Investigation of the use of meshfree methods for haptic thermal management of design and simulation of MEMS

This thesis presents a novel approach of using haptic sensing technology combined with virtual environment (VE) for the thermal management of Micro-Electro-Mechanical-Systems (MEMS) design. The goal is to reduce the development cycle by avoiding the costly iterative prototyping procedure. In this regard, we use haptic feedback with virtua lprototyping along with an immersing environment. We also aim to improve the productivity and capability of the designer to better grasp the phenomena operating at the micro-scale level, as well as to augment computational steering through haptic channels. To validate the concept of haptic thermal management, we have implemented a demonstrator with a user friendly interface which allows to intuitively "feel" the temperature field through our concept of haptic texturing. The temperature field in a simple MEMS component is modeled using finite element methods (FEM) or finite difference method (FDM) and the user is able to feel thermal expansion using a combination of different haptic feedback. In haptic application, the force rendering loop needs to be updated at a frequency of 1Khz in order to maintain continuity in the user perception. When using FEM or FDM for our three-dimensional model, the computational cost increases rapidly as the mesh size is reduced to ensure accuracy. Hence, it constrains the complexity of the physical model to approximate temperature or stress field solution. It would also be difficult to generate or refine the mesh in real time for CAD process. In order to circumvent the limitations due to the use of conventional mesh-based techniques and to avoid the bothersome task of generating and refining the mesh, we investigate the potential of meshfree methods in the context of our haptic application. We review and compare the different meshfree formulations against FEM mesh based technique. We have implemented the different methods for benchmarking thermal conduction and elastic problems. The main work of this thesis is to determine the relevance of the meshfree option in terms of flexibility of design and computational charge for haptic physical model

Partition of unity interpolation using stable kernel-based techniques

In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each PU subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient

Meshless Collocation Methods for the Numerical Solution of Elliptic Boundary Valued Problems and the Rotational Shallow Water Equations on the Sphere

This dissertation thesis has three main goals: 1) To explore the anatomy of meshless collocation approximation methods that have recently gained attention in the numerical analysis community; 2) Numerically demonstrate why the meshless collocation method should clearly become an attractive alternative to standard finite-element methods due to the simplicity of its implementation and its high-order convergence properties; 3) Propose a meshless collocation method for large scale computational geophysical fluid dynamics models. We provide numerical verification and validation of the meshless collocation scheme applied to the rotational shallow-water equations on the sphere and demonstrate computationally that the proposed model can compete with existing high performance methods for approximating the shallow-water equations such as the SEAM (spectral-element atmospheric model) developed at NCAR. A detailed analysis of the parallel implementation of the model, along with the introduction of parallel algorithmic routines for the high-performance simulation of the model will be given. We analyze the programming and computational aspects of the model using Fortran 90 and the message passing interface (mpi) library along with software and hardware specifications and performance tests. Details from many aspects of the implementation in regards to performance, optimization, and stabilization will be given. In order to verify the mathematical correctness of the algorithms presented and to validate the performance of the meshless collocation shallow-water model, we conclude the thesis with numerical experiments on some standardized test cases for the shallow-water equations on the sphere using the proposed method

Fundamental solution based numerical methods for three dimensional problems: efficient treatments of inhomogeneous terms and hypersingular integrals

In recent years, fundamental solution based numerical methods including the meshless method of fundamental solutions (MFS), the boundary element method (BEM) and the hybrid fundamental solution based finite element method (HFS-FEM) have become popular for solving complex engineering problems. The application of such fundamental solutions is capable of reducing computation requirements by simplifying the domain integral to the boundary integral for the homogeneous partial differential equations. The resulting weak formulations, which are of lower dimensions, are often more computationally competitive than conventional domain-type numerical methods such as the finite element method (FEM) and the finite difference method (FDM). In the case of inhomogeneous partial differential equations arising from transient problems or problems involving body forces, the domain integral related to the inhomogeneous solutions term will need to be integrated over the interior domain, which risks losing the competitive edge over the FEM or FDM. To overcome this, a particular treatment to the inhomogeneous term is needed in the solution procedure so that the integral equation can be defined for the boundary. In practice, particular solutions in approximated form are usually applied rather than the closed form solutions, due to their robustness and readiness. Moreover, special numerical treatment may be required when evaluating stress directly on the domain surface which may give rise to hypersingular integral formulation. This thesis will discuss how the MFS and the BEM can be applied to the three-dimensional elastic problems subjected to body forces by introducing the compactly supported radial basis functions in addition to the efficient treatment of hypersingular surface integrals. The present meshless approach with the MFS and the compactly supported radial basis functions is later extended to solve transient and coupled problems for three-dimensional porous media simulation

Error Estimation and Adaptive Refinement of Finite Element Thin Plate Spline

The thin plate spline smoother is a data fitting and smoothing technique that captures important patterns of potentially noisy data. However, it is computationally expensive for large data sets. The finite element thin plate spline smoother (TPSFEM) combines the thin plate spline smoother and finite element surface fitting to efficiently interpolate large data sets. When the TPSFEM uses uniform finite element grids, it may require a fine grid to achieve the desired accuracy. Adaptive refinement uses error indicators to identify sensitive regions and adapts the precision of the solution dynamically, which reduces the computational cost to achieve the required accuracy. Traditional error indicators were developed for the finite element method to approximate partial differential equations and may not be applicable for the TPSFEM. We examined techniques that may indicate errors for the TPSFEM and adapted four traditional error indicators that use different information to produce efficient adaptive grids. The iterative adaptive refinement process has also been adjusted to handle additional complexities caused by the TPSFEM. The four error indicators presented in this thesis are the auxiliary problem error indicator, recovery-based error indicator, norm-based error indicator and residual-based error indicator. The auxiliary problem error indicator approximates the error by solving auxiliary problems to evaluate approximation quality. The recovery-based error indicator calculates the error by post-processing discontinuous gradients of the TPSFEM. The norm-based error indicator uses an error bound on the interpolation error to indicate large errors. The residual-based error indicator computes interior element residuals and jumps of gradients across elements to estimate the energy norm of the error. Numerical experiments were conducted to evaluate the error indicators' performance on producing efficient adaptive grids, which are measured by the error versus the number of nodes in the grid. A set of one and two-dimensional model problems with various features are chosen to examine the effectiveness of the error indicators. As opposed to the finite element method, error indicators of the TPSFEM may also be affected by noise, data distribution patterns, data sizes and boundary conditions, which are assessed in the experiments. It is found that adaptive grids are significantly more efficient than uniform grids for two-dimensional model problems with difficulties like peaks and singularities. While the TPSFEM may not recover the original solution in the presence of noise or scarce data, error indicators still produce more efficient grids. We also learned that the difference is less obvious when the data has mostly smooth or oscillatory surfaces. Some error indicators that use data may be affected by data distribution patterns and boundary conditions, but the others are robust and produce stable results. Our error indicators also successfully identify sensitive regions for one-dimensional data sets. Lastly, when errors of the TPSFEM cannot be further reduced due to factors like noise, new stopping criteria terminate the iterative process aptly

Finite Element Methods in Smart Materials and Polymers

Functional polymers show unique physical and chemical properties, which can manifest as dynamic responses to external stimuli such as radiation, temperature, chemical reaction, external force, and magnetic and electric fields. Recent advances in the fabrication techniques have enabled different types of polymer systems to be utilized in a wide range of potential applications in smart structures and systems, including structural health monitoring, anti‐vibration, and actuators. The progress in these integrated smart structures requires the implementation of finite element modelling using a multiphysics approach in various computational platforms. This book presents finite element methods applied in modeling of the smart structures and materials with particular emphasis on hydrogels, metamaterials, 3D-printed and anti-vibration constructs, and fibers

On the Selection of a Good Shape Parameter for RBF Approximation and Its Application for Solving PDEs

Meshless methods utilizing Radial Basis Functions~(RBFs) are a numerical method that require no mesh connections within the computational domain. They are useful for solving numerous real-world engineering problems. Over the past decades, after the 1970s, several RBFs have been developed and successfully applied to recover unknown functions and to solve Partial Differential Equations (PDEs).However, some RBFs, such as Multiquadratic (MQ), Gaussian (GA), and Matern functions, contain a free variable, the shape parameter, c. Because c exerts a strong influence on the accuracy of numerical solutions, much effort has been devoted to developing methods for determining shape parameters which provide accurate results. Most past strategies, which have utilized a trail-and-error approach or focused on mathematically proven values for c, remain cumbersome and impractical for real-world implementations.This dissertation presents a new method, Residue-Error Cross Validation (RECV), which can be used to select good shape parameters for RBFs in both interpolation and PDE problems. The RECV method maps the original optimization problem of defining a shape parameter into a root-finding problem, thus avoiding the local optimum issue associated with RBF interpolation matrices, which are inherently ill-conditioned.With minimal computational time, the RECV method provides shape parameter values which yield highly accurate interpolations. Additionally, when considering smaller data sets, accuracy and stability can be further increased by using the shape parameter provided by the RECV method as the upper bound of the c interval considered by the LOOCV method. The RECV method can also be combined with an adaptive method, knot insertion, to achieve accuracy up to two orders of magnitude higher than that achieved using Halton uniformly distributed points