A novel meshfree smoothed least squares(SLS) method with applications to dielectrophoresis simulations

Acknowledgements Acknowledgements I would like to express my deepest gratitude and sincerest appreciation to my supervisor, Professor Liu Gui-Rong, for his invaluable guidance, dedicated support and continuous encouragement throughout my four years Ph.D. study. His passion and enthusiasm in research has inspired me enormously and will continue to influence me for a life time. I would also like to extend my gratitude to my co-supervisor, Assistant Professor Li Hua for his great help and valuable guidance in my research work. Many thanks are conveyed to my fellow colleagues and friends in Center for ACES, Dr. Gu Yuan Tong, Dr. Dai Keyang, Dr. Zhang Guiyong, Dr. Zhao Xin, Dr. Deng Bin, Mr. Li Zirui, Mr Bernard Kee Buck Tong, Mr. Zhang Jian, Mr. Khin Zaw, Ms. Chen Yuan, Mr. Trung, and Mr. George Xu. I would like to thank them all for their helpful discussions, constructive suggestions, as well as their inspirations and encouragement throughout the course of my Ph.D study. I sincerely appreciate their friendship and support. I am grateful to every one of my family members, my parents, my younger sister and younger brother, for their continuous support and encouragement which made my Ph.D years meaningful and happy. Last but not least, I must thank the National University of Singapore for granting me research scholarship. Many thanks are due to Mechanical department and Center for ACES for their material support to every aspect of this work. i Table of Content Table of contents Acknowledgements i Table of contents i Summary vi Nomenclature .ix List of Figures .xi List of Tables . xviii Chapter 1.1 Introduction .1 Background 1.1.1 Meshfree methods 1.1.2 Classification of meshfree method .6 1.1.3 Dielectrophoresis background .8 1.2 Literature review 10 1.2.1 A review of meshfree methods 10 1.2.1.1 SPH and RKPM method 10 1.2.1.2 The EFG method 11 1.2.1.3 The MLPG method 12 1.2.1.4 Point interplolation method (PIM) .12 1.2.2 1.3 Studies of dielectrophoresis .13 A review of meshfree shape functions .17 1.3.1 Moving least-squares (MLS) approximation .18 1.3.1.1 Formulation procedure of MLS .18 1.3.1.2 Weight functions 22 i Table of Content 1.3.1.3 1.3.2 Properties of MLS shape functions 24 Polynomial point interpolation method (Polynomial PIM) .25 1.3.2.1 Formulation procedure of Polynomial PIM .25 1.3.2.2 Properties of polynomial PIM Shape Functions 28 1.3.2.3 Techniques for overcoming singularity in moment matrix 31 1.3.3 Radial point interpolation method (RPIM) 33 1.3.3.1 Formulation procedure of RPIM 33 1.3.3.2 Property of RPIM shape function 36 1.3.3.3 Implementation Issues .38 1.4 Objectives of the thesis 39 1.5 Organization of the thesis 41 Chapter Development of a novel meshfree smoothed least-squares (SLS) method .45 2.1 Introduction 45 2.2 Meshfree smoothed least-squares (SLS) formulation 47 2.2.1 General least-squares formulations 48 2.2.2 Gradient smoothing 50 2.3 Numerical Examples 52 2.3.1 One-dimensional problems 52 2.3.1.1 Convection-diffusion problem .52 2.3.1.2 Pure convection problem .54 2.3.2 2.4 Two-dimensional problems 56 Remarks .61 Chapter Validation of the developed meshfree smoothed least-squares (SLS) method for linear elasticity 72 3.1 Introduction 72 ii Table of Content 3.2 The SLS formulation for linear elasticity problem 73 3.3 Elasticity problems .77 3.3.1 2-D Standard patch test 77 3.3.2 Cantilever beam subjected to a parabolic shear traction at the end .78 3.3.3 An infinite plate subjected to uniaxial traction along horizontal direction …………………………………………………………………… 80 3.4 Remarks .82 Chapter Validation of the developed meshfree smoothed least-squares (SLS) method for steady incompressible flow 92 4.1 Introduction 92 4.2 The Navier-Stokes equations in the velocity-pressure-vorticity formulation …………………………………………………………………………… 94 4.3 The SLS formulation for Navier-Stokes equations 97 4.4 Steady incompressible flow problems .99 4.4.1 A model problem for Stokes equations 99 4.4.2 Driven cavity flow problem for Stokes equations .101 4.4.3 Driven cavity flow problem for Navier -Stokes equations 102 4.4.4 Backward-facing step flow problem 103 4.5 Remarks .103 Chapter Application of the meshfree smoothed least-squares (SLS) method for dielectrophoresis 117 5.1 Introduction 117 5.2 Dielectrophoresis theory 118 5.3 Meshfree smoothed least-squares formulation for dielectrophoresis 119 5.4 Dielectrophoresis simulation .122 iii Table of Content 5.5 Remarks .126 Chapter Simulation of an extruded quadrupolar dielectrophoretic trap .133 6.1 Introduction 133 6.2 Radial point collocation method (RPCM) .134 6.3 Meshless finite difference method .138 6.4 Simulation of extruded quadruple trap 143 6.4.1 Governing equations and boundary conditions 143 6.4.2 Determination of dielectrophoretic forces .145 6.4.3 Determination of hydrodynamic forces .146 6.4.4 Determination of the total resultant force 147 6.4.5 Validation with experimental results 147 6.4.5.1 Comparison between RPCM and MFD .148 6.4.5.2 Comparison between numerical prediction and experimental results ……………………………………………………………………148 6.4.6 6.5 Results and discussion .150 6.4.6.1 Results for resultant force field 150 6.4.6.2 Variation of holding characteristic with trap geometry 152 6.4.6.3 Variation of holding characteristic with particle radius .154 6.4.6.4 Variation of holding characteristic with Clausius-Mossotti factor 155 Remarks .155 Chapter Simulation of an interdigitated dielectrophoretic array 166 7.1 Introduction 166 7.2 Additional dielectrophoresis theories 167 7.3 Linearly conforming point interpolation method (LC-PIM) .169 iv Table of Content 7.3.1 Node selection 169 7.3.2 Gradient smoothing 171 7.3.3 Variational form .173 7.4 Results and discussion .175 7.4.1 Simulation of the DEP array 176 7.4.1.1 Linear potential change in the gap .177 7.4.1.2 Exact boundary condition in the gap .178 7.4.2 Simulation of the traveling wave DEP array .179 7.4.2.1 7.4.3 7.5 Study of the traveling wave DEP array 179 Simulation results using RPIM shape function 180 Remarks .182 Chapter Conclusion and future work .201 8.1 Conclusion remarks .201 8.2 Recommendations for future research .205 References 207 Publications arising from thesis 216 v Summary Summary Mesh-based numerical methods, such as finite element method (FEM), and finite difference method (FDM), have been the primary numerical techniques in engineering computations. Due to mesh related problems of these methods, a new group of numerical techniques called meshfree methods have been proposed and developed in recent years. Many different methods and techniques have been developed for applications in different engineering fields. It has been a standard practice to employ different numerical schemes for different types of differential equations in engineering problems. This thesis focuses on the development and application of a unified meshfree method applicable for all types of differential equations that govern practical engineering problems. The objectives of the present study are two-fold: One is to develop new meshfree method with a unified formulation so that it can be potentially applied to all engineering problems; the other is to apply the developed and existing meshfree methods to simulations of dielectrophoresis (DEP) based devices, which have attracted great attention in recent Micro-Electro-Mechanical Systems (MEMS) researches. The first contribution of this thesis is development of the meshfree smoothed least-squares (SLS) method based on first-order least-squares formulation. The meshfree SLS method uses a unified formulation for all types of partial differential equations: elliptic, parabolic, hyperbolic or mixed. As long as the equations are well vi Summary posed and have a unique solution, the SLS method can always produce a good approximate solution. The properties of the SLS method have been studied in details. The SLS method is found particularly effective for solving non-self-adjoint system such as the convection dominated problem, which is difficult to solve by conventional Galerkin methods. The SLS method always leads to symmetric positive-definite matrices which can be efficiently solved by iterative methods. Using the SLS method, no special treatments, such as upwinding, artificial dissipation, staggered grid or non-equal-order elements, operator-preconditioning, etc are needed. In the second part, the SLS method is devoted to numerical analysis of various engineering problems, including linear elastic problems, incompressible steady flow problem, and dielectrophoresis problem, etc. It is found that the SLS method achieves better accuracy and convergence rate, comparing with other methods based on Galerkin formulation. The SLS method is based on first-order least-squares method, so that the primary variables and the derivatives can be solved simultaneously and with the same order of accuracy. This unique feature is of very importance for many practical problems where it is essential to obtain accurate solutions in the derivatives, such as strain and stress in elasticity problems, flux in fluid problems. The last part of the thesis deals with simulations of DEP based systems using meshfree techniques. A strong-form meshfree method termed radial point collocation method (RPCM) is used to simulate the extruded quadrupolar DEP trap. Compared to weak-form methods, strong-form methods are easy to implement and have lower computational cost. The model developed is able to approximate the strength of the trap, and it can also be used for design optimization purpose. The model is validated vii Summary with good accuracy by comparing with experimental data. Another meshfree technique, linear conforming point interpolation method (LC-PIM) is used for simulation of the dielectrophoretic array as well as the traveling wave dielectrophoretic array. LC-PIM has been found to be very effective to capture the high gradient feature of the electric field, and can produce accurate results for derivatives of the shape functions, which are important for computing the DEP forces in DEP related simulations. The results have been compared with the analytical solution obtained using Fourier series analysis, good accuracy has been demonstrated. viii Nomenclature Nomenclature a Coefficient vector A Linear differential operator B Boundary algebraic operator dc Characteristic length (average nodal spacing) E Young’s modulus εm Permittivity of medium εp Permittivity of particle fCM Clausius-Mossotti (CM) factor f Force vector K Stiffness matrix L( ), B( 　 ) Differential operator L (Ω) Hilbert space n Vector of unit outward normal n Number of supporting nodes N(x) Vector of shape functions p ( x) Polynomial basis function Pm Polynomial moment matrix q Shape parameter of MQ radial basis function r Distance rw Size of weight function domain R Residual function R (x) Radial basis function RQ Moment matrix of radial basis function ix Chapter Conclusion and Future Work Chapter Conclusion and future work 8.1 Conclusion remarks This study has focused on two main aspects. One is to develop a universal meshfree method which can be used to solve all types of partial differential equations using one unified formulation. Another is to use meshfree techniques to develop modeling tools for dielectrophoresis simulations. Through the current research, the following conclusions can be drawn. (1) The SLS method developed in this thesis is a universal meshfree method can be used for solving any first-order partial differential equations. Many practical problems are governed by first-order system, such as the convective transport problems in fluid dynamics and Maxwell equations in electromagnetic. However, it is difficult to deal with first-order differential operators which are often non-self-adjoint by conventional Galerkin based methods, because Galerkin methods generally lead to non-symmetric matrices for first-order systems. For other problems governed by high-order partial differential equations, we can always turn the equations to first-order by introducing the 201 Chapter Conclusion and Future Work derivatives as dual variables, such that the primary variables and the derivatives can be solved simultaneously and with the same order of accuracy. In many of the practical problems, it is essential to obtain the derivatives accurately, such as strain and stress in elasticity problems, flux in fluid problems. (2) The SLS method is based on least-squares formulation, and always leads to symmetrical and positive-definite linear system equations which can be efficiently solved by iterative methods, such as preconditioned conjugate gradient method. Consequently, large-scale and three-dimensional problems can be solved efficiently using the SLS method. (3) The SLS method has a unified formulation in one mathematical framework for numerical solution of all types of partial differential equations regardless whether the equations are elliptic, parabolic, hyperbolic or mixed. As long as the equations have a unique solution, the SLS method always gives good approximated solutions. Since the SLS method uses a unified formulation for approximate solution of differential equations governing various physical phenomena, one-algorithm or one-code for concurrent analysis of different disciplines can be developed. The SLS method can be programmed systematically so that for a new application, one needs only add simple subroutines to supply the coefficients, the load vector, and the boundary 202 Chapter Conclusion and Future Work conditions for the first-order system. (4) When the SLS method is employed, special treatments, such as upwinding, artificial dissipation, staggered grid or non-equal-order elements, operator-preconditioning, etc are unnecessary. (5) The SLS method has been applied to various engineering problems in this thesis to demonstrate the above-mentioned advantages. The SLS method has been used to solve the convection dominated problem with excellent results and no special treatment is needed. Other conventional methods failed to deal with this type of problem without using any special treatments. The SLS method has been further verified for applications in linear elasticity problem. It is proven that the SLS method is effective and accurate for solving linear elasticity problems and it can avoid the incompressible locking problems, which is very difficult to deal with using other meshfree methods. The SLS method can be naturally applied for solving steady incompressible Navier-Stokes equations in the velocity-pressure-vorticity form. Since the resulting linear system is symmetric and positive-definite, the SLS method is a good method for large scale computations in fluid dynamics. (6) Since the SLS method uses a unified formulation for all types of partial differential equations, it is a great method to develop modeling tools for 203 Chapter Conclusion and Future Work simulation of dielectrophoresis based systems, which involves multiple physical phenomena in fluid dynamics, electromagnetics, and thermodynamics. It is possible to develop one general code for simulation of all phenomena in the DEP systems. One needs only provide the different coefficients and boundary conditions of the first-order system for different phenomenon. The SLS method has been applied to DEP simulation in a simple case. The most superior advantage of using SLS method is that the electric field intensity can be obtained simultaneously with the electric potential and with the same order of accuracy. (7) In SLS method, there are more variables involved, so the computational time is higher. For problems in which the dual variables are not important, SLS method loses in efficiency. (8) Meshfree technique has superior advantages over traditional FEM in developing modeling tools for DEP device due to its unnecessary of remeshing during design optimizations. The radial point collocation method has been used for modeling extruded quadrupolar trap. The model developed in this work enables us to understand more detailed information on the behavior of the particles within the extruded quadrupolar trap. It simulates the interaction among the particle, electric field and fluidic field. The model is able to approximate the strength of the trap, which is one of the main concerns in practical DEP device 204 Chapter Conclusion and Future Work design. It can also be used for design optimization purpose. (9) Another meshfree method, LC-PIM has been used for simulation of the dielectrophoretic array as well as the traveling wave dielectrophoretic array, which are most commonly used in DEP researches. LC-PIM shows superior advantage over the conventional finite element method in solving DEP problems, due to its capability of computing the second derivatives with good accuracy. LC-PIM can be used as a robust numerical tool in various DEP designs and modeling. 8.2 Recommendations for future research Based on the work done in this thesis, the following recommendations are made for future research: (1) The development of Meshfree methods is still in its infant stage, although many meshfree methods have been developed in the past decades, there are still rooms for improving the current methods to develop more efficient and robust methods. (2) The SLS method developed in this thesis has many attractive properties. It should be extended to the applications in more engineering fields, such as nonlinear problems in solid mechanics, including material and geometrical nonlinear problems (3) The application of SLS method should be extended to large scale 205 Chapter Conclusion and Future Work problems and three-dimensional problems, since the fast interative solvers can be used to solve the linear system equations obtained from the SLS method. 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Zhang GY, Liu GR, Wang YY, Huang HT, Zhong ZH, Li GY and Han X (2007) A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems. International Journal for Numerical Methods in Engineering, 72: 1524-1543. Zhang GY, Liu GR (2007). An efficient adaptive analysis procedure for certified solutions with exact bounds of strain energy for elasticity problems. Finite Elements in Analysis and Design, (submitted) 215 Publications arising from thesis Publications arising from thesis Journal papers: 1. Song CX, Liu GR, Li H (2006) Simulation of an extruded quadrupolar dielectrophoretic trap using meshfree approach, Engineering analysis with boundary elements, 30 (11): 994-1005. 2. Song CX, Liu GR, Li H, Han X (2007). Numerical simulation of dielectrophoresis using linearly conforming RPIM, Journal of computational physics (Submitted) 3. Song CX, Liu GR, Li H, Zhang GY (2007). A meshfree smoothed least-squares (SLS) method for solving first order partial differential equations, Computers & structures (Submitted) 4. Song CX, Liu GR, Li H, George Xu (2007). A meshfree smoothed least-squares (SLS) method for simulating steady incompressible viscous flow, International journal for numerical methods in fluids (Revised) 5. Kee BBT, Liu GR, Song CX, Zhang J, Zhang GY (2007) A study on the effect of the number of local nodes for meshfree methods based on radial basis functions, Computer modeling in engineering & science (Submitted) 6. Zhang GY, Liu GR, Nguyen TT, Song CX, Han X, Zhong ZH, Li GY (2007), The upper bound property for solid mechanics of the linearly conforming radial point interpolation method (LC-RPIM). International journal of computational methods, 4(3): 521-541. Book chapter: Liu G.R and Song CX, (2008) Numerical simulation of BioMEMS with dielectrophoresis, in: Advances in multiphysics simulation and experimental testing of MEMS edited by: Frangi A., Cercignani C, Mukherjee S., Aluru N., Imperial College Press. Conference paper: Song CX, Liu GR, Li H, A meshfree smoothed least-squares (SLS) method for solving first order partial differential equations, International conference on computational methods, Hiroshima, Japan, April 2007. 216 [...]... local behavior of the solution One of the advantages of the FEM is that it is essentially independent of geometry, and many domains of complex shapes can be handled by the FEM with ease The clear structure of the FEM makes it possible to construct general purpose software, many commercial software packages are made available nowadays e.g ABAQUS, ANSYS, etc The FEM has a solid mathematical basis due to. .. separating such distinctly different particles as bacteria from blood cells, but inadequate for many mammalian cell applications b) Field-flow fractionation Field-flow fractionation (FFF) (Wang et al., 1998; Huang et al., 1999; Yang et al., 2000; Wang et al., 2000; Muller et al., 2000; Markx and Tethig, 1995) is a family of methods in which force fields are applied to particles to position them characteristically... manipulating particles in micrometer scale, DEP has a wide variety of applications in micro electromechanical system (MEMS), especially in biomedical field It has been used for trapping, focusing, translation, fractionation of chemical and biological particles in fluid medium It is particularly suitable for applications at microscale fluidic device that can be fabricated by inexpensive fabrication methods... Hugh et al., 1996) that by changing the frequency of the traveling field, it is possible to switch between conventional and travelling wave DEP to enhance separation The discovery and utilization of traveling wave dielectrophoresis have received a great deal of attention in laboratory-on -a- chip systems application, since the force exerted can be made to act in a direction parallel to the plane of the... post-processing techniques are required to restore the accuracy of the derivatives 3) Difficulty in adaptive analysis Adaptive analysis is an important step in numerical analysis to improve the accuracy of the solution In using the FEM, re-meshing is necessary at each adaptive process to ensure the proper connectivity, and add additional expensive computational cost The mapping of field variables between meshes... found in Masuda’s work (Masuda et al., 1987; Masuda et al., 1988) The traveling fields were generated by applying three-phase voltages of frequency 0.1-100 Hz to a series of bar-shaped electrodes Masuda et al proposed that such traveling fields could eventually find application in the separation of particles according to their size or electrical charge It has been shown in Huang’s study (Huang et al., 1993)... that traveling fields of frequency between 1 kHz and 10 MHz can be used to manipulate yeast cells and to separate them selectively when they are mixed with bacteria It was shown by Fuhr (Fuhr et al., 1991; Hagedorn et al., 1992) traveling fields of frequency between 10 kHz and 30 MHz are capable of imparting linear motion to pollen and cellulose particles It has been shown in later work (Talary et al.,... 2004) Comparing to strong-form methods, meshfree weak-from methods are more stable and accurate, and have been applied successfully to problems in many engineering fields such as solid and structure mechanics In meshfree weak-form methods, the Neumann boundary conditions can be imposed naturally However, most of the above-mentioned weak-form methods still have to use a background mesh for 6 Chapter 1... methods DEP methods are applicable to purification and characterization of a wide variety of biological and clinical components 9 Chapter 1 Introduction 1.2 Literature review 1.2.1 A review of meshfree methods 1.2.1.1 SPH and RKPM method The smooth particle hydrodynamics (SPH) method (Lucy, 1977; Gingold and Managhan, 1977) is one of the earliest developed meshfree methods, which was originally used... Flow separation is the simplest method of practical dielectrophoretic separation The separation is carried out in a chamber which has an electrode array on the bottom, and is enclosed by sides and a lid There is a single inlet and outlet The mixture that is to be separated is pumped into the chamber by using a syringe pump Then the electrodes are energized and the mixture will be separated due to the . made my Ph.D years meaningful and happy. Last but not least, I must thank the National University of Singapore for granting me research scholarship. Many thanks are due to Mechanical department. series analysis, good accuracy has been demonstrated. Nomenclature ix Nomenclature a Coefficient vector A Linear differential operator B Boundary algebraic operator c d Characteristic. better accuracy and convergence rate, comparing with other methods based on Galerkin formulation. The SLS method is based on first-order least- squares method, so that the primary variables and the