Current Search: radial basis functions (x)
View All Items
- Title
- AUTOMATED ADAPTIVE DATA CENTER GENERATION FOR MESHLESS METHODS.
- Creator
-
Mitteff, Eric, Divo, Eduardo, University of Central Florida
- Abstract / Description
-
Meshless methods have recently received much attention but are yet to reach their full potential as the required problem setup (i.e. collocation point distribution) is still significant and far from automated. The distribution of points still closely resembles the nodes of finite volume-type meshes and the free parameter, c, of the radial-basis expansion functions (RBF) still must be tailored specifically to a problem. The localized meshless collocation method investigated requires a local...
Show moreMeshless methods have recently received much attention but are yet to reach their full potential as the required problem setup (i.e. collocation point distribution) is still significant and far from automated. The distribution of points still closely resembles the nodes of finite volume-type meshes and the free parameter, c, of the radial-basis expansion functions (RBF) still must be tailored specifically to a problem. The localized meshless collocation method investigated requires a local influence region, or topology, used as the expansion medium to produce the required field derivatives. Tests have shown a regular cartesian point distribution produces optimal results, however, in order to maintain a locally cartesian point distribution a recursive quadtree scheme is herein proposed. The quadtree method allows modeling of irregular geometries and refinement of regions of interest and it lends itself for full automation, thus, reducing problem setup efforts. Furthermore, the construction of the localized expansion regions is closely tied up to the point distribution process and, hence, incorporated into the automated sequence. This also allows for the optimization of the RBF free parameter on a local basis to achieve a desired level of accuracy in the expansion. In addition, an optimized auto-segmentation process is adopted to distribute and balance the problem loads throughout a parallel computational environment while minimizing communication requirements.
Show less - Date Issued
- 2006
- Identifier
- CFE0001321, ucf:47032
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0001321
- Title
- MESHFREE APPROXIMATION METHODS FOR FREE-FORM OPTICAL SURFACES WITH APPLICATIONS TO HEAD-WORN DISPLAYS.
- Creator
-
Cakmakci, Ozan, Rolland, Jannick, University of Central Florida
- Abstract / Description
-
Compact and lightweight optical designs achieving acceptable image quality, field of view, eye clearance, eyebox size, operating across the visible spectrum, are the key to the success of next generation head-worn displays. The first part of this thesis reports on the design, fabrication, and analysis of off-axis magnifier designs. The first design is catadioptric and consists of two elements. The lens utilizes a diffractive optical element and the mirror has a free-form surface described...
Show moreCompact and lightweight optical designs achieving acceptable image quality, field of view, eye clearance, eyebox size, operating across the visible spectrum, are the key to the success of next generation head-worn displays. The first part of this thesis reports on the design, fabrication, and analysis of off-axis magnifier designs. The first design is catadioptric and consists of two elements. The lens utilizes a diffractive optical element and the mirror has a free-form surface described with an x-y polynomial. A comparison of color correction between doublets and single layer diffractive optical elements in an eyepiece as a function of eye clearance is provided to justify the use of a diffractive optical element. The dual-element design has an 8 mm diameter eyebox, 15 mm eye clearance, 20 degree diagonal full field, and is designed to operate across the visible spectrum between 450-650 nm. 20% MTF at the Nyquist frequency with less than 3% distortion has been achieved in the dual-element head-worn display. An ideal solution for a head-worn display would be a single free-form surface mirror design. A single surface mirror does not have dispersion; therefore, color correction is not required. A single surface mirror can be made see-through by machining the appropriate surface shape on the opposite side to form a zero power shell. The second design consists of a single off-axis free-form mirror described with an x-y polynomial, which achieves a 3 mm diameter exit pupil, 15 mm eye relief, and a 24 degree diagonal full field of view. The second design achieves 10% MTF at the Nyquist frequency set by the pixel spacing of the VGA microdisplay with less than 3% distortion. Both designs have been fabricated using diamond turning techniques. Finally, this thesis addresses the question of "what is the optimal surface shape for a single mirror constrained in an off-axis magnifier configuration with multiple fields?" Typical optical surfaces implemented in raytrace codes today are functions mapping two dimensional vectors to real numbers. The majority of optical designs to-date have relied on conic sections and polynomials as the functions of choice. The choice of conic sections is justified since conic sections are stigmatic surfaces under certain imaging geometries. The choice of polynomials from the point of view of surface description can be challenged. A polynomial surface description may link a designer's understanding of the wavefront aberrations and the surface description. The limitations of using multivariate polynomials are described by a theorem due to Mairhuber and Curtis from approximation theory. This thesis proposes and applies radial basis functions to represent free-form optical surfaces as an alternative to multivariate polynomials. We compare the polynomial descriptions to radial basis functions using the MTF criteria. The benefits of using radial basis functions for surface description are summarized in the context of specific head-worn displays. The benefits include, for example, the performance increase measured by the MTF, or the ability to increase the field of view or pupil size. Even though Zernike polynomials are a complete and orthogonal set of basis over the unit circle and they can be orthogonalized for rectangular or hexagonal pupils using Gram-Schmidt, taking practical considerations into account, such as optimization time and the maximum number of variables available in current raytrace codes, for the specific case of the single off-axis magnifier with a 3 mm pupil, 15 mm eye relief, 24 degree diagonal full field of view, we found the Gaussian radial basis functions to yield a 20% gain in the average MTF at 17 field points compared to a Zernike (using 66 terms) and an x-y polynomial up to and including 10th order. The linear combination of radial basis function representation is not limited to circular apertures. Visualization tools such as field map plots provided by nodal aberration theory have been applied during the analysis of the off-axis systems discussed in this thesis. Full-field displays are used to establish node locations within the field of view for the dual-element head-worn display. The judicious separation of the nodes along the x-direction in the field of view results in well-behaved MTF plots. This is in contrast to an expectation of achieving better performance through restoring symmetry via collapsing the nodes to yield field-quadratic astigmatism.
Show less - Date Issued
- 2008
- Identifier
- CFE0002479, ucf:47674
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0002479
- Title
- APPLICATION OF TRAINED POD-RBF TO INTERPOLATION IN HEAT TRANSFER AND FLUID MECHANICS.
- Creator
-
Ashley, Rebecca A, Kassab, Alain, University of Central Florida
- Abstract / Description
-
To accurately model or predict future operating conditions of a system in engineering or applied mechanics, it is necessary to understand its fundamental principles. These may be the material parameters, defining dimensional characteristics, or the boundary conditions. However, there are instances when there is little to no prior knowledge of the system properties or conditions, and consequently, the problem cannot be modeled accurately. It is therefore critical to define a method that can...
Show moreTo accurately model or predict future operating conditions of a system in engineering or applied mechanics, it is necessary to understand its fundamental principles. These may be the material parameters, defining dimensional characteristics, or the boundary conditions. However, there are instances when there is little to no prior knowledge of the system properties or conditions, and consequently, the problem cannot be modeled accurately. It is therefore critical to define a method that can identify the desired characteristics of the current system without accumulating extensive computation time. This thesis formulates an inverse approach using proper orthogonal decomposition (POD) with an accompanying radial basis function (RBF) interpolation network. This method is capable of predicting the desired characteristics of a specimen even with little prior knowledge of the system. This thesis first develops a conductive heat transfer problem, and by using the truncated POD - RBF interpolation network, temperature values are predicted given a varying Biot number. Then, a simple bifurcation problem is modeled and solved for velocity profiles while changing the mass flow rate. This bifurcation problem provides the data and foundation for future research into the left ventricular assist device (LVAD) and implementation of POD - RBF. The trained POD - RBF inverse approach defined in this thesis can be implemented in several applications of engineering and mechanics. It provides model reduction, error filtration, regularization and an improvement over previous analysis utilizing computational fluid dynamics (CFD).
Show less - Date Issued
- 2018
- Identifier
- CFH2000279, ucf:45782
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFH2000279
- Title
- A Localized Blended RBF Collocation Method for Effective Shock Capturing.
- Creator
-
Harris, Michael, Kassab, Alain, Moslehy, Faissal, Divo, Eduardo, Chopra, Manoj, University of Central Florida
- Abstract / Description
-
Solving partial differential equations (PDEs) can require numerical methods, especially for non-linear problems and complex geometry. Common numerical methods used today are the finite difference method (FDM), finite element method (FEM) and the finite volume method (FVM). These methods require a mesh or grid before a solution is attempted. Developing the mesh can require expensive preprocessing time and the quality of the mesh can have major effects on the solution. In recent years, meshless...
Show moreSolving partial differential equations (PDEs) can require numerical methods, especially for non-linear problems and complex geometry. Common numerical methods used today are the finite difference method (FDM), finite element method (FEM) and the finite volume method (FVM). These methods require a mesh or grid before a solution is attempted. Developing the mesh can require expensive preprocessing time and the quality of the mesh can have major effects on the solution. In recent years, meshless methods have become a research interest due to the simplicity of using scattered data points. Many types of meshless methods exist stemming from the spectral or pseudo-spectral methods, but the focus of this research involves a meshless method using radial basis function (RBF) interpolation. Radial basis functions (RBF) interpolation is a class of meshless method and can be used in solving partial differential equations. Radial basis functions are impressive because of the capability of multivariate interpolation over scattered data, even for data with discontinuities. Also, radial basis function interpolation is capable of spectral accuracy and exponential convergence. For infinitely smooth radial basis functions such as the Hardy Multiquadric and inverse Multiquadric, the RBF is dependent on a shape parameter that must be chosen properly to obtain accurate approximations. The optimum shape parameter can vary depending on the smoothness of the field. Typically, the shape parameter is chosen to be a large value rendering the RBF flat and yielding high condition number interpolation matrix. This strategy works well for smooth data and as shown to produce phenomenal results for problems in heat transfer and incompressible fluid dynamics. The approach of flat RBF or high condition matrices tends to fail for steep gradients and shocks. Instead, a low-value shape parameter rendering the RBF steep and the condition number of the interpolation matrix small should be used in the presence of steep gradients or shocks. This work demonstrates a method to capture steep gradients and shocks using a blended RBF approach. The method switches between flat and steep RBF interpolation depending on the smoothness of the data. Flat RBF or high condition number RBF interpolation is used for smooth regions maintaining high accuracy. Steep RBF or low condition number RBF interpolation provides stability for steep gradients and shocks. This method is demonstrated using several numerical experiments such as 1-D advection equation, 2-D advection equation, Burgers equation, 2-D inviscid compressible Euler equations, and the Navier-Stokes equations.
Show less - Date Issued
- 2018
- Identifier
- CFE0007332, ucf:52108
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0007332
- Title
- A MODEL INTEGRATED MESHLESS SOLVER (MIMS) FOR FLUID FLOW AND HEAT TRANSFER.
- Creator
-
Gerace, Salvadore, Kassab, Alain, University of Central Florida
- Abstract / Description
-
Numerical methods for solving partial differential equations are commonplace in the engineering community and their popularity can be attributed to the rapid performance improvement of modern workstations and desktop computers. The ubiquity of computer technology has allowed all areas of engineering to have access to detailed thermal, stress, and fluid flow analysis packages capable of performing complex studies of current and future designs. The rapid pace of computer development, however,...
Show moreNumerical methods for solving partial differential equations are commonplace in the engineering community and their popularity can be attributed to the rapid performance improvement of modern workstations and desktop computers. The ubiquity of computer technology has allowed all areas of engineering to have access to detailed thermal, stress, and fluid flow analysis packages capable of performing complex studies of current and future designs. The rapid pace of computer development, however, has begun to outstrip efforts to reduce analysis overhead. As such, most commercially available software packages are now limited by the human effort required to prepare, develop, and initialize the necessary computational models. Primarily due to the mesh-based analysis methods utilized in these software packages, the dependence on model preparation greatly limits the accessibility of these analysis tools. In response, the so-called meshless or mesh-free methods have seen considerable interest as they promise to greatly reduce the necessary human interaction during model setup. However, despite the success of these methods in areas demanding high degrees of model adaptability (such as crack growth, multi-phase flow, and solid friction), meshless methods have yet to gain notoriety as a viable alternative to more traditional solution approaches in general solution domains. Although this may be due (at least in part) to the relative youth of the techniques, another potential cause is the lack of focus on developing robust methodologies. The failure to approach development from a practical perspective has prevented researchers from obtaining commercially relevant meshless methodologies which reach the full potential of the approach. The primary goal of this research is to present a novel meshless approach called MIMS (Model Integrated Meshless Solver) which establishes the method as a generalized solution technique capable of competing with more traditional PDE methodologies (such as the finite element and finite volume methods). This was accomplished by developing a robust meshless technique as well as a comprehensive model generation procedure. By closely integrating the model generation process into the overall solution methodology, the presented techniques are able to fully exploit the strengths of the meshless approach to achieve levels of automation, stability, and accuracy currently unseen in the area of engineering analysis. Specifically, MIMS implements a blended meshless solution approach which utilizes a variety of shape functions to obtain a stable and accurate iteration process. This solution approach is then integrated with a newly developed, highly adaptive model generation process which employs a quaternary triangular surface discretization for the boundary, a binary-subdivision discretization for the interior, and a unique shadow layer discretization for near-boundary regions. Together, these discretization techniques are able to achieve directionally independent, automatic refinement of the underlying model, allowing the method to generate accurate solutions without need for intermediate human involvement. In addition, by coupling the model generation with the solution process, the presented method is able to address the issue of ill-constructed geometric input (small features, poorly formed faces, etc.) to provide an intuitive, yet powerful approach to solving modern engineering analysis problems.
Show less - Date Issued
- 2010
- Identifier
- CFE0003299, ucf:48489
- Format
- Document (PDF)
- PURL
- http://purl.flvc.org/ucf/fd/CFE0003299