Numerical Analysis and Computation Commons^™

Stabilized Weak-Gradient Discontinuous Finite Elements With Optimal Error Estimates For Second-Order Elliptic Pdes, Aymen Laadhari

Mathematical Modelling and Numerical Simulation with Applications

This work introduces an accurate finite element approach employing a new stabilized discrete weak gradient, designed for second-order elliptic problems on arbitrary conforming meshes. We formulate the approach within a discontinuous Galerkin framework and derive a consistent and coercive bilinear form. Appropriate error analysis on a model problem confirms optimal convergence. Building on the core analysis, we extend the method to more challenging settings, including time-dependent heterogeneous scenarios and a biophysically realistic optimal-control model of photobleaching in the budding yeast cell. We further illustrate the versatility of the weak-gradient construction by applying it to an unsteady level-set equation relevant to …

Trefftz Finite Elements On Curvilinear Polygons, Akash Anand, Jeffrey S. Ovall, Samuel E. Reynolds, Steffen Weisser

Mathematics and Statistics Faculty Publications and Presentations

We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be "polynomial" of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomial spaces in our local …

Spectral Discretization Errors In Filtered Subspace Iteration, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall

Mathematics and Statistics Faculty Publications and Presentations

We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters …

Full Field Computing For Elastic Pulse Dispersion In Inhomogeneous Bars, A. Berezovski, R. Kolman, M. Berezovski, D. Gabriel, V. Adamek

Publications

In the paper, the finite element method and the finite volume method are used in parallel for the simulation of a pulse propagation in periodically layered composites beyond the validity of homogenization methods. The direct numerical integration of a pulse propagation demonstrates dispersion effects and dynamic stress redistribution in physical space on example of a one-dimensional layered bar. Results of numerical simulations are compared with analytical solution constructed specifically for the considered problem. Analytical solution as well as numerical computations show the strong influence of the composition of constituents on the dispersion of a pulse in a heterogeneous bar and …

The Auxiliary Space Preconditioner For The De Rham Complex, Jay Gopalakrishnan, Martin Neumüller, Panayot S. Vassilevski

Portland Institute for Computational Science Publications

We generalize the construction and analysis of auxiliary space preconditioners to the n-dimensional finite element subcomplex of the de Rham complex. These preconditioners are based on a generalization of a decomposition of Sobolev space functions into a regular part and a potential. A discrete version is easily established using the tools of finite element exterior calculus. We then discuss the four-dimensional de Rham complex in detail. By identifying forms in four dimensions (4D) with simple proxies, form operations are written out in terms of familiar algebraic operations on matrices, vectors, and scalars. This provides the basis for our implementation of …

Prandtl Number Effect On Assisted Convective Heat Transfer Through A Solar Collector, Rehen Nasrin, Salma Parvin, M. A. Alim

Applications and Applied Mathematics: An International Journal (AAM)

Numerical study of the influence of Prandtl number on forced convective heat transfer through a riser pipe of a flat plate solar collector is done. The working fluid is Al2O3/water nanofluid. By Finite Element Method the governing partial differential equations are solved. The effect of the Prandtl number on the temperature and velocity field has been depicted. Comprehensive average Nusselt number, average bulk temperature, mean velocity, mid-height temperature inside the pipe, mean output temperature and collector efficiency are presented for the governing parameter mentioned above. Nu increases by 16% with the variation of Pr from 4.6 to 6.6 using nanofluid. …

Multiphysics Modeling To Enhance Understanding Of Microwave Heating Of Food In Domestic Ovens, Jiajia Chen

Department of Agricultural and Biological Systems Engineering: Dissertations, Theses, and Student Research

Nonuniform heating is the biggest issue in the microwave heating of prepared meals. Multiphysics based models are promising tools to improve microwave heating uniformity by properly designing the food product. However, limited availability of accurate temperature-dependent material properties, inadequate model prediction accuracy, and high computational power and complexity in model development are three gaps that greatly limited the application of these models in the food industry.

To fill in the gaps, firstly, we developed a multitemperature calibration protocol to measure temperature-dependent dielectric properties (dielectric constant and loss factor). The temperature-dependent dielectric and thermal (thermal conductivity and specific heat capacity) properties …

Comparison Of Mesh And Meshless Methods For Partial Differential Equations Of Galerkin Form, Wallace F. Atterberry

UNLV Theses, Dissertations, Professional Papers, and Capstones

There are two purposes of this research project. The first purpose is to compare two types of Galerkin methods: The finite element mesh method and moving least sqaures meshless Galerkin (EFG) method. The second purpose of this project is to determine if a hybrid between the mesh and meshless method is beneficial.

This manuscript will be divided into three main parts. The first part is chapter one which develops the finite element method. The second part (Chapter two) will be developing the meshless method. The last part will provide a method for combining the mesh and meshless methods for a …

Finite Element Analysis In Porous Media For Incompressible Flow Of Contamination From Nuclear Waste, Abbas Al-Bayati, Saad A. Manaa, Ekhlass S. Ahmed

Applications and Applied Mathematics: An International Journal (AAM)

A non-linear parabolic system is used to describe incompressible nuclear waste disposal contamination in porous media, in which both molecular diffusion and dispersion are considered. The Galerkin method is applied for the pressure equation. For the brine, radionuclide and heat, a kind of partial upwind finite element scheme is constructed. Examples are included to demonstrate certain aspects of the theory and illustrate the capabilities of the kind of partial upwind finite element approach.

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 …

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.

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 …

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.

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 …

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 …