This thesis reports onreview and research work carried out on the numerical analysis of elastomers. The two numerical techniques investigated for this purpose are the finite and boundary element methods. The finite element method is studied so that existing theory is used to develop a finite element code both to review the finite element method as applied to the stress analysis of elastomers and to provide a comparison of results and numerical approach with the boundary element method. The research work supported on in this thesis covers the application of the boundary element method to the stress analysis of elastomers. To this end a simplified regularization approach is discussed for the removal of strong and hypersingularities generated in the system on non-linear boundary integral equations. The necessary programming details for the implementation of the boundary element method are discussed based on the code developed for this research. Both the finite and boundary element codes developed for this research use the Mooney-Rivlin material model as the strain energy based constitutive stress strain function. For validation purposes four test cases are investigated. These are the uni-axial patch test, pressurized thick wall cylinder, centrifugal loading of a rotating disk and the J-Integral evaluation for a centrally cracked plate. For the patch test and pressurized cylinder, both plane stress and strain have been investigated. For the centrifugal loading and centrally cracked plate test cases only plane stress has been investigated. For each test case the equivalent results for an equivalent FEM program mesh have been presented. The test results included in this thesis prove that the FE and BE derivations detailed in this work are correct. Specifically the simplified domain integral singular and hyper-singular regularization approach was shown to lead to accurate results for the test cases detailed. Various algorithm findings specific to the BEM implementation of the theory are also discussed.

Made available in DSpace on 2014-10-09T12:33:52Z (GMT). No. of bitstreams: 0 / Made available in DSpace on 2014-10-09T14:03:56Z (GMT). No. of bitstreams: 0 / Tese (Doutoramento) / IPEN/T / Instituto de Pesquisas Energeticas e Nucleares - IPEN-CNEN/SP

Este trabalho aborda a solução numérica das equações lineares de águas rasas. O método dos elementos finitos e utilizado para a discretização espacial das equações que modelam o problema, e para a discretização temporal, o esquema semi-implícito de Crank-Nicolson é empregado. Além de alguns conceitos comuns quando se trabalha com escoamentos geofísicos, são descritas também a formulação das equações de águas rasas, sua linearização e uma solução analítica para um caso onde o parâmetro de Coriolis é nulo. A escolha adequada de pares de elementos finitos é a principal dificuldade quando se trabalha com esse método para a resolução da equação de águas rasas. Assim, é discutido o uso de quatro pares de elementos finitos e técnicas de estabilização para contornar o surgimento de modos espúrios na solução discreta. Os resultados numéricos são realizados com auxílio do software FreeFem++, onde se pode notar a capacidade dos pares de elementos de reproduzirem o escoamento, através da solução discreta, além das propriedades de conservação de massa e energia de cada discretização. / This work is about the numerical solution of the linear shallow water equations. The finite element method is used for spatial discretization of the equations that model the problem and for the time discretization the semi-implicit Crank-Nicolson scheme is used. Besides the concepts related to geophysical flows, the formulation of the shallow water equations, their linearization and an analytical solution for a case where the Coriolis parameter is zero are also described. The appropriate choice of a pair of finite elements is the main difficulty when working with this method for solving the shallow water equations. The use of four pairs of finite elements and stabilization techniques to circumvent the appearance of spurious modes in the discrete solution are discussed. The numerical results are performed using the software FreeFem++, where one can notice the ability of the elements to represent the discrete solution and mass and energy conservation of each discretization.

Orientador: Renato Pavanello / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-10T06:16:16Z (GMT). No. of bitstreams: 1 RomeroCueto_Liliana_M.pdf: 2272152 bytes, checksum: 792e6fbefc070243b8ea9c5f015ace8a (MD5) Previous issue date: 2007 / Resumo: O comportamento de sistemas acoplados fluido-estrutura pode ser modificado pela presença de defeitos estruturais. O estudo desta modificações, usando metodos numericos de simulação, permite aos engenheiros, estimar o novo estado do sistema a partir a avaliação da variação da resposta dinâmica e vibroacustica. Desta forma, a analise de sensibilidade da resposta do sistema, permite verificar a viabilidade da aplicação das tecnicas apresentadas, em tecnicas de detecção de falhas, controle de ruido e otimização estrutural. Este trabalho apresenta um estudo do comportamento vibroacustico de paineis aeronauticos reforçados e a analise de sensibilidade da resposta, destes sistemas, na presença de falhas estruturais. È elaborado um modelo estrutural baseado na teoria de cascas e vigas que é acoplado a um meio fluido acustico interior bi e tridimensional. O metodo de Elementos Finitos é usado para a solução das equações que governam o comportamento acustico-estrutural do sistema acoplado, adotando a formulação não simetrica, com os deslocamentos e pressões como variaveis incognitas dos dominios estrutural e acustico respectivamente. São apresentadas tecnicas analiticas de analise de sensibilidade da resposta em frequencia, estrutural e acustica, em relação as modificações estruturais. A precisão numerica dos metodos de sensibilidade, propostas para o caso vibroacustico, é avaliada usando-se a tecnica da Diferenças Finitas como referencia. Com a finalidade de validar os modelos implementados, são apresentados alguns testes representativos. Na modelagem de paineis aeronauticos, dois modelos de fuselagem de aeronave são apresentados. São calculadas resposta estrutural e vibroacustica e analise de sensibilidade da resposta para modelos de fuselagem 2D e 3D na presença de modificações estruturais / Abstract: The behavior of coupled fluid-structural systems can be modified due to presence of located structural defects. The study of these modifications, by simulation, using numerical methods, allows engineers to predict the new state of the system from the evaluation of structural and acoustic responses. Sensitivity analysis of acoustic and structural responses, with respect to structural geometry modifications, allow to verify the viability to apply techniques to be presented in this work for damage detection, noise control and structural optimization. In this work a formulation for vibro-acoustic sensitivity analysis of aircraft stiffener panels under structural defects is presented. Shells and beams structural elements for two and three-dimensional models coupled with an interior fluid domain are formulated. The Finite Element Method is used to solve the structural-acoustic coupled governing equations, adopting structural displacements and fluid pressure as variables. Structural and vibro-acoustic sensitivity techniques, with respect to structural modifications are presented. The accuracy of sensitivity methods is estimated using the Finite Difference technique as a reference. Respect to aircraft panels models, two fuselage models are described. Dynamic and Vibro-acoustic response and response sensitivity with respect to structural defects are calculated. / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

Este trabalho aborda a solução numérica das equações lineares de águas rasas. O método dos elementos finitos e utilizado para a discretização espacial das equações que modelam o problema, e para a discretização temporal, o esquema semi-implícito de Crank-Nicolson é empregado. Além de alguns conceitos comuns quando se trabalha com escoamentos geofísicos, são descritas também a formulação das equações de águas rasas, sua linearização e uma solução analítica para um caso onde o parâmetro de Coriolis é nulo. A escolha adequada de pares de elementos finitos é a principal dificuldade quando se trabalha com esse método para a resolução da equação de águas rasas. Assim, é discutido o uso de quatro pares de elementos finitos e técnicas de estabilização para contornar o surgimento de modos espúrios na solução discreta. Os resultados numéricos são realizados com auxílio do software FreeFem++, onde se pode notar a capacidade dos pares de elementos de reproduzirem o escoamento, através da solução discreta, além das propriedades de conservação de massa e energia de cada discretização. / This work is about the numerical solution of the linear shallow water equations. The finite element method is used for spatial discretization of the equations that model the problem and for the time discretization the semi-implicit Crank-Nicolson scheme is used. Besides the concepts related to geophysical flows, the formulation of the shallow water equations, their linearization and an analytical solution for a case where the Coriolis parameter is zero are also described. The appropriate choice of a pair of finite elements is the main difficulty when working with this method for solving the shallow water equations. The use of four pairs of finite elements and stabilization techniques to circumvent the appearance of spurious modes in the discrete solution are discussed. The numerical results are performed using the software FreeFem++, where one can notice the ability of the elements to represent the discrete solution and mass and energy conservation of each discretization.

The experimental program ¨SPC-PM Po 3D¨ is part of the ongoing research of the Chemnitz research group Scientific Parallel Computing (SPC) into finite element methods for problems over three dimensional domains. The package in its version 2.0 is documented in two manuals. The User's Manual provides an overview over the program, its capabilities, its installation, and handling. Moreover, test examples are explained. The aim of the Programmer's Manual is to provide a description of the algorithms and their realization. It is written for those who are interested in a deeper insight into the code, for example for improving and extending. In Version 2.0 the program can solve the Poisson equation and the Lam'e system of linear elasticity with in general mixed boundary conditions of Dirichlet and Neumann type. The domain $\Omega\subset\R^3$ can be an arbitrarily bounded polyhedron. The input is a coarse mesh, a description of the data and some control parameters. The program distributes the elements of the coarse mesh to the processors, refines the elements, generates the system of equations using linear or quadratic shape functions, solves this system and offers graphical tools to display the solution. Further, the behavior of the algorithms can be monitored: arithmetic and communication time is measured, the discretization error is measured, different preconditioners can be compared. We plan to extend the program in the next future by including a multigrid solver, an error estimator and adaptive mesh refinement, as well as the treatment of coupled thermo-elastic problems. The program has been developed for MIMD computers; it has been tested on Parsytec machines (GCPowerPlus-128 with Motorola Power PC601 processors and GCel-192 on transputer basis) and on workstation clusters using PVM. The special case of only one processor is included, that means the package can be compiled for single processor machines without any change in the source files.

No description available.

In this thesis, techniques for efficient utilization of modern computer hardwarefor numerical simulation are considered. In particular, we study techniques for improving the performance of computations using the finite element method. One of the main difficulties in finite-element computations is how to perform the assembly of the system matrix efficiently in parallel, due to its complicated memory access pattern. The challenge lies in the fact that many entries of the matrix are being updated concurrently by several parallel threads. We consider transactional memory, an exotic hardware feature for concurrent update of shared variables, and conduct benchmarks on a prototype multicore processor supporting it. Our experiments show that transactions can both simplify programming and provide good performance for concurrent updates of floating point data. Secondly, we study a matrix-free approach to finite-element computation which avoids the matrix assembly. In addition to removing the need to store the system matrix, matrix-free methods are attractive due to their low memory footprint and therefore better match the architecture of modern processors where memory bandwidth is scarce and compute power is abundant. Motivated by this, we consider matrix-free implementations of high-order finite-element methods for execution on graphics processors, which have seen a revolutionary increase in usage for numerical computations during recent years due to their more efficient architecture. In the implementation, we exploit sum-factorization techniques for efficient evaluation of matrix-vector products, mesh coloring and atomic updates for concurrent updates, and a geometric multigrid algorithm for efficient preconditioning of iterative solvers. Our performance studies show that on the GPU, a matrix-free approach is the method of choice for elements of order two and higher, yielding both a significantly faster execution, and allowing for solution of considerably larger problems. Compared to corresponding CPU implementations executed on comparable multicore processors, the GPU implementation is about twice as fast, suggesting that graphics processors are about twice as power efficient as multicores for computations of this kind.

<div> <div> <div> <p>This dissertation contains two parts: one part is about the error estimate for the finite element approximation to elliptic PDEs with discontinuous Dirichlet boundary data, the other is about the error estimate of the DG method for elliptic equations with low regularity. </p> <p>Elliptic problems with low regularities arise in many applications, error estimate for sufficiently smooth solutions have been thoroughly studied but few results have been obtained for elliptic problems with low regularities. Part I provides an error estimate for finite element approximation to elliptic partial differential equations (PDEs) with discontinuous Dirichlet boundary data. Solutions of problems of this type are not in H1 and, hence, the standard variational formulation is not valid. To circumvent this difficulty, an error estimate of a finite element approximation in the W1,r(Ω) (0 < r < 2) norm is obtained through a regularization by constructing a continuous approximation of the Dirichlet boundary data. With discontinuous boundary data, the variational form is not valid since the solution for the general elliptic equations is not in H1. By using the W1,r (1 < r < 2) regularity and constructing continuous approximation to the boundary data, here we present error estimates for general elliptic equations. </p> <p>Part II presents a class of DG methods and proves the stability when the solution belong to H1+ε where ε < 1/2 could be very small. we derive a non-standard variational formulation for advection-diffusion-reaction problems. The formulation is defined in an appropriate function space that permits discontinuity across element </p> </div> </div> <div> <div> <p>viii </p> </div> </div> </div> <div> <div> <div> <p>interfaces and does not require piece wise Hs(Ω), s ≥ 3/2, smoothness. Hence, both continuous and discontinuous (including Crouzeix-Raviart) finite element spaces may be used and are conforming with respect to this variational formulation. Then it establishes the a priori error estimates of these methods when the underlying problem is not piece wise H3/2 regular. The constant in the estimate is independent of the parameters of the underlying problem. Error analysis presented here is new. The analysis makes use of the discrete coercivity of the bilinear form, an error equation, and an efficiency bound of the continuous finite element approximation obtained in the a posteriori error estimation. Finally a new DG method is introduced i to over- come the difficulty in convergence analysis in the standard DG methods and also proves the stability. </p> </div> </div> </div>

Machine learning techniques promise to greatly accelerate structural design and optimization. In this thesis, deep learning and active learning techniques are applied to different non-convex structural optimization problems. Finite Element Analysis (FEA) based standard optimization methods for aircraft panels with bio-inspired curvilinear stiffeners are computationally expensive. The main reason for employing many of these standard optimization methods is the ease of their integration with FEA. However, each optimization requires multiple computationally expensive FEA evaluations, making their use impractical at times. To accelerate optimization, the use of Deep Neural Networks (DNNs) is proposed to approximate the FEA buckling response. The results show that DNNs obtained an accuracy of 95% for evaluating the buckling load. The DNN accelerated the optimization by a factor of nearly 200. The presented work demonstrates the potential of DNN-based machine learning algorithms for accelerating the optimization of bio-inspired curvilinearly stiffened panels. But, the approach could have disadvantages for being only specific to similar structural design problems, and requiring large datasets for DNNs training. An adaptive machine learning technique called active learning is used in this thesis to accelerate the evolutionary optimization of complex structures. The active learner helps the Genetic Algorithms (GA) by predicting if the possible design is going to satisfy the required constraints or not. The approach does not need a trained surrogate model prior to the optimization. The active learner adaptively improve its own accuracy during the optimization for saving the required number of FEA evaluations. The results show that the approach has the potential to reduce the total required FEA evaluations by more than 50%. Lastly, the machine learning is used to make recommendations for modeling choices while analyzing a structure using FEA. The decisions about the selection of appropriate modeling techniques are usually based on an analyst's judgement based upon their knowledge and intuition from past experience. The machine learning-based approach provides recommendations within seconds, thus, saving significant computational resources for making accurate design choices. / Doctor of Philosophy / This thesis presents an innovative application of artificial intelligence (AI) techniques for designing aircraft structures. An important objective for the aerospace industry is to design robust and fuel-efficient aerospace structures. The state of the art research in the literature shows that the structure of aircraft in future could mimic organic cellular structure. However, the design of these new panels with arbitrary structures is computationally expensive. For instance, applying standard optimization methods currently being applied to aerospace structures to design an aircraft, can take anywhere from a few days to months. The presented research demonstrates the potential of AI for accelerating the optimization of an aircraft structures. This will provide an efficient way for aircraft designers to design futuristic fuel-efficient aircraft which will have positive impact on the environment and the world.