Applied Mathematics Commons^™

Polynomial Extension Operators. Part Ii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such extensions are a fundamental ingredient of high order finite element analysis.

An H-Adaptive Finite-Element Technique For Constructing 3d Wind Fields, Darrell Pepper, Xiuling Wang

Mechanical Engineering Faculty Research

An h-adaptive, mass-consistent finite-element model (FEM) has been developed for constructing 3D wind fields over irregular terrain utilizing sparse meteorological tower data. The element size in the computational domain is dynamically controlled by an a posteriori error estimator based on the L2 norm. In the h-adaptive FEM algorithm, large element sizes are typically associated with smooth flow regions and small errors; small element sizes are attributed to fast-changing flow regions and large errors. The adaptive procedure employed in this model uses mesh refinement–unrefinement to satisfy error criteria. Results are presented for wind fields using sparse data obtained from two regions …

Polynomial Extension Operators. Part I, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

In this series of papers, we construct operators that extend certain given functions on the boundary of a tetrahedron into the interior of the tetrahedron, with continuity properties in appropriate Sobolev norms. These extensions are novel in that they have certain polynomial preservation properties important in the analysis of high order finite elements. This part of the series is devoted to introducing our new technique for constructing the extensions, and its application to the case of polynomial extensions from H^½(∂K) into H¹(K), for any tetrahedron K.

Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

The finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither Hermitian nor definite when the coefficient function is complex valued. For such problems, under a uniqueness assumption, we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero. Then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number, providing an efficient solver for computing discrete approximations. Numerical experiments, while confirming the theory, …

Locally Conservative Fluxes For The Continuous Galerkin Method, Bernardo Cockburn, Jay Gopalakrishnan, Haiying Wang

Mathematics and Statistics Faculty Publications and Presentations

The standard continuous Galerkin (CG) finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations, a much needed quantity in many applications. We show how to overcome this shortcoming by using a two step postprocessing. The first step is the computation of a numerical flux trace defined on element inter- faces and is motivated by the structure of the numerical traces of discontinuous Galerkin methods. This computation is non-local in that it requires the solution of a symmetric positive definite system, but the system is well conditioned independently of mesh size, so it …

Can We Have Superconvergent Gradient Recovery Under Adaptive Meshes?, Haijun Wu, Zhimin Zhang

Mathematics Research Reports

We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two dimensional Poisson equation. Results of this paper are two folds. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered.gradient by the Polynomial Preserving Recovery (PPR) is superconvergent. Secondly, we demonstrate by numerical examples that an adaptive procedure with a posteriori error estimator based on PPR does produce adaptive meshes satisfy our mesh density assumption, and the recovered gradient by PPR is indeed supercoveregent in the adaptive process.

Validation Of Finite Element Beam Models, Pavan R. Madhineni

Mechanical & Aerospace Engineering Theses & Dissertations

Finite Element (FE) methods these days are becoming very popular to predict the responses of structures in industries, such as the aerospace and automobile industries. When there is a new design or modification of an existing design, the product must be tested to fulfill the criterion projected by the industry or external agencies before it can be launched. The product must be checked for each and every aspect, and every minute detail of each of the external forces. Hence, it becomes very important to carry out the model validation on the product.

This thesis describes validation of finite element (FE) …

Error Analysis Of Variable Degree Mixed Methods For Elliptic Problems Via Hybridization, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

A new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree Raviart-Thomas method for second order elliptic problems. The approach gives error estimates for the Lagrange multipliers without using error estimates for the other variables. Error estimates for the primal and flux variables then follow from those for the Lagrange multipliers. In contrast, traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the Lagrange multipliers. The new approach not only gives new error estimates for the new …

Incompressible Finite Elements Via Hybridization. Part Ii: The Stokes System In Three Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

We introduce a method that gives exactly incompressible velocity approximations to Stokes ow in three space dimensions. The method is designed by extending the ideas in Part I (http://archives.pdx.edu/ds/psu/10914) of this series, where the Stokes system in two space dimensions was considered. Thus we hybridize a vorticity-velocity formulation to obtain a new mixed method coupling approximations of tangential velocity and pressure on mesh faces. Once this relatively small tangential velocity-pressure system is solved, it is possible to recover a globally divergence-free numerical approximation of the fluid velocity, an approximation of the vorticity whose tangential component is continuous across …

Incompressible Finite Elements Via Hybridization. Part I: The Stokes System In Two Space Dimensions, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we introduce a new and efficient way to compute exactly divergence-free velocity approximations for the Stokes equations in two space dimensions. We begin by considering a mixed method that provides an exactly divergence-free approximation of the velocity and a continuous approximation of the vorticity. We then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique. This novel formulation bypasses the difficult task of constructing an exactly divergence-free basis for velocity approximations. Moreover, the discrete system resulting from our method has fewer degrees …

Natural Superconvergent Points Of Triangular Finite Elements, Zhimin Zhang, Runchang Lin

Mathematics Research Reports

In this work, we analytically identify natural superconvergent points of function values and gradients for triangular elements. Both the Poisson equation and the Laplace equation are discussed for polynomial finite element spaces (with degrees up to 8) under four different mesh patterns. Our results verify computer findings of [2], especially, we confirm that the computed data have 9 digits of accuracy with an exception of one pair (which has 8-7 digits of accuracy). In addition, we demonstrate that the function value superconvergent points predicted by the symmetry theory [14] are the only superconvergent points for the Poisson equation. Finally, we …

Validation Of The A Posteriori Error Estimator Based On Polynomial Preserving Recovery For Linear Elements, Zhimin Zhang, Ahmed Naga

Mathematics Research Reports

In this paper the quality of the error estimator based on the Polynomial Preserving Recovery (PPR) is investigated using the computer-based approach proposed by Babiiska et al. A comparison is made between the error estimator based on the PPR and the one based on the Superconvergence Patch Recovery (SPR). It was found that the PPR is at least as good as the SPR.

A Multilevel Discontinuous Galerkin Method, Jay Gopalakrishnan, Guido Kanschat

Mathematics and Statistics Faculty Publications and Presentations

A variable V-cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented. An analysis under a mild regularity assumption shows that the preconditioner is uniform. The interior penalty method is then combined with a discontinuous Galerkin scheme to arrive at a discretization scheme for an advection-diffusion problem, for which an error estimate is proved. A multigrid algorithm for this method is presented, and numerical experiments indicating its robustness with respect to diffusion coefficient are reported.

A Schwarz Preconditioner For A Hybridized Mixed Method, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we provide a Schwarz preconditioner for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods. The preconditioner is for the linear equation for Lagrange multipliers arrived at by eliminating the ux as well as the primal variable. We also prove a condition number estimate for this equation when no preconditioner is used. Although preconditioners for the lowest order case of the Raviart-Thomas method have been constructed previously by exploiting its connection with a nonconforming method, our approach is different, in that we use a new variational characterization of the Lagrange multiplier equation. This allows us to …

A Posteriori Error Estimates Based On Polynomial Preserving Recovery, Zhimin Zhang, Ahmed Naga

Mathematics Research Reports

Superconvergence of order O(h^1+rho), for some rho is greater than 0, is established for gradients recovered using Polynomial Preserving Recovery technique when the mesh is mildly structured. Consequently this technique can be used in building a posteriori error estimator that is asymptotically exact.

Gradient Recovery And A Posteriori Estimate For Bilinear Element On Irregular Quadrilateral Meshes, Zhimin Zhang

Mathematics Research Reports

A polynomial preserving gradient recovery method is proposed and analyzed for bilinear element under general quadrilateral meshes. It has been proven that the recovered gradient converges at a rate O(h^{1+rho) for rho = min(alpha, 1) when the mesh is distorted O(h1+alpha) (alpha > 0) from a regular one. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact.}

A Meshless Gradient Recovery Method Part I: Superconvergence Property, Zhiming Zhang, Ahmed Naga

Mathematics Research Reports

A new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz-Zhu patch recovery method. In addition, under uniform triangular meshes, the method is superconvergent for the Chevron pattern, and ultraconvergence at element edge centers for the regular pattern.

Ultraconvergence Of Zz Patch Recovery At Mesh Symmetry Points, Zhimin Zhang, Runchang Lin

Mathematics Research Reports

Ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed. Superconvergence recovery for the Q8 element is proved and ultraconvergence numerical examples are demonstrated.

An Efficient Method For Band Structure Calculations In 3d Photonic Crystals, David C. Dobson, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

A method for computing band structures for three-dimensional photonic crystals is described. The method combines a mixed finite element discretization on a uniform grid with a fast Fourier transform preconditioner and a preconditioned subspace iteration algorithm. Numerical examples illustrating the behavior of the method are presented.

Finite Element Model Updating Using Antiresonant Frequencies, Keith W. Jones

Theses and Dissertations

The applications of antiresonant frequencies to finite element (FE) model updating are few and usually limited to numerical examples. This work uses antiresonant frequencies in the model updating of an experimental structure and analyzes the physical correctness of the updated model by using it to detect damage. Antiresonant frequencies were used in the FE model updating of a six-meter aluminum truss. The model used rigid links to model welded and bolted joints. Rigid link dimensions were used as parameters in an iterative update based on eigenvalue and antiresonance sensitivities. The first update used 11 natural frequencies and 21 antiresonant frequencies …

Mortar Estimates Independent Of Number Of Subdomains, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

The stability and error estimates for the mortar finite element method are well established. This work examines the dependence of constants in these estimates on shape and number of subdomains. By means of a Poincar´e inequality and some scaling arguments, these estimates are found not to deteriorate with increase in number of subdomains.

Three-Dimensional Calculation Of Contaminant Transport In Groundwater At A Dover Afb Site, Tariq O. Hashim

Theses and Dissertations

Macroscale rate-limited sorption modeling was tested using a production transport code, the GMS/FEMWATER ground-water modeling package. The code (Version 1.1 of FEMWATER. dated 1 August 1995) was applied to a 3D conceptual model developed from a field site at Dover AFB, DE. A simulation was performed of a 200 hour contaminant injection pulse followed by clean water flushing. A moment analysis performed on the resulting breakthrough curve validated code self-consistency. Another injection pulse simulation showed that retardation temporally delays the breakthrough peak. Transport simulations of pulsed clean water pumping of the test cell with a prescribed initial contaminant distribution demonstrated …

Correlation Of Finite Element Prediction And Experimental Measurements Of Aircraft Panel Structural Vibration, Olaf L. Storaasli

Mechanical & Aerospace Engineering Theses & Dissertations

With the rapid development of computational capabilities and the implementation of complex tools for commercial finite element programs, predicting structural dynamics and related acoustics of complex structures is presently achievable. Structural finite element modeling is a design tool widely utilized for analysis of problems of all kinds. However, high frequency vibration of complex structures, such as the reinforced aircraft panel used for this study, currently does not correlate well with experimental measurements using conventional modeling methods. An investigation into the commonly used modeling techniques as well as the new tools of commercial finite element analysis packages for the prediction of …

Finite-Element Ray Tracing, Yong-Chun Liu

Theses

The interesting acoustic modeling problems often push the practical limits of full-wave models. For instance, in acoustic tomography one needs to be able to predict the propagation of an acoustic pulse for successive realizations of 31) environments. For these types of problems ray methods continue to be attractive because of their speed. Unfortunately, existing codes are prone to a number of implementation difficulties which often degrade their accuracy.

As a result most ray models are actually incapable of producing the ray theoretic result. We discuss a. new method for implementing ray theory that uses a. finite-clement formulation. This method is …

Existence, Uniqueness, And Numerical Analysis Of Solutions Of A Quasilinear Parabolic Problem, Dongming Wei

Mathematics Faculty Publications

A quasilinear parabolic problem is studied. By using the method of lines, the existence and uniqueness of a solution to the initial boundary value problem with sufficiently smooth initial conditions are shown. Also given are L² error estimates for the error between the extended fully discrete finite element solutions and the exact solution.

Hierarchical Flux-Based Thermal-Structural Finite Element Analysis Method, Sandra P. Polesky

Mechanical & Aerospace Engineering Theses & Dissertations

A hierarchical flux-based finite element method is developed for both one- and two-dimensional thermal-structural analyses. Derivation of the finite element equations is presented. The resulting finite element matrices associated with the flux-based formulation are evaluated in closed-form. The hierarchical finite elements include additional degrees of freedom in the approximation of the element variable distributions by the use of nodeless variables. The nodeless variables offer increased solution accuracy without the need for defining actual nodes and rediscretizing the finite element model. Thermal and structural responses obtained using the hierarchical flux-based method are compared with results obtained from a conventional linear finite …

Finite Element Thermal-Structural Analysis Of Cable-Stiffened Space Structures, Ajay Kumar Pandey

Mechanical & Aerospace Engineering Theses & Dissertations

Finite element thermal-structural analyses of cable-stiffened space structures are presented. A computational scheme for calculation of prestresses in the cable-stiffened structures is also described. The determination of thermal loads on orbiting space structures due to environmental heating is described briefly. Three finite element structural analysis techniques are presented for the analysis of prestressed structures. Linear, stress stiffening and large displacement analysis techniques are investigated.

The three techniques are employed for structural analysis of prestressed cable structures at different prestress levels. The three analyses produce similar results at small prestress. but at higher prestress differences between the results become significant. For …