Shantanu S. Mulay

Meshless methods and their numerical properties

Introduction Formulation of Classical Meshless Methods Introduction Fundamentals of Meshless Methods Common Steps of Meshless Method Classical Meshless Methods Summary Recent Development of Meshless Methods Introduction Hermite-Cloud Method Point Weighted Least-Squares Method Local Kriging (LoKriging) Method Variation of Local Point Interpolation Method (vLPIM) Random Differential Quadrature (RDQ) Method Summary Convergence and Consistency Analyses Introduction to Convergence Analysis Development of Superconvergence Condition Convergence Analysis Application of RDQ Method for Solving Fixed-Fixed and Cantilever Microswitches under Nonlinear Electrostatic Loading Introduction to Consistency Analysis of RDQ Method Consistency Analysis of Locally Applied DQ Method Effect of Uniform and Cosine Distributions of Virtual Nodes on Convergence of RDQ Method Summary Stability Analyses Introduction Stability Analysis of First Order Wave Equation by RDQ Method Stability Analysis of Transient Heat Conduction Equation Stability Analysis of the Transverse Beam Deflection Equation Summary Adaptive Analysis Introduction Error Recovery Technique in ARDQ Method Adaptive RDQ Method Convergence Analysis in ARDQ Method Summary Engineering Applications Introduction Application of Meshless Methods to Microelectromechanical System Problems Application of Meshless Method in Submarine Engineering Application of RDQ Method for 2-D Simulation of pH-Sensitive Hydrogel Summary Appendix A: Derivation of Characteristic Polynomial PHI(Z) Appendix B: Definition of Reduced Polynomial PHI1(Z) Appendix C: Derivation of Discretization Equation by Taylor Series Appendix D: Derivation of Ratio of Successive Amplitude Reduction Values for Fixed-Fixed Beam using Explicit and Implicit Approaches Appendix E: Source Code Development Index

Analysis of Microelectromechanical Systems Using the Meshless Random Differential Quadrature Method

Several devices of microelectromechanical systems (MEMS) are analyzed in the presented work, using a novel numerical meshless method called the random differential quadrature (RDQ) method. The differential quadrature (DQ) is an effective derivative discretization technique but it requires all the field nodes to be arranged in a collinear manner with a pre-defined pattern. This limitation of the DQ method is overcome in the RDQ method using the interpolation function by the fixed reproducing kernel particle method (fixed RKPM). The RDQ method extends the applicability of the DQ method over a regular as well as an irregular domain discretized by uniform or randomly distributed field nodes. Due to the strong-form nature, RDQ method captures well the local high gradients. These features of the RDQ method enable it to efficiently solve the MEMS problems with different boundary conditions. In the presented work, several MEMS devices that are governed by the nonlinear electrostatic force are analyzed using the RDQ method, and their results are compared with the other simulation results presented in the existing literature. It is seen that the RDQ method effectively and accurately solves the MEMS devices problems.

Influence of Young's modulus and geometrical shapes on the 2D simulation of pH-sensitive hydrogels by the meshless random differential quadrature method

In this paper, 2D simulation of a pH-sensitive hydrogel is performed by a novel strong-form meshless method called the random differential quadrature (RDQ) method. So far the simulations of pH-responsive hydrogels have been performed over 1D hydrogel domains by simplification, where the hydrogel is allowed to deform in one direction only with a constant axis-symmetric cross-section. However, for an irregular cross-section, in which the hydrogel swells unevenly in different directions, it truly becomes the 2D problem. The RDQ method is a novel meshless technique based on the fixed reproducing kernel particle method and the differential quadrature method.The diffusion of mobile ionic species between the hydrogel and solution is simulated by the system of the Poisson–Nernst–Planck equations, and the hydrogel swelling is captured by mechanical equilibrium equations. The analytical expressions of displacements in the x and y directions are derived for a constant osmotic pressure at field nodes located along the interface between the hydrogel and solution domains. The numerical values of the displacements are verified with the corresponding analytical values obtained from the derived expressions for a hydrogel with square geometry. It is shown from the simulation results that the RDQ method is capable of capturing the jumps in the values of field variables across the interface between the multiple domains. The hydrogel swelling is studied by changing Youngs modulus and the geometrical shape, and the simulation results are found qualitatively in good agreement with the physics of the problem.