A spectral element solver program using MATLAB is written for the solution of Poisson and Helmholtz equations. The accuracy of spectral methods (p-type high order) and the geometric flexibility of the low-order h-type finite elements are combined in spectral element methods. Rectangular elements are used to solve Poisson and Helmholtz equations with Dirichlet and Neumann boundary conditions which are homogeneous or non homogeneous. Robin (mixed) boundary conditions are also implemented. Poisson equation is also solved by discretising the domain with curvilinear quadrilateral elements so that the accuracy of both isoparametric quadrilateral and rectangular element stiffness matrices and element mass matrices are tested. Quadrilateral elements are used to obtain the stream functions of the inviscid flow around a cylinder problem. Nonhomogeneous Neumann boundary conditions are imposed to the quadrilateral element stiffness matrix to solve the velocity potentials.

No / The Helmholtz integral equation formulation is used to produce the solution for the acoustic field reflected from a finite, thin, poroelastic plate in a rigid baffle with simply supported edges. The acoustic properties of the porous material are predicted using the effective fluid assumption. The solutions for the displacement of the plate and for the loading acoustic pressures are given in the form of the sine transform. The sine transform coefficients are obtained from the solution of a system of linear equations resulting from three integral Helmholtz formulations which relate the displacement of the plate and the acoustic pressures on the front and on the back of the plate. The effect of an air gap behind the plate in the front of a rigid wall is also considered. A parametric study is performed to predict the effect of variations in the parameters of the poroelastic plate. It is shown that thin, light, poroelastic plates can provide high values of the acoustic absorption even for low frequency sound. This effect can be exploited to design compact noise control systems with improved acoustic performance.

Submitted by Maria Cristina (library@lncc.br) on 2017-04-12T20:03:02Z No. of bitstreams: 1 Dissertacao_Juliano_Abimael.pdf: 1562533 bytes, checksum: 72a2f22f7a5dd247b98bf5da9985fc3e (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-04-12T20:03:23Z (GMT) No. of bitstreams: 1 Dissertacao_Juliano_Abimael.pdf: 1562533 bytes, checksum: 72a2f22f7a5dd247b98bf5da9985fc3e (MD5) / Made available in DSpace on 2017-04-12T20:03:59Z (GMT). No. of bitstreams: 1 Dissertacao_Juliano_Abimael.pdf: 1562533 bytes, checksum: 72a2f22f7a5dd247b98bf5da9985fc3e (MD5) Previous issue date: 2016-08-24 / Agencia Nacional de Pesquisa (ANP) / The classical methods of finite differences and Galerkin finite element are unable to eliminate the error of pollution effect for high wave numbers. Methods such as Galerkin Least Square (GLS) and Quasi Stabilized Finite Element Method (QSFEM) are methods that minimize error pollution is feasible, however, only in uniform grids. An important step to be taken is the study and development of methodologies that minimize the error pollution effect on non-uniform grids. In this line, the formulation Quasi Optimal Finite Difference (QOFD) obtained by numerical minimization of the functional truncation error for plane waves in an arbitrary direction, and has minimal pollution to stencils for uniform grids is a reliable method in more general meshes. In this work, and describe the methods mentioned above, we propose an approach that generates the same QOFD coefficients through the use of a radial basis functions, composed of the Bessel functions of the first kind and order zero. Furthermore, for wave equation in the time domain, we propose finite difference approximations to the high-order wave equation. This methodology will use a polynomial base constructed from the characteristic functions of this equation. / As metodologias clássicas de diferenças finitas e elementos finitos de Galerkin se mostram incapazes de eliminar o efeito de poluição do erro para altos números de onda. Métodos como Galerkin Least Square (GLS) e Quasi Stabilized Finite Element Method (QSFEM) são métodos que minimizam a poluição do erro sendo factíveis, contudo, apenas em malhas uniformes. Um passo importante a ser dado é o estudo e desenvolvimento de metodologias que minimizem o efeito de poluição do erro em malhas não-uniformes. Nessa linha, a formulação Quasi Optimal Finite Difference (QOFD), obtida numericamente pela minimização do funcional do erro de truncamento para ondas planas em direção arbitrária, além de ter mínima poluição para stencils sobre malhas uniformes é um método factível em malhas mais gerais. Neste trabalho, além de descrevermos os métodos citados anteriormente, propomos uma aproximação que gera os mesmos coeficientes do QOFD por meio do emprego de uma base radial de funções, composta pelas funções de Bessel de primeiro tipo e ordem zero. Além disso, para a equação da onda no domínio do tempo, propomos aproximações por diferenças finitas de alta ordem para a equação da onda. Tal metodologia fará uso de uma base polinomial construída a partir das funções características desta equação.