Thesis (M.S.) -- Mississippi State University. Department of Computational Engineering. / Title from title screen. Includes bibliographical references.

Thesis (Ph. D.)--Mississippi State University. Department of Engineering. / Title from title screen. Includes bibliographical references.

This study deals with the use of high-order spectral/hp approximation functions in the finite element models of various nonlinear boundary-value and initial-value problems arising in the fields of structural mechanics and flows of viscous incompressible fluids. For many of these classes of problems, the high-order (typically, polynomial order p greater than or equal to 4) spectral/hp finite element technology offers many computational advantages over traditional low-order (i.e., p < 3) finite elements. For instance, higher-order spectral/hp finite element procedures allow us to develop robust structural elements for beams, plates, and shells in a purely displacement-based setting, which avoid all forms of numerical locking. The higher-order spectral/hp basis functions avoid the interpolation error in the numerical schemes, thereby making them accurate and stable. Furthermore, for fluid flows, when combined with least-squares variational principles, such technology allows us to develop efficient finite element models, that always yield a symmetric positive-definite (SPD) coefficient matrix, and thereby robust direct or iterative solvers can be used. The least-squares formulation avoids ad-hoc stabilization methods employed with traditional low-order weak-form Galerkin formulations. Also, the use of spectral/hp finite element technology results in a better conservation of physical quantities (e.g., dilatation, volume, and mass) and stable evolution of variables with time in the case of unsteady flows. The present study uses spectral/hp approximations in the (1) weak-form Galerkin finite element models of viscoelastic beams, (2) weak-form Galerkin displacement finite element models of shear-deformable elastic shell structures under thermal and mechanical loads, and (3) least-squares formulations for the Navier-Stokes equations governing flows of viscous incompressible fluids. Numerical simulations using the developed technology of several non-trivial benchmark problems are presented to illustrate the robustness of the higher-order spectral/hp based finite element technology.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 169-180).

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 114-118).

Scientists' ability to generate and store simulation results is outpacing their ability to analyze them via ad hoc programs. We observe that these programs exhibit an algebraic structure that can be used to facilitate reasoning and improve performance. In this dissertation, we present a formal data model that exposes this algebraic structure, then implement the model, evaluate it, and use it to express, optimize, and reason about data transformations in a variety of scientific domains. Simulation results are defined over a logical grid structure that allows a continuous domain to be represented discretely in the computer. Existing approaches for manipulating these gridded datasets are incomplete. The performance of SQL queries that manipulate large numeric datasets is not competitive with that of specialized tools, and the up-front effort required to deploy a relational database makes them unpopular for dynamic scientific applications. Tools for processing multidimensional arrays can only capture regular, rectilinear grids. Visualization libraries accommodate arbitrary grids, but no algebra has been developed to simplify their use and afford optimization. Further, these libraries are data dependent—physical changes to data characteristics break user programs. We adopt the grid as a first-class citizen, separating topology from geometry and separating structure from data. Our model is agnostic with respect to dimension, uniformly capturing, for example, particle trajectories (1-D), sea-surface temperatures (2-D), and blood flow in the heart (3-D). Equipped with data, a grid becomes a gridfield. We provide operators for constructing, transforming, and aggregating gridfields that admit algebraic laws useful for optimization. We implement the model by analyzing several candidate data structures and incorporating their best features. We then show how to deploy gridfields in practice by injecting the model as middleware between heterogeneous, ad hoc file formats and a popular visualization library. In this dissertation, we define, develop, implement, evaluate and deploy a model of gridded datasets that accommodates a variety of complex grid structures and a variety of complex data products. We evaluate the applicability and performance of the model using datasets from oceanography, seismology, and medicine and conclude that our model-driven approach offers significant advantages over the status quo.

A new methodology for the determination of target element sizes for the construction of finite element meshes applicable to the simulation of tidal flow in coastal and oceanic domains is developed and tested. The methodology is consistent with the discrete physics of tidal flow, and includes the effects of bottom stress. The method enables the estimation of the localized truncation error of the nonconservative momentum equations throughout a triangulated data set of water surface elevation and flow velocity. The method's domain-wide applicability is due in part to the formulation of a new localized truncation error estimator in terms of complex derivatives. More conventional criteria that are often used to determine target element sizes are limited to certain bathymetric conditions. The methodology developed herein is applicable over a broad range of bathymetric conditions, and can be implemented efficiently. Since the methodology permits the determination of target element size at points up to and including the coastal boundary, it is amenable to coastal domain applications including estuaries, embayments, and riverine systems. These applications require consideration of spatially varying bottom stress and advective terms, addressed herein. The new method, called LTEA-CD (localized truncation error analysis with complex derivatives), is applied to model solutions over the Western North Atlantic Tidal model domain (the bodies of water lying west of the 60° W meridian). The convergence properties of LTEACD are also analyzed. It is found that LTEA-CD may be used to build a series of meshes that produce converging solutions of the shallow water equations. An enhanced version of the new methodology, LTEA+CD (which accounts for locally variable bottom stress and Coriolis terms) is used to generate a mesh of the WNAT model domain having 25% fewer nodes and elements than an existing mesh upon which it is based; performance of the two meshes, in an average sense, is indistinguishable when considering elevation tidal signals. Finally, LTEA+CD is applied to the development of a mesh for the Loxahatchee River estuary; it is found that application of LTEA+CD provides a target element size distribution that, when implemented, outperforms a high-resolution semi-uniform mesh as well as a manually constructed, existing, documented mesh.

"This thesis presents an implementation of the Finite Difference Time Domain (FDTD) method on a massively parallel computer system, for the analysis of electromagnetic phenomenon. In addition, the implementation of an efficient mesh generator is also presented. For this research we selected the MasPar system, as it is a relatively low cost, reliable, high performance computer system. In this thesis we are primarily concerned with the selection of an efficient algorithm for each of the programs written for our selected application, and devising clever ways to make the best use of the MasPar system. This thesis has a large emphasis on examining the application performance."

A set of algorithms building a program package for the generation of twoandthree-dimensional unstructured/hybrid grids around complex geometrieshas been developed. The unstructured part of the grid generator is based on the advancing frontalgorithm. Tetrahedra of variable size, as well as directionally stretched tetrahedracan be generated by specification of a proper background grid, initiallygenerated by a Delaunay algorithm. A marching layer prismatic grid generation algorithm has been developedfor the generation of grids for viscous flows. The algorithm is able to handleregions of narrow gaps, as well as concave regions. The body surface is describedby a triangular unstructured surface grid. The subsequent grid layers in theprismatic grid are marched away from the body by an algebraic procedurecombined with an optimization procedure, resulting in a semi-structured gridof prismatic cells. Adaptive computations using remeshing have been done with use of a gradientsensor. Several key-variables can be monitored simultaneously. The sensorindicates that only the key-variables with the largest gradients give a substantialcontribution to the sensor. The sensor gives directionally stretched grids. An algorithm for the surface definition of curved surfaces using a biharmonicequation has been developed. This representation of the surface canbe used both for projection of the new surface nodes in h-refinement, and theinitial generation of the surface grid. For unsteady flows an algorithm has been developed for the deformationof hybrid grids, based on the solution of the biharmonic equation for the deformationfield. The main advantage of the grid deformation algorithm is that itcan handle large deformations. It also produces a smooth deformation distributionfor cells which are very skewed or stretched. This is necessary in orderto handle the very thin cells in the prismatic layers. The algorithms have been applied to complex three-dimensional geometries,and the influence of the grid quality on the accuracy for a finite volumeflow solver has been studied for some simpler generic geometries. / QC 20100812

For the numerical simulation of many problems of engineering interest, it is desirable to have an automated mesh adaption tool. This is important especially for problems characterized by anisotropic features and require mesh clustering in the direction of high gradients. Another significant issue in meshing emerges in unsteady simulations with moving boundaries, where the boundary motion has to be accommodated by deforming the computational grid. Similarly, there exist problems where current mesh needs to be adapted to get more accurate solutions. To solve these problems, we propose three novel procedures. In the first part of this work, we present an optimization procedure for three-dimensional anisotropic tetrahedral grids based on metric-driven h-adaptation. Through the use of topological and geometrical operators, the mesh is iteratively adapted until the final mesh minimizes a given objective function. We propose an optimization process based on an ad-hoc application of the simulated annealing technique, which improves the likelihood of removing poor elements from the grid. Moreover, a local implementation of the simulated annealing is proposed to reduce the computational cost. Many challenging unsteady multi-physics problems are characterized by moving boundaries and/or interfaces. When the boundary displacements are large, degenerate elements are easily formed in the grid such that frequent remeshing is required. We propose a new r-adaptation technique that is valid for all types of elements (e.g., triangle, tet, quad, hex, hybrid) and deforms grids that undergo large imposed displacements at their boundaries. A grid is deformed using a network of linear springs composed of edge springs and a set of virtual springs. The virtual springs are constructed in such a way as to oppose element collapsing. Both frequent remeshing, and exact-pinpointing of clustering locations are great challenges of numerical simulations, which can be overcome by adaptive meshing algorithms. Therefore, we conclude this work by defining a novel mesh adaptation technique where the entire mesh is adapted upon application of a force field in order to comply with the target mesh or to get more accurate solutions. The method has been tested for two-dimensional problems of a-priori metric definitions as well as for oblique shock clusterings.