Implementing and using high-order finite element methods

We present theoretical analyses of and detailed timings for two programs which use high-order finite element methods to solve the Navier- Strokes equations in two and three dimensions. The analyses show that algorithms popular in low-order finite element implementations are not always appropriate for high-order methods. The timings show that with the proper algorithms high-order finite element methods are viable for solving the Navier-Stokes equations. We show that it is more efficient, both in time and storage, not to precompute element matrices as the degree of approximating functions increases. We also study the cost of assembling the stiffness matrix versus directly evaluating bilinear forms in two and three dimensions. We show that it is more efficient not to assemble the full stiffness matrix for high-order methods in some cases. We consider the computational issues with regard to both Euclidean and isoparametric elements. We show that isoparametric elements may be used with higher-order elements without increasing the order of computational complexity.     

In search of the Griffiths shield region

The van der Waals equation of state for binary mixtures has been used to determine the location and shape of the Griffiths shield region (where three tricritical lines intersect). If one takes the geometric mean fora ₁₂, the arithmetic mean forb ₁₂, and the configurational free energy as ideal, the center of the Griffiths shield region is found only when the ratio of molecular sizes is infinite. When the Flory equation for the configurational free energy for mixtures of chain molecules is substituted for the ideal form, the results appear to be somewhat different. However, for all the cases studied, with systems which approach geometric mean behavior one finds the shield region only when the ratio of molecular size is very large and when the internal pressure of the small molecule is very much greater than that of the long-chain molecule.This paper is dedicated to our colleague Howard Reiss on the occasion of his 66th birthday.     

Non-linear vibrations of three-layer beams with viscoelastic cores,II: Experiment

Experimental results are presented for large amplitude, forced motion of damped, three-layer beams. The beams are constructed with a viscoelastic material constrained between stiff, elastic, outer layers. The sandwich beam is axially restrained; therefore large amplitude displacements cause non-linear response. When the beam is forced at one-half of the lateral vibration resonant frequency, superharmonic response occurs. The experiment is briefly described and frequency response characteristics, spatial shapes and a measure of superharmonic response are presented. The results are compared with predictions from a previously developed theory.     

Absorption spectra and magneto-optic figures of merit in the Bi x Sm3-x Fe5-y Ga y O12 system

The absorption (α) and Faraday rotation (θ) spectra of 14 garnets belonging to the series Bi _x Sm_3-x Fe_5-y Ga _y O₁₂ (0<x<1.05, 0.8<y<1.15) have been measured between 15 000 cm⁻¹ and 19 000 cm⁻¹. The figure of merit (θ/α) at 17 850 cm⁻¹ (560 nm) increases linearly with increasing bismuth concentration up tox∼0.6 where it begins to increase less rapidly. For operation of magneto-optic display devices at 17 850 cm⁻¹ there is no advantage in using garnets in this series withx>0.8. The Faraday rotation at 17 850 cm⁻¹ increases linearly with bismuth concentration whereas the absorption coefficient increases more rapidly. The presence of Bi³⁺ increases the intensity of all Fe³⁺ pair transitions in the garnet system as a result of the increased superexchange induced by Bi³⁺. This is in keeping with the observation that the intensity of the⁶A_1g (S)→⁴T_1g (G) transition in (RE)₃Fe₅O₁₂ (RE=Er, Y, Dy, Gd, Eu) increases on traversing the above RE series as do the Curie temperatures of these iron garnets.     

Magnetic resonance imaging of microvascular leakage induced by myocardial contrast echocardiography in rats

The extent and magnitude of microvascular leakage induced by myocardial contrast echocardiography (MCE) were characterized with contrast-aided magnetic resonance imaging (MRI). Evans blue dye, Definity ultrasound contrast agent and Omniscan magnetic resonance contrast agent were injected intravenously in anesthetized rats suspended in a water bath. Diagnostic ultrasound B mode scans with 1:4 end-systolic triggering were performed at 1.5 MHz using a cardiac phased array scanhead to provide a short axis view of the left ventricle. The in situ peak rarefactional pressure amplitude (PRPA) was 2.0 MPa. Microvascular leakage was characterized by extraction of the dye from tissue samples and by imaging the distribution and concentration of Omniscan within the myocardium. The extracted Evans blue was 2.3 times greater than in shams (P<.05) for heart samples perfused with heparin saline, and 1.6 times greater than shams (not significant) for unperfused samples. The MRI showed the penetration of the ultrasound-induced capillary leakage throughout much of the scan plane. The overall gadolinium content measured by MR showed the same trends as the extracted Evans blue, but was more variable. For pooled data (perfused and unperfused), the exposed samples were significantly increased (P<.05) relative to the sham samples for both Evans blue and gadolinium content. Omniscan leakage was also discernable in two of four MRIs from intact rats (after sacrifice). These results demonstrate a potential for MR mapping of capillary leakage induced by contrast-aided ultrasound, with a possible application to spatial characterization of local drug delivery.     

Dynamic shim updating on the human brain

Dynamic alteration of shim settings during a multi-slice imaging experiment can improve static magnetic-field homogeneity over extended volumes. In this report, a pre-emphasized dynamic shim updating (DSU) system capable of rapidly updating all non-degenerate zeroth through second-order shims is presented and applied to high-field multi-slice imaging studies on the human brain. DSU is utilized in both non-oblique and oblique slicing geometries while updating in-plane and through-slice shims. Image-based magnetic-field maps are used to quantify homogeneity improvements and comparisons are made on a slice-specific basis between static global shimming and increasing orders of shim inclusion utilized DSU. The influence of oblique slicing geometry on DSU-utilized global homogeneity is also quantified computationally. Finally, the effect of DSU on susceptibility artifact reduction in single-shot axial-sliced EPI is analyzed using experimental acquisitions.     

Grassmann bundles

Summary In the classical theory of the Grassmann Variety there are three principal results. The Basis Theorem asserts that the Chow ring has a selfdual linear basis of classes. Determinantal Formulawhich expresses any basic class as a determinant in the special classes. Finally the ring structure is elucidated by Pieri's Formulawhich expresses the intersection of a basic class and a special class in terms of the basic classes. Here we show how all these results can be established also for the Chow ring of a Grassmann bundle. There are however some differences. In the classical case the basic classes are Schubert classes: this is impossible in the general case as there need not be enough Schubert classes to provide a basis and in the general case there is a pair of dual bases which both reduce to the Schubert basis in the classical case. In addition to these generalizations of the classical results we also enlarge on the theory of Schubert classes developed in the important paper of Kempf and Laksov [4].Following them we shall henceforth use the phrase « determinantal formula » to mean their formula for Schubert classes and our generalization of it to « improper » Schubert classes.