Single-field Formulation Research Articles

Implementation of ADE-CFS-PML for the Single Field WCS-FDTD Method

In this letter, the auxiliary differential equation (ADE) form of the complex-frequency-shifted perfectly matched layer (CFS-PML) is implemented for the single field (SF) weakly conditionally stable (WCS) finite difference time domain (FDTD) method. The magnetic field terms in the auxiliary variables of ADE-CFS-PML are eliminated, conforming to the SF formulation. The reflection error and stability of the proposed method is verified with numerical examples. Results show that the proposed method is stable and features low reflection error when the maximum time step is chosen for the SF-WCS-FDTD method. The higher multithread computation efficiency of the proposed method is also demonstrated.

Modeling of interfacial mass transfer based on a single-field formulation and an algebraic VOF method considering non-isothermal systems and large volume changes

This work presents a consistent numerical model for the prediction of interfacial heat and mass transfer with high volume changes based on a single-field formulation and an algebraic Volume-of-Fluid (VOF) method. The model is based on the Continuous Species Transfer (CST) model for chemical species transport, and the compressive CICSAM scheme is used to advect both volume fraction and species concentration in order to ensure consistency. The computation of the interfacial mass transfer rate (ṁ) is based on the variables solved by the single-field formulation and four different procedures for its discretisation are analysed. It is shown that the discretisation of this term has a significant impact on the predictive capabilities of the model. The use of the compressive velocity term, which is based on the relative velocity between the phases (ur) is also investigated for both the algebraic VOF method and the Compressive Continuous Species Transfer (C-CST) model. The results show that the use of ur is not necessary if the CICSAM scheme is used for the discretisation of the advection terms of both the volume fraction and the concentration, even for advection-dominated cases with negligible diffusion. The modeling approach is validated against several test cases and applied to the heating and evaporation of droplets. The proposed numerical model provides accurate results for droplet evaporation and is able to handle the high concentration jump at the droplet interface and the high volume changes resulting from droplet evaporation.

A unified single-field Volume-of-Fluid-based formulation for multi-component interfacial transfer with local volume changes

This paper presents a novel unified single-field formulation for Volume-Of-Fluid simulation of interfacial mass transfer with local volume changes. By comparison with the previous models referred as Continuous Species Transfer in the literature, our improved model uses a single-field formulation of the local mass transfer across the interface, enabling us to take into account local volume changes induced by non-dilute species transfer. The numerical model, implemented in our in-house OpenFOAM-based simulator, is validated by comparison with analytical solutions in 1D and 2D, and a semi-analytical solution in 3D. The implemented approach is first applied to investigate competing mass transfer in an infinite cylinder. We then simulate the shrinking of a single-component rising bubble at low Schmidt number. The numerical model is shown to be well adapted to investigate Sherwood numbers and existing correlation for mass transfer at fluid interfaces.

A numerical framework for diffusion-controlled bimolecular-reactive systems to enforce maximum principles and the non-negative constraint

We present a novel computational framework for diffusive–reactive systems that satisfies the non-negative constraint and maximum principles on general computational grids. The governing equations for the concentration of reactants and product are written in terms of tensorial diffusion–reaction equations. We restrict our studies to fast irreversible bimolecular reactions. If one assumes that the reaction is diffusion-limited and all chemical species have the same diffusion coefficient, one can employ a linear transformation to rewrite the governing equations in terms of invariants, which are unaffected by the reaction. This results in two uncoupled tensorial diffusion equations in terms of these invariants, which are solved using a novel non-negative solver for tensorial diffusion-type equations. The concentrations of the reactants and the product are then calculated from invariants using algebraic manipulations. The novel aspect of the proposed computational framework is that it will always produce physically meaningful non-negative values for the concentrations of all chemical species. Several representative numerical examples are presented to illustrate the robustness, convergence, and the numerical performance of the proposed computational framework. We will also compare the proposed framework with other popular formulations. In particular, we will show that the Galerkin formulation (which is the standard single-field formulation) does not produce reliable solutions, and the reason can be attributed to the fact that the single-field formulation does not guarantee non-negative solutions. We will also show that the clipping procedure (which produces non-negative solutions but is considered as a variational crime) does not give accurate results when compared with the proposed computational framework.

Local and global error estimations in linear structural dynamics

The study discusses the concept of error estimation in linear elastodynamics. Two different types of error estimators are presented. First ‘classical’ methods based on post-processing techniques are discussed starting from a semidiscrete formulation. The temporal error due to the finite difference discretization is measured independently of the spatial error of the finite element discretization. The temporal error estimators are applied within one time step and the spatial error estimators at a time point. The error is measured in the global energy norm. The temporal evolution of the error cannot be reflected. Furthermore the estimators can only evaluate the mean error of the whole spatial domain. As the second scheme local error estimators are presented. These estimators are designed to evaluate the error of local variables in a certain region by applying duality techniques. Local estimators are known from linear elastostatics and have later on been extended to nonlinear problems. The corresponding dual problem represents the influence of the local variable on the initial problem and may be related to the reciprocal theorem of Betti–Maxwell. In the present study this concept is transferred to linear structural dynamics. Because the dual problem is established over the total space–time domain, the spatial and temporal error of all time steps can be accumulated within one procedure. In this study the space–time finite element method is introduced as a single field formulation.

Computation of Microsegregation and Microstructure in Solidification with Fluid Convection

AbstractThe effects of fluid flow on the dendritic solidification morphology of pure materials is studied using a computational methodology based on a two-dimensional front tracking/finite difference method. A general single-field formulation is presented for the full coupling of phase change, fluid flow, and heat transport in both the solid and liquid phases. This formulation accounts for interfacial rejection/absorption of latent heat, interfacial anisotropies, discontinuities in material properties between the liquid and solid phases, shrinkage/expansion upon solidification and motion and deformation of the solid. Numerical results are presented for the dendritic solidification of pure succinonitrile in a shear flow. Comparison with solidification into a quiescent liquid indicates that fluid convection increases the overall rate of solidification while the growth rate of the leading dendrite tip is unchanged.

Finite element analysis of reinforced rubber shells

The application of numerical techniques to the solution of practical problems which exist in rubber technology is described. Structures and components in the form of reinforced rubber shells are widely used in industry and prediction of their performance is complicated by both the anisotropic nature of composite construction and the incompressible behaviour of the basic material. A layered shell element is developed for the solution of such problems with general anisotropic behaviour independently permitted in each layer. The approach adopted permits the easy location of reinforcement patterns. Numerical solution is based on a single field formulation by eliminating at integrating point level the Lagrange multiplier imposing the incompressible constraint. Large deformation, including large rotation, behaviour is accommodated and a total Lagrangian solution process is adopted. The code developed permits the simulation of non‐conservative loading and its versatility is demonstrated by the solution of some practical examples.