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. (4) Software packages could be developed using the SLS method for modeling complicated DEP phenomena in which multiple disciplines are involved. The unified formulation of SLS method makes it easier to develop a general code for multiple phenomena. The software packages could be used for designing new DEP based devices for bioMEMs applications. (5) The SLS method or other meshfree techniques could be used to develop more sophisticated DEP models which involve more complicated phenomena such as particle interaction, multipole DEP forces, thermal heating, Brownian motion, etc. 206 References References Atluri SN, Cho JY and Kim HG (1999) Analysis of thin beams, using the meshless local Petrov-Galerkin (MLPG) method, with generalized moving least squares interpolation. Comput. Mech., 24: 334-347. Atluri SN and Zhu T (1998) A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech., 22:117-127. Atluri SN and Zhu T (2000) The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics. Comput. Mech., 25, 169-179. Becker FF, Wang XB, Huang Y, Pethig R, Vykoukal J, Gascoyne PR (1995) Separation of human breast cancer cells from blood by differential dielectric affinity. Proc. Natl. Acad. Sci. USA 92, 860-864 Belytschko T, Krongauz Y, Organ D, Fleming M and Krysl P (1996), Meshless method: an overview and recent development. Comput. Meth. Appl. Mech. Eng., 139, 3-47., 74:111-126. Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int. J. Numer. Methods Eng., 37: 229–256. Cai Z, Manteuffel TA, McCormick SF (1997) First-order system least squares for the Stokes equations, with application to linear elasticity. SIAM. J. Numer.Anal. 34:1727-1741. Chang C.L and Jiang B.N. (1990)An error analysis of least-squares finite element method of velocity-pressure-vorticity formulation for stokes problem. Comput. Methods Appl. Mech. Engrg., 84:247-255. Chen DF, Du HJ, Li WH, Shu C (2005) Numerical modeling of dielectrophoresis using a meshless approach. J. Micromech. Microeng., 15:1040-1048. Cheng J, Sheldon EL, Wu L, Heller MJ, O'Connell JP (1998a) Isolation of cultured cervical carcinoma cells mixed with peripheral blood cells on a bioelectronic chip. Analytical Chemistry, 70:2321-2326. Cheng J, Sheldon EL, WuL, Uribe A, Gerrue LO, Carrino J, Heller MJ, O'Connell J P (1998b) Preparation and hybridization analysis of DNA/RNA from E-coli on microfabricated bioelectronic chips. Nature Biotechnology 16: 541-546. Chen JS, Wu CT, Yoon S, and You Y (2001).A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Methods Eng., 50:435-466 207 References Chen JS, Yoon S and Wu CT (2002) Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods, Int. J. Numer. Methods Eng. 53: 2587-2615. Ching HK and Batra RC (2001), Determination of crack tip fields in linear elastostatics by meshless local Petrov-Galerkin (MLPG) method. CMES, 2(2), 273-290. Cho JY, Kim HG and Atluri SN (2001), Analysis of shear flexible beams, using the meshless local Petrov-Galerkin method based on locking-free formulation. Engineering Computations, 18(1/2), 215-240. Fuhr G, Arnold WM, Hagedorn R, Muller T, Benecke W, Wagner B, Zimmermann U (1992) Levitation, holding, and rotation of cells within traps made by high-frequency fields. Biochim. Biophys. Acta, 1108: 215-223. Fuhr G, Schelle T, Muller T, Hitzler H, Monajembashi S, and Creulich KO(1998) Force measurement of optical tweezers in electro-optical cages. App. Phys. A Mater. Sci. Process, 67:385-390. Gascoyne P, Mahidol C. Ruchirawat M, Satayavivad J, Watcharasit P, Becker F (2002) Microsample preparation by dielectrophoresis: Isolation of malaria. Lab on a chip, 2:70–75. Gascoyne PRC, Wang X-B, Huang Y, Becker FF (1997) Dielectrophoretic separation of cancer cells from blood. IEEE Transactions on Industry Applications, 33: 670-678. Gingold RA, Monaghan JJ (1977) Smooth particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 181(2):375-389. Green NG, Morgan H (1997) Dielectrophoretic separation of nano-particles. Journal of Physics D-Applied Physics, 30: L41-L44. Green NG, Morgan H, Milner JJ (1997) Manipulation and trapping of sub-micron bioparticles using dielectrophoresis. J.Biophys. Methods, 35: 89-102. Green NG, Ramos A, Morgan H (2002) Numerical solution of the dielectric and traveling wave forces for interdigitated electrode arrays using the finite element method. Journal of Electrastatics ,56: 235-254. Gu YT, Liu GR (2003) A boundary radial point interpolation method (BRPIM) for 2-D structural analyses. Structural Engineering and Mechanics, 15 (5): 535-550. Hagedorn R, Fuhr G, Muller T and Gimsa J (1992) Traveling-wave dielectrophoresis of microparticles. Electrophoresis, 13:49-54. 208 References Hartley LF, Kaler KVIS, Paul R (1999) Quadrupole levitation of microscopic dielectric particles. Journal of Electrostatics, 46: 233-246. Huang Y, Yang J, Wang XB, Becker FF, Gascoyne PRC (1999) Cutting Edge Communication: the removal of human breast cancer cells from hematopoietic CD34+ stem cells by dielectrophoretic field-flow-fractionation. J. Hematother. Stem Cell Res. (5):481-490. Huang Y, Wang XB, Tame JA and Pethig R (1993) J. Phys. D: Appl. Phys. 26: 1526-1535 Hughes MP, Pethig R., and Wang XB (1996) Dielectrophoretic forces on particles in traveling electric fields. J. Phys. D: Appl. Phys., 26:312-22. Hughes T.J.R. (1987) Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations. Comput. Methods.Appl. Mech. Engrg. 7:1261-1295. Hughes T.J.R., Liu W.K and Brooks A. (1979) Finite element analysis of incompressible viscous flows by the penalty function formulation. J. Comput. Phys. 30:1-75. Jiang BN (1998) The least-squares finite element method – theory and applications in computational fluid dynamics and electromagnetics, Springer, Berlin. Jiang, B. N. (2002), The least-squares finite element method in elasticity- part II: bending of thin plates. Int. J. Numer. Meth. Engng. 54: 1459-1475. Jiang BN, Lin TL and Povinelli LA, (1994) Large-scale computation of incompressible viscous flow by least-squares finite element method, Compu. Methods Appl. Mech. Engrg. 114: 213-231. Jiang, B.N and Wu, J. (2002), The least-squares finite element method in elasticity – part I: plane stress or strain with drilling degrees of freedom. Int. J. Numer. Meth. Engng. 53:621-636. Jones TB (1995) Electomecahnics of Particles. Cambridge University Press, Cambridge. Kansa EJ (1990a) A scattered data approximation scheme with application to computational fluid dynamics. I & II, Comput. Math. Appl., 19:127-161. Kansa EJ (1990b) Multiquadrics-a scattered data approximation scheme with application to computational fluid dynamics. Comput. Math. Appl., 19(8/9): 127-145. 209 References Kansa EJ and Hon YC (2000) Circumventing the ill-conditioning problem with multiquadric radial basis functions: application to elliptic partial differential equations. Comput. Math. Appl., 39:123-137. Kee BBT, Liu GR, Lu C (2007a) A regularized least-square radial point collocation method (RLS-RPCM) for adaptive analysis, Comp. Mech., (Available Online). Kee BBT, Liu GR, Lu C (2007b) A Least-square Radial Point Collocation Method for Adaptive Analysis in Linear Elasticity, Engineering Analysis with Boundary Elements, (Accepted) Kenneth HH, Donald LD, Douglas ES and Ted GB (2001) The Finite Element Method for Engineers . Wiley-Interscience, New York. Krongauz Y and Belytschko T (1996), Enforcement of essential boundary conditions in meshless approximations using finite element. Comput. Methods in Appl. Mech. And Engrg., 131:133-145 Li Y, Liu G.R, Luan MT, Dai KY, Zhong ZH, Li GY, and Han X (2006).Contact analysis for solids based on linearly conforming RPIM, Comput. Mech. 39: 537-554, 2007 Lin H and Atluri SN (2001), Analysis of incompressible Navier-Stokes flows by the meshless MLPG method. Computer Modeling in Engineering & Sciences, 2(2): 117-142. Lin H and Atluri SN (2000), Meshless local Petrov-Galerkin (MLPG) method for convection-diffusion problems. Computer Modeling in Engineering & Sciences, 1(2) :45-60. Liszka T, Orkisz J (1979) The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comp. Struct., 11:83-95. Liszka TJ, Duarte CAM, Tworzydlo WW (1996) Hp-Meshless cloud method, Comput. Methods Appl. Mech. Engrg., 139:263-288. Liu GR and Gu YT (2001) A point interpolation method for two-dimensional solids. Int. J. Numer. Methods Eng. 50 937-951. Liu GR (2003) Mesh Free Methods: Moving beyond the Finite Element Method. CRC Press. Liu GR, Dai KY, Lim KM, Gu YT (2003) A radial point interpolation method for simulation of two-dimensional piezoelectric structures. Smart Materials and Structures, 12:171-180. 210 References Liu GR, Gu YT (2001a), A point interpolation method for two-dimensional solids. International Journal for Numerical Methods in Engineering, 50: 937-951. Liu GR, Gu YT (2001b), A local point interpolation method for stress analysis of two-dimensional solids. Structural Engineering and Mechanics, 11(2): 221-236. Liu GR, Gu YT (2001c) A Local Radial Point Interpolation Method (LRPIM) For Free Vibration Analyses of 2-D Solids. Journal of Sound and Vibration, 246 (1): 29-46 Liu GR, Gu YT (2003a), A matrix triangularization algorithm for point interpolation method. Computer Methods in Applied Mechanics and Engineering, 192(19): 2269-2295 Liu GR, Gu YT (2003b), A meshfree method: Meshfree Weak-Strong (MWS) form method, for 2-D solids. Computational Mechanics, 33(1): 2-14. Liu GR, Gu YT (2005) An Introduction to Meshfree Methods and Their Programming. Springer Dordrecht, The Netherlands. Liu GR, Kee BBT, Lu C (2006a) A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis, Comput. Methods Appl. Mech. Engrg., 195: 4843-4861 Liu GR, Li Y, Dai KY, Luan MT and Xue W (2006b) A Linearly conforming radial point interpolation method for solid mechanics problems, International Journal of Computational Methods, 3(4): 401-428. Liu GR, Liu MB (2003) Smoothed Particle Hydrodynamics: a Meshfree Particle Method. World Scientific. Liu GR, Wu YL, Ding H (2004), Meshfree Weak-Strong (MWS) form method and its application to incompressible flow problems. International Journal for Numerical Methods in Fluids, 46: 1025-1047. Liu GR, Yan L, Wang JG, et al. (2002) Point interpolation method based on local residual formulation using radial basis functions. Struct. Eng. Mech., 14 (6): 713-732. Liu GR, Zhang GY and Dai KY (2005) A linear conforming point interpolation method(LC-PIM) for 2D solid mechanics problems. Int. J. Comput. Methods 2(4) 645-665. Liu GR and Zhang GY (2007) Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM). International Journal for Numerical Methods in Engineering, (Published on line; DOI: 10.1002/nme.2204) 211 References Liu WK, Adee J, Jun S (1993) Reproducing kernel and wavelet particle methods for elastic and plastic problems. Advanced computational methods for material modeling, (Eds. D.J. Benson 180/PVP 268 ASME): 175-190. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. International Journal for Numerical Methods in Engineering, 20: 1081-1106. Lucy L (1977) A numerical approach to testing the fission hypothesis. Astron. J., 82: 1013-1024. Markx GH, Pethig R (1995) Dielectrophoretic separation of cells: Continuous separation. Biotechnol. Bioeng, 45:337-343. Markx GH, Pethig R, Rousselet J (1997) The dielectrophoretic levitation of latex beads with reference to field-flow fractionation. Journal of Physics D (Applied Physics), 30:2470-2477. Markx GH, Talary MS, Pethig R (1994) Separation of viable and non-viable yeast using dielectrophoresis. J.Biotech, 32:29-37 Monaghan JJ (1982), Why particle methods work. SIAM Journal on Scientific and Statistical Computing, 3(4): 422-433 Monaghan JJ (1988), An introduction to SPH. Comput. Phys. Commum, 48 :89-96. Morgan H, Hughes MP, Green NG (1999) Separation of submicron bioparticles by dielectrophoresis. Biophysical Journal, 77:516-525. Morgan H, Izquierdo AG, Bakewell D, Green NG and Ramos A (2001) The dielectrophoretic and traveling wave forces generated by interdigitated electrode arrays: analytical solution using Fourier Series. J. Phys. D: Appl. Phys., 34:1553-61. Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Computational Mechanics, 10: 307-318. Oden J.T. and Jacquotte O.P. (1984) Stability of some mixed finite element methods for Stokesian flows. Comput. Methods.Appl. Mech. Engrg. 43: 231-248. Onate E, Idelsohn S, Zienkiewicz OC, Taylor RL (1996) A finite point method in computational mechanics--Application to convective transport and fluid flow. International Journal for Numerical Methods in Engineering, 39(22): 3839-3866. Park, S.H., Kwon, K.C. and Youn S.K. (2003), A study on the convergence of least-squares meshfree method under inaccurate integration. Int. J. Numer. Meth. Engng. 56: 1397-1419. 212 References Park, S.H. and Youn, S.K. (2001), The least-squares meshfree method. Int. J. Numer. Meth. Engng. 52: 997-1012. Pethig R (1996). Dielectrphoresis: Using Inhomogeneous ac Electrical Fields to Separate and Manipulate Cells, Crit. Rev. Biotechnol. 16:331-48 Pethig R, Huang Y, Wang XB, and Burt JPH (1992) Positive and negative dielectrophoretic collection of colloidal particles using interdigitated castellated microelectrodes. J. Phys. D: Appl. Phys., 25:881-8. Pohl HA (1978) Dielectrophoresis. Cambridge University Press, Cambridge. Powell MJD (1992) The theory of radial basis function approximation in 1990. Advances in Numerical Analysis, W. Light, ed., Oxford: Oxford Science Publications, pp. 105-210. Prabhakar V. and Reddy J.N (2006) Spectral/hp penalty least-squares finite element formulation for the steady incompressible Navier-Stokes equations, J.Comput. Phys. 215:274-297. Reichle C, Muller T, Schnelle T, Fuhr G (1999) Electro-rotation in octopole micro cages. J. Phys. D, 32: 2128-2135. Schaback R (1994) Approximation of polynomials by radial basis functions. Wavelets, image and surface fitting. (Eds. Laurent P.J., Mehaute Le and Schumaker L.L., Wellesley Mass.), 445-453. Schnelle T, Hagedorn R, Fuhr G, Fiedler S, Muller T (1993) 3-dimensional electric-field traps for manipulation of cells-calculation an experimental verification. BioChim. Biophys. Acta, 1157:127-140. Stephens M, Talary MS, Pethig R, Burnett, AK, Mills KI (1996) 1996. The dielectrophoresis enrichment of CD34+ cells from peripheral stem cell harvests. Bone Marrow Trans, 18: 777-782. Talary MS, Burt JPH, Tame JA, and Rethig R (1996) Electromanipulation and Separation of Cells Using Traveling Electric Fields. J. Phys. D: Appl. Phys., 29:2198-2203 Talary MS, Mills KI, Hoy T, Burnett AK, Pethig R (1995) Dielectrophoretic separation and enrichment of CD34+ cell subpopulation from bone marrow and peripheral blood stem cells. Medical & Biological Engineering & Computing, 33: 235-237. Timoshenko S. P. and Goodier J. N. (1970), Theory of Elasticity. 3rd edition, McGraw, New York. 213 References Voldman J, Gray ML, Toner M, Schmidt MA ( 2002) A microfabrication-based dynamic array cytometer. Anal. Chem., 74: 3984-3990. Voldman J, Toner M, Gray ML, Schmidt MA (2003) Design and analysis of extruded quadrupolar dielectrophoretic traps. Journal of Electrastatics, 57: 69-90. Wang JG, Liu GR (2002a) A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering, 54: 1623-1648. Wang JG, Liu GR (2002b) On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Computer Methods in Applied Mechanics and Engineering, 191: 2611-2630. Wang JG, Liu GR, Lin P (2002) Numerical analysis of biot's consolidation process by radial point interpolation method. International Journal of Solids and Structures, 39(6): 1557 - 1573. Wang X, Wang XB, Gascoyne PRC (1997) General expressions for dielectrophoretic force and electrorotational torque derived using the Maxwell stress tensor method. Journal of Electrostatics, 39: 277-295. Wang XB, Huang Y, Burt JPH, Markx G H, Pethig R (1993) Selective dielectrophoretic confinement of bioparticles in potential energy wells. Journal of Physics D (Applied Physics), 26: 1278-1285. Wang XB, Vykoukal J, Becker FF, Gascoyne PRC (1998). Separation of polystyrene microbeads using dielectrophoretic/gravitational field-flow-fractionation. Biophysical Journal, 74: 2689-2701. Wang XB, Yang J, Huang Y, Vykoukal J, Becker FF, Gascoyne PRC, Cell separation by dielectrophoretic field-flow-fractionation. Anal. Chem, 72:832-839. Wang XJ, Wang XB, Becker FF and Gascoyne PRC (1996) A theoretical method of electrical field analysis for dielectrophoretic electrode arrays using Green’s Theorem. J. Phys. D: Appl. Phys, 29:1649-60. Wang XJ, Wang XB, and Gascoyne PRC (1997) General expression for dielectrophoretic force and electrorotational torque derived using the maxwell stress tensor method. J. Electrostat., 39:277-95. Wendland H (1995) Piecewise polynomial positive definite and compactly supported radial basis functions of minimal degree. Adv. Comput. Math. 4:389-396. Yang J, Huang Y, Gascoyne PRC (2000) Differential analysis of human leukocytes by dielectrophoretic field-flow-fractionation. Biophysical Journal, 78: 2680-2689. 214 References Zienkiewicz O.C., Taylor R.L. (2000), The Finite Element Method. 5th edition, Butterworth Heinemann, Oxford. UK. 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