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SUMMARYA general approach to the dimensional reduction of non-linear ÿnite element models of solid dynamics is presented. For the Newmark implicit time-discretization, the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments. To deal with this, it is shown how the problem can be formulated in an approximation (Ritz) basis of much smaller dimension. Similarly, the explicit Newmark algorithm can be also written in a reduced-dimension basis, and the computation time savings in that case follow from an increase in the stable time step length. In addition, the empirical eigenvectors are proposed as the basis in which to expand the incremental problem. This basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense. Because of this 'global' time viewpoint, the basis need not be updated as with reduced bases computed from a linearization of the full ÿnite element model. If the dynamics of a ÿnite element model is expressed in terms of a small number of basis vectors, the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved. At the same time, numerical experiments indicate that by using reduced models, substantial savings can be achieved even in the pre-asymptotic range. Furthermore, the algorithm parallelizes very e ciently. The method we present is expected to become a useful tool in applications requiring a large number of repeated non-linear solid dynamics simulations, such as convergence studies, design optimization, and design of controllers of mechanical systems.

This study uses remote imaging technology to quantify, compare, and contrast the cephalic anatomy between a neonate female and a young adult male Cuvier's beaked whale. Primary results reveal details of anatomic geometry with implications for acoustic function and diving. Specifically, we describe the juxtaposition of the large pterygoid sinuses, a fibrous venous plexus, and a lipid-rich pathway that connects the acoustic environment to the bony ear complex. We surmise that the large pterygoid air sinuses are essential adaptations for maintaining acoustic isolation and auditory acuity of the ears at depth. In the adult male, an acoustic waveguide lined with pachyosteosclerotic bones is apparently part of a novel transmission pathway for outgoing biosonar signals. Substitution of dense tissue boundaries where we normally find air sacs in delphinoids appears to be a recurring theme in deep-diving beaked whales and sperm whales. The anatomic configuration of the adult male Ziphius forehead resembles an upside-down sperm whale nose and may be its functional equivalent, but the homologous relationships between forehead structures are equivocal.

The finite element modeling (FEM) space reported here contains the head of a simulated whale based on CT data sets as well as physical measurements of sound-propagation characteristics of actual tissue samples. Simulated sound sources placed inside and outside of an adult male Cuvier's beaked whale (Ziphius cavirostris) reveal likely sound propagation pathways into and out of the head. Two separate virtual sound sources that were located at the left and right phonic lips produced beams that converged just outside the head. This result supports the notion that dual sound sources can interfere constructively to form a biologically useful and, in fact, excellent sonar beam in front of the animal. The most intriguing FEM results concern pathways by which sounds reach the ears. The simulations reveal a previously undescribed 'gular pathway' for sound reception in Ziphius. Propagated sound pressure waves enter the head from below and between the lower jaws, pass through an opening created by the absence of the medial bony wall of the posterior mandibles, and continue toward the bony ear complexes through the internal mandibular fat bodies. This new pathway has implications for understanding the evolution of underwater hearing in odontocetes. Our model also provides evidence for receive beam directionality, off-axis acoustic shadowing and a plausible mechanism for the long-standing orthodox sound reception pathway in odontocetes. The techniques developed for this study can be used to study acoustic perturbation in a wide variety of marine organisms.

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This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen-Loève expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

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Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piecewise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.