The extended finite element method for rigid particles in Stokes flow

A new method for the simulation of particulate flows, based on the extended finite element method (X‐FEM), is described. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The near field form of the fluid flow about each particle is built into the finite element basis using a partition of unity enrichment, allowing the simple enforcement of boundary conditions and improved accuracy over other methods on a coarse mesh. We present a weak form of the equations of motion useful for the simulation of freely moving particles, and solve example problems for particles with prescribed and unknown velocities. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical Solutions for Heat Transfer of An Unsteady Cavity with Viscous Heating

The mechanism of viscous heating of a Newtonian fluid filled inside a cavity under the effect of an external applied force on the top lid is evaluated numerically in this exploration. The investigation is carried out by assuming a two-dimensional laminar in-compressible fluid flow subject to Neumann boundary conditions throughout the numerical iterations in a transient analysis. All the walls of the square cavity are perfectly insulated and the top moving lid produces a constant finite heat flux even though the fluid flow attains the steady-state condition. The objective is to examine the effects of viscous heating in the fully insulated lid-driven cavity under no-slip and free-slip Neumann boundary conditions coupled with variations in Reynolds and Prandtl numbers. The partial differential equations of time-dependent vorticity-stream function and thermal energy are discretized and solved using a self-developed finite difference code in MATLAB^® environment. Time dependence of fluid thermodynamics is envisaged through contour and image plots. A commercial simulation software, Ansys Fluent^® utilizing a finite element code is employed to verify the finite difference results produced. Although the effect of viscous heating is very minimal, Neumann no-slip and free-slip boundary conditions are able to trap the heat inside the fully insulated cavity as the heat flux is constantly supplied at the top lid. A lower Reynolds number and a greater Prandtl number with free-slip effects reduce temperature distribution in the cavity with a faster velocity than in the no-slip condition as the free-slip behaves as a lubricant.     

A three-step finite element method for unsteady incompressible flows

This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. The stability analysis of the one-dimensional purely convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax-Wendroff finite element method. The method is cost effective for incompressible flows, because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present three-step finite element method does not contain any new higher-order derivatives, and is suitable for solving non-linear multi-dimensional problems and flows with complicated outlet boundary conditions. The three-step finite element method has been used to simulate unsteady incompressible flows, such as the vortex pairing in mixing layer. The properties of the flow fields are displayed by the marker and cell technique. The obtained numerical results are in good agreement with the literature.     

BOUNDARY ELEMENT–FINITE ELEMENT COUPLED EIGENANALYSIS OF FLUID–STRUCTURE SYSTEMS

The eigenanalysis of acoustical cavities with flexible structure boundaries, such as a fluid-filled container or an automobile cabin enclosure, is considered. An algebraic eigenvalue problem formulation for the fluid–structure problem is presented by combining the acoustic fluid boundary element eigenvalue analysis method and the structural finite elements. For many practical eigenproblems, use of finite elements to discretize the fluid domain leads to large stiffness and mass matrices. Since the acoustic boundary element discretization requires putting nodes only on the wetted surface of the structure, the size of the eigenproblem is reduced considerably, thus reducing the eigenvalue extraction effort. Futhermore, unlike in ordinary cases, the finite element discretization of pressure–displacement based fluid–structure problem gives rise to unsymmetric matrices. Therefore, the fact that the boundary element formulation produces unsymmetric matrices does not introduce additional difficulties here compared to the finite element case in the choice of an eigenvalue extraction procedure. Examples are included to demonstrate the fluid–structure eigenanalysis using boundary elements for the fluid domain and finite elements for the structure.     

A conservative embedded boundary method for an inviscid compressible flow coupled with a fragmenting structure

We present an embedded boundary method for the interaction between an inviscid compressible flow and a fragmenting structure. The fluid is discretized using a finite volume method combining Lax–Friedrichs fluxes near the opening fractures, where the density and pressure can be very low, with high‐order monotonicity‐preserving fluxes elsewhere. The fragmenting structure is discretized using a discrete element method based on particles, and fragmentation results from breaking the links between particles. The fluid‐solid coupling is achieved by an embedded boundary method using a cut‐cell finite volume method that ensures exact conservation of mass, momentum, and energy in the fluid. A time explicit approach is used for the computation of the energy and momentum transfer between the solid and the fluid. The embedded boundary method ensures that the exchange of fluid and solid momentum and energy is balanced. Numerical results are presented for two‐dimensional and three‐dimensional fragmenting structures interacting with shocked flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Lagrangian finite element analysis of Newtonian fluid flows

A fully Lagrangian finite element method for the analysis of Newtonian flows is developed. The approach furnishes, in effect, a Lagrangian implementation of the compressible Navier–Stokes equations. As the flow proceeds, the mesh is maintained undistorted through continuous and adaptive remeshing of the fluid mass. The principal advantage of the present approach lies in the treatment of boundary conditions at material surfaces such as free boundaries, fluid/fluid or fluid/solid interfaces. In contrast to Eulerian approaches, boundary conditions are enforced at material surfaces ab initio and therefore require no special attention. Consistent tangents are obtained for Lagrangian implicit analysis of a Newtonian fluid flow which may exhibit compressibility effects. The accuracy of the approach is assessed by comparison of the solution for a sloshing problem with existing numerical results and its versatility demonstrated through a simulation of wave breaking. The finite element mesh is maintained undistorted throughout the computation by recourse to frequent and adaptive remeshing © 1998 John Wiley & Sons, Ltd.     

Moving Particles Through a Finite Element Mesh

We present a new numerical technique for modeling the flow around multiple objects moving in a fluid. The method tracks the dynamic interaction between each particle and the fluid. The movements of the fluid and the object are directly coupled. A background mesh is designed to fit the geometry of the overall domain. The mesh is designed independently of the presence of the particles except in terms of how fine it must be to track particles of a given size. Each particle is represented by a geometric figure that describes its boundary. This figure overlies the mesh. Nodes are added to the mesh where the particle boundaries intersect the background mesh, increasing the number of nodes contained in each element whose boundary is intersected. These additional nodes are then used to describe and track the particle in the numerical scheme. Appropriate element shape functions are defined to approximate the solution on the elements with extra nodes. The particles are moved through the mesh by moving only the overlying nodes defining the particles. The regular finite element grid remains unchanged. In this method, the mesh does not distort as the particles move. Instead, only the placement of particle-defining nodes changes as the particles move. Element shape functions are updated as the nodes move through the elements. This method is especially suited for models of moderate numbers of moderate-size particles, where the details of the fluid-particle coupling are important. Both the complications of creating finite element meshes around appreciable numbers of particles, and extensive remeshing upon movement of the particles are simplified in this method.     

Particulate flow simulations using lubrication theory solution enrichment

A technique for the numerical simulation of suspensions of particles in fluid based on the extended finite element method (X‐FEM) is developed. In this method, the particle surfaces need not conform to the finite element boundaries, so that moving particles can be simulated without remeshing. The finite element basis is enriched with the Stokes flow solution for flow past a single particle and the lubrication theory solution for flow between particles. The latter enrichment allows the simulation of particles that come arbitrarily close together without refining the mesh in the gap between them. Example problems illustrating both types of enrichment are shown, along with a study of a 50% solution in channel flow. Copyright © 2003 John Wiley & Sons, Ltd.     

Dynamic response of flexible container during the impact with the ground

In order to understand and improve the crashworthiness of airdropped flexible containers, this paper presents the work of numerical simulations about the dynamic behaviour of the container under impact loading. A finite element model of the container is constructed and used to predict the dynamic response of the container. Water movement and sloshing in the container are modelled using the multi-material ALE method; interaction between the container and the fluid is studied by the penalty method. Mechanical properties of the fabric composite are obtained by applying periodic boundary conditions on the unit cell model. Numerical results are compared with experiment data for the validation of the model and the approaches. With a range of system parameters, numerical tests show that the shoulder of the container is the most vulnerable part during the impact. Both the drop heights and the amount of water have great effects on the crashworthiness of the container. The maximum dynamic stress of the container is proportional to the drop height. The amount of water in the container influences the failure of the container.     

Specifications of boundary conditions for Reissner/Mindlin plate bending finite elements

Plate bending finite elements based on the Reissner/Mindlin theory offer improved possibilities to pursue reliable finite element analyses. The physical behaviour near the boundary can be modelled in a realistic manner and inherent limitations in the Kirchhoff plate bending elements when modelling curved boundaries can easily be avoided. However, the boundary conditions used are crucial for the quality of the solution. We identify the part of the Reissner/Mindlin solution that controls the boundary layer and examine the behaviour near smooth edges and corners. The presence of boundary layers of different strengths for different sets of boundary conditions is noted. For a corner with soft simply supported edges the boundary layer removes the singular behaviour of Kirchhoff type from the stress resultants. We demonstrate the theoretical results by some numerical studies on simple plate structures which are discretized by an accurate, higher-order plate element. The results provide guidance in choosing efficient meshes and appropriate boundary conditions in finite element analyses.     

The CIP method embedded in finite element discretizations of incompressible fluid flows

Quite effective low‐order finite element and finite volume methods for incompressible fluid flows have been established and are widely used. However, higher‐order finite element methods that are stable, have high accuracy and are computationally efficient are still sought. Such discretization schemes could be particularly useful to establish error estimates in numerical solutions of fluid flows. The objective of this paper is to report on a study in which the cubic interpolated polynomial (CIP) method is embedded into 4‐node and 9‐node finite element discretizations of 2D flows in order to stabilize the convective terms. To illustrate the capabilities of the formulations, the results obtained in the solution of the driven flow square cavity problem are given. Copyright © 2006 John Wiley & Sons, Ltd.     

基于单胞模型复合材料性能预测的储液容器抗冲击性

基于复合材料的性能预测, 对空投柔性储液容器的抗冲击性和动态响应进行了研究。针对该类材料结构具有周期性的特点, 在单胞有限元模型基础上, 应用周期性边界条件加载方法, 通过数值模拟预测了复合材料的宏观性能和力学参数。以此为基础, 采用基于任意拉格朗日欧拉(ALE)描述的流固耦合方法, 通过数值仿真研究了装水量80%(体积分数)的柔性容器空投冲击的动态响应, 并将获得的结果与物理实验进行了对比, 证明了本文中采用的性能预测方法、 有限元模型和流固耦合技术的合理性和有效性。      

In‐plane progressive matrix cracking analysis of symmetric cross‐ply laminates with holes

Most of the previously performed damage analyses in composite laminates have been restricted to model the plain laminates without geometry discontinuities. In this study, a micromechanical damage model is combined with the finite element formulation and is implemented in the integration points to perform progressive damage analyses of composite laminates. A micromechanical damage model based on the stress transfer method is used to find the degradation of mechanical properties of composite laminates. Crack density is also used as an only state variable representing the damage in each Gauss point of every layer of the laminate. The strain energy and critical energy release rate criterion is also used to predict the damage initiation and evolution in each layer. A finite element discretization is used in conjunction with the user element definition capability of ABAQUS commercial software. To verify the developed procedure, a single element is analysed, and the obtained results are compared with available results in the literature. Progressive damage analyses are also performed for several symmetric cross‐ply laminates with and without geometry discontinuity subjected to matrix cracking damage mechanism under in‐plane loading conditions. The obtained mechanical response and variations of matrix crack density versus the applied load are also discussed.     

The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems

This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.     

Conditions for similitude between the fluid velocity and electric field in electroosmotic flow

Electroosmotic flow is fluid motion driven by an electric field acting on the net fluid charge produced by charge separation at a fluid-solid interface. Under many conditions of practical interest, the resulting fluid velocity is proportional to the local electric field, and the constant of proportionality is everywhere the same. Here we show that the main conditions necessary for this similitude are a steady electric field, uniform fluid and electric properties, an electric Debye layer that is thin compared to any physical dimension, and fluid velocities on all inlet and outlet boundaries that satisfy the Helmholtz-Smoluchowski relation normally applicable to fluid-solid boundaries. Under these conditions, the velocity field can be determined directly from the Laplace equation governing the electric potential, without solving either the continuity or momentum equations. Three important consequences of these conditions are that the fluid motion is everywhere irrotational, that fluid velocities in two-dimensional channels bounded by parallel planes are independent of the channel depth, and that such flows exhibit no dependence on the Reynolds number. Similitude is demonstrated by comparing measured and computed fluid streamlines with computed electric flux lines.     

A three dimensional immersed smoothed finite element method (3D IS-FEM) for fluid–structure interaction problems

A three-dimensional immersed smoothed finite element method (3D IS-FEM) using four-node tetrahedral element is proposed to solve 3D fluid–structure interaction (FSI) problems. The 3D IS-FEM is able to determine accurately the physical deformation of the nonlinear solids placed within the incompressible viscous fluid governed by Navier-Stokes equations. The method employs the semi-implicit characteristic-based split scheme to solve the fluid flows and smoothed finite element methods to calculate the transient dynamics responses of the nonlinear solids based on explicit time integration. To impose the FSI conditions, a novel, effective and sufficiently general technique via simple linear interpolation is presented based on Lagrangian fictitious fluid meshes coinciding with the moving and deforming solid meshes. In the comparisons to the referenced works including experiments, it is clear that the proposed 3D IS-FEM ensures stability of the scheme with the second order spatial convergence property; and the IS-FEM is fairly independent of a wide range of mesh size ratio.     

Defects in colloidal crystal

Colloidal crystal of polystyrene spheres of 0·53 μm diameter dispersed in water is prepared in a quartz container. Optical microscope, video camera and image processor are used to investigate the defects, their dynamics and the effect of shear in this colloidal crystal. Six layers of the crystal from the surface of the container are imaged sequentially, one after another, and the crystal is reconstructed. FCC structure with (111) planes parallel to the container surface is stable at high volume fraction whereas, at low volume fraction (100) planes also coexist. Vacancies, dislocations, grain boundaries, voids and particle aggregates are identified. The vibration of aggregates is sluggish as compared to that of a single particle. The nearest layer of particles to the container surface is better ordered than those in the interior layers. The container wall provides a stabilizing potential to the nearby layers of particles. The 2-D crystalline-like characteristics of the nearby layers are probed. The colloidal crystal is shear-melted by shaking the container. On resting it, the crystal-flow slows down. The flow is constrained to be parallel to the container surface and occurs along the layer in quantum jumps. During the flow, both (111) and (100) planes coexist in the layer. Flow at low rates anneals the defects created by the shear-melting.     

Droplet Deformation and 2-D/3-D Marangoni Flow Phenomena in Droplets Levitated by Electric Fields

An integrated numerical model is presented for free surface phenomena and Marangoni fluid flows in electrically levitated droplets under both terrestrial and microgravity conditions. The model development is based on the boundary element solution of the Maxwell equations simplified for electrostatic levitation applications and the free surface deformation that is primarily caused from the surface Maxwell stresses resulting from the applied electric fields. The electric and free surface model is further integrated with a finite element model for the surface-tension-induced fluid flows in the levitated droplets. Both 2-D and 3-D fluid flow structures may be developed in the electrically levitated droplets depending on the applied laser heating sources. The integrated model is applied to study the electric field distribution, free surface deformation, and 2-D and 3-D internal fluid flow structures in normal and microgravity for single, symmetric two-beam, four-beam, and six-beam laser heating arrangements. Among these arrangements, the six-beam arrangement with equal heating intensity gives the smallest temperature difference and the smallest maximum velocity.     

Direct numerical simulation of two-dimensional channel flows of electro-rheological fluids

Direct numerical simulation (DNS) of electro-rheological (ER) fluid flows in two-dimensional (2D) electrode channel has been performed by adopting a combined finite element method (FEM). Hydrodynamic interactions between the particles and the fluid are described by the Navier-Stokes equations for the fluid in combination with the equations of motion for the particles, while the multi-body electrostatic interaction is represented by the point-dipole model.ER effects on the plane channel flow for a given pressure gradient have been studied by varying the Mason number and volume fraction of the particles, and interrogating the motion of the particles in views of the formation of ER chain structures, the fluid velocity profile in the channel, and the shear stress versus the shear rate. As the Mason number decreases and volume fraction increases, the tendency that particles align to form chain structures becomes stronger. The yield stress of the ER fluid increases with the electric field intensity and the particle concentration. The quadratic correlation between the yield stress and the electric field intensity has been extracted from the present direct numerical simulation. Lastly, it has been shown that the yield stress linearly increases with the volume fraction in the intermediate range.     

地震载荷下岸桥跳轨相似模型研究

为了提高地震载荷下岸桥跳轨研究结果的准确率问题,基于简化的岸桥跳轨运动方程推导了岸桥原型与模型之间的相似关系,建立了岸桥的动力学相似模型,制作并开展了1∶20岸桥比例模型的振动台试验,研究了不同地震波载荷条件下的试验模型监测点的响应值,分析了不同工况条件下岸桥轮轨压力随时间变化的规律,测试了岸桥比例模型在振动试验中的跳轨响应结果,最后结合有限元方法将试验结果与仿真结果进行了对比。结果表明:振动台试验中模型跳轨响应结果与有限元仿真中出现的跳轨的情况吻合,基于跳轨运动方程建立的相似关系是准确可靠的。该研究为大型结构相似模型设计提供了一种理论参考。