abstract: Divergence-free vector field interpolants properties are explored on uniform and scattered nodes, and also their application to fluid flow problems. These interpolants may be applied to physical problems that require the approximant to have zero divergence, such as the velocity field in the incompressible Navier-Stokes equations and the magnetic and electric fields in the Maxwell's equations. In addition, the methods studied here are meshfree, and are suitable for problems defined on complex domains, where mesh generation is computationally expensive or inaccurate, or for problems where the data is only available at scattered locations. The contributions of this work include a detailed comparison between standard and divergence-free radial basis approximations, a study of the Lebesgue constants for divergence-free approximations and their dependence on node placement, and an investigation of the flat limit of divergence-free interpolants. Finally, numerical solvers for the incompressible Navier-Stokes equations in primitive variables are implemented using discretizations based on traditional and divergence-free kernels. The numerical results are compared to reference solutions obtained with a spectral method. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

In this paper the obstacle description with Radial Basis Functions for contact problems in three dimensional elasticity will be done. A short Introduction of the idea of Radial Basis Functions will be followed by the usage of Radial Basis Functions for approximation of isosurfaces. Then these isosurfaces are used for the obstacle-description in three dimensional elasticity contact problems. In the last part some computational examples will be shown.

Radial basis function (RBF) methods for partial differential equations (PDEs), either in bulk domains, on surfaces, or in a combination of the formers, arise in a wide range of practical applications. This thesis proposes numerical approaches of RBF-based meshless techniques to solve these three kinds of PDEs on stationary and nonstationary surfaces and domains. In Chapter 1, we introduce the background of RBF methods, some basic concepts, and error estimates for RBF interpolation. We then provide some preliminaries for manifolds, restricted RBFs on manifolds, and some convergence properties of RBF interpolation. Finally, implicit-explicit time stepping schemes are briefly presented. In Chapter 2, we propose methods to implement meshless collocation approaches intrinsically to solve elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, that is, Kansa methods. Our main theoretical contribution is a robust convergence analysis under some standard smoothness assumptions for high-order convergence. We simulate reaction-diffusion equations to generate Turing patterns and solve shallow water problems on manifolds. In Chapter 3, we consider convective-diffusion problems that model surfactants or heat transport along moving surfaces. We propose two time-space algorithms by combining the methods of lines and kernel-based meshless collocation techniques intrinsic to surfaces. We use a low-order time discretization for fair comparison, and higher-order schemes in time are possible. The proposed methods can achieve second-order convergence. They use either analytic or approximated spatial discretization of the surface operators, which do not require regeneration of point clouds at each temporal iteration. Thus, they are alternatively applied to handle models on two types of evolving surfaces, which are defined as prescribed motions and governed by geometric evolution laws, respectively. We present numerical examples on various evolving surfaces for the performance of our algorithms and apply the approximated one to merging surfaces. In Chapter 4, a kernel-based meshless method is developed to solve coupled second-order elliptic PDEs in bulk domains and on surfaces, subject to Robin boundary conditions. It combines a least-squares kernel-based collocation method with a surface-type intrinsic approach. We can thus use each pair for discrete point sets, RBF kernels (globally and restrictedly), trial spaces, and some essential assumptions, to search for least-squares solutions in bulks and on surfaces, respectively. We first analyze error estimates for a domain-type Robin-boundary problem. Based on this analysis and the existing results for surface PDEs, we discuss the theoretical requirements for the Sobolev kernels used. We then select the orders of smoothness for the kernels in bulks and on surfaces. Finally, several numerical experiments are demonstrated to test the robustness of the coupled method in terms of accuracy and convergence rates under different settings.

Distributed modal filters based on piezoelectric polymer have especially become popular in the field of active vibration control to reduce the problem of spillover. While distributed modal filters for one-dimensional structures can be found analytically based on the orthogonality between the mode shapes, the design for two-dimensional structures is not straightforward. It requires a continuous gain variation in two dimensions, which is not realizable from the current manufacturing point of view. In this thesis, a structural optimization problem is considered to approximate distributed modal sensors for two-dimensional plate structures, where the thickness is constant but the polarization can switch between positive and negative. The problem is solved through an explicit parametric level set method. In this framework, the boundary of a domain is represented implicitly by the zero isoline of a level set function. This allows simultaneous shape and topology changes. The level set function is approximated by a linear combination of Gaussian radial basis functions. As a result, the structural optimization problem can be directly posed in terms of the parameters of the approximation. This allows to apply standard optimization methods and bypasses the numerical drawbacks, such as reinitialization, velocity extension and regularization, which are associated with the numerical solution of the Hamilton-Jacobi equation in conventional methods.Since the level set method based on the shape derivative formally only allows shape but not topology transformation, the optimization problem is firstly tackled with a derivative-free optimization algorithm. It is shown that the approach is able to find approximate modal sensor designs with only few design variables. However, this approach becomes unsuitable as soon as the number of optimization variables is growing. Therefore, a sensitivity-based optimization approach is being applied, based on the parametric shape derivative which is with respect to the parameters of the radial basis functions. Although the shape derivatives does not exist at points where the topology changes, it is demonstrated that an optimization routine based on a SQP solver is able to perform topological changes during the optimization and finds optimal designs even from poor initial designs. In order to include the sensors' distribution as design variable, the parametric level set approach is extended to multiple level sets. It turns out that, despite the increased design space, optimal solutions always converge to full-material polarization designs. Numerical examples are provided for a simply supported as well as a cantilever square plate. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

Current methods for solving thennoelasticity problems involve using finite element analysis, boundary element analysis, or other meshed-type methods to determine the deflections under an imposed temperature/stress field. This thesis will detail a new approach using meshless methods to solve these types of thermoelasticity problems in which the solution is independent of boundary and internal meshing. With the rapidly increasing availability and performance of computer workstations and clusters, the major time requirement for solving a thermoelasticity model is no longer the computation time, but rather the problem setup. Defining the required mesh for a complex geometry can be extremely complicated and time consuming, and new methods are desired that can reduce this model setup time. The proposed meshless methods completely eliminate the need for a mesh, and thus, eliminate the need for complicated meshing procedures. Although the savings gain due to eliminating the meshing process would be more than sufficient to warrant further study, the localized meshless method can also be comparable in computational speed to more traditional finite element solvers when analyzing complex problems. The reduction of both setup and computational time makes the meshless approach an ideal method of solving coupled thermoelasticity problems. Through the development of these methods it can be determined whether they are feasible as potential replacements for more traditional solution methods. More specifically, two methods will be covered in depth from the development to the implementation. The first method covered will be the global meshless method and the second will be the improved localized method. Although they both produce similar results in terms of accuracy, the localized method greatly improves upon the stability and computation time of the global method.

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 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.

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 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.

Yes / Metamodelling or surrogate modelling techniques are frequently used across the engineering disciplines in conjunction with expensive simulation models or physical experiments. With the proliferation of metamodeling techniques developed to provide enhanced performance for specific problems, and the wide availability of a diverse choice of tools in engineering software packages, the engineering task of selecting a robust metamodeling technique for practical problems is still a challenge. This research introduces a framework for describing the typology of engineering problems, in terms of dimensionality and complexity, and the modelling conditions, reflecting the noisiness of the signals and the affordability of sample sizes, and on this basis presents a systematic evaluation of the performance of frequently used metamodeling techniques. A set of metamodeling techniques, selected based on their reported use for engineering problems (i.e. Polynomial, Radial Basis Function, and Kriging), were systematically evaluated in terms of accuracy and robustness against a carefully assembled set of 18 test functions covering different types of problems, sampling conditions and noise conditions. A set of four real-world engineering case studies covering both computer simulation and physical experiments were also analysed as validation tests for the proposed guidelines. The main conclusions drawn from the study are that Kriging model with Matérn 5/2 correlation function performs consistently well across different problem types with smooth (i.e. not noisy) data, while Kriging model with Matérn 3/2 correlation function provides robust performance under noisy conditions, except for the very high noise conditions, where the Kriging model with nugget appears to provide better models. These results provide engineering practitioners with a guide for the choice of a metamodeling technique for problem types and modelling conditions represented in the study, whereas the evaluation framework and benchmarking problems set will be useful for researchers conducting similar studies.

Discontinuous Galerkin methods form a class of numerical methods to find a solution of partial differential equations by combining features of finite element and finite volume methods. Methods are defined using a weak form of a particular model problem, allowing for discontinuities in the discrete trial and test spaces. Using a discontinuous discrete space mesh provides proper flexibility and a compact discretisation pattern, allowing a multidomain and multiphysics simulation. Discontinuous Galerkin methods with a higher approximation polynomial order, the socalled p-version, performs better in terms of convergence rate, compared with the low order h-version with smaller element sizes and bigger mesh. However, the condition number of the Galerkin system grows subsequently. This causes surge in the amount of required storage, computational complexity and in the time required for computation. We use the following three approaches to keep the advantages and eliminate the disadvantages. The first approach will be a specific choice of basis functions which we call C1 polynomials. These ensure that the majority of integrals over the edge of the mesh elements disappears. This reduces the total number of non-zero elements in the resulting system. This decreases the computational complexity without loss in precision. This approach does not affect the number of iterations required by chosen Conjugate Gradients method when compared to the other choice of basis functions. It actually decreases the total number of algebraic operations performed. The second approach is the introduction of suitable preconditioners. In our case, the Additive two-layer Schwarz method, developed in [4], for the iterative Conjugate Gradients method is considered. This directly affects the spectral condition number of the system matrix and decreases the number of iterations required for the computation. This approach, however, increases the total number of algebraic operations and might require more operational time. To tackle the rise in the number of algebraic operations, we introduced a modified Additive two-layer non-overlapping Schwarz method with a Multigrid process. This using a fixed low-order approximation polynomial degree on a coarse grid. We show that this approach is spectrally equivalent to the first preconditioner, and requires less time for computation. The third approach is a development of an efficient mathematical framework for distributed data structure. This allows a high performance, massively parallel, implementation of the discontinuous Galerkin method. We demonstrate that it is possible to exploit properties of the system matrix and C1 polynomials as basis functions to optimize the parallel structures. The previously mentioned parallel data structure allows us to parallelize at the same time both the matrix-vector multiplication routines for the Conjugate Gradients method, as well as the preconditioner routines on the solver level. This minimizes the transfer ratio amongst the distributed system. Finally, we combined all three approaches and created a framework, which allowed us to successfully implement all of the above.

Made available in DSpace on 2016-12-23T14:08:15Z (GMT). No. of bitstreams: 1 Atila Lupim Cruz.pdf: 1394501 bytes, checksum: 0954b2c5b1fdcb864ee81cef7d14e9e5 (MD5) Previous issue date: 2012-10-19 / A pesquisa envolvida na presente dissertação se baseou no uso de funções de base radial para gerar uma nova formulação integral, que interpola diretamente o termo não homogêneo da equação diferencial de governo, no contexto do Método dos Elementos de Contorno (MEC). Emprega-se o uso de funções primitivas das funções de interpolação originais no núcleo da integral de domínio, permitindo a transformação desta última numa integral de contorno, evitando assim a discretização do domínio por meio de células, semelhante ao realizado na Dupla Reciprocidade. Para melhor avaliação das potencialidades da formulação, os testes numéricos apresentados abordaram apenas a solução de problemas governados pela Equação de Poisson. Os problemas escolhidos dentro desta categoria possuem solução analítica, o que permitiu aferir com mais rigor a precisão dos resultados. Para melhor balizamento da eficiência da formulação proposta, todos os problemas abordados também foram resolvidos pela formulação com Dupla Reciprocidade. O custo computacional dispendido para cada uma dessas formulações também foi comparado. Para ambas as formulações também foram testados esquemas de ajuste da interpolação realizada, visando avaliar seus efeitos na precisão dos resultados e também propositando obter economia computacional em futuras aplicações em simulações na área de propagações de ondas / This research was based on the use of radial basis functions to generate a new integral formulation that interpolates directly the domain action, related to the inhomogeneous term of the governing differential equation, using the Boundary Element Method (BEM). The use of primitive functions of the original interpolation functions in the kernel of the inhomogeneous integral is proposed, allowing its transformation into a boundary integral, thus avoiding the domain discretization through cells, similar to that conducted in the Dual Reciprocity. To better evaluation of the capability of the proposed formulation, the numerical tests presented only solved problems governed by the Poisson Equation. Test problems chosen have known analytical solution, which allowed a better evaluation of the numerical accuracy. To better check the efficiency of the proposed formulation, all the problems were also solved by the Dual Reciprocity Boundary Element Formulation. The computational cost expended for each of these formulations was also compared. Fitting interpolation schemes for both formulations were also tested in order to evaluate their effects on the accuracy of the results and also looking for economy in future computational applications related to wave propagation problems