Acoustics & Electromagnetics Group

Hearing impairment remains the primary disability among military personnel. Sound pressure levels caused by proximity to aircraft engines or impulse noise from large caliber weapons may easily exceed the pain threshold value of 100 dB.

The focus of this project is to to develop a reliable numerical model for investigating the bone-conducted sound in the human head. The problem is difficult because of a lack of fundamental knowledge regarding the transmission of acoustic energy through non-airborne pathways to the cochlea. A fully coupled model based on the acoustic/elastic interaction problem with a detailed resolution of the cochlea region and its interface with the skull and the air pathways, should provide an insight into this fundamental, long standing research problem.

The project builds on an interaction of experts in numerical wave propagation -- Drs. Elisabeth and Marek Bleszynski from Monopole Research with a team at the University of Texas headed by Dr. Leszek Demkowicz and including two experts on wave propagation and hearing science: Dr. Mark Hamilton and Dr. Craig Champlin.

Distribution of pressure over a head model

Accurate numerical simulation of borehole acoustic measurements is of great relevance to improving the efficacy of acoustic logging techniques and to computationally estimating elastic formation properties. Such simulations require sound physical modeling combined with accurate and efficient numerical discretization and solution techniques.

Monopole source in a borehole surrounded with a homogeneous formation with no tool present. Final coarse hp mesh in the formation.

Railgun is a multiphysical device which propels a rigid object (the armature) with the electromagnetic force. The electric current flowing in rails forms an electromagnetic field that depends upon material properties (conductivity) of the railgun and the surrounding air. As the armature moves, the conductivity changes which makes the electromagnetic field evolve along with the induce force. This project is concerned with the study of a two-dimensional model of the railgun in which the eddy current version of Maxwell equations, expressed in terms of potentials, is coupled with the Newton's law of motion for the armature. The coupled problems is discretized using hp-adaptive edge finite elements and a one-step implicit Euler method. The project is financed by Institute of Advanced Technology (IAT).

A short course on:

      hp-ADAPTIVE FINITE ELEMENT METHODS
       FOR ELLIPTIC AND MAXWELL PROBLEMS

  L. Demkowicz, J. Kurtz and W. Rachowicz

             Institute for
 Computational Engineering and Sciences
   The University of Texas at Austin

Traditional, low order, finite element discretizations are well suited to resolve complex topologies and curvilinear geometries. The corresponding rates of convergence are limited by the polynomial order, and the regularity of the solution. Those include not only singularities coming from non-convex geometries and material interfaces but also regions with high gradients, (e.g. boundary layers) perceived by the computer in the preasymptotic range as singularities. In presence of problems with large geometrical or material contrasts, they "lock" (100 % error). For wave propagation problems, they suffer from large dispersion (phase) errors making solution of problems with large wave numbers impossible.

Spectral methods do not lock for singularly perturbed problems, and deliver exponential convergence, provided the solution is analytic up to boundary, i.e. no singularities are present on the boundary, They do not suffer from dispersion error for wave propagation. If the solution is, however, singular on the boundary or material interfaces, the advantage of using spectral methods is lost - the convergence slows down to algebraic rates again. They also behave very badly in the preasymptotic range if the meshes do not reflect well the structure of the solution. For complex curvilinear geometries, meshes are difficult to generate.

hp Finite Element Methods combine advantages of low order and spectral methods. The short course presents fundamental technological components of hp Finite Element Methods, including:

• Mathematical foundations (variational formulations, construction of H^1 and Hcurl-conforming elements, convergence estimates).
• Construction of hierarchical shape functions.
• hp data structures.
• Constrained approximation.
• Geometry modeling. Exact geometry and isoparametric elements.
• Projection-based interpolation.

The hp technology culminates in a fully automatic hp-adaptive strategy in which element size h and polynomial degree p are automatically selected to construct a sequence of optimal meshes which deliver exponential convergence for both regular and singular solutions. The methodology is based on a coarse- fine grid paradigm where the fine grid solution guides optimal hp-refinements of the coarse grid. We shall discuss two versions of the hp-algorithm:

• The energy-driven hp-algorithm.
• The goal-driven hp-algorithm.

The presentation includes a hands-on presentation of 1D and 2D hp codes, and a large number of 2D and 3D numerical examples, for both elliptic and Maxwell problems, focusing on wave propagation applications. For details on the subject, see [1,2]. We help to install the 1D and 2D codes on the participants' laptops. The laptops have to operate under LINUX and must have the Intel Fortran 90 and C compilers preinstalled.

[1] L. Demkowicz, L., ``Computing with $hp$ Finite Elements. I.One- and Two-Dimensional Elliptic and Maxwell Problems'', Chapman & Hall/CRC Press, Taylor and Francis, October 2006.

[2] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz and A. Zdunek, ``Computing with hp Finite Elements. II. Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications'', Chapman & Hall/CRC Press, Taylor and Francis, October 2007.

An optimal hp mesh for the Fichera problem. Colors indicate different polynomial order.

The project focuses on studying acoustic wave propagation in shallow reservoirs