Nonlinear Spherical Caps and Associated Plate and Membrane Problems

This paper deals with the existence and multiplicity problem of the equilibrium solutions of an elastic spherical cap within nonlinear strain theory. We pose the problem in the form of a three parameter bifurcation problem, one parameter being related to the load, the others to the geometry. When the geometrical parameters are different from zero, the so-called generic case, we revisit the nonuniqueness results, and explore the solutions in the parameter space. Then we analyze the formal limits as the geometrical parameters tend to zero. When the curvature tends to zero, we obtain from the nonlinear shell a von Kármán plate, a new, although natural, result. When the thickness parameter tends to zero, we get a nonlinear membrane problem. A study of the latter gives infinitely many solutions, and we discuss the construction, shapes, and stability in detail. This revised version was published online in August 2006 with corrections to the Cover Date.     

Numerical model of a pine in a wind

Steady-state nonlinear differential equations govering the stem curve of a wind-loaded pine are derived and solved numerically. Comparison is made between the results computed and the data from photographs of a pine stem during strong wind. The pine breaking is solved at the end.     

Coupled Models and Parallel Simulations for Three-Dimensional Full-Stokes Ice Sheet Modeling

A three-dimensional full-Stokes computational model is considered for determining the dynamics,temperature,and thickness of ice sheets.The goveming thermomechanical equations consist of the three-dimensional full-Stokes system with nonlinear rheology for the momentum,an advective-diffusion energy equation for temperature evolution,and a mass conservation equation for ice-thickness changes.Here,we discuss the variable resolution meshes,the finite element discretizations,and the parallel algorithms employed by the model components.The solvers are integrated through a well-designed coupler for the exchange of parametric data between components.The discretization utilizes high-quality,variable-resolution centroidal Voronoi Delaunay triangulation meshing and existing parallel solvers.We demonstrate the gridding technology,discretization schemes,and the efficiency and scalability of the parallel solvers through computational experiments using both simplified geometries arising from benchmark test problems and a realistic Greenland ice sheet geometry.     

Crystal structure and the role of cation group factor in the third‐order nonlinear optical properties of cadmium complexes constructed by 1,3‐dithiole‐2‐thione‐4,5‐dithiolato ligand

A cadmium complex bis(benzyltriethylammonium) bis(1,3‐dithiole‐2‐thione‐4,5‐dithiolato)‐cadmium(II) ((TEBA)₂[Cd(DMIT)₂]) has been synthesized and its crystal structure has been determined by means of X‐ray single‐crystal diffraction. The central cadmium(II) ion coordinates with two DMIT, which constructed a distorted tetrahedron environment. Its third‐order nonlinear optical properties have been studied using Z‐scan technique with 20 ps pulses at wavelength 1064 nm. Its third‐order nonlinear susceptibility χ⁽³⁾ value was determined to be 1.24 × 10⁻¹⁹ m² V⁻², the figure of merit, χ⁽³⁾/α₀, was estimated to be 2.64 × 10⁻²⁰ m³ V⁻². Copyright © 2011 John Wiley & Sons, Ltd.     

Convergence of some asynchronous nonlinear multisplitting methods

Frommer's nonlinear multisplitting methods for solving nonlinear systems of equations are extended to the asynchronous setting. Block methods are extended to include overlap as well. Several specific cases are discussed. Sufficient conditions to guarantee their local convergence are given. A numerical example is presented illustrating the performance of the new approach.     

On Weak Solutions of a Class of Variational Inequalities of Parabolic Type7

§ 1.Ｐｒｅｌｉｍｉｎａｒｙ　ＬｅｔＥｂｅａｒｅｆｌｅｘｉｖｅＢａｎａｃｈｓｐａｃｅｗｉｔｈａｎｏｒｍ‖·‖ ,Ｅ′ｉｔｓｄｕａｌｓｐａｃｅａｎｄ〈· ,·〉ｔｈｅｐａｉｒｂｅｔｗｅｅｎＥ′ａｎｄＥ .ＳｕｐｐｏｓｅｔｈａｔＨｉｓａＨｉｌｂｅｒｔｓｐａｃｅｗｉｔｈｔｈｅｎｏｒｍ |·|ａｎｄｉｎｎｅｒｐｒｏｄｕｃｔ (· ,·)ｓｕｃｈｔｈａｔＥ Ｈ Ｅ′ ,( 1°)ｗｈｅｒｅｔｈｅｆｏｒｍｅｒｓａｒｅｄｅｎｓｅｉｎｔｈｅｌａｔｔｅｒｓａｎｄｔｈｅｃｏｒｒｅｓｐｏｎｄｉｎｇｉｄｅｎｔｉｃａｌｍａｐｓａｒｅｃｏｎｔ…     

Autoresonant asymptotics in an oscillating system with weak dissipation

We study a system of two first-order differential equations arising in averaging nonlinear systems over fast single-frequency oscillations. We consider the situation where the original system contains weak dissipative terms. We construct the asymptotic form of a two-parameter solution with an unbounded increasing amplitude. This result gives a key for understanding autoresonance in weak dissipative systems as a phenomenon of significant increase in the forced nonlinear oscillation initiated by a small external pumping. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 102–111, July, 2009.     

Application of Volterra and Wiener Theories for Nonlinear Parameter Estimation in a Rotor-Bearing System

Volterra and Wiener theories provide the concepts of linear,bilinear, tri-linear, etc., kernels, which upon convolution with theexcitation force, can be employed to represent the response of anonlinear system. Based on these theories, higher-order frequencyresponse functions (FRFs) are employed to estimate the nonlinearstiffness of rolling element bearings, supporting a rigid rotor. Therotor-bearing assembly is idealized as a single-degree-freedom system,with cubic nonlinearity. The analysis involves a third-order kernelrepresentation of the system response. The first and third-order kerneltransforms are extracted from the measurements of the appliedwhite-noise excitation and the resultant response. A third-order kernelfactor is synthesized from this first-order kernel and is processedalong with the third-order kernel for estimation of the nonlinearparameter. Damping is assumed to be linear in the analysis. Theprocedure is demonstrated through measurements on a laboratory test rig.