Ulrich Hetmaniuk

A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime

We present a discontinuous Galerkin method (DGM) for the solution of the Helmholtz equation in the mid-frequency regime. Our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non-conforming field that contains free-space solutions of the homogeneous partial differential equation to be solved. Hence, for the Helmholtz equation, the enrichment field is chosen here as the superposition of plane waves. We enforce a weak continuity of these plane waves across the element interfaces by suitable Lagrange multipliers. Preliminary results obtained for two-dimensional model problems discretized by uniform meshes reveal that the proposed DGM enables the development of elements that are far more competitive than both the standard linear and the standard quadratic Galerkin elements for the discretization of Helmholtz problems.

The discontinuous enrichment method for multiscale analysis

Computation naturally separates scales of a problem according to the mesh size. A variety of improved numerical methods are described by multiscale considerations, differing in the treatment of the unresolved, fine scales. The discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework. The method thus combines relative ease of implementation with improved numerical performance. These properties are demonstrated for both multiscale wave and transport problems, pointing to the potential of considerable savings in computational resources.

Anasazi software for the numerical solution of large-scale eigenvalue problems

Anasazi is a package within the Trilinos software project that provides a framework for the iterative, numerical solution of large-scale eigenvalue problems. Anasazi is written in ANSI C++ and exploits modern software paradigms to enable the research and development of eigensolver algorithms. Furthermore, Anasazi provides implementations for some of the most recent eigensolver methods. The purpose of our article is to describe the design and development of the Anasazi framework. A performance comparison of Anasazi and the popular FORTRAN 77 code ARPACK is given.

A linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems in all flow regimes

Abstract We present a computational fluid dynamics (CFD)-based linearized method for the frequency analysis of three-dimensional fluid/structure interaction problems. This method is valid in the subsonic, transonic, and supersonic flow regimes, and is insensitive to the frequency or damping level of the sought-after coupled eigenmodes. It is based on the solution by an orthogonal iteration procedure of a complex eigenvalue problem derived from the linearization of a three-field fluid/structure/moving mesh formulation. The key computational features of the proposed method include the reuse of existing unsteady flow solvers, a second-order approximation of the flux Jacobian matrix, and a parallel domain decomposition-based iterative solver for the solution of large-scale systems of discretized fluid/structure equations. While the frequency analysis method proposed here is primarily targeted at the extraction of the eigenpairs of a wet structure, we validate it with the flutter analysis of the AGARD Wing 445.6, for which experimental data is available.

Basis selection in LOBPCG

The purpose of our paper is to discuss basis selection for Knyazevs locally optimal block preconditioned conjugate gradient (LOBPCG) method. An inappropriate choice of basis can lead to ill-conditioned Gram matrices in the Rayleigh-Ritz analysis that can delay convergence or produce inaccurate eigenpairs. We demonstrate that the choice of basis is not merely related to computing in finite precision arithmetic. We propose a representation that maintains orthogonality of the basis vectors and so has excellent numerical properties.

Local absorbing boundary conditions for elliptical shaped boundaries

We compare several local absorbing boundary conditions for solving the Helmholtz equation, by a finite difference or finite element method, exterior to a general scatterer. These boundary conditions are imposed on an artificial elliptical or prolate spheroid outer surface. In order to compare the computational solution with an analytical solution, we consider, as an example, scattering about an ellipse. We solve the Helmholtz equation with both finite differences and finite elements. We also introduce a new boundary condition for an ellipse based on a modal expansion.

On a parallel multilevel preconditioned Maxwell eigensolver

We report on a parallel implementation of the Jacobi–Davidson (JD) to compute a few eigenpairs of a large real symmetric generalized matrix eigenvalue problem

Multilevel Methods for Eigenspace Computations in Structural Dynamics

A \mathbf{x} = \lambda M \mathbf{x}, \qquad C^T \mathbf{x} = \mathbf{0}.