Freezing time of an infinite cylinder and sphere using the method of lines

An energy balance model has been developed to simulate freezing processes for infinite cylinders and spheres. The mathematical model (1) was numerically solved using the method of lines. In this method, spatial derivatives are discretized by the finite difference method and the resulting system of ordinary differential equations in time is integrated using an appropriate solver. Freezing times obtained with the proposed model were compared to experimental data and results calculated by different published methods. The freezing times predicted by the proposed model agreed well with the published experimental results and predictions by other published methods. Model (1) gives a percentage error in the range −4.61 ≤ E (%) ≤ 6.81, which includes the experimental data for the 123 spheres analyzed and 30 infinite cylinders, within range −2.96 ≤ E (%) ≤ 3.34.     

Freezing of foods under surface boiling boundary conditions

The surface boiling boundary condition is encountered in the freezing of foods when foods are immersed in boiling freezants such as R12. This phenomenon was incorporated in a mathematical model of the freezing process as a surface temperature dependent convective boundary condition. A finite difference numerical scheme was formulated to solve the model for one- and two-dimensional geometries. The pool boiling characteristic for R12 was obtained by inverse heat transfer analysis of the experimental quenching curves of a transient calorimeter. The model was used to simulate the experimental freezing processes with reasonable agreement.     

含缺陷平纹机织复合材料拉伸力学行为数值模拟

基于平纹机织复合材料的细观结构单胞模型, 考虑其制备过程中产生的孔隙缺陷为随机分布的特征, 通过引入两参数Weibull分布函数, 应用Python语言实现了ABAQUS的二次开发, 并采用Linde等提出的失效准则, 建立了含孔隙缺陷平纹机织复合材料的渐进损伤模型, 利用有限元数值方法模拟了其拉伸应力-应变行为, 针对该模型, 讨论了孔隙缺陷对材料拉伸应力-应变行为的影响, 并阐述了该平纹机织复合材料单胞模型在经向拉伸载荷作用下其纤维束的损伤及演化过程。结果表明, 该模型给出的数值模拟结果与实验数据吻合较好, 证明了模型的有效性, 为该类材料的优化设计及其力学性能分析提供了一种有效方法。      

A two-dimensional frost-heave model for buried pipelines

The rigid-ice model of frost heave is one of the most comprehensive frost-heave models but is restricted to one-dimensional cases in its present form. In this paper, the model is extended to two-dimensional problems. The complete formulation of the partial differential equations governing heat, moisture and ice transport in freezing soils is provided. The equations are subsequently solved using the Galerkin finite element method in space and the finite difference method in time. A computer program is developed for the two-dimensional rigid-ice model. A case of freezing around chilled gas pipeline is solved and the numerical results are compared with experimental values, with good agreement between the two sets of results.     

Nonlinear Compression Behavior of Warp-Knitted Spacer Fabric: Effect of Sandwich Structure

Compressibility of warp-knitted spacer fabrics is one of their important mechanical properties with regard to many special applications such as body protection, cushion and mattresses. Due to specific structural features of the fabric and a non-linear mechanical behavior of monofilaments, the compression properties of this kind of fabrics are very complicated. Although several studies have been performed to investigate their compression behavior, its mechanism has not well been understood yet. This work is concerned with a study of compression mechanism of a selected warp-knitted spacer fabric with a given sandwich structure. Both experimental and numerical methods are used to study the effect of the material's structure on the overall compression mechanism. Compression tests are conducted to obtain force-displacement relationships of the fabric. A micro-computed tomography system is used to analyze specimens under different levels of compression displacement to investigate the change in material's structure during the compression process. At the same time, finite element models are developed separately to simulate the initial geometric structure and the compression behavior of the fabric. Three finite element models based on beam elements are firstly developed to simulate the effect of manufacturing process on shapes of monofilaments within the fabric and to determine their morphologies, which are used to assemble a geometry part of the finite element model of the overall fabric. Then the finite-element model is developed using beam and shell elements to describe the compression behavior of the fabric by introducing the effect of its complex microstructure and real non-linear mechanical properties of the monofilaments. A comparison of the obtained experimental and CT data, and results of simulation is carried out, demonstrating a good agreement. With this study, a compression mechanism of the warp-knitted spacer fabric can be better understood.     

Comparison of method of lines and finite difference solutions of 2‐D Navier–Stokes equations for transient laminar pipe flow

Performances of method of lines (MOL) and finite difference method (FDM) were tested from the viewpoints of solution accuracy and central processing unit (CPU) time by applying them to the solution of time‐dependent 2‐D Navier–Stokes equations for transient laminar flow without/with sudden expansion and comparing their results with steady‐state numerical predictions and measurements previously reported in the literature. Predictions of both methods were obtained on the same computer by using the same order of spatial discretization and the same uniform grid distribution. Axial velocity and pressure distribution in pipe flow and steady‐state reattachment lengths in sudden expansion flow on uniform grid distribution predicted by both methods were found to be in excellent agreement. Transient solutions of both methods for pipe flow problem show favourable comparison and are in accordance with expected trends. However, non‐physical oscillations were produced by both methods in the transient solution of sudden expansion pipe flow. MOL was demonstrated to yield non‐oscillatory solutions for recirculating flows when intelligent higher‐order discretization scheme is utilized for convective terms. MOL was found to be superior to FDM with respect to CPU and set‐up times and its flexibility for incorporation of other conservation equations. Copyright © 2001 John Wiley & Sons, Ltd.     

Generalized model for the simulation of food refrigeration. Development and validation of the predictive numerical method

A generalized mathematical model was developed to simulate food refrigeration in air. The model takes into account surface water evaporation. It allows food physical properties to be considered as functions of temperature and/or composition, while providing means for including internal heat generation, composite materials (for instance, flesh and skin) and time-varying external temperature and humidity. Wetted surfaces and packaging can also be accounted for by the model, which can be used for spheres, infinite or finite cylinders, slabs. The numerical method was developed using the Crank-Nicolson scheme, and compared with exact analytical solutions as well as with experimental data on the refrigeration of various foods.     

树脂传递模塑工艺中的非饱和流动过程模拟与实验研究

针对编织类纤维增强体的纤维束之间与纤维束内孔隙的双尺度特点,建立了平纹织物的细观结构模型,并推导了汇函数的数学表达式。建立了局部细观流动特征的非饱和流动控制方程,利用有限元/控制体积方法求解,得到了局部饱和度分布。与实验进行比较,吻合较好。      

Modeling frost heave in freezing soils

A generalized model for secondary frost heave in freezing fine-grained soils is presented. The cryostatic suction effect, which causes an increase in upward water permeation, ice-lens growth during freezing, and, as a consequence, the increase of soil heave, is considered to be the main mechanism of moisture transfer. Although the model in this paper has a number of approaches in common with the model of Fowler and Krantz (1994) it differs in at least several important respects. We recognize the need to determine the distribution of the moisture within the frozen fringe by approximation of the experimental data for the equilibrium unfrozen water content. This distribution is the result of the complicated interaction between water, ice and the mineral skeleton during the freezing process. The generalization of the Clapeyron relation, which is used in the work cited above, only estimates the drop in initial freezing temperature and does not define the connection with the external temperature gradient ∇T, which is responsible for the frost heave process. This very important aspect is discussed in detail in the Introduction to our paper. Another difference is the fact that our solution is based on a dimensionless system of equations. We take into account the ratio Pe/Ste ≠ 1 (where Pe << 1). This approach allows us to obtain both a more general solution as well as analyze frost heave and propagation of the freezing front as they depend upon the convective (Pe) and phase transition (Ste) characteristics (criteria) of the process. The theoretical results derived from our solution of the analysis for fine-grained soils are compared, in good agreement, with experimental investigations and numerical models. A singularity of the solution at the initial point in time is discussed. In this respect the asymptotic solution for short and long times is obtained. The results are compared with both solutions (modeling and asymptotic). The model presented predicts the frost heave and freezing processes in porous media with reasonable accuracy and satisfactorily reflects observed phenomena, and thus can be suitable for engineering practice.     

聚合物基复合材料模压成型过程固化度与非稳态温度场的数值模拟

聚合物基复合材料模压成型过程固化度与温度的动态变化为强耦合关系。本文作者根据固化动力学和传热学理论,建立了非稳态温度场与固化动力学数学模型。通过DSC实验分析确定模型中固化动力学参数。利用有限单元与有限差分相结合的方法,建立了温度场和固化度数值模型。应用Euler逐步迭代法实现解耦。对聚合物基复合材料模压成型过程固化度与非稳态温度场动态变化进行计算机数值模拟,与试验测定结果吻合。为优化模压成型工艺提供理论依据。     

Numerical simulation of individual quick freezing of spherical foods

A one-dimensional three time level implicit finite difference computer program was developed to predict temperature profiles during the individual quick freezing of spherically shaped foods. Temperature varying thermal properties necessary for these predictions were calculated based upon properties of the unfrozen product. The centre temperatures of individual peas within a bed of peas being frozen under various conditions of fluidization were recorded. Comparison of experimental results and published data with predicted freezing curves showed good agreement.     

An X‐FEM and level set computational approach for image‐based modelling: Application to homogenization

The advances in material characterization by means of imaging techniques require powerful computational methods for numerical analysis. The present contribution focuses on highlighting the advantages of coupling the extended finite elements method and the level sets method, applied to solve microstructures with complex geometries. The process of obtaining the level set data starting from a digital image of a material structure and its input into an extended finite element framework is presented. The coupled method is validated using reference examples and applied to obtain homogenized properties for heterogeneous structures. Although the computational applications presented here are mainly two‐dimensional, the method is equally applicable for three‐dimensional problems. Copyright © 2010 John Wiley & Sons, Ltd.     

A meshless method of lines for the numerical solution of KdV equation using radial basis functions

In this paper, a meshless method of lines (MOL) is presented for the numerical solution of the Korteweg–de Vries (KdV) equation. This novel method has an advantage over the traditional method of lines which approximates the spatial derivatives using finite difference method (FDM) or finite element method (FEM), because it does not need the mesh in the domain, and it approximates the solution using the radial basis functions (RBFs) on a set of node scattered in problem domain. A comparison among some RBFs is made in numerical examples. Numerical examples demonstrate the accuracy and easy implementation of this novel method and it is an efficient method for the nonlinear time-dependent partial differential equations (PDEs).     

不同离散化方法在正冻土水热耦合模型中的应用

为了研究土体冻结过程中温度传导与水分迁移的变化,在不饱和土理查德斯方程的基础上考虑相变作用建立了一维土体冻结过程的水热耦合模型。分别应用有限差分与有限元方法对模型进行离散化,得到两种方法各自的数值计算程序。同时对土体室内单向冻结实验进行模拟,将两种数值方法得到的土体冻结深度、冻结速率与温度变化分别与实验结果进行对比。结果表明:对于一维冻结问题,有限差分模拟更接近实验结果,有限元模拟具有更高的计算稳定性。     

A NEW TWO-DIMENSIONAL FINITE ELEMENT MODEL FOR THE SHALLOW WATER EQUATIONS USING A LAGRANGIAN FRAMEWORK CONSTRUCTED ALONG FLUID PARTICLE TRAJECTORIES

In this paper we describe a new finite element model for the tidal hydrodynamics in estuaries. The mathematical model is based on the solution of the two-dimensional shallow water equations in a Lagrangian framework which is defined along the trajectories of fluid particles. This method gives a flexible and robust numerical scheme for moving boundary flows encountered in tidal water systems. In order to validate the developed model we have, at first instance, compared our numerical results with analytical solutions obtained for domains with simple geometries. Further tests are then conducted to demonstrate the model's ability to cope with conditions such as hydraulic shock, abrupt changes in the flow domain geometry and gradual changes of water surface breadth. The change in the water surface breadth corresponds to the drying and wetting of the plains along the banks of a typical tidal river/estuary reach. The drying and wetting of flood plains result in the existence of very shallow depth of water at some sections of the flow domain during a tidal cycle. The flow equations under these conditions are strongly convection dominated. Previously published tidal models rely on either, some form of upwinding or the use of extremely fine meshes to give stable results for the convection dominated very shallow depth computations in estuaries. We show that our model can yield stable and accurate results for very shallow depths in the tidal flow domains without using any kind of artifical damping or excessive mesh refinement. Computational costs of simulating hydrodynamical conditions in a natural water course, even using a depth averaged two-dimensional approach, can be very high. The ability of our scheme to cope with convection dominated conditions has enabled us to economize the computational efforts by using coarse meshes in our finite element calculations. After the validation stage, the developed model is applied to simulate the tidal conditions in a real estuary. The comparison of the model results with the field observations shows a close agreement between these sets of data     

家用制冰机结冰过程的理论与实验研究

根据家用制冰机的工作原理和结构利用焓法对其制冰过程建立数学物理模型，采用全隐式格式开展有限差分数值模拟研究。在一台家用制冰样机上进行实验测试，通过改变环境温度观测冰层厚度与制冰时间的关系，并将实验结果与数值解进行对比。结果表明，数值解与实测值之间相对误差的绝对值小于3％。升高环境温度，在同样的时间内成品冰厚度减小，质量下降。     

Modelling liquid–solid phase changes with melt convection

Methods are described for modelling of phase change processes using the finite element method to simulate freezing and melting including convection in the melt. Evaluation of several enthalply/specific heat methods and time marching schemes is also included. Suppression of velocities in the solid region is described, and example problems are given. Comparison is made to simulations performed by other researchers using finite difference methods. Substantially different results were found for one of these problems, and this result is shown to be caused by numerical problems in the earlier work.     

Freezing of a parallelepiped food product. Part 2. Comparison of experimental and calculated results

The objective of the project reported here was to test the adequacy and applicability of existing mathematical models when used to predict the freezing time of a small parallelepiped food product. The freezing time of french fries was determined experimentally by individually freezing the samples in an air blast; this work was presented in a previous paper. In this paper the experimental results are compared with estimates from 19 mathematical models. Three empirical models and one approximate model yielded estimates within 10% of the experimental freezing time and most of the other empirical and approximate models overestimated by more than 10%. Neither of the two exact models examined was applicable to the freezing of a small food product when the surface heat transfer coefficient was finite.     

Low-velocity impact-induced damage of continuous fiber-reinforced composite laminates. Part I. An FEM numerical model

A numerical model for simulating the process of low-velocity impact damage in composite laminates using the finite element method is presented in this paper, i.e. Part I of this two part series on the study of impact. In this model, the 9-node Lagrangian element of the Mindlin plate with consideration of large deformation analysis is employed. To analyze the transient response of the laminated plates, a modified Newmark time integration algorithm previously proposed by the authors is adopted here. We also proved that the impact process between a rigid ball and laminated plates is a stiff system, therefore a kind of A(α) stable method has been advocated here to solve the motion equation of the rigid ball. Furthermore, various types of damages including delamination, matrix cracking and fiber breakage, etc. and their mutual influences are modeled and investigated in detail. To overcome the difficulty of numerical oscillation or instability in the analysis of the dynamic contact problem between delaminated layers using the traditional penalty methods, we have employed dynamic spring constraints to simulate the contact effect, which are added to the numerical model by a kind of continuous penalty function. Moreover, an effective technique to calculate the strain energy release rate based on the Mindlin plate model is proposed, which can attain high precision. Finally, some techniques of adaptive analyses have been realized for improving the computational efficiency. Based on this model, a program has been developed for numerically simulating the damage process of cross-ply fiber-reinforced carbon/epoxy composite laminates under low-velocity impact load. In Part II, this numerical model will be verified by comparing with the experimental results. Also the impact damage will be investigated in detail using this numerical approach.     

Numerical modelling of the boundary conditions on beams stuck transversely by a mass

In this paper, numerical results of a drop weight impact test examining the dynamic response of a clamped steel beams struck transversely at the centre by a mass with a rectangular indenter are presented. The numerical simulation showed that the results are very sensitive to the way in which the supports are modelled and thus this paper studies the modelling of the structural supports used to clamp the specimen beam in the experiment. The first approach simplifies the structural support as boundary condition on the nodes; the second model represents the entire support as the real approximation; the last approach uses simplified support plate model acting with the beam to simulate the experimental set up. The complete force-displacement curve is used to compare the plastic deformation of the impacted beams, thus the best approximation is selected for further calculations. The finite element model was performed using the LS-DYNA non-linear, dynamic finite element software. The numerical calculations can predict accurately the response of deflections, forces and absorbed energies, when proper boundary conditions are applied, in this case with shell elements representing the support plates.