Shortcomings of discontinuous‐pressure finite element methods on a class of transient problems

Past studies that have compared LBB stable discontinuous‐ and continuous‐pressure finite element formulations on a variety of problems have concluded that both methods yield solutions of comparable accuracy, and that the choice of interpolation is dictated by which of the two is more efficient. In this work, we show that using discontinuous‐pressure interpolations can yield inaccurate solutions at large times on a class of transient problems, while the continuous‐pressure formulation yields solutions that are in good agreement with the analytical solution. Copyright © 2009 John Wiley & Sons, Ltd.     

Synthesis of niobocene imido cations: X-ray crystal structure of [Nb(NBu)(η-C5H4SiMe3)2(CNBu)][BPh4]

The reduction of [Nb(NBu^t)(η⁵-C₅H₄SiMe₃)₂Cl] by sodium amalgam followed by oxidation by [Fe(η⁵-C₅H₅)₂][BPh₄] in the presence of CNBu^t gave [Nb(NBu^t)(η⁵-C₅H₄SiMe₃)₂(CNBu^t)][BPh₄] (1). In a similar manner, [Nb(NPh)(η⁵-C₅H₄SiMe₃)₂(CNBu^t)][BPh₄] (2), [Nb(NPh)(η⁵-C₅H₄SiMe₃)₂(CO)][BPh₄] (3) and [Nb(NBu^t){Me₂Si(η⁵-C₅Me₄)(η⁵-C₅H₄)}(CNBu^t)][BPh₄] (4), were prepared. The reduction of [Nb(NBu^t){Me₂Si(η⁵-C₅H₄)₂}Cl] gave, depending on the experimental conditions, either the d¹-d¹ dimer [(Nb{Me₂Si(η⁵-C₅H₄)₂}(μ-NBu^t))₂] (5) or the hydride derivative [Nb(NBu^t){Me₂Si(η⁵-C₅H₄)₂}H] (6). The reaction of 5 with I₂ led to the formation of [Nb(NBu^t){Me₂Si(η⁵-C₅H₄)₂}I] (7). The molecular structure of 1 was determined by single-crystal X-ray diffraction studies.     

Derivatives of the Stretch, Rotation and Exponential Tensors in n-Dimensional Vector Spaces

We present a solution for the tensor equation TX + XT ^T = H, where T is a diagonalizable (in particular, symmetric) tensor, which is valid for any arbitrary underlying vector space dimension n. This solution is then used to derive compact expressions for the derivatives of the stretch and rotation tensors, which in turn are used to derive expressions for the material time derivatives of these tensors. Some existing expressions for n = 2 and n = 3 are shown to follow from the presented solution as special cases. An alternative methodology for finding the derivatives of diagonalizable tensor-valued functions that is based on differentiating the spectral decomposition is also discussed. Lastly, we also present a method for finding the derivatives of the exponential of an arbitrary tensor for arbitrary n.     

The Explicit Determination of the Logarithm of a Tensor and Its Derivatives

The logarithm of a tensor is often used in nonlinear constitutive relations of elastic materials. Here we show how the logarithm of an arbitrary tensor can be explicitly evaluated for any underlying space dimension n. We also present a method for the explicit evaluation of the derivatives of the logarithm of a tensor.      

A finite element method for compressible viscous fluid flows

This work presents a mixed three‐dimensional finite element formulation for analyzing compressible viscous flows. The formulation is based on the primitive variables velocity, density, temperature and pressure. The goal of this work is to present a ‘stable’ numerical formulation, and, thus, the interpolation functions for the field variables are chosen so as to satisfy the inf–sup conditions. An exact tangent stiffness matrix is derived for the formulation, which ensures a quadratic rate of convergence. The good performance of the proposed strategy is shown in a number of steady‐state and transient problems where compressibility effects are important such as high Mach number flows, natural convection, Riemann problems, etc., and also on problems where the fluid can be treated as almost incompressible. Copyright © 2010 John Wiley & Sons, Ltd.     

A Concise Proof of the Representation Theorem for Fourth-Order Isotropic Tensors

We present a new proof of the representation theorem for fourth-order isotropic tensors that does not assume the tensor to have major or minor symmetries at the outset.     

Poly(1‐butene)/clay nanocomposites: Preparation and properties

The preparation and properties of poly(1‐butene) (PB)/clay nanocomposites are described for the first time. Nanocomposites were prepared with the melt‐intercalation technique, using organically modified clay. The X‐ray diffraction patterns portrayed well‐defined diffraction peaks at higher d‐spacing than pristine clay, confirming the intercalation of polymer in silicate layers. Because PB exhibits time‐dependent polymorphism, the effect of clay on the phase transformation of PB was examined with thermal analysis. The phase transformation from a metastable tetragonal form to a stable hexagonal form was enhanced because of incorporation of layered silicates in the polymer matrix. The nanocomposites exhibited about a 40–140% increase in storage modulus depending on the clay content and significantly lower coefficient of thermal expansion. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1014–1021, 2003