Academic literature on the topic 'Scalar curvature problem'

AbstractIt is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In our main result we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. We call these conditions the scalar flag curvature (SFC) test. The proof of the main result provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. Hilbert’s fourth problem asks to determine the Finsler functions with rectilinear geodesics. A Finsler function that is a solution to Hilbert’s fourth problem is necessarily of constant or scalar flag curvature. Therefore, we can use the constant flag curvature (CFC) test, which we developed in our previous paper, Bucataru and Muzsnay [‘Sprays metrizable by Finsler functions of constant flag curvature’, Differential Geom. Appl.31 (3)(2013), 405–415] as well as the SFC test to decide whether or not the projective deformations of a flat spray, which are isotropic, are metrizable by Finsler functions of constant or scalar flag curvature. We show how to use the algorithms provided by the CFC and SFC tests to construct solutions to Hilbert’s fourth problem.

Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore, it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. This paper discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray [Formula: see text] on a manifold is of vanishing [Formula: see text]-curvature, but [Formula: see text] has not isotropic curvature, then [Formula: see text] is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain [Formula: see text] with scalar curvature and vanishing [Formula: see text]-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that [Formula: see text] is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature.

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

In this paper, we study some geometric properties on doubly or singly warped­ product manifolds. In general, on a fixed topological product manifold, the problem for finding warped-product metrics satisfying certain curvature conditions are finally reduced to find positive solutions of linear or non-linear differential equations. Here, we are mainly interested in the following problems on essentially warped-product manifolds: one is the sufficient and necessary conditions for conformal flatness, and the other is to find warped-product metrics so that their scalar curvatures are contants.

AbstractIn this paper, a newnotion of scalar curvature for a Finslermetric F is introduced, and two conformal invariants Y(M, F) and C(M, F) are deûned. We prove that there exists a Finslermetric with constant scalar curvature in the conformal class of F if the Cartan torsion of F is suõciently small and Y(M, F)C(M, F) < Y(Sn) where Y(Sn) is the Yamabe constant of the standard sphere.

Foi provado por Byde que é possível adicionar um fim do tipo Delaunay a uma variedade compacta não degenerada de curvatura escalar constante positiva; desde que ela seja localmente conformemente plana em alguma vizinhança do ponto de colagem. A variedade resultante é não-compacta e possui a mesma curvatura escalar constante. O principal objetivo desta tese é generalizar este resultado. Construiremos uma família a um parâmetro de soluções para o problema de Yamabe singular positivo em qualquer variedade compacta não degenerada cujo tensor de Weyl anula-se até uma ordem suficientemente grande no ponto singular. Se a dimensão da variedade é no máximo 5; nenhuma condição sobre o tensor de Weyl é necessária. Usaremos técnicas de pertubação e o método de colagem. _________________________________________________________________________________________ ABSTRACT: It has been showed by Byde [5] that it is possible to attach a Delaunay type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this thesis is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to suciently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.

SANTOS, A. S. L. Problema de Yamabe modificado em variedades compactas de dimensão quatro e métricas críticas do funcional curvatura escalar. 2017. 58 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.
Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-25T19:34:47Z No. of bitstreams: 1 2017_tese_aslsantos.pdf: 535461 bytes, checksum: 8c3ddbdd33d74c4eb7b265354b3bafb3 (MD5)
Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Eu revisei a Tese de ALEX SANDRO LOPES SANTOS, e encontrei um pequeno erro na capa, ele colocou os seguintes elementos: UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIAS PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA DOUTORADO EM MATEMÁTICA Mas deve ser alterado para: UNIVERSIDADE FEDERAL DO CEARÁ CENTRO DE CIÊNCIAS DEPARTAMENTO DE MATEMÁTICA PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA Com os demais elementos da Tese, não há nenhum problema de formatação. Atenciosamente, on 2017-05-26T15:06:03Z (GMT)
Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-05-29T13:47:44Z No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5)
Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-05-29T14:08:17Z (GMT) No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5)
Made available in DSpace on 2017-05-29T14:08:17Z (GMT). No. of bitstreams: 1 2017_tese_aslsantos.pdf: 536279 bytes, checksum: f37ece7d8035a2d9c788c45d2e7807ae (MD5) Previous issue date: 2017-05-19
In the fisrt part of this work we investigate the modified Yamabe problem on four-dimensional manifolds whose the modifiers invariants depending on the eigenvalues of the Weyl curvature tensor and they are described in terms of maximum and minimum of the biorthogonal (sectional) curvature. We provide some geometrical and topological properties on four-dimensional manifolds in terms of these invariants. In the second part we investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in the 1980’s that every CPE metric must be Einstein. We prove that such a conjecture is true under a second-order vanishing condition on the Weyl tensor.
Na primeira parte deste trabalho investigamos o problema de Yamabe modificado em variedades de dimensão quatro cujos invariantes modificadores dependem dos autovalores do tensor de Weyl e são descritos em termos do máximo e mínimo da curvatura biortogonal (seccional). Fornecemos algumas propriedades geométricas e topológicas para tais variedades em termos destes invariantes. Na segunda parte investigamos os pontos críticos do funcional curvatura escalar total restrito ao espaço de métricas com curvatura escalar constante e volume unitário, abreviadamente chamamos de métricas CPE. Conjecturou-se na década de 1980 que toda métrica CPE deve ser Einstein. Provamos que tal conjectura é verdadeira sob uma condição de nulidade sobre o divergente de segunda ordem do tensor de Weyl.

[Introduction]: In questi ultimi due decenni le tecniche di somma connessa, basate essenzialmente su strumenti di natura analitica, hanno permesso di fare importanti progressi nella comprensione di svariati problemi non lineari derivati dalla geometria (studio di metriche a curvatura scalare costante in geometria Riemanniana, metriche autoduali, metriche con gruppi di olonomia speciali, metriche estremali in geometria K¨ahleriana, equazioni di Yang-Mills, studio di ipersuperfici minime e di superfici a curvatura media costante, metriche di Einstein,...). Queste tecniche si sono rivelate essere uno strumento potente per dimostrare l’esistenza di soluzioni di problemi altamente non lineari. La somma connessa (ossia l’aggiunta di un manico) è un’operazione topologica che consiste nel prendere due varietà M1 e M2, rimuovere da ciascuna di esse una piccola palla geodetica e identificare i bordi (i.e., due sfere) che si sono formati al fine di ottenere una nuova variet`a M1♯M2 che, in generale, sar`a topologicamente diversa dalle due variet`a iniziali. Pi`u in generale si può considerare la sommma connessa di due varietà M1 ed M2 lungo una sottovarietà K (somma connessa generalizzata). In questo caso si rimuove un piccolo intorno tubolare di K nelle due varietà iniziali e si identificano i bordi ottenuti per costruire M1 ♯K M2. Osserviamo che per effettuare una tale costruzione bisogna richiedere che i fibrati normali di K in M1 ed M2 siano diffeomorfi. Le cose si complicano quando le due variet`a iniziali sono munite di una particolare struttura (come nel caso di variet`a con metriche a curvatura scalare costante, o varietà che sono superfici minime,...) e si vuole preservare questa struttura, o quando sulle varietà iniziali esistono soluzioni di certe equazioni non lineari e si vogliono risolvere le stesse equazioni sulla somma connessa delle due variet`a M1 e M2 (come ad esempio le equazioni di Yang-Mills). Se da un lato le tecniche che permettono di effettuare le somme connesse in punti isolati sono state ben comprese e frequentemente utilizzate, dall’altro non si ha ancora un’effettiva padronanza delle tecniche che permettono di effettuare la somma connessa lungo sottovarietà. Il principale obiettivo di questo lavoro `e quello di colmare (parzialmente) questa lacuna, sviluppando questo tipo di tecnologie nel quadro delle metriche a curvatura scalare costante e nel quadro delle equazioni di vincolo di Einstein, in relatività generale.

Dans tout ce qui suit, nous considérons une variété riemannienne compacte de dimension au moins égale à 2. A tout domaine (suffisamment régulier) , on peut associer la première valeur propre ?Ù de l’opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous dirons qu’un domaine est extrémal (sous entendu, pour la première valeur propre de l’opérateur de Laplace-Beltrami) si est un point critique de la fonctionnelle Ù? ?O sous une contrainte de volume V ol(Ù) = c0. Autrement dit, est extrémal si, pour toute famille régulière {Ot}te (-t0,t0) de domaines de volume constant, telle que Ù 0 = Ù, la dérivée de la fonction t ? ?Ot en 0 est nulle. Rappelons que les domaines extrémaux sont caractérisés par le fait que la fonction propre, associée à la première valeur propre sur le domaine avec condition de Dirichlet au bord, a une donnée de Neumann constante au bord. Ce résultat a été démontré par A. El Soufi et S. Ilias en 2007. Les domaines extrémaux sont donc des domaines sur lesquels peut être résolu un problème elliptique surdéterminé. L’objectif principal de cette thèse est la construction de domaines extrémaux pour la première valeur propre de l’opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous donnons des résultats d’existence de domaines extrémaux dans le cas de petits volumes ou bien dans le cas de volumes proches du volume de la variété. Nos résultats permettent ainsi de donner de nouveaux exemples non triviaux de domaines extrémaux. Le premier résultat que nous avons obtenu affirme que si une variété admet un point critique non dégénéré de la courbure scalaire, alors pour tout volume petit il existe un domaine extrémal qui peut être construit en perturbant une boule géodésique centrée en ce point critique non dégénéré de la courbure scalaire. La méthode que nous utilisons pour construire ces domaines extrémaux revient à étudier l’opérateur (non linéaire) qui à un domaine associe la donnée de Neumann de la première fonction propre de l’opérateur de Laplace-Beltrami sur le domaine. Il s’agit d’un opérateur (hautement non linéaire), nonlocal, elliptique d’ordre 1. Dans Rn × R/Z, le domaine cylindrique Br × R/Z, o`u Br est la boule de rayon r > 0 dans Rn, est un domaine extrémal. En étudiant le linéarisé de l’opérateur elliptique du premier ordre défini par le problème précédent et en utilisant un résultat de bifurcation, nous avons démontré l’existence de domaines extrémaux nontriviaux dans Rn × R/Z. Ces nouveaux domaines extrémaux sont proches de domaines cylindriques Br × R/Z. S’ils sont invariants par rotation autour de l’axe vertical, ces domaines ne sont plus invariants par translations verticales. Ce deuxi`eme r´esultat donne un contre-exemple à une conjecture de Berestycki, Caffarelli et Nirenberg énoncée en 1997. Pour de grands volumes la construction de domaines extrémaux est techniquement plus difficile et fait apparaître des phénomènes nouveaux. Dans ce cadre, nous avons dû distinguer deux cas selon que la première fonction propre Ø0 de l’opérateur de Laplace-Beltrami sur la variété est constante ou non. Les résultats que nous avons obtenus sont les suivants : 1. Si Ø0 a des points critiques non dégénérés (donc en particulier n’est pas constante), alors pour tout volume assez proche du volume de la variété, il existe un domaine extrémal obtenu en perturbant le complément d’une boule géodésique centrée en un des points critiques non dégénérés de Ø0. 2. Si Ø0 est constante et la variété admet des points critiques non dégénérés de la courbure scalaire, alors pour tout volume assez proche du volume de la variété il existe un domaine extrémal obtenu en perturbant le complément d’une boule géodésique centrée en un des points critiques non dégénérés de la courbure scalaire
In what follows, we will consider a compact Riemannian manifold whose dimension is at least 2. Let Ù be a (smooth enough) domain and ?O the first eigenvalue of the Laplace-Beltrami operator on Ù with 0 Dirichlet boundary condition. We say that Ù is extremal (for the first eigenvalue of the Laplace-Beltrami operator) if is a critical point for the functional Ù? ?O with respect to variations of the domain which preserve its volume. In other words, Ù is extremal if, for all smooth family of domains { Ù t}te(-t0,t0) whose volume is equal to a constant c0, and Ù 0 = Ù, the derivative of the function t ? ?Ot computed at t = 0 is equal to 0. We recall that an extremal domain is characterized by the fact that the eigenfunction associated to the first eigenvalue of the Laplace-Beltrami operator over the domain with 0 Dirichlet boundary condition, has constant Neumann data at the boundary. This result has been proved by A. El Soufi and S. Ilias in 2007. Extremal domains are then domains over which can be solved an elliptic overdeterminated problem. The main aim of this thesis is the construction of extremal domains for the first eigenvalue of the Laplace-Beltrami operator with 0 Dirichlet boundary condition. We give some existence results of extremal domains in the cases of small volume or volume closed to the volume of the manifold. Our results allow also to construct some new nontrivial exemples of extremal domains. The first result we obtained states that if the manifold has a nondegenerate critical point of the scalar curvature, then, given a fixed volume small enough, there exists an extremal domain that can be constructed by perturbation of a geodesic ball centered in that nondegenerated critical point of the scalar curvature. The methode used is based on the study of the operator that to a given domain associes the Neumann data of the first eigenfunction of the Laplace-Beltrami operator over the domain. It is a highly nonlinear, non local, elliptic first order operator. In Rn × R/Z, the circular-cylinder-type domain Br × R/Z, where Br is the ball of radius r > 0 in Rn, is an extremal domain. By studying the linearized of the elliptic first order operator defined in the previous problem, and using some bifurcation results, we prove the existence of nontrivial extremal domains in Rn × R/Z. Such extremal domains are closed to the circular-cylinder-type domains Br × R/Z. If they are invariant by rotation with respect to the vertical axe, they are not invariant by vertical translations. This second result gives a counterexemple to a conjecture of Berestycki, Caffarelli and Nirenberg stated in 1997. For big volumes the construction of extremal domains is technically more difficult and shows some new phenomena. In this context, we had to distinguish two cases, according to the fact that the first eigenfunction Ø0 of the Laplace-Beltrami operator over the manifold is constant or not. The results obtained are the following : 1. If Ø0 has a nondegenerated critical point (in particular it is not constant), then, given a fixed volume closed to the volume of the manifold, there exists an extremal domain obtained by perturbation of the complement of a geodesic ball centered in a nondegenerated critical point of Ø0. 2. If Ø0 is constant and the manifold has some nondegenerate critical points of the scalar curvature, then, for a given fixed volume closed to the volume of the manifold, there exists an extremal domain obtained by perturbation of the complement of a geodesic ball centered in a nondegenerate critical point of the scalar curvature

AbstractDouble curvature enables elegant and material-efficient shell structures, but their construction typically relies on heavy machining, manual labor, and the additional use of material wasted as one-off formwork. Using a material’s intrinsic properties for self-shaping is an energy and resource-efficient solution to this problem. This research presents a fabrication approach for self-shaping double-curved shell structures combining the hygroscopic shape-changing and scalability of wood actuators with the tunability of 3D-printed metamaterial patterning. Using hybrid robotic fabrication, components are additively manufactured flat and self-shape to a pre-programmed configuration through drying. A computational design workflow including a lattice and shell-based finite element model was developed for the design of the metamaterial pattern, actuator layout, and shape prediction. The workflow was tested through physical prototypes at centimeter and meter scales. The results show an architectural scale proof of concept for self-shaping double-curved shell structures as a resource-efficient physical form generation method.

Chapter 8 focuses on nonlinear optimal control and its applications. The chapter begins by introducing the fundamentals of optimal control and prototypical problem formulations. This is followed by the treatment of first-order necessary conditions including the Pontryagin minimum principle, dynamic programming, and the Hamilton–Jacobi–Bellman equation. Singular arcs and bang–bang controls are relevant in the solution of many minimum-time and minimum-fuel problems and so these issues are discussed with the help of examples that have been worked out in detail.This chapter then turns towards direct and indirect numericalmethods suitable for solving large-scale optimal control problems numerically.The chapter concludes with an example relating to the calculation of an optimal track curvature estimate from global positioning system (GPS) data.

This chapter has two purposes; to describe a few elements of differential geometry that are required in different places in this work, and to provide, for completeness, a short introduction to general relativity (GR) and the problem of its quantization. A few concepts related to reparametrization (more accurately, diffeomorphism) of Riemannian manifolds, like parallel transport, affine connection, or curvature, are recalled. To define fermions on Riemannian manifolds, additional mathematical objects are required, the vielbein and the spin connection. Einstein–Hilbert's action for classical gravity GR is defined and the field equations derived. Some formal aspects of the quantization of GR, following the lines of the quantization of non-Abelian gauge theories, are described. Because GR is not renormalizable in four dimensions (even in its extended forms like supersymmetric gravity), at present time, a reasonable assumption is that GR is the low-energy, large-distance remnant of a more complete theory that probably no longer has the form of a quantum field theory (QFT) (strings, non-commutative geometry?). In the terminology of critical phenomena, GR belongs to the class of irrelevant interactions: due to the presence of the massless graviton, GR can be compared with an interacting theory of Goldstone modes at low temperature, in the ordered phase. The scale of this new physics seems to be of the order of 10¹⁹ GeV (Planck's mass). Still, because the equations of GR follow from varying Einstein–Hilbert action, some regularized form is expected to be relevant to quantum gravity. In the framework of GR, the presence of a cosmological constant, generated by the quantum vacuum energy, is expected, but it is extremely difficult to account for its extremely small, measured value.

Gradient–transport models for the turbulent scalar fluxes have proved to be deficient in a number of important practical flows, especially in those where the effects of body forces and extra rates of strain are significant. There are numerous examples in this respect, most notably in the prediction of the heat transfer rates in flows where the effects of stabilizing streamline curvature are sufficiently strong to completely extinguish turbulent mixing. The cause of the problem appears to be in the omission from these models of an explicit dependence of these fluxes on the mean strain field — a dependence that is required by the exact equations that govern the evolution of the turbulent scalar fluxes. The purpose of this paper is to determine whether the incorporation of such dependence in an algebraic model for the turbulent scalar fluxes is on its own sufficient to capture the main effects of streamline curvature. The model used here is an explicit, non–linear model that was developed in collaboration with the late Professor Speziale (Younis, Speziale & Clark, 1996). It is first validated in this paper by comparisons with data from one– and two–dimiensional heated free shear flows and is then used to predict the main features of a heated homogeneous shear layer in local equilibrium. Comparisons made with experimental data and with other models demonstrate the validity of the present approach.

This work presents a numerical study on the turbulent Schmidt numbers in jets in crossflow. This study contains two main parts. In the first part the problem of the proper choice of the turbulent Schmidt number in the Reynolds-Averaged Navier-Stokes (RANS) jet in crossflow mixing simulations is outlined. The results of RANS employing the shear-stress transport (SST) model of Menter and its curvature correction modification and different turbulent Schmidt number values are validated against experimental data. The dependence of the “optimal” value of the turbulent Schmidt number on the dynamic RANS model is studied. Furthermore a comparison is made with the large-eddy simulation (LES) results obtained using the WALE (Wall-Adapted Local Eddy Viscosity) model. The accuracy given by LES is superior in comparison to RANS results. This leads to the second part of the current study, in which the time-averaged mean and fluctuating velocity and scalar fields from LES are used for the evaluation of the turbulent viscosities, turbulent scalar diffusivities, and the turbulent Schmidt numbers in a jet in crossflow configuration. The values obtained from the LES data are compared with those given by the RANS modeling. The deviations are discussed and the possible ways for the RANS model improvements are outlined.

Tensegrity structures have a great value in the academia and in industry, in particular for adjustable tensegrity structures that can sustain external forces when deployed. The main problem with the latter systems is checking their stability during deployment. One of the most famous methods for checking stability was developed 20 years ago by two mathematicians [1]. They showed that if the tensegrity structure is redundant then the check is simple. But if it is a determinate tensegrity structure then there is a need to calculate the velocities of all the joints and then after matrix multiplications a scalar is obtained. If the scalar is negative then it is concluded that the tensegrity system is unstable without knowing which element causes the problem and what should be done in order to stabilize it. This paper proves that if the structure is a minimal rigid determinate structure, named Assur Graph, then there is a simple method for checking the stability. The proposed method suggests to remove a cable, calculate the curvature radius of one its inner joint and then conclude whether the structure is stable or not. In case that it was concluded that the system is unstable, then to shorten the cable so it becomes stable. The main topic from the combinatorial method being used in this paper is the special properties of Assur Graphs, in particular their singular positions. It is proved that from all the determinate structures only the Assur Graphs have these special singular properties, upon which the proposed method and the proof relies on.

Anisotropy of surface texture can in many practical cases significantly affect the interaction between the surface and phenomena that influence or are influenced by the topography. Tribological contacts in sheet forming, wetting behavior or dental wear are good examples. This article introduces and exemplifies a method for quantification and visualization of anisotropy using the newly developed 3D multi-scale curvature tensor analysis. Examples of a milled steel surface, which exhibited an evident anisotropy, and a ruby contact probe surface, which was the example of isotropic surface, were measured by the confocal microscope. They were presented in the paper to support the proposed approach. In the method, the curvature tensor T is calculated using three proximate unit vectors normal to the surface. The multi-scale effect is achieved by changing the size of the sampling interval for the estimation of the normals. Normals are estimated from regular meshes by applying a covariance matrix method. Estimation of curvature tensor allows determination of two directions around which surface bends the most and the least (principal directions) and the bending radii (principal curvatures). The direction of the normal plane, where the curvature took its maximum, could be plotted for each analyzed region and scale. In addition, 2D and 3D distribution graphs could be provided to visualize anisotropic or isotropic characteristics. This helps to determine the dominant texture direction or directions for each scale. In contrast to commonly used surface isotropy/anisotropy determination techniques such as Fourier transform or autocorrelation, the presented method provides the analysis in 3D and for every region at each scale. Thus, different aspects of the studied surfaces could clearly be seen at different scales.

Superharmonic resonances of nonlinear forced bending vibrations of moderately large curvature of nonuniform cantilever beams of rectangular cross section and a sharp end are reported. Cantilevers of constant width and parabolic thickness variation are considered in this paper. Method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zero- and first-order problem, result. Solving the zero-order problem, the linear modes are obtained in terms of hypergeometric functions by using the factorization method. The first-order problem provides the amplitude and phase evolution equation and consequently the superharmonic frequency response of the nonlinear system.

Simultaneous resonances, superharmonic and subharmonic, of two-term excitation nonlinear bending vibrations in the case of moderately large curvature of nonuniform cantilever beams are reported. Cantilevers of constant width and parabolic thickness variation are considered in this research. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zero- and first-order, result. Using factorization method, the linear modes of the zero-order problem are obtained in terms of hypergeometric functions. The first-order problem provides the amplitude-phase evolution relationship and consequently the simultaneous resonances response.

Subharmonic resonances of nonlinear forced bending vibrations in the case of moderately large curvature of nonuniform cantilever beams of rectangular cross section and a sharp end are reported. Cantilevers of constant width and parabolic thickness variation are considered in this research. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zeroth- and first-order, result. Using factorization method, the linear modes of the zeroth-order problem are obtained in terms of hypergeometric functions. The first-order problem provides the amplitude and phase evolution equation and consequently the regions where subharmonic responses exist.

Analytical solutions for the lateral buckling of pipelines exist for the case when the pipe material remains in the linearly elastic range. However for truly high temperatures and/or heavier flowlines, plastic deformation cannot be excluded. One then has to resort to finite element analyses, as no analytical solutions are available. This paper does not provide such an analytical solution, but it does show that if the finite element solution has been calculated once, then that solution can be scaled so that it applies for any other values of the design parameters. Thus the finite element solution need only be calculated once and for all. Thereafter, other solutions can be calculated by scaling the finite element solution using simple analytical formulae. The only significant limitation is that the shape of the moment-curvature relation must not change. I.e. the moment-curvature relation for the problem to be solved must be a scaled version of the moment-curvature relation for the reference problem, where different scale factors may be applied to the moment and curvature. This paper goes beyond standard dimensional analysis (as justified by the Bucklingham Π theorem), to establish a stronger scalability result, and uses it to develop simple formulae for the lateral buckling of any pipeline made of elastic-plastic material. The paper includes the derivation of the scaling result, the application procedure, the reference solution for an elastic-perfectly plastic pipe, and an example to illustrate how this reference solution can be used to calculate the lateral buckling response for any elastic-perfectly plastic pipe.

The role played by the curvature of the body surface on the hydrodynamics of water entry with high horizontal velocity component is investigated experimentally. The study is a part of a research activity finalized at the understanding of the aircraft ditching problem. In order to avoid scaling effects which may prevent the development of ventilation/cavitation phenomena, the study is carried out at full scale velocity. Measurements are presented in terms of pressures and loads whereas some underwater visualizations are used for the interpretation of the data. Both a convex and concave body surface are considered and comparisons with the flat plate data are established. In the case of a concave shape, a quite complicated flow with large air entrainment develops beneath the plate. The air entrainment causes a general reduction of the pressure peak at the middle, whereas the pressure peaks recorded at the side probes are about in line with those found for the flat plate in the same conditions. The total hydrodynamic load acting normal to the plate grows more regularly but the maximum load is essentially the same as that measured in the flat plate case. For the convex shape, the pressure probes located in the middle of the plate get wetted well before the ones at the side and the pressure peaks at the sides are much lower than those in the middle. The reduced pressures at the sides cause a reduction of the total loading in the normal direction compared to flat and concave plates.

The pressure distribution in the tip clearance region of a 2D turbine cascade was examined with reference to unknown factors which cause high heat transfer rates and burnout along the edge of the pressure surface of unshrouded cooled axial turbines. Using a special micro-tapping technique, the pressure along a very narrow strip of the blade edge was found to be 2.8 times lower than the cascade outlet pressure. This low pressure, coupled with a thin boundary layer due to the intense acceleration at gap entry, are believed to cause blade burnout. The flow phenomena causing the low pressure are of very small scale and do not appear to have been previously reported. The ultra low pressure is primarily caused by the sharp flow curvature demanded of the leakage flow at gap entry. The curvature is made more severe by the apparent attachement of the flow around the corner instead of immediately separating to increase the radius demanded of the flow. The low pressures are intensified by a depression in the suction corner and by the formation of a separation bubble in the clearance gap. The bubble creates a venturi action. The suction corner depression is due to the mainstream flow moving round the leakage and secondary vortices.