Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems

[EN] In this paper, a parametric family of seventh-order of iterative method to solve systems of nonlinear equations is presented. Its local convergence is studied and quadratic polynomials are used to investigate its dynamical behavior. The study of the fixed and critical points of the rational function associated to this class allows us to obtain regions of the complex plane where the method is stable. By depicting parameter planes and dynamical planes we obtain complementary information of the analytical results. These results are used to solve some nonlinear problems. (C) 2017 Elsevier Inc. All rights reserved.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P and by Generalitat Valenciana PROMETEO/2016/089.Amiri, A.; Cordero Barbero, A.; Darvishi, M.; Torregrosa Sánchez, JR. (2018). Stability analysis of a parametric family of seventh-order iterative methods for solving nonlinear systems. Applied Mathematics and Computation. 323:43-57. https://doi.org/10.1016/j.amc.2017.11.040S435732

Stability analysis of a family of optimal fourth-order methods for multiple roots

[EN] Complex dynamics tools applied on the rational functions resulting from a parametric family of roots solvers for nonlinear equations provide very useful results that have been stated in the last years. These qualitative properties allow the user to select the most efficient members from the family of iterative schemes, in terms of stability and wideness of the sets of convergent initial guesses. These tools have been widely used in the case of iterative procedures for finding simple roots and only recently are being applied on the case of multiplicity m >1. In this paper, by using weight function procedure, we design a general class of iterative methods for calculating multiple roots that includes some known methods. In this class, conditions on the weight function are not very restrictive, so a large number of different subfamilies can be generated, all of them are optimal with fourth-order of convergence. Their dynamical analysis gives us enough information to select those with better properties and test them on different numerical experiments, showing their numerical properties.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C2-2-P, Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.Zafar, F.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2019). Stability analysis of a family of optimal fourth-order methods for multiple roots. Numerical Algorithms. 81(3):947-981. https://doi.org/10.1007/s11075-018-0577-0S94798181

Estudio sobre convergencia y dinámica de los métodos de Newton, Stirling y alto orden

Las matemáticas, desde el origen de esta ciencia, han estado al servicio de la sociedad tratando de dar respuesta a los problemas que surgían. Hoy en día sigue siendo así, el desarrollo de las matemáticas está ligado a la demanda de otras ciencias que necesitan dar solución a situaciones concretas y reales. La mayoría de los problemas de ciencia e ingeniería no pueden resolverse usando ecuaciones lineales, es por tanto que hay que recurrir a las ecuaciones no lineales para modelizar dichos problemas (Amat, 2008; véase también Argyros y Magreñán, 2017, 2018), entre otros. El conflicto que presentan las ecuaciones no lineales es que solo en unos pocos casos es posible encontrar una solución única, por tanto, en la mayor parte de los casos, para resolverlas hay que recurrir a los métodos iterativos. Los métodos iterativos generan, a partir de un punto inicial, una sucesión que puede converger o no a la solución

Positivity-preserving discontinuous spectral element methods for compressible multi-species flows

We introduce a novel positivity-preserving, parameter-free numerical stabilisation approach for high-order discontinuous spectral element approximations of compressible multi-species flows. The underlying stabilisation method is the adaptive entropy filtering approach (Dzanic and Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative formulation of the multi-species flow equations. We show that the straightforward enforcement of entropy constraints in the filter yields poor results around species interfaces and propose an adaptive, parameter-free switch for the entropy bounds based on the convergence properties of the pressure field which drastically improves its performance for multi-species flows. The proposed approach is shown in a variety of numerical experiments applied to the multi-species Euler and Navier--Stokes equations computed on unstructured grids, ranging from shock-fluid interaction problems to three-dimensional viscous flow instabilities. We demonstrate that the approach can retain the high-order accuracy of the underlying numerical scheme even at smooth extrema, ensure the positivity of the species density and pressure in the vicinity of shocks and contact discontinuities, and accurately predict small-scale flow features with minimal numerical dissipation.Comment: Submitted for revie

Model-based approaches to support process improvement in complex product development

The performance of product development processes is important to the commercial success of new products. The improvement of these processes is thus a strategic imperative for many engineering companies — the aero-engine is one example of a complex product for which market pressures necessitate ever-shorter development times. This thesis argues that process modelling and simulation can support the improvement of complex product development processes. A literature review identified that design process modelling is a well-established research area encompassing a diverse range of approaches. However, most existing tools and methods are not widely applied in industry. An extended case study was therefore conducted to explore the pragmatic utility of process modelling and simulation. It is argued that iteration is a key driver of design process behaviour which cannot be fully reflected in a mechanistic model. Understanding iteration can help select an appropriate representation for a given process domain and modelling objective. A model-based approach to improve the management of iterative design processes was developed. This approach shows that design process simulation models can support practice despite their limited fidelity. The modelling and simulation framework resulting from this work was enhanced for application to a wider range of process improvement activities. A robust and extensible software platform was also developed. The framework and software tool have made significant contribution to research projects investigating process redesign, process robustness and process optimisation. These projects are discussed to validate the framework and tool and to highlight their applicability beyond the original approach. The research results were disseminated in academia and industry — 72 copies of the software were distributed following requests in the first three months of its release

Updating the Lambda modes of a nuclear power reactor

[EN] Starting from a steady state configuration of a nuclear power reactor some situations arise in which the reactor configuration is perturbed. The Lambda modes are eigenfunctions associated with a given configuration of the reactor, which have successfully been used to describe unstable events in BWRs. To compute several eigenvalues and its corresponding eigenfunctions for a nuclear reactor is quite expensive from the computational point of view. Krylov subspace methods are efficient methods to compute the dominant Lambda modes associated with a given configuration of the reactor, but if the Lambda modes have to be computed for different perturbed configurations of the reactor more efficient methods can be used. In this paper, different methods for the updating Lambda modes problem will be proposed and compared by computing the dominant Lambda modes of different configurations associated with a Boron injection transient in a typical BWR reactor. (C) 2010 Elsevier Ltd. All rights reserved.This work has been partially supported by the Spanish Ministerio de Educacion y Ciencia under projects ENE2008-02669 and MTM2007-64477-AR07, the Generalitat Valenciana under project ACOMP/2009/058, and the Universidad Politecnica de Valencia under project PAID-05-09-4285.González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ. (2011). Updating the Lambda modes of a nuclear power reactor. Mathematical and Computer Modelling. 54(7):1796-1801. https://doi.org/10.1016/j.mcm.2010.12.013S1796180154

A new computational approach to topology optimization in solid mechanics problems

Tesi en modalitat compendi de publicacionsThis doctoral thesis addresses topology optimization problems at a single scale. Based on this purpose, a new topology optimization approach is developed in order to improve existing and widespread techniques in the research community on the topic. The proposed technique presents several characteristics that overcome some of the well-known difficulties in topological optimization while maintaining a considerable degree of simplicity. In the first place, the formulation of the topological optimization technique is presented, as well as its algorithm. The method is based on 4 fundamental features: (1) the use of a 1-0 characteristic function, as well as the precise identification of the material boundaries from a discrimination function (0-level-set function), (2) the definition of a topological derivative consistent with the ersatz method (used in the state problem), as an approximation to the exact topological derivative, (3) the inclusion of a Laplacian regularization with minimum size control of the different components, and (4) the formulation of an analytical optimality condition aiming at the optimal topology solution. The approach is applied to different topology optimization problems, well-reported in the literature and used as numerical benchmarks (in structural and thermal problems), to examine their performance. In these fields, stiffness and conductivity maximization problems are considered for validation, respectively. In addition, different topological optimization problems of major engineering interest are tackled, including the design of compliant mechanisms within the structural field and thermal cloaking devices within the thermal field. Finally, a comparison of the formulation with other existing topology optimization techniques is performed, including (1) SIMP, (2) ESO/BESO, and (3) Level-set with Hamilton-Jacobi as the updating equation. The analysis of the results provides a comparison in terms of the quality of the topology of each method, the computational cost of the optimal solutions, as well as the simplicity of implementation. The resulting study reveals the potential of the developed methodology in these specific comparison terms. In an attempt to bring the method closer to other researchers and to promote its use, an educational version of the method (written in MATLAB) has been published in an online repository, together with documentation, facilitating its dissemination and subsequent use in other applications of interest.El objetivo de esta tesis doctoral es abordar el problema de optimización topológica a una única escala. En base a este propósito, se desarrolla una nueva técnica de optimización capaz de competir con técnicas ya existentes y extendidas entre la comunidad investigadora sobre el tema. Esta técnica presenta características que superan algunas de las dificultades bien conocidas en optimización topológica manteniendo un buen grado de simplicidad. En primer lugar, se presenta la formulación de la técnica de optimización topológica, así como su algoritmia. El método se fundamenta en 4 aspectos básicos: (1) la utilización de una función característica 1-0, así como la definición precisa de las fronteras materiales a partir de una función de discriminación (isonivel 0 de la función level-set), (2) la definición de una derivada topológica coherente con el método ersatz (utilizado en la ecuación de estado), como aproximación a la derivada topológica exacta, (3) la inclusión de una regularización Laplaciana con control de tamaño mínimo de los diferentes componentes, y (4) la definición de una condición de optimalidad analítica para la determinación de la solución óptima de la topología. La metodología se aplica a diferentes problemas de optimización topológica bien detallados en la literatura y utilizados como ensayos numéricos para examinar su respuesta frente a problemas estructurales y térmicos. En estos campos, se incluyen problemas de maximización de la rigidez y de la conductividad, respectivamente. Además, se resuelven diferentes problemas de optimización topológica con gran interés ingenieril en los campos estructurales con el diseño de mecanismos y térmicos con el diseño de dispositivos de camuflaje térmicos. Finalmente, se realiza una comparación de la formulación con otras técnicas ya existentes, por ejemplo: (1) SIMP, (2) ESO/BESO, y (3) Level-set con Hamilton-Jacobi como ecuación de evolución. El análisis de los resultados permite comparar la calidad de la topología de cada método, el coste computacional de las soluciones óptimas, así como la simplicidad de implementación, demostrando el potencial de la metodología desarrollada principalmente en estos términos de comparación. Con la finalidad de acercar el método a otros investigadores y de promover su utilización, se ha publicado una versión educativa del mismo (en MATLAB) en un repositorio online, junto a documentación, permitiendo así la divulgación del mismo y la posible utilización en otras aplicaciones de interés.Postprint (published version

Development of Methods to Predict the Effects of Test Media in Ground-Based Propulsion Testing

This report discusses work that began in mid-2004 sponsored by the Office of the Secretary of Defense (OSD) Test & Evaluation/Science & Technology (T&E/S&T) Program. The work was undertaken to improve the state of the art of CFD capabilities for predicting the effects of the test media on the flameholding characteristics in scramjet engines. The program had several components including the development of advanced algorithms and models for simulating engine flowpaths as well as a fundamental experimental and diagnostic development effort to support the formulation and validation of the mathematical models. This report provides details of the completed work, involving the development of phenomenological models for Reynolds averaged Navier-Stokes codes, large-eddy simulation techniques and reduced-kinetics models. Experiments that provided data for the modeling efforts are also described, along with with the associated nonintrusive diagnostics used to collect the data