Parametric study of the forces on microspheres held by optical tweezers

Optical-trapping forces exerted on polystyrene microspheres are predicted and measured as a function of sphere size, laser spot size, and laser beam polarization. Axial and transverse forces are in good and excellent agreement, respectively, with a ray-optics model when the sphere diameter is ≥ 10 μm. Results are compared with results from an electromagnetic model when the sphere size is ≤ 1 μm. Axial trapping performance is found to be optimum when the numerical aperture of the objective lens is as large as possible, and when the trapped sphere is located just below the chamber cover slip. Forces in the transverse direction are not as sensitive to parametric variations as are the axial forces. These results are important as a first-order approximation to the forces that can be applied either directly to biological objects or by means of microsphere handles attached to the biological specimen.     

Initial conditions contribution in a BEM formulation based on the convolution quadrature method

This work presents a two‐dimensional boundary element method (BEM) formulation for the analysis of scalar wave propagation problems. The formulation is based on the so‐called convolution quadrature method (CQM) by means of which the convolution integral, presented in time‐domain BEM formulations, is numerically substituted by a quadrature formula, whose weights are computed using the Laplace transform of the fundamental solution and a linear multistep method. This BEM formulation was initially developed for scalar wave propagation problems with null initial conditions. In order to overcome this limitation, this work presents a general procedure that enables one to take into account non‐homogeneous initial conditions, after replacing the initial conditions by equivalent pseudo‐forces. The numerical results included in this work show the accuracy of the proposed BEM formulation and its applicability to such kind of analysis. Copyright © 2006 John Wiley & Sons, Ltd.     

The calculation of drag on nano‐cylinders

The study of molecular flows at low Knudsen numbers (～0.1–0.5), over nano‐scaled objects of 20–100 nm size is becoming an important area of research. The simulation of fluid–structure interaction at nano‐scale is important for understanding the adsorption and drag resistance characteristics of nano‐devices in the fields of drug delivery, surface cleaning and protein movement. A novel formulation has been proposed that calculates localised values for both the kinetic and configurational parts of the Irving–Kirkwood stress tensor at given fixed positions within the computational domain. Macroscopic properties, such as streaming velocity, pressure and drag coefficients, are predicted by modelling the fluid–structure interaction using a moving least‐squares method. The gravitation‐driven molecular flow is examined over three different cross‐sectional shapes—i.e. diamond‐, circular‐ and square‐shaped cylinders—confined within parallel walls and has been simulated for rough and smooth surfaces. The molecular dynamics formulation has allowed, for the first time, the calculation of localised drag forces over nano‐cylinders. The computational simulation has shown that existing methods, including continuum‐based approaches, significantly underestimate drag coefficients over nano‐cylinders. The proposed molecular dynamics formulation has been verified on simulation based tests, as experimental and analytical results are unavailable at this scale. Copyright © 2016 John Wiley & Sons, Ltd.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A 3-D indirect boundary element method for bounded creeping flow of drops

The simulation of the flow of emulsions in porous media presents formidable challenges, due to the extremely complex evolving geometry. Methods based on boundary integral equations, suitable for creeping flows, reduce the effort dedicated to geometry representation, but can become computationally expensive. An efficient indirect boundary integral formulation representing deformable drops in a bounded Stokes flow, resulting in a set of Fredholm integral equations of the second kind, is presented. The boundary element method (BEM) based on the formulation employs an accurate numerical integration scheme for the singular kernels involved, an effective and accurate curvature and normal calculation method, and an adaptive remeshing method to simulate interfacial deformation of drops. Two benchmark problems are used to assess the accuracy of the method, and to investigate its behavior for large problems. The method is found to provide accurate results combined with well-posedness, making it suitable for use in accelerated fast multipole method algorithms.     

The first-kind and the second-kind boundary integral equation systems for solution of frictionless contact problems

An original approach to the numerical solution of displacement boundary integral equation (BIE) and traction hypersingular boundary integral equation (HBIE) by the boundary element method (BEM) for contact problems is given. The main point is to show, how the contact conditions are used to formulate the first-kind and the second-kind BIE systems in the case of frictionless two-body elastic contact. The solution of the first-kind BIE is performed by symmetric Galerkin BEM; the second-kind BIE is solved by an appropriate collocation BEM. The contact problem in itself is solved by the method of subsequent approximations of contact region. Both forms of BIE system are compared in several numerical examples. This comparison is made for different kinds of contact problem. The major emphasis is put on the evaluation of contact pressure. The obtained results are compared with referenced numerical and with the analytical ones.     

A coupled BEM‐flexibility force method for bending analysis of internally supported plates

This paper presents a new method for the analysis of plates in bending with internal supports. The proposed method can be regarded as an extension of the well‐known force method (the flexibility matrix method) in the matrix analysis of structures. The solution is performed through two phases: the released plate phase, in which the plate is released from all internal supports and solved using the Boundary Element Method (BEM). The effect of internal supports is considered in the second phase, where a series of unit virtual loads is placed instead of the unknown redundant reactions at internal supports. The flexibility matrix is formed and compatibility of deformations at the locations of internal supports is satisfied. Hence, the corresponding system of equations is solved for the unknown redundant forces at internal supports. The final solution of the problem consists of the summation of two phases: the released plate phase and the cases of virtual unit loads phase. An efficient solution algorithm is developed to solve both phases simultaneously. The main advantages of the present formulation are: (1) the present formulation increases the versatility of the BEM as it allows the re‐usability of standard BEM codes for solution of plates in bending to be used in solving problems having internal supports, with even no modifications; and (2) the two solution phases are completely uncoupled; therefore it is easy to trace behaviour of the plate due to failure of one or more of the internal supports without re‐analysis. Several numerical examples are analysed. The results are compared to those of analytical and finite element models to demonstrate the accuracy and the validity of the present formulation. The present formulation is used also to study the differences between the finite element and boundary element modelling for building slabs. Copyright © 2002 John Wiley & Sons, Ltd.     

Flow past a cylinder with and without an appurtenance

This paper constitutes a numerical study of flow around a cylinder for cases with and without an appurtenance, and throughout a range of Reynolds numbers. This is achieved by solving the two-dimensional evolutionary Navier-Stokes equations using a pressure-velocity finite element method. The particular form of scheme employed is a combination of Taylor-Galerkin and pressure-correction methods. This scheme delivers second-order accuracy in time and introduces three fractional stages within each time-step. One of the practical applications of this work is the prediction of flows around gas and oil platforms at sea, and of concern to off-shore engineers are the effects of adding a sacrificial anode to the cylindrical legs of a platform. Such an appurtenance added to the basic cylindrical shape induces different drag forces on the legs depending on the angle of attack of the motion of the sea relative to the appurtenance location. Unidirectional and oscillatory flow solutions are reported, concentrating on flow patterns and forces experienced on the body surface, and the vortex shedding process.     

Numerical simulation of dispersed gas-liquid flows

This paper reviews the available information on numerical simulation of dispersed gas-liquid flows. Emphasis is on informing the reader about various aspects of constructing simulation models rather than giving an exhaustive literature review. The information is organised in a way so as to provide answers to the following questions: how to formulate model equations? how to select suitable algorithms and numerical techniques to solve these model equations? how to translate these into workable computer codes? how to use such codes for simulating flows in industrial equipment? Though greater emphasis is given to dispersed gas-liquid flows, the methodology can be applied to any multi-phase problem. Special features of multi-phase flow simulation over single-phase simulation are highlighted. A case of gas-liquid flow in a bubble column is presented as an illustration for the general methodology. The simulated mean flow field agrees reasonably with the experimental data. Properly validatedcfd codes thus can serve as a useful tool for design engineers of multi-phase systems. Some of the common pitfalls in using simulation codes for design are also discussed. This review is expected to give an overall idea about the present state-of-art of two-phase simulation in industrial equipment.     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two‐dimensional elasticity

For a plane elasticity problem, the boundary integral equation approach has been shown to yield a non‐unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not embedded in the influence matrix for any case. Copyright © 2002 John Wiley & Sons, Ltd.     

Boundary element analysis of three‐dimensional mixed‐mode thermoelastic crack problems using the interaction and energy domain integrals

A three‐dimensional boundary element method (BEM) implementation of the interaction integral methodology for the numerical analysis of mixed‐mode three‐dimensional thermoelastic crack problems is presented in this paper. The interaction integral is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains, temperatures and derivatives of displacements and temperatures at internal points can be evaluated using the appropriate boundary integral equations. Several examples are analysed and the results compared with those available in the literature to demonstrate the efficiency and accuracy of the implementation to solve straight and curved crack‐front problems. Copyright © 2007 John Wiley & Sons, Ltd.     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients

This paper proposed numerical methods to determine the multiple stress singularities (including the oscillatory stress singularities) and the related stress intensity coefficients, by the use of common numerical solutions (stresses or displacements) obtained by an ordinary numerical tool such as finite element method (FEM) or boundary element method (BEM). To verify the efficiency of the present methods, two models of bonded dissimilar materials under the plane strain state are analyzed by BEM, and the orders of the stress singularities and the related stress intensity coefficients are examined numerically. The results show that all the orders of the stress singularities at an interface edge can be determined precisely by the present method, and the related stress intensity coefficients can be determined by the extrapolation method with a very good linearity. It is found that the methods presented in this paper are very simple and efficient. Moreover, they can be easily extended to any singular problem.     

Boundary integral equation analysis for radiation and scattering of elastic waves in three dimensions

Abstract

In this paper, the boundary integral equation (BIE) method is employed to investigate the radiation and scattering of time‐harmonic elastic waves by obstacles of arbitrary shape embedded in an infinite medium. Based on the vector BIE, entirely free of Cauchy principal value integrals, an efficient numerical scheme using quadratic isoparametric boundary elements is proposed. Furthermore, the difficulty of non‐uniquess of a solution inherent with BIE formulations for exterior elastodynamic problems is studied numerically and analytically. The counterparts of the combined Helmholtz integral formulation method for elastodynamics together with the least‐square or Lagrange‐multiplier technique are derived and applied to overcome this difficulty successfully. In addition, the elastic‐wave fields radiated or scattered by either a spherical cavity or a rigid sphere in an infinite medium are calculated and the results are compared with the analytical solutions to demonstrate the accuracy and versatility of the proposed numerical scheme.     

风雪流滞空颗粒受力及运动行为的数值模拟

为研究稀疏风雪流系统下滞空颗粒在介质中的变加速运动行为及非线性气动特性,以运动学微分方程为基础,建立了微观层面下不同粒径、密度颗粒的三维动网格计算模型。采用时间-空间离散的数值积分方法,求解了小时空尺度下,单个雪颗粒在静止空气及梯度风场中的自由沉降与强迫运动问题。通过对比不同颗粒参数及流场环境的模拟结果发现：颗粒自由沉降所能达到的终端线速度及对应稳定时间均随平均粒径及密度的增大而增大,粒径确定的情况下,相同时间内大密度颗粒沉降距离相对更远,自由沉降初期的非线性变速运动行为可近似考虑为小时空尺度问题;剪切流条件下,较小的风速梯度可能引起颗粒在来流及自重方向运动速度的波动,所受气动外力的改变总是滞后于运动速度的变化,强风环境下颗粒将具有更好的流场跟随性,其非线性变速阶段仍可视为小时空尺度问题。滞空雪颗粒在流场中的运动行为基本满足多相流理论的局部短时空尺度均衡假定。     

An improved moving‐least‐squares reconstruction for immersed boundary method

The current work presents an improved immersed boundary method based on the ideas proposed by Vanella and Balaras (M. Vanella, E. Balaras, A moving‐least‐squares reconstruction for embedded‐boundary formulations, J. Comput. Phys. 228 (2009) 6617–6628). In the method, an improved moving‐least‐squares approximation is employed to build the transfer functions between the Lagrangian points and discrete Eulerian grid points. The main advantage of the improved method is that there is no need to obtain the inverse matrix, which effectively eliminates numerical instabilities caused by matrix inversion and reduces the computational cost significantly. Several different flow problems (Taylor‐Green decaying vortices, flows past a stationary circular cylinder and a sphere, and the sedimentation of a free‐falling sphere in viscous fluid) are simulated to validate the accuracy and efficiency of the method proposed in the present paper. The simulation results show good agreement with previous numerical and experimental results, indicating that the improved immersed boundary method is efficient and reliable in dealing with the fluid–solid interaction problems. Copyright © 2015 John Wiley & Sons, Ltd.     

3D multidomain BEM for a Poisson equation

This paper deals with the efficient 3D multidomain boundary element method (BEM) for solving a Poisson equation. The integral boundary equation is discretized using linear mixed boundary elements. Sparse system matrices similar to the finite element method are obtained, using a multidomain approach, also known as the ‘subdomain technique’. Interface boundary conditions between subdomains lead to an overdetermined system matrix, which is solved using a fast iterative linear least square solver. The accuracy, efficiency and robustness of the developed numerical algorithm are presented using cube and sphere geometry, where the comparison with the competitive BEM is performed. The efficiency is demonstrated using a mesh with over 200,000 hexahedral volume elements on a personal computer with 1 GB memory.     

Crack–surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two‐symmetric cracks emanating from a circular hole in an infinite plate for use in strip‐yield crack‐closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral‐equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary‐element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack‐tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral‐equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life‐prediction methods based on the strip‐yield crack‐closure models.     

Two‐fluid model with interface sharpening

Two‐fluid models are applicable for simulations of all types of two‐phase flows ranging from separated flows with large characteristic interfacial length scales to highly dispersed flows with very small characteristic interfacial length scales. The main drawback of the two‐fluid model, when used for simulations of stratified flows, is the numerical diffusion of the interface. Stratified flows can be easily and more accurately solved with interface tracking methods; however, these methods are limited to the flows, that do not develop into dispersed types of flows. The present paper describes a new approach, where the advantage of the two‐fluid model is combined with the conservative level set method for interface tracking. The advection step of the volume fraction transport equation is followed by the interface sharpening, which preserves the thickness of the interface during the simulation. The proposed two‐fluid model with interface sharpening was found to be more accurate than the existing two‐fluid models. The mixed flow with both: stratified and dispersed flow, is simulated with the coupled model in this paper. In the coupled model, the dispersed two‐fluid model and two‐fluid model with interface sharpening are used locally, depending on the parameter which recognizes the flow regime. Copyright © 2010 John Wiley & Sons, Ltd.