galerkin finite difference method: A multi-bases approach

At present there are many general methods of approximating a possibly nonlinear operator equation by a finite equation system. The most commonly applied methods are the finite element and the finite difference. Numerous papers (e.g. References 9 and 10) have dealt with a comparison of these methods. In Reference 12 the Galerkin finite difference method (GFDM) is developed. The GFDM is a special finite element method designed t o solve nonlinear and possibly coupled partial differential equations numerically. It consists of a finite difference scheme derived from a Galerkin finite element method through the use of special local basis functions and a special grid. In this paper, we are concerned with extending the GFDM to derive ‘normal’ difference schemes. e.g. five-point schemes on two-dimensional domains for a general class of operators, even in nonlinear cases. Using GFDM or the finite element method on two-dimensional regions generally leads to at least 7- or 9-point schemes as well as expensive approximations of the nonlinear terms. Often, numerical integration is necessary. These computational costs are due to the non-orthogonality of the continuous and differentiable local basis functions that are needed in this case. The basic idea of the multi-bases approaches, which are the major concern in this paper¹¹, is to reduce the smoothnessproperties ofthelocal basis functions in favour oftheir orthogonality. Thiscan beachieved with the help of so-called transfer operators which map the non-orthogonal and differentiable basis onto an only bounded but orthogonal basis.     

A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation

We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.     

Some numerical schemes using curvilinear coordinate grids for incompressible and compressible Navier-Stokes equations

In this review paper some numerical schemes recently developed by the authors and their coworkers for analysing the cascade flows of turbomachinery are described. These schemes use the curvilinear coordinate grid and solve the momentum equations of contravariant velocities (volume flux). The compressible flow schemes are based on the delta-form approximate-factorization finite-difference scheme, and are improved by using the diagonalization, the flux difference splitting and thetvd schemes to save computational effort and to increase stability and resolvability. Furthermore, using higher-order compacttvd muscl schemes, we can capture not only shock waves but also contact surfaces very sharply. On the other hand, the incompressible flow schemes are based on the well-knownSMAC scheme, and are extended to the curvilinear coordinate grid and further to the implicit scheme to reduce computations. These schemes, like thesmac scheme, satisfy the continuity condition identically, and suppress the occurrence of spurious errors. In both the compressible and incompressible schemes, for the turbulent flow thek-ɛ turbulence model with the law of the wall or considering the low Reynolds number effects is employed, and for the unsteady flow the Crank-Nicholson method is employed and the solution at each time step is obtained by the Newton iteration. Use of the volume flux instead of the physical velocity is inevitable for theMAC type schemes, and makes it easy to impose boundary conditions. Finally, some calculated results using the present schemes are shown.     

An analysis of the convection-diffusion problems using meshless and meshbased methods

The numerical solution of the convection-diffusion equation represents a very important issue in many numerical methods that need some artificial methods to obtain stable and accurate solutions. In this article, a meshless method based on the local Petrov-Galerkin method is applied to solve this equation. The essential boundary condition is enforced by the transformation method, and the MLS method is used for the interpolation schemes. The streamline upwind Petrov-Galerkin (SUPG) scheme is developed to employ on the present meshless method to overcome the influence of false diffusion. In order to validate the stability and accuracy of the present method, the model is used to solve two different cases and the results of the present method are compared with the results of the upwind scheme of the MLPG method and the high order upwind scheme (QUICK) of the finite volume method. The computational results show that fairly accurate solutions can be obtained for high Peclet number and the SUPG scheme can very well eliminate the influence of false diffusion.     

An extended finite element method for fluid flow in partially saturated porous media with weak discontinuities; the convergence analysis of local enrichment strategies

In this paper, a numerical model is developed for the fully coupled analysis of deforming porous media containing weak discontinuities which interact with the flow of two immiscible, compressible wetting and non-wetting pore fluids. The governing equations involving the coupled solid skeleton deformation and two-phase fluid flow in partially saturated porous media are derived within the framework of the generalized Biot theory. The solid phase displacement, the wetting phase pressure and the capillary pressure are taken as the primary variables of the three-phase formulation. The other variables are incorporated into the model via the experimentally determined functions that specify the relationship between the hydraulic properties of the porous medium, i.e. saturation, permeability and capillary pressure. The spatial discretization by making use of the extended finite element method (XFEM) and the time domain discretization by employing the generalized Newmark scheme yield the final system of fully coupled non-linear equations, which is solved using an iterative solution procedure. Numerical convergence analysis is carried out to study the approximation error and convergence rate of several enrichment strategies for bimaterial multiphase problems exhibiting a weak discontinuity in the displacement field across the material interface. It is confirmed that the problems which arise in the blending elements can have a significant effect on the accuracy and convergence rate of the solution.     

An efficient solver of the saturation equation in liquid composite molding processes

A major issue in Liquid Composite Molding Process (LCM) concerns the reduction of voids formed during the resin filling process. Reducing the void content increases the quality of the composite and improves its mechanical properties. Most of modeling efforts on process simulation of mold filling has been focused on the single phase Darcy’s law, with resin as the only phase, ignoring the formation and transport of voids. The resin flow in a partially saturated region can be characterized as two phase flow through a porous medium. The mathematical formulation of saturation in LCM takes into account the interaction between resin and air as it occurs in a two phase flow. This model leads to the introduction of relative permeabilities as a function of saturation. The modified saturation equation is obtained as a result, which is a non-linear advection-diffusion equation with viscous and capillary phenomena. In this work, a flux limiter technique has been used to solve a modified saturation equation for the LCM process. The implemented algorithm allows a numerical optimization of the injected flow rate which minimizes the micro/macroscopic void formation during mold filling. Some preliminary numerical results are presented here in order to validate the proposed mathematical model and the numerical scheme. This formulation opens up new opportunities to improve LCM flow simulations and optimize injection molds.     

Eikonal equation‐based front propagation for arbitrary complex configurations

This paper presents a front propagation method using the Eikonal equation, ?? ? ?? = 1, in which, ? represents the smallest Euclidean distance field to the front to be propagated. The offset capturing approach consists in first calculating the ? field over a uniform Cartesian grid fully covering the front to be propagated, and then constructing the iso‐? curves or surfaces as the propagated result. The calculation of ? uses a 3D numerical scheme, the Fast Sweeping Scheme. Validation for accuracy of the method is presented using academic test cases. A real 3D industry application, draft tube with two piers, is successfully propagated and demonstrated using special boundary conditions to cope with inlet and outlet planes during front propagtion. This application involves the propagation of a front that exhibits both concave and convex shape regions, sharp corners, and local tangent plane surface discontinuities as well as a multi‐connected domain. Copyright © 2007 John Wiley & Sons, Ltd.     

Improved guaranteed computable bounds on homogenized properties of periodic media by the Fourier–Galerkin method with exact integration

Moulinec and Suquet introduced FFT‐based homogenization in 1994, and 20years later, their approach is still effective for evaluating the homogenized properties arising from the periodic cell problem. This paper builds on the author's (2013) variational reformulation approximated by trigonometric polynomials establishing two numerical schemes: Galerkin approximation (Ga) and a version with numerical integration (GaNi). The latter approach, fully equivalent to the original Moulinec–Suquet algorithm, was used to evaluate guaranteed upper–lower bounds on homogenized coefficients incorporating a closed‐form double‐grid quadrature. Here, these concepts, based on the primal and dual formulations, are employed for the Ga scheme. For the same computational effort, the Ga outperforms the GaNi with more accurate guaranteed bounds and more predictable numerical behaviors. The quadrature technique leading to block‐sparse linear systems is extended here to materials defined via high‐resolution images in a way that allows for effective treatment using the FFT. Memory demands are reduced by a reformulation of the double‐grid scheme to the original grid scheme using FFT shifts. Minimization of the bounds during iterations of conjugate gradients is effective, particularly when incorporating a solution from a coarser grid. The methodology presented here for the scalar linear elliptic problem could be extended to more complex frameworks. Copyright © 2015 John Wiley & Sons, Ltd.     

A mixed-variable continuously deforming finite element method for parabolic evolution problems. Part III: Numerical implementation and computational results

In Part I of this paper,¹ the conceptual framework of a rate variational least squares formulation of a continuously deforming mixed-variable finite element method was presented for solving a single evolution equation. In Part II² a system of ordinary differential equations with respect to time was derived for solving a system of three coupled evolution equations by the deforming grid mixed-variable least squares rate variational finite element method. The system of evolution equations describes the coupled heat flow, fluid flow and trace species transport in porous media under conditions when the flow velocities and constituent phase transitions induce sharp fronts in the solution domain. In this paper, we present the method we have adopted to integrate with respect to time the resulting spatially discretized system of non-linear ordinary differential equations. Next, we present computational results obtained using the code in which this deforming mixed finite element method was implemented. Because several features of the formulation are novel and have not been previously attempted, the problems were selected to exercise these features with the objective of demonstrating that the formulation is correct and that the numerical procedures adopted converge to the correct solutions.     

CONDIF: A modified central-difference scheme for convective flows

For most practical purposes, the central-difference scheme (CDS) would be ideal only if it were unconditionally stable. It is a simple and second-order scheme which is easy to implement. It does not introduce any second-order ‘diffusion’ like truncation error. However, for grid Peclet numbers larger than 2, the CDS leads to over- and under-shoots and is unstable. This paper presents a method, called CONDIF, which eliminates this undesirable feature of the CDS. It modifies the CDS by introducing a controlled amount of numerical diffusion based on the local gradients. The numerical diffusion can be adjusted to be negligibly low for most problems. CONDIF has been used to solve a number of test problems which have been widely used for comparative study of numerical schemes in the published literature. For all these problems the CONDIF results are significantly more accurate than those obtained from the hybrid scheme when the Peclet number is very high (→∞) and the flow is at large angles (→45 degrees) to the grid. In general the computational effort for CONDIF is comparable (within 20 per cent) to that for the hybrid scheme. However, in one instance the rate of convergence was found to be significantly slower.     

Boundary domain integral method for the study of double diffusive natural convection in porous media

The main purpose of this paper is to present a boundary domain integral method (BDIM) for the solution of natural convection in porous media driven by combining thermal and solutal buoyancy forces. The Brinkman extension of the classical Darcy equation is used for the momentum conservation equation. The numerical scheme was tested on a natural convection problem within a square porous cavity, where different temperature and concentration values are applied on the vertical walls, while the horizontal walls are adiabatic and impermeable. The results for different governing parameters (Rayleigh number, Darcy number, buoyancy ratio and Lewis number) are presented and compared with published work. There is a good agreement between those results obtained using the presented numerical scheme and reported studies using other numerical methods.