The block gauss-seidel method in sound transmission problems

Sound transmission through partitions can be modelled as an acoustic fluid-elastic structure interaction problem. The block Gauss-Seidel iterative method is used in order to solve the finite element linear system of equations. The blocks are defined in a natural way, respecting the fluid and structural domains. The convergence criterion (spectral radius of iteration matrix smaller than one) is analysed and interpreted in physical terms by means of simple one-dimensional problems. This analysis highlights the negative influence on the convergence of a strong degree of coupling between the acoustic domains. A selective coupling strategy has been developed and successfully applied to problems with strong coupling (i.e. sound transmission through double walls)

Galerkin projected residual method applied to diffusion–reaction problems

A stabilized finite element method is presented for scalar and linear second-order boundary value problems. The method is obtained by adding to the Galerkin formulation multiple projections of the residual of the differential equation at element level. These multiple projections allow the generation of appropriate number of free stabilization parameters in the element matrix depending on the local space of approximation and on the differential operator. The free parameters can be determined imposing some convergence and/or stability criteria or by postulating the element matrix with the desired stability properties. The element matrix of most stabilized methods (such as, GLS and GGLS methods) can be obtained using this new method with appropriate choices of the stabilization parameters. We applied this formulation to diffusion–reaction problems. Optimal rates of convergency are numerically observed for regular solutions.Indisponível

Restricted Overlapping Balancing Domain Decomposition Methods and Restricted Coarse Problems for the Helmholtz Problem

Abstract Overlapping balancing domain decomposition methods and their combination with restricted additive Schwarz methods are proposed for the Helmholtz equation. These new methods also extend previous work on non-overlapping balancing domain decomposition methods toward simplifying their coarse problems and local solvers. They also extend restricted Schwarz methods, originally designed to overlapping domain decomposition and Dirichlet local solvers, to the case of non-overlapping domain decomposition and/or Neumann and Sommerfeld local solvers. Finally, we introduce coarse spaces based on partitions of unity and planes waves, and show how oblique projection coarse problems can be designed from restricted additive Schwarz methods. Numerical tests are presented

A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems

International audienceA non-overlapping domain decomposition method (DDM) is proposed for the parallel finite-element solution of large-scale time-harmonic wave problems. It is well-known that the convergence rate of this kind of method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) have proved to be well-suited, as a good compromise between basic impedance conditions, which lead to suboptimal convergence, and conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain — which are too expensive to compute. However, a direct application of this approach for configurations with interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) is suboptimal and, in some cases, can lead to incorrect results.In this work, we extend a non-overlapping DDM with HABC-based transmission conditions approach to efficiently deal with cross-points for lattice-type partitioning. We address the question of the cross-point treatment when the HABC operator is used in the transmission condition, or when it is used in the exterior boundary condition, or both. The proposed cross-point treatment relies on corner conditions developed for Padé-type HABCs. Two-dimensional numerical results with a nodal finite-element discretization are proposed to validate the approach, including convergence studies with respect to the frequency, the mesh size and the number of subdomains. These results demonstrate the efficiency of the cross-point treatment for settings with regular partitions and homogeneous media. Numerical experiments with distorted partitions and smoothly varying heterogeneous media show the robustness of this treatment

Asymptotic bounds to outputs of the exact weak solution of the three-dimensional Helmholtz equation

In engineering practice, the design is based on certain design quantities or "outputs of interest" which are functionals of field variables such as displacement, velocity field, or pressure. In order to gain confidence in the numerical approximation of "outputs," a method of obtaining sharp, rigorous upper and lower bounds to outputs of the exact solution have been developed for symmetric and coercive problems (the Poisson equation and the elasticity equation), for non-symmetric coercive problems (advection-diffusion-reaction equation), and more recently for certain constrained problems (Stokes equation). In this thesis we develop the method for the Helmholtz equation. The common approach relies on decomposing the global problem into independent local elemental sub-problems by relaxing the continuity along the edges of a partitioning of the entire domain, using approximate Lagrange multipliers. The method exploits the Lagrangian saddle point property by recasting the output problem as a constrained minimization problem. The upper and lower computed bounds then hold for all levels of refinement and are shown to approach the exact solution at the same rate as its underlying finite element approach. The certificate of precision can then determine the best as well as the worst case scenario in an engineering design problem. This thesis addresses bounds to outputs of interest for the complex Helmholtz equation. The Helmholtz equation is in general non-coercive for high wave numbers and therefore, the previous approaches that relied on duality theory of convex minimization do not directly apply. Only in the asymptotic regime does the Helmholtz equation become coercive, and reliable (guaranteed) bounds can thus be obtained. Therefore, in order to achieve good bounds, several new ingredients have been introduced. The bounds procedure is firstly formulated with appropriate extension to complex-valued equations. Secondly, in the computation of the inter-subdomain continuity multipliers we follow the FETI-H approach and regularize the system matrix with a complex term to make the system non-singular. Finally, in order to obtain sharper output bounds in the presence of pollution errors, higher order nodal spectral element method is employed which has several computational advantages over the traditional finite element approach. We performed verification of our results and demonstrate the bounding properties for the Helmholtz problem

Towards a more efficient spectrum usage: spectrum sensing and cognitive radio techniques

The traditional approach of dealing with spectrum management in wireless communications has been through the definition on a license user granted exclusive exploitation rights for a specific frequency.Peer ReviewedPostprint (published version