Mesh-free approximations via the error reproducing kernel method and applications to nonlinear systems developing shocks

Error estimates for the error reproducing kernel method (ERKM) are provided. The ERKM is a mesh-free functional approximation scheme [A. Shaw, D. Roy, A NURBS-based error reproducing kernel method with applications in solid mechanics, Computational Mechanics (2006), to appear (available online)], wherein a targeted function and its derivatives are first approximated via non-uniform rational B-splines (NURBS) basis function. Errors in the NURBS approximation are then reproduced via a family of non-NURBS basis functions, constructed using a polynomial reproduction condition, and added to the NURBS approximation of the function obtained in the first step. In addition to the derivation of error estimates, convergence studies are undertaken for a couple of test boundary value problems with known exact solutions. The ERKM is next applied to a one-dimensional Burgers equation where, time evolution leads to a breakdown of the continuous solution and the appearance of a shock. Many available mesh-free schemes appear to be unable to capture this shock without numerical instability. However, given that any desired order of continuity is achievable through NURBS approximations, the ERKM can even accurately approximate functions with discontinuous derivatives. Moreover, due to the variation diminishing property of NURBS, it has advantages in representing sharp changes in gradients. This paper is focused on demonstrating this ability of ERKM via some numerical examples. Comparisons of some of the results with those via the standard form of the reproducing kernel particle method (RKPM) demonstrate the relative numerical advantages and accuracy of the ERKM.

Bull sperm plasma and acrosomal membranes: Fluorescence studies of lipid phase fluidity

Bull sperm plasma and outer acrosomal membranes have been isolated by discontinuous sucrose density gradient centrifugation and Ca2+-ATPase activity has been determined for both the membranes. Pyrene excimer fluorescence and diphenylhexatriene fluorescence polarization studies show that the lipid phase of the sperm plasma membranes is more fluid than the lipids of the outer acrosomal membranes. Approximately, a three fold increase in the cholesterol content has been found in the outer acrosomal membranes as compared to that in the plasma membranes.

Regional differentiationnext term in previous termbull sperm plasma membranesnext term

previous termBull spermnext term heads and tails have been separated by proteolytic digestion (trypsin) and previous termplasma membranesnext term have been isolated, using discontinuous sucrose density gradient centrifugation. previous termPlasma membranenext term bound Ca2+-ATPase is shown to be associated mostly with the tail previous termmembranes.next term Pyrene excimer fluorescence and diphenylhexatriene fluorescence polarization experiments indicate a more fluid lipid phase in the tail region. Differences in surface charge distribution have been found, using 1-anilinonaphthalene-8-sulfonate and Tb3+ as fluorescent probes.

Stress Concentration in Post-Tensioned Prestressed Concrete Beams

The use of the multiple Fourier method to analyses the stress distribution in the and regions of as a post-tensioned prestressed concrete beam had shown. The multiple Fourier method demonstrated have is a relatively new method for solving those problems for which the “Saint Vansant principle” is not applicable, The actual three-dimensional problem and a two-dimensional simplified representation of it are treated. The two-dimensional case is treated first and rather completely to gain further experience with multiple Fourier procedure, the appropriate Galerkin Vector for the three-dimensional case is found and the required relations between the arbitrary functions are stated.

3-D kinematical conservation laws (KCL): Evolution of a surface in R-3 - in particular propagation of a nonlinear wavefront

3-D KCL are equations of evolution of a propagating surface (or a wavefront) Omega(t), in 3-space dimensions and were first derived by Giles, Prasad and Ravindran in 1995 assuming the motion of the surface to be isotropic. Here we discuss various properties of these 3-D KCL.These are the most general equations in conservation form, governing the evolution of Omega(t) with singularities which we call kinks and which are curves across which the normal n to Omega(t) and amplitude won Omega(t) are discontinuous. From KCL we derive a system of six differential equations and show that the KCL system is equivalent to the ray equations of 2, The six independent equations and an energy transport equation (for small amplitude waves in a polytropic gas) involving an amplitude w (which is related to the normal velocity m of Omega(t)) form a completely determined system of seven equations. We have determined eigenvalues of the system by a very novel method and find that the system has two distinct nonzero eigenvalues and five zero eigenvalues and the dimension of the eigenspace associated with the multiple eigenvalue 0 is only 4. For an appropriately defined m, the two nonzero eigenvalues are real when m > 1 and pure imaginary when m < 1. Finally we give some examples of evolution of weakly nonlinear wavefronts.

Esr study of (SO4)2- dynamics in relation to the ferroelectric phase transition in ammonium sulfate

Ferroelectric phase transition in ammonium sulfate has been studied by ESR of CrO43- radical substituting for SO42- ion in (NH4)2SO4. In addition to discontinuous changes at Tc, certain continuous changes are observed in ESR parameters of this probe below Tc, which reflect the role of the sulfate ion in the phase transition. A microscopic mechanism of the phase transition is proposed and discussed in terms of the change of orientation of the sulfate tetrahedron through a finite angle. The degree of the change of orientation below Tc is thought to be the possible order parameter of the phase transition.

On a linear steady-state system of equations in microcontinuum fluid mechanics

Galerkin representations and integral representations are obtained for the linearized system of coupled differential equations governing steady incompressible flow of a micropolar fluid. The special case of 2-dimensional Stokes flows is then examined and further representation formulae as well as asymptotic expressions, are generated for both the microrotation and velocity vectors. With the aid of these formulae, the Stokes Paradox for micropolar fluids is established.

Generalized Burgers equations and Euler-Painlev transcendents. I

Initial-value problems for the generalized Burgers equation (GBE) ut+u betaux+lambdaualpha =(delta/2)uxx are discussed for the single hump type of initial data both continuous and discontinuous. The numerical solution is carried to the self-similar ``intermediate asymptotic'' regime when the solution is given analytically by the self-similar form. The nonlinear (transformed) ordinary differential equations (ODE's) describing the self-similar form are generalizations of a class discussed by Euler and Painlevé and quoted by Kamke. These ODE's are new, and it is postulated that they characterize GBE's in the same manner as the Painlev equations categorize the Kortweg-de Vries (KdV) type. A connection problem for some related ODE's satisfying proper asymptotic conditions at x=±[infinity], is solved. The range of amplitude parameter is found for which the solution of the connection problem exists. The other solutions of the above GBE, which display several interesting features such as peaking, breaking, and a long shelf on the left for negative values of the damping coefficient lambda, are also discussed. The results are compared with those holding for the modified KdV equation with damping. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Regional differentiation in bull sperm plasma-membranes

Bull sperm heads and tails have been separated by proteolytic digestion (trypsin) and plasma membranes have been isolated, using discontinuous sucrose density gradient centrifugation. Plasma membrane bound Ca2+-ATPase is shown to be associated mostly with the tail membranes. Pyrene excimer fluorescence and diphenylhexatriene fluorescence polarization experiments indicate a more fluid lipid phase in the tail region. Differences in surface charge distribution have been found, using 1-anilinonaphthalene-8-sulfonate and Tb3+ as fluorescent probes.

A Branch and Bound Algorithm for a Transportation Type Problem with Piecewise Linear Convex Objective Function

A branch and bound type algorithm is presented in this paper to the problem of finding a transportation schedule which minimises the total transportation cost, where the transportation cost over each route is assumed to be a piecewice linear continuous convex function with increasing slopes. The algorithm is an extension of the work done by Balachandran and Perry, in which the transportation cost over each route is assumed to beapiecewise linear discontinuous function with decreasing slopes. A numerical example is solved illustrating the algorithm.

Phase transitions in (NH4)2SO4 and (NH4)2SO4---K2SO4 mixed crystals as studied by EPR of CrO3- radicals

A study of the phase transitions in (NH4)2SO4 and (NH4)2SO4---K2SO4 mixed crystals by EPR of the CrO3- ion is reported. The results indicate a bilinear coupling of the order parameter with spontaneous polarization and a crossover from a discontinuous to a continuous nature of the phase transition in mixed crystals.

Evidence for a new metastable crystalline compound in Ge---Te system

Pressure dependence of the electrical resistivity of bulk, melt quenched GexTe100−x glasses (15 less-than-or-equals, slant x less-than-or-equals, slant 28) has been studied up to 8GPa pressure. All the glasses exhibit a sharp, discontinuous glass to crystal transition under pressure. The high pressure crystalline phases are identified to have a face centered cubic structure. The value of the cell constant is 0.779nm for 15 less-than-or-equals, slant x less-than-or-equals, slant 17, 0.642nm for x=20 and 0.55lnm for 22 ≤ x ≤ 28 samples respectively. The cell constants of the high pressure crystalline phases suggest the possible existance of a new metastable crystalline compound in the Ge---Te system with F.C.C. structure and cell constant equal to 1.109nm as reported by Moore et al.

Transverse vibrations of hybrid laminated plates

In this paper, an attempt is made to obtain the free vibration response of hybrid, laminated rectangular and skew plates. The Galerkin technique is employed to obtain an approximate solution of the governing differential equations. It is found that this technique is well suited for the study of such problems. Results are presented in a graphical form for plates with one pair of opposite edges simply supported and the other two edges clamped. The method is quite general and can be applied to any other boundary conditions.

Deflection analysis of hybrid laminated composite plates

In this paper an attempt is made to obtain deflections of hybrid, laminated, rectangular and skew composite plates. Analysis is performed by employing the Galerkin technique. Numerical results have been obtained for two types of layups employing Kevlar/epoxy and Boron/epoxy laminae. It is observed that for a given aspect ratio the rigidity of the skew plate increases with an increase in the skew angle. Further, for a specified deflection, the hybrid laminates turn out to be lighter.

Pressure-induced polymorphous crystallization in bulk Si20Te80 glass

The effect of pressure on the electrical resistivity of bulk Si20Te80 glass has been studied up to a pressure of 8 GPa. A discontinuous transition occurs at a pressure of 7 GPa. The X-ray diffraction studies on the pressure quenched sample show that the high pressure phase is crystalline with hexagonal structure (c/a = 1.5). On heating, the high pressure hexagonal phase has on exothermic decomposition atT = 586 K into two crystalline phases, which are the stable phases tellurium and SiTe2 obtained by simple heating of the glass.