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A spectral element algorithm for solution of the incompressible Navier-Stokes and heat transfer equations is developed, with an emphasis on extending the classical conforming Galerkin formulations to nonconforming spectral elements. The new algorithm employs both the Constrained Approximation Method (CAM), and the Mortar Element Method (MEM) for p-and h-type nonconforming elements. Detailed descriptions, and formulation steps for both methods, as well as the performance comparisons between CAM and MEM, are presented. This study fills an important gap in the literature by providing a detailed explanation for treatment of p-and h-type nonconforming interfaces. A comparative eigenvalue spectrum analysis of diffusion and convection operators is provided for CAM and MEM. Effects of consistency errors due to the nonconforming formulations on the convergence of steady and time dependent problems are studied in detail. Incompressible flow solvers that can utilize these nonconforming formulations on both p- and h-type nonconforming grids are developed and validated. Engineering use of the developed solvers are demonstrated by detailed parametric analyses of oscillatory flow forced convection heat transfer in two-dimensional channels.

<p>The goal of this work is to develop a practical and comprehensive methodology to study the response and the stability of various delay differential equations (DDEs). The development of these new analysis techniques is motivated by the existence of delays in the governing equations of many physical systems such as turning and milling processes. </p><p>Delay differential equations appear in many models in science in engineering either as an intrinsic component (e.g. machining dynamics) or as a modeling decision (biology related dynamics). However, the infinite dimensionality of DDEs significantly complicates the resulting analysis from both an analytical and numerical perspective. Since the delay results in an infinite dimensional state-space, it is often necessary to use an approximate procedure to study DDEs and ascertain their stability.</p><p>Several approximate techniques appeared in literature to study the stability of DDEs. However, a large number of these techniques---such as D-subdivision, Cluster Treatment of Characteristic Roots and Continuous Time Approximation---are limited to autonomous DDEs. Moreover, the methods that are suitable for non-autonomous DDEs, e.g. the Semi-discretization approach, often result in a very large system of algebraic equations that can cause computational difficulties. Collocation-type methods, such as Chebyshev-collocation approach, have also been used to study DDEs. One major limitation of the conventional Chebyshev collocation approach is that it is strictly applicable to DDEs with continuous coefficients. An alternative approach that can handle DDEs with piecewise continuous coefficients is the Temporal Finite Element Analysis (TFEA). However, TFEA has only linear rates of convergence and is limited to h-convergence schemes. The limited rate of convergence in TFEA has prohibited its application to a wide class of DDEs such as DDEs with complicated coefficients or with distributed and multiple delays. </p><p>In this thesis, I develop a spectral element method for the stability analysis of DDEs. The spectral element method is a Galerkin-type approach that discretizes the infinite dimensional DDE into a finite set of algebraic equations (or a dynamic map). The stability of the system is then studied using the eigenvalues of the map. </p><p>In contrast to TFEA, the spectral element method was shown to have exponential rates of convergence and hp-refinement capabilities. Further, a comparison with the widely-used collocation methods showed that our approach can often yield higher rates of convergence. The higher rates of convergence of the developed approach enabled extending it to DDEs with multiple and distributed delays. I further extended this approach to calculating the periodic orbits of DDEs and their stability. </p><p>As an application of the methods developed in this thesis, I studied the stability of turning and milling models. For example, a distributed force model was proposed to characterize cutting forces in turning. The stability of the resulting delay integro-differential equation was studied using the methods developed in this study and they were shown to agree with practical observations. As another example, the stability of a milling process--- whose model contains piecewise coefficients---was investigated. The effect of multiple-flute engagement, which contributed to the complexity of the coefficients, was also investigated. The resulting stability charts revealed new stability observations in comparison to typical analysis methods. Specifically, I was able to show that unstable regions appear in what was deemed a stable region by prior analysis techniques.</p> / Dissertation

<p>In this work, we proposed a spectral integral method (SIM)-spectral element method (SEM)- finite element method (FEM) domain decomposition method (DDM) for solving inhomogeneous multi-scale problems. The proposed SIM-SEM-FEM domain decomposition algorithm can efficiently handle problems with multi-scale structures, </p><p>by using FEM to model electrically small sub-domains and using SEM to model electrically large and smooth sub-domains. The SIM is utilized as an efficient boundary condition. This combination can reduce the total number of elements used in solving multi-scale problems, thus it is more efficient than conventional FEM or conventional FEM domain decomposition method. Another merit of the proposed method is that it is capable of handling arbitrary non-conforming elements. Both geometry modeling and mesh generation are totally independent for different sub-domains, thus the geometry modeling and mesh generation are highly flexible for the proposed SEM-FEM domain decomposition method. As a result, the proposed SIM-SEM-FEM DDM algorithm is very suitable for solving inhomogeneous multi-scale problems.</p> / Dissertation

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step.

A spectral element solver program using MATLAB is written for the solution of Poisson and Helmholtz equations. The accuracy of spectral methods (p-type high order) and the geometric flexibility of the low-order h-type finite elements are combined in spectral element methods. Rectangular elements are used to solve Poisson and Helmholtz equations with Dirichlet and Neumann boundary conditions which are homogeneous or non homogeneous. Robin (mixed) boundary conditions are also implemented. Poisson equation is also solved by discretising the domain with curvilinear quadrilateral elements so that the accuracy of both isoparametric quadrilateral and rectangular element stiffness matrices and element mass matrices are tested. Quadrilateral elements are used to obtain the stream functions of the inviscid flow around a cylinder problem. Nonhomogeneous Neumann boundary conditions are imposed to the quadrilateral element stiffness matrix to solve the velocity potentials.

The aim of this thesis is to develop a flow solver that has the ability to obtain an accurate numerical solution fast and efficiently with minimum user intervention. In this study, a two-dimensional viscous, laminar, incompressible flow solver based on Least-Squares Spectral Element Method (LSSEM) is developed. The LSSEM flow solver can work on hp-type nonconforming grids and can perform p-type adaptive refinement. Several benchmark problems are solved in order to validate the solver and successful results are obtained. In particular, it is demonstrated that p-type adaptive refinement on hp-type non-conforming grids can be used to improve the quality of the solution. Moreover, it is found that mass conservation performance of LSSEM can be enhanced by using p-type adaptive refinement strategies while keeping computational costs reasonable.

A spectral element algorithm for solution of the incompressible Navier-Stokes and heat transfer equations is developed, with an emphasis on extending the classical conforming Galerkin formulations to nonconforming spectral elements. The new algorithm employs both the Constrained Approximation Method (CAM), and the Mortar Element Method (MEM) for p-and h-type nonconforming elements. Detailed descriptions, and formulation steps for both methods, as well as the performance comparisons between CAM and MEM, are presented. This study fills an important gap in the literature by providing a detailed explanation for treatment of p-and h-type nonconforming interfaces. A comparative eigenvalue spectrum analysis of diffusion and convection operators is provided for CAM and MEM. Effects of consistency errors due to the nonconforming formulations on the convergence of steady and time dependent problems are studied in detail. Incompressible flow solvers that can utilize these nonconforming formulations on both p- and h-type nonconforming grids are developed and validated. Engineering use of the developed solvers are demonstrated by detailed parametric analyses of oscillatory flow forced convection heat transfer in two-dimensional channels.

Spectral element based numerical solvers are developed to simulate electrokinetically driven flows for micro-fluidic applications. Based on these numerical solvers, basic phenomena and devices for electrokinetic applications in micro and nano flows are systematically studied. As a first application, flow and species transport control in a grooved micro-channel using local electrokinetic forces are studied. Locally applied electric fields, zeta potential patterned grooved surfaces, and geometry are manipulated to control mixed electroosmotic/pressure driven flow in the grooved micro-channel. The controlled flow pattern enables entrapment and release of prescribed amounts of scalar species in the grooves. As another application, hydrodynamic/ electrokinetic focusing in a micro-channel is studied. External electric field, flow rate of pressure driven flow, and geometry in the micro-channel are manipulated to obtain the focusing point, which led to determination of the electrophoretic mobility and (relative) concentration of dilute mixtures. This technique can be used to identify and detect species in dilute mixtures.

abstract: A moving overlapping mesh methodology that achieves spectral accuracy in space and up to second-order accuracy in time is developed for solution of unsteady incompressible flow equations in three-dimensional domains. The targeted applications are in aerospace and mechanical engineering domains and involve problems in turbomachinery, rotary aircrafts, wind turbines and others. The methodology is built within the dual-session communication framework initially developed for stationary overlapping meshes. The methodology employs semi-implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally-accurate spatial interpolation and high-order accurate temporal extrapolation, and requires few, if any, iterations, yet maintains the global accuracy and stability of the underlying flow solver. Mesh movement is enabled through the Arbitrary Lagrangian-Eulerian formulation of the governing equations, which allows for prescription of arbitrary velocity values at discrete mesh points. The stationary and moving overlapping mesh methodologies are thoroughly validated using two- and three-dimensional benchmark problems in laminar and turbulent flows. The spatial and temporal global convergence, for both methods, is documented and is in agreement with the nominal order of accuracy of the underlying solver. Stationary overlapping mesh methodology was validated to assess the influence of long integration times and inflow-outflow global boundary conditions on the performance. In a turbulent benchmark of fully-developed turbulent pipe flow, the turbulent statistics are validated against the available data. Moving overlapping mesh simulations are validated on the problems of two-dimensional oscillating cylinder and a three-dimensional rotating sphere. The aerodynamic forces acting on these moving rigid bodies are determined, and all results are compared with published data. Scaling tests, with both methodologies, show near linear strong scaling, even for moderately large processor counts. The moving overlapping mesh methodology is utilized to investigate the effect of an upstream turbulent wake on a three-dimensional oscillating NACA0012 extruded airfoil. A direct numerical simulation (DNS) at Reynolds Number 44,000 is performed for steady inflow incident upon the airfoil oscillating between angle of attack 5.6 and 25 degrees with reduced frequency k=0.16. Results are contrasted with subsequent DNS of the same oscillating airfoil in a turbulent wake generated by a stationary upstream cylinder. / Dissertation/Thesis / Doctoral Dissertation Aerospace Engineering 2016