Non-uniqueness and stability of two-family fiber-reinforced incompressible hyper-elastic sheet under equibiaxial loading

The problems on the non-uniqueness and stability of a two-family fiber- reinforced anisotropic incompressible hyper-elastic square sheet under equibiaxial tensile dead loading are examined within the framework of finite elasticity. For a two-family fiber-reinforced square sheet, which is in-plane symmetric and subjected to the in-plane symmetric tension in dead loading on the edges, three symmetrically deformed configu- rations and six asymmetrically deformed configurations are possible for any values of the loading. Moreover, another four bifurcated asymmetrically deformed configurations are possible for the loading beyond a certain criticM value. The stability of all the solutions is discussed in comparison with the energy of the sheet. It is shown that only one of the symmetric solutions is stable when the loading is less than the critical value. However, this symmetric solution will become unstable when the loading is larger than the critical value, while one of the four bifurcated asymmetric solutions will be stable.

Study on Non-Uniqueness of Taylor Vortex Flow Changing Inner Cylinder Acceleration Time

In concentrically rotating double cylinders consisting of a stationary outer cylinder a rotating inner cylinder, Taylor vortex flow appears. Taylor vortex flow occurs in journal bearings, various fluid machineries, containers for chemical reaction, and other rotating components. Therefore, the analysis of the flow structure of Taylor vortex flow is highly effective for its control. The main parameters that determine the modes of Taylor vortex flow of a finite length are the aspect ratio, Reynolds number Re. Aspect ratio is defined as the ratio of the cylinder length to the gap length between cylinders, and Re is determined on the basis of the angular speed of the inner cylinder. Aspect ratio was set to be 4.0, and Re to be values in the range from 100 to 1000 at intervals of 100. Thus far, a large number of studies on Taylor vortex flow have been carried out;however, the effects of the differences in initial conditions have not yet been sufficiently clarified. In this study, we changed the acceleration time of the inner cylinder in a numerical analysis, and examined the resulting changes in the mode formation and bifurcation processes. The acceleration time was changed from 1.0 s to 10.0 s. As a result, a difference was observed in the final mode depending on the difference in the acceleration time. From this finding, non-uniqueness, which is a major characteristic of Taylor vortex flow, was confirmed. However, no regularities regarding the difference in mode formation were found and the tendency of the mode formation process was not specified. Moreover, the processes of developing the vortex resulting in different final modes were monitored over time by visual observation. Similar flow behaviors were initially observed after the start of the calculation. Then, a bifurcation point, at which the flow changed to a mode depending on the acceleration time observed, and finally the flow became steady. In addition, there was also a difference in the time taken for the well-developed flow to reach the steady state. Both EFD (Experimental Fluid Dynamics) and CFD (Computational Fluid Dynamics) results show good agreement qualitatively.

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures,and the solutions admit the concentration of mass.It is found that under the requirement of satisfying the over-compressing entropy condition:(i)there is a unique delta shock solution,corresponding to the case that has two strong classical Lax shocks;(ii)for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave,or two shocks with one being weak,there are infinitely many solutions,each consists of a delta shock and a rarefaction wave;(iii)there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves.These solutions are self-similar.Furthermore,for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data,there always exists a unique delta shock for at least a short time.It could be prolonged to a global solution.Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass(particle).Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.We also discuss possible physical interpretations and applications of these new solutions.

A NOTE FOR ANALYSIS OF THERMOMECHANICAL CONTACT PROBLEMS

A discussion about the bifurcation and non-uniqueness of solutions in theanalysis of thermo-mechanical contact problems with initial gap is given. Without loss ofgenerality, a mechanical contact problem coupled with steady heat transfer is studied and an exampleof non-uniqueness of solutions caused by the thermo-mechanical mechanism is presented. Theimportant work is that the non-uniqueness of solutions, which is different from that found in theanalysis of the traditional frictional contact problems, is studied in detail. The possibleoscillation and non-convergence problems in the iteration process of the numerical computation arediscussed, and an enhanced algorithm is put forward to overcome the difficulties.

SELF-SIMILAR SOLUTIONS OF A CLASS OF NONLINEAR PAPTIAL DIFFERENTIAL EQUATION

In this paper, we investigate the following partial differential equation, ut , where a > 0 and p> 1. When n(p-1)/2 > 1 andp > 3, we obtained a nontrivial non-negative global solution. Furthermore, on Sobolev space W1,s(W2,s) with s > 1. a nonuniqueness result is established which shows that there exists a positive solution u(t,x) with u(t,x)→0 as t→0 in W1,s(W2,s). On the other hand, our result can be regarded as a generalization of conclusion of Haraux, A.and Weissler, F.B. in [5].

Construction of Equivalent Functions in Anisotropic Radon Tomography

We consider a real-valued function on a plane of the form m(x,y,θ)=A(x,y)+Bc(x,y)cos(2θ)+Bs(x,y)sin(2θ)+Cc(x,y)cos(4θ)Cs(x,y)sin(4θ) that models anisotropic acoustic slowness (reciprocal velocity) perturbations. This “slowness function” depends on Cartesian coordinates and polar angle θ. The five anisotropic “component functions” A (x,y), Bc(x,y), Bs(x,y), Cc(x,y) and Cs(x,y) are assumed to be real-valued Schwartz functions. The “travel time” function d(u, θ) models the travel time perturbations on an indefinitely long straight-line observation path, where the line is parameterized by perpendicular distance u from the origin and polar angle θ;it is the Radon transform of m ( x, y, θ). We show that: 1) an A can always be found with the same d(u, θ) as an arbitrary (Bc,Bs) and/or an arbitrary (Cc,Cs);2) a (Bc,Bs) can always be found with the same d(u, θ) as an arbitrary A, and furthermore, infinite families of them exist;3) a (Cc,Cs) can always be found with the same d(u, θ) as an arbitrary A, and furthermore, infinite families of them exist;4) a (Bc,Bs) can always be found with the same d(u, θ) as an arbitrary (Cc,Cs) , and vice versa;and furthermore, infinite families of them exist;and 5) given an arbitrary isotropic reference slowness function m0(x,y), “null coefficients” (Bc,Bs) can be constructed for which d(u, θ) is identically zero (and similarly for Cc,Cs ). We provide explicit methods of constructing each of these “equivalent functions”.

Uncertainty analysis of gravity data inversion

The traditional gravity modelling methods fail to deal with ambiguity zones. These zones are likely to be associated with uncertain boundaries. Source boundaries can be divided into "hard "and "soft "types."Hard"boundary stands for distinguished margin of discontinuity of source distributions. Traditional modelling method is based on this type of boundary. Uncertain boundaries of ambiguity zones belong to "soft"boundary.The authors put forward the uncertainty analysis because lots of uncertainty factors need to be taken into consideration. Any powerful system designed to interpret gravity data for a real source distribution must be able to deal with uncertain information. Uncertainty analysis( UA) is a useful tool to estimate the ambiguity of the data interpretation. It is often applied in conjunction with a specific modelling or inversion method. In this study,we use an iterative inversion and its results are evaluated using uncertainty analysis and proved to be effective.

Equivalent-Source from 3D Inversion Modeling for Magnetic Data Transformation

The well-known non-uniqueness in modeling of potential-field data results in an infinite number of models that fit the data almost equally. This non-uniqueness concept is exploited to devise a method to transform the magnetic data based on their equivalent-source. The unconstrained 3D magnetic inversion modeling is used to obtain the anomalous sources, i.e. 3D magnetization distribution in the subsurface. Although the 3D model fitting the data is not geologically feasible, it can serve as an equivalent-source. The transformations, which are commonly applied to magnetic data (reduction to the pole, reduction to the equator, upward and downward continuation), are the response of the equivalent-source with appropriate kernel functions. The application of the method to both synthetic and field data showed that the transformation of magnetic data using the 3D equivalent-source gave satisfactory results. The method is relatively more stable than the filtering technique, with respect to the noise present in the data.

Experimental Study on Vortex Growth during the Early Development of Rotating Disks

In previous studies, the effects of radial clearance on the flow of a rotating disk in a cylindrical vessel have been investigated by using rotating disks of different shapes. As a result, different flow phases were observed in each disk due to the difference in disk dimensions. In this study, we focus on the end-face effect and conduct experiments to visualize the vortex growth process and elucidate the generation mechanism of the vortex structure. From the experiment results, at Re = 4000, 7000, and 9000, four types of vortex flow modes appeared in the vortex development process. However, at Re = 4000, only regular 2-cells and regular 4-cells appeared, and at Re = 9000, only mutated 2-cells and mutated 3-cells appeared. In addition, it was found that only one type appeared depending on the rotational ascent time ts. When Re = 4000, the rotational ascent time ts = 0, 2, 7, and 8 was stable at regular 4-cells, while the others were finally stable in regular 2-cells. This study revealed the influence of the acceleration of the rotating disk on the non-unique flows in the cylindrical casing.

Robust Reconstruction of Sensor Swarms Floating through Enclosed Environments

A novel type of application for the exploration of enclosed or otherwise difficult to access environments requires large quantities of miniaturized sensor nodes to perform measurements while they traverse the environment in a “go with the flow” approach. Examples of these are the exploration of underground cavities and the inspection of industrial pipelines or mixing tanks, all of which have in common that the environments are difficult to access and do not allow position determination using e.g. GPS or similar techniques. The sensor nodes need to be scaled down towards the millimetre range in order to physically fit through the narrowest of parts in the environments and should measure distances between each other in order to enable the reconstruction of their positions relative to each other in offline analysis. Reaching those levels of miniaturization and enabling reconstruction functionality requires: 1) novel reconstruction algorithms that can deal with the specific measurement limitations and imperfections of millimetre-sized nodes, and 2) improved understanding of the relation between the highly constraint hardware design space of the sensor nodes and the reconstruction algorithms. To this end, this work provides a novel and highly robust sensor swarm reconstruction algorithm and studies the effect of hardware design trade-offs on its performance. Our findings based on extensive simulations, which push the reconstruction algorithm to its breaking point, provide important guidelines for the future development of millimetre-sized sensor nodes.

NON-UNIQUE SOLUTIONS FROM SURFACE ELASTICITY FOR FUNCTIONALLY GRADED MATERIALS

This paper considers the unusual behavior of functionally graded materials/structures when the surface effect is involved. It is found that on the assumption that the surface energy is not positive semi-definite, the solution can be non-unique. Several examples are given for simple spherically-symmetric and axisymmetric FGM bodies with surface effects characterized by Gurtin-Murdoch surface elasticity. The results show that the conditions for non-uniqueness of solution emerge when the magnitude of negative effective surface modulus is of the order of a characteristic dimension of the problem multiplied by the bulk modulus near the surface, which is quite different from that for homogeneous materials.

The Riemann Problem for a Blood Flow Model in Arteries

In this paper,the Riemann solutions of a reduced 6×6 blood flow model in mediumsized to large vessels are constructed.The model is nonstrictly hyperbolic and non-conservative in nature,which brings two difficulties of the Riemann problem.One is the appearance of resonance while the other one is loss of uniqueness.The elementary waves include shock wave,rarefaction wave,contact discontinuity and stationary wave.The stationary wave is obtained by solving a steady equation.We construct the Riemann solutions especially when the steady equation has no solution for supersonic initial data.We also verify that the global entropy condition proposed by C.Dafermos can be used here to select the physical relevant solution.The Riemann solutions may contribute to the design of numerical schemes,which can apply to the complex blood flows.

A robust technique to reconstruct sound field for all wave-numbers

A robust technique has been proposed for reconstructing sound field for all wave-numbers, which is suitable for the radiation analysis on an arbitrarily-shaped body. According to the wave superposition formulation, the spatial sound field of an actual radiator can be represented equivalently by superposing the fields generated by some fictitious sources interior to the actual radiator. Meanwhile, the strength density on the fictitious sources can be obtained by matching the sound pressures of a finite number of field points. Once the strength density is calculated, the acoustic information at any field point can be obtained subsequently. Finally, some typical numerical examples are given to validate the theoretical analysis, and some investigations are made to analyze the influence of sound frequency and the fictitious source position on the reconstruction precision, which are helpful to the application of the proposed technique to real industrial measurements.

A New DGM(1,1)Model with a Grey Parameter and Its Application

In the DGM(1, 1) model modeling process, the influencing factors are uncertain. But the solution of DGM(1, 1) model with uncertain information is unique, which conflicts with the nonuniqueness principle of solution in grey theory. In view of this situation, this paper makes an in-depth analysis of the meaning of grey action quantity β_(2) in DGM(1, 1) model and regards β_(2) as an interval grey number. The maximum possibility whitenization value is given to estimate the kernel of grey number,and the typical possibility function is constructed to describe the possibility of grey number taking different values. A new DGM(1, 1) model with a grey parameter is then proposed, whose simulation results are interval grey numbers. The proposed model is compatible with the DGM(1, 1) model in model structure and simulation results. Finally, the practical example results show the applicability and effectiveness of the proposed model.

Numerical study of bifurcation solutions of spherical Taylor-Couette flow

The steady bifurcation flows in a spherical gap (gap ratio =0.18) with rotating inner and stationary outer spheres are simulated numerically for Reci【Re【1500 1500 by solving steady axisymmetric incompressible Navier-Stokes equations using a finite difference method. The simulation shows that there exist two steady stable flows with 1 or 2 vortices per hemisphere for 775【Re【1220 and three steady stable flows with 0, 1, or 2 vortices for 1 220【Re【l500. The formation of different flows at the same Reynolds number is related with different initial conditions which can be generated by different accelerations of the inner sphere. Generation of asro-or two-vortex flow depends mainly on the acceleration, but that of one-vortex flow also depends on the perturbation breaking the equatorial symmetry. The mechanism of development of a saddle point in the meridional plane at higher Re number and its role in the formation of two-vortex flow are analy2Ed.

Numerical study of multiple periodic flow states in spherical Couette flow

The supercritical flow states of the spherical Couette flow between two concentric spheres with the inner sphere rotating are investigated via direct numerical simulation using a three-dimensional finite difference method. For comparison with experiments of Nakabayashi et al. And Wimmer, a narrow gap and a medium gap with clearance ratio β=0.06 and 0.18 respectively areconsidered for the Reynolds number range covering the first Hopfbifurcation point. With adequate initial conditions and temporaryimposition of small wave-type perturbation, multiple periodicflow states with three different pair numbers of spiralTaylor-G{ortler (TG) vortices have been simulated successfullyfor β=0.06, of which the 1-pair and 2-pair of spiral Tgvortices are newly obtained. Three different periodic flow stateswith shear waves, Stuart vortices or wavy outflow boundary, have been obtained for β=0.18. Analysis of the numerical resultsreveals these higher flow modes in terms of fundamental frequency,wave number and spatial structure.