Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell's equations

The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be used in order to avoid the pollution effect well known for the Helmholtz equation, and second the large scale systems obtained from the vector valued equations in three spatial dimensions need to be solved by iterative methods, since direct factorizations are not feasible any more at that scale. As for the Helmholtz equation, classical iterative methods applied to discretized Maxwell equations have severe convergence problems.We explain in this paper a family of domain decomposition methods based on well chosen transmission conditions. We show that all transmission conditions proposed so far in the literature, both for the first and second order formulation of Maxwell's equations, can be written and optimized in the common framework of optimized Schwarz methods, independently of the first or second order formulation one uses, and the performance of the corresponding algorithms is identical. We use a decomposition into transverse electric and transverse magnetic fields to describe these algorithms, which greatly simplifies the convergence analysis of the methods. We illustrate the performance of our algorithms with large scale numerical simulations

Advanced techniques in scientific computing: application to electromagnetics

Mención Internacional en el título de doctorDurante los últimos años, los componentes de radiofrecuencia que forman parte de un sistema de comunicaciones necesitan simulaciones cada vez más exigentes desde el punto de vista de recursos computacionales. Para ello, se han desarrollado diferentes técnicas con el método de los elementos finitos (FEM) como la conocida como adaptatividad hp, que consiste en estimar el error en el problema electromagnético para generar mallas de elementos adecuadas al problema que obtienen una aproximación de forma más efectiva que las mallas estándar; o métodos de descomposición de dominios (DDM), basado en la división del problema original en problemas más pequeños que se pueden resolver en paralelo. El principal problema de las técnicas de adaptatividad es que ofrecen buenas prestaciones para problemas bidimensionales, mientras que en tres dimensiones el tiempo de generación de las mallas adaptadas es prohibitivo. Por otra parte, DDM se ha utilizado satisfactoriamente para la simulación de problemas eléctricamente muy grandes y de gran complejidad, convirtiéndose en uno de los temas más actuales en la comunidad de electromagnetismo computacional. El principal objetivo de este trabajo es estudiar la viabilidad de algoritmos escalables (en términos de paralelización) combinando DDM no conformes y adaptatividad automática en tres dimensiones. Esto permitir ía la ejecución de algoritmos de adaptatividad independiente en cada subdominio de DDM. En este trabajo se presenta y discute un prototipo que combina técnicas de adaptatividad y DDM, que aún no se han tratado en detalle en la comunidad científica. Para ello, se implementan tres bloques fundamentales: i) funciones de base para los elementos finitos que permitan órdenes variables dentro de la misma malla; ii) DDM no conforme y sin solapamiento; y iii) algoritmos de adaptatividad en tres dimensiones. Estos tres bloques se han implementado satisfactoriamente en un código FEM mediante un método sistemático basado en el método de las soluciones manufacturadas (MMS). Además, se ha llevado a cabo una paralelización a tres niveles: a nivel de algoritmo, con DDM; a nivel de proceso, con MPI (Message Passing Interface); y a nivel de hebra, con OpenMP; todo en un código modular que facilita el mantenimiento y la introducción de nuevas características. Con respecto al primer bloque fundamental, se ha desarrollado una familia de funciones base con un enfoque sistemático que permite la expansión correcta del espacio de funciones. Por otra parte, se han introducido funciones de base jerárquicas de otros autores (con los que el grupo al que pertenece el autor de la tesis ha colaborado estrechamente en los últimos años) para facilitar la introducción de diferentes órdenes de aproximación en el mismo mallado. En lo relativo a DDM, se ha realizado un estudio cuantitativo del error generado por las disconformidades en la interfaz entre subdominios, incluidas las discontinuidades generadas por un algoritmo de adaptatividad. Este estudio es fundamental para el correcto funcionamiento de la adaptatividad, y no ha sido evaluado con detalle en la comunidad científica. Además, se ha desarrollado un algoritmo de adaptatividad con prismas triangulares, haciendo especial énfasis en las peculiaridades debidas a la elección de este elemento. Finalmente, estos tres bloques básicos se han utilizado para desarrollar, y discutir, un prototipo que une las técnicas de adaptatividad y DDM.In the last years, more and more accurate and demanding simulations of radiofrequency components in a system of communications are requested by the community. To address this need, some techniques have been introduced in finite element methods (FEM), such as hp adaptivity (which estimates the error in the problem and generates tailored meshes to achieve more accuracy with less unknowns than in the case of uniformly refined meshes) or domain decomposition methods (DDM, consisting of dividing the whole problem into more manageable subdomains which can be solved in parallel). The performance of the adaptivity techniques is good up to two dimensions, whereas for three dimensions the generation time of the adapted meshes may be prohibitive. On the other hand, large scale simulations have been reported with DDM becoming a hot topic in the computational electromagnetics community. The main objective of this dissertation is to study the viability of scalable (in terms of parallel performance) algorithms combining nonconformal DDM and automatic adaptivity in three dimensions. Specifically, the adaptivity algorithms might be run in each subdomain independently. This combination has not been detailed in the literature and a proof of concept is discussed in this work. Thus, three building blocks must be introduced: i) basis functions for the finite elements which support non-uniform approximation orders p; ii) non-conformal and non-overlapping DDM; and iii) adaptivity algorithms in 3D. In this work, these three building blocks have been successfully introduced in a FEM code with a systematic procedure based on the method of manufactured solutions (MMS). Moreover, a three-level parallelization (at the algorithm level, with DDM; at the process level, with message passing interface (MPI), and at the thread level, with OpenMP) has been developed using the paradigm of modular programming which eases the software maintenance and the introduction of new features. Regarding first building block, a family of basis functions which follows a sound mathematical approach to expand the correct space of functions is developed and particularized for triangular prisms. Also, to ease the introduction of different approximation orders in the same mesh, hierarchical basis functions from other authors are used as a black box. With respect to DDM, a thorough study of the error introduced by the non-conformal interfaces between subdomains is required for the adaptivity algorithm. Thus, a quantitative analysis is detailed including non-conformalities generated by independent refinements in neighbor subdomains. This error has not been assessed with detail in the literature and it is a key factor for the adaptivity algorithm to perform properly. An adaptivity algorithm with triangular prisms is also developed and special considerations for the implementation are explained. Finally, on top of these three building blocks, the proof of concept of adaptivity with DDM is discussed.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Daniel Segovia Vargas.- Secretario: David Pardo Zubiaur.- Vocal: Romanus Dyczij-Edlinge

The EMCC / DARPA Massively Parallel Electromagnetic Scattering Project

The Electromagnetic Code Consortium (EMCC) was sponsored by the Advanced Research Program Agency (ARPA) to demonstrate the effectiveness of massively parallel computing in large scale radar signature predictions. The EMCC/ARPA project consisted of three parts

Numerical modelling of the deformation of elastic material by the TLM method

The transmission line matrix (TLM) method is a numerical tool for the solution of wave and diffusion type equations. The application of TLM to physical phenomena such as heat flow and electromagnetic wave propagation is well established. A previous attempt to apply TLM models to the area of elastic wave propagation and elastic deformation had limited success. The work of this thesis extends the application base of TLM to the area of elastic deformation modelling and validates the model for several two-dimensional situations. In doing this it has been necessary to develop new nodal structures which facilitate the scaling of differential coefficients and incorporation of cross derivatives. Nodal structures which allow the modelling of two and three-dimensional, and anisotropic, elastic deformation are described.The technique is demonstrated by applying the elastic deformation model to several elastic problems. These include two-dimensional isotropic models and models of anisotropic elastic deformation. Provision is also made for the application of various boundary conditions which include displacement, force and frictional boundaries

2004 Graduate Bulletin

After 2003 the University of Dayton Bulletin went exclusively online. This copy was printed from the web and scanned by the Registrar’s Office. For general information about the university please see the Undergraduate Bulletin.https://ecommons.udayton.edu/bulletin_grad/1000/thumbnail.jp

The application of dynamic relaxation to the design of modular space structures

This thesis is concerned with the development and assessment of computer techniques for the formfinding and sizing of large modular building space structures suitable for urban development. The contents of the Chapters are summarised as follows: 1: An introduction to conceptual and computer aided design of large building space structures. 2: A review of topological computer design methods. 3: A comparison of a dynamic relaxation method, for the formfinding of modularly constrained structures subject to a dominant design loading case, with linear programming and fully stressed design methods. This comparison shows that the dynamic relaxation method is efficient and particularly suitable for interactive use. 4: A parametric study of the effects of the iteration parameters on the stability and rate of convergence is presented. No general rules appear to be possible regarding the effects of these parameters on stability. It is noted, however, that the number of structure modifications before the solution becomes apparent is independent of the parameters. The dynamic relaxation formfinding procedure is generalised to cater for different stress constraints in tension and compression members and, for the problem considered, derives a lighter form than the fully stressed design technique. The optimum form of the D.R. solution is verified by the linear programming technique. 5: An intuitive dynamic relaxation method for the sizing of structures of fixed topology subject to multiple loading cases ,and stress constraints is presented. The method also caters for maximum member area sizes and deflection constraints by the use of parallel elastic and elasto-plastic analyses. Solutions derived using this method are compared with solutions derived using the non-linear program algorithm. They are shown to be of similar weight and to require similar solution times. 6: Two dynamic relaxation methods are presented for the formfinding and sizing of multiply loaded space structures. The first or parallel method is suitable for deriving and sizing forms of optimum or near optimum weight by deleting members which are small in area size and reducing in size. The second or series method is particularly suitable for interactive use and consists of testing the efficiency of each member with respect to each loading case. The final topology is then sized considering all loading cases simultaneously. These methods are both applied to a bridging ground structure subject to multiple loads and compared with solutions derived using linear, nonlinear programming and topological design methods. The parallel dynamic relaxation method is then extended to cater for cable members allowing for on-off non-linearities and prestress effects. The bridging structure is subsequently redesigned using internal cable members and adjusting the prestress level to ensure that the bridge deck does not deflect vertically under the action of the primary loading case. 7: A summary of conclusions

Fifteenth NASTRAN (R) Users' Colloquium

Numerous applications of the NASA Structural Analysis (NASTRAN) computer program, a general purpose finite element code, are discussed. Additional features that can be added to NASTRAN, interactive plotting of NASTRAN data on microcomputers, mass modeling for bars, the design of wind tunnel models, the analysis of ship structures subjected to underwater explosions, and buckling analysis of radio antennas are among the topics discussed

3D parallel computations of turbofan noise propagation using a spectral element method

A three-dimensional code has been developed for the simulation of tone noise generated by turbofan engine inlets using computational aeroacoustics. The governing equations are the linearized Euler equations, which are further simplified to a set of equations in terms of acoustic potential, using the irrotational flow assumption, and subsequently solved in the frequency domain.Due to the special nature of acoustic wave propagation, the spatial discretization is performed using a spectral element method, where a tensor product of the nth-degree polynomials based on Chebyshev orthogonal functions is used to approximate variations within hexahedral elements. Non-reflecting boundary conditions are imposed at the far-field using a damping layer concept. This is done by augmenting the continuity equation with an additional term without modifying the governing equations as in PML methods.Solution of the linear system of equations for the acoustic problem is based on the Schur complement method, which is a nonoverlapping domain decomposition technique. The Schur matrix is first solved using a matrix-free iterative method, whose convergence is accelerated with a novel local preconditioner. The solution in the entire domain is then obtained by finding solutions in smaller subdomains.The 3D code also contains a mean flow solver based on the full potential equation in order to take into account the effects of flow variations around the nacelle on the scattering of the radiated sound field.All aspects of numerical simulations, including building and assembling the coefficient matrices, implementation of the Schur complement method, and solution of the system of equations for both the acoustic and mean flow problems are performed on multiprocessors in parallel using the resources of the CLUMEQ Supercomputer Center. A large number of test cases are presented, ranging in size from 100 000-2 000 000 unknowns for which, depending on the size of the problem, between 8-48 CPU's are used.The developed code is demonstrated to be robust and efficient in simulating acoustic propagation for a large number of problems, with an excellent parallel performance

[Research activities in applied mathematics, fluid mechanics, and computer science]

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period April 1, 1995 through September 30, 1995

Finite element-based non-linear dynamic soil-structure interaction.

The modelling of unbounded domains is an important consideration in many engineering problems, for example in fluid flow, electro-magnetics, acoustics and solid mechanics. This thesis focuses on the problem of modelling elastic solids to infinity, with the specific purpose of modelling dynamic soil-structure interaction (DSSI). However, the reader should be aware that the techniques presented may also be adapted to address those other physical phenomena. The need for techniques to model the soil domain to infinity and a qualitative introduction into the problems associated with dynamic soil-structure interaction are outlined in chapter 1. This is done to illustrate why such an abstract mathematical concept of modelling infinite domains has an important role to play within the design process of large, safety critical, civil engineering structures. A brief review of a number of alternative ways of addressing this problem is given in chapter 2. Their relative strengths and weaknesses along with the typical applicability of the techniques is discussed. A consequence of this review is the identification of a very promising rigorous approach [59] which is singled-out for further study. A detailed explanation of this (Consistent Infinitesimal Finite Element Cell Method, CIFECM) method is then given in chapter 3. Attention is restricted to the use of the technique for solving the 3-D vector wave equation in the time domain. The features of the non-linear dynamic finite element code, into which the CIFECM has been incorporated, is highlighted in chapter 4. The non-linear (microplane) material model for quasi-brittle materials is described along with the solution strategy employed. It should be mentioned that the soil is treated within this thesis as drained linear elastic medium. The method of coupling the CIFECM into the dynamic equation of force equilibrium for both directly applied and transmitted loading regimes is detailed. Application of the code follows in chapter 5; firstly by introducing the simplest test problem of one finite element coupled with one CIFECM element to model a surface foundation. Comparisons are made between the dynamic displacements resulting from the method and standard FE solutions obtained from the use of extended meshes and fixed boundary conditions, along with a study of the influence input variables. Following these examples a larger (more realistic) engineering problem is tacked involving the simulation of an aircraft impact on a reinforced concrete nuclear containment vessel. This represents the first use of the method in a 3-D nonlinear structural analysis problem. The results illustrate the practical implications of including DSSI in the analysis. III In chapter 6, a series of general observations on the method are made with an assessment of its value together with a discussion on its wider application to other engineering fields. Possible future developments to make the method more computationally efficient are finally suggested