Finite Rectangle Research Articles

Flutter of a Plate at High Supersonic Speeds

The vibrations of plate structures placed in a supersonic flow was considered. The undisturbed fluid flow was parallel to the plate. This type of problem is especially important in the aerospace industry, where it is named panel flutter. It has been noticed for a long time that panel flutter may be problematic at high speeds. In this article, two specific problems were treated: in the first one, the plate was in the form of an infinite strip and the flow was in the direction of its finite length. Rigid walls indefinitely extended from the sides of the plate. In the second problem, the plate was a finite rectangle and the flow was parallel to one of its sides. The rest of the plane of the rectangle was again rigid. The first problem was a limiting case of the second problem. The flow was modeled by piston theory, which assumes that the fluid pressure on the plate is proportional to its local slope. This approximation is widely used at high speeds (supersonic speeds in the range of M > 1), and reduces the interaction between the fluid flow and the vibrations of the plate to an additional term in the vibration equation. The resulting problem can be solved by assumed mode methods. In this study, the solution was also found by using the collocation method. The contribution of this study is the correlation between the flutter velocity and the other parameters of the plate. The main result is the flutter velocity of the free fluid flow under which the plate vibrations become unstable. Finally, simple expressions are proposed between the various non-dimensional parameters that allows for the quick estimation of the flutter velocity. These simple expressions were deduced by least squares fits to the computed flutter velocities.

Rate of coalescence of lineage pairs in the Spatial [formula omitted]-Fleming–Viot process

We revisit the Spatial Λ-Fleming–Viot process introduced in Barton and Kelleher (2010). Particularly, we are interested in the time T0 to the most recent common ancestor for two lineages. We distinguish between the cases where the process acts on the two-dimensional plane and on a finite rectangle. Utilizing a differential equation linking T0 with the physical distance between the lineages, we arrive at computationally efficient and reasonably accurate approximation schemes for both cases. Furthermore, our analysis enables us to address the question of whether the genealogical process of the model “comes down from infinity”, which has been partly answered before in Véber and Wakolbinger (2015).

Quantum Integral Inequalities with Respect to Raina’s Function via Coordinated GeneralizedΨ-Convex Functions with Applications

In accordance with the quantum calculus, we introduced the two variable forms of Hermite-Hadamard- (HH-) type inequality over finite rectangles for generalizedΨ-convex functions. This novel framework is the convolution of quantum calculus, convexity, and special functions. Taking into account theq^1q^2-integral identity, we demonstrate the novel generalizations of theHH-type inequality forq^1q^2-differentiable function by acquainting Raina’s functions. Additionally, we present a different approach that can be used to characterizeHH-type variants with respect to Raina’s function of coordinated generalizedΨ-convex functions within the quantum techniques. This new study has the ability to generate certain novel bounds and some well-known consequences in the relative literature. As application viewpoint, the proposed study for changing parametric values associated with Raina’s functions exhibits interesting results in order to show the applicability and supremacy of the obtained results. It is expected that this method which is very useful, accurate, and versatile will open a new venue for the real-world phenomena of special relativity and quantum theory.

Post Quantum Integral Inequalities of Hermite-Hadamard-Type Associated with Co-Ordinated Higher-Order Generalized Strongly Pre-Invex and Quasi-Pre-Invex Mappings

By using the contemporary theory of inequalities, this study is devoted to proposing a number of refinements inequalities for the Hermite-Hadamard’s type inequality and conclude explicit bounds for two new definitions of ( p 1 p 2 , q 1 q 2 ) -differentiable function and ( p 1 p 2 , q 1 q 2 ) -integral for two variables mappings over finite rectangles by using pre-invex set. We have derived a new auxiliary result for ( p 1 p 2 , q 1 q 2 ) -integral. Meanwhile, by using the symmetry of an auxiliary result, it is shown that novel variants of the the Hermite-Hadamard type for ( p 1 p 2 , q 1 q 2 ) -differentiable utilizing new definitions of generalized higher-order strongly pre-invex and quasi-pre-invex mappings. It is to be acknowledged that this research study would develop new possibilities in pre-invex theory, quantum mechanics and special relativity frameworks of varying nature for thorough investigation.

Radiation View Factor from Rectangular Plate to Coaxial Disk: A Model Correlation

A robust correlation is proposed for calculation of radiation view factor (VF) from a rectangular plate to a coaxially-parallel round disk. It is nearly impossible to develop an exact, closed-form VF formula for this basic configuration. It seems that at present there are only two relevant studies that report generalized results for VF between: (a) disk and coaxial square; and (b) finite rectangle and coaxial disk. A well-known online catalog of radiation VFs documents only the latter. Contemporary heat transfer textbooks or handbooks lack ready-to-use tools to estimate VF for this arrangement. The available prescriptions for this configuration have some limitations. Therefore, an alternative VF calculation tool is desirable. In this paper, a single VF correlation is developed for use over a wide range of plate and disk sizes. Compared to the existing correlations, this generalized model correlation offers a smaller margin of error in its predicted VF-values. The proposed tool can also be used for VF calculations when a rectangular plate and a parallel circular disk are non-coaxial.

Parametrized homology via zigzag persistence

This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or, equivalently, as a persistence diagram. By defining persistence in terms of finite rectangle measures, we classify barcode intervals into four classes. Each of these conveys how the homological features perish at both ends of the interval over which they are defined.

Some q-analogues of Hermite–Hadamard inequality of functions of two variables on finite rectangles in the plane

Preliminaries of q-calculus for functions of two variables over finite rectangles in the plane are introduced. Some q-analogues of the famous Hermite–Hadamard inequality of functions of two variables defined on finite rectangles in the plane are presented. A q1q2-Hölder inequality for functions of two variables over finite rectangles is also established to provide some quantum estimates of trapezoidal type inequality of functions of two variables whose q1q2-partial derivatives in absolute value with certain powers satisfy the criteria of convexity on co-ordinates.

Random infinite squarings of rectangles

Un papier récément publié (Electron. Commun. Probab. 19 (2014) 1–12) introduit une procédure pour générer une suite de cartes aléatoires qui a presque sûrement comme limite la « carte 3-connexe infinie uniforme » . La construction classique de Brooks, Smith, Stone et Tutte (Duke Math. J. 7 (1940) 312–340) associe à chaque carte finie, avec une arête racine qui n’est pas un isthme, un rectangle pavé par des carrés. Nous utilisons ces deux procédures afin de générer une suite aléatoire de rectangles pavé par des carrés. Nous démontrons que cette suite a presque surement une limite qui est un rectangle aléatoire infiniment pavés par des carrés, et que cet objet a presque surement un seul point d’accumulation. Ceci fournit un plongement canonique de la carte 3-connexe infinie uniforme dans le plan.

Analysis of Sheet Fracture Failure Based on XFEM

Extended finite element method (XFEM) is the most effective numerical method to solve discrete mechanical problem. Crack growth problem of two-dimension finite length rectangle panel is researched based on Abaqus XFEM frame. Stress intensity factor is obtained respectively by theoretical calculation and XFEM simulation, which proves reliability of XFEM and the software.

Dynamics of geodesic flows on hyperbolic compact surfaces with some elliptic points

We extend the results in [R. Adler, L. Flatto, Geodesic flows, interval maps and symbolic dynamics, Bull Am Math Soc. 1975;25(42): 229–334.] to compact surfaces of genus greater than one and with some elliptic points. Similar to hyperbolic case, a map TR conjugate to the cross-section map and defined on R, a union of finite rectangles in the plane, exists. This gives rise to a subshift of finite type and a sofic shift and their projections to axis known as Bowen–Series maps. Additionally, we will give a simple geometrical presentation of synchronizing and Markov words for , and will show that when considered as a sofic map, it is not of almost finite type. Moreover, if h is the entropy of any of those shift spaces, then ln (8g + 2r − 7) ≤ h ≤ ln (8g + 2r − 5), where g is the genus and r is the number of elliptic points.

Analytical solutions for determining residual stresses in two-dimensional domains using the contour method

The contour method is one of the most prevalent destructive techniques for residual stress measurement. Up to now, the method has involved the use of the finite-element (FE) method to determine the residual stresses from the experimental measurements. This paper presents analytical solutions, obtained for a semi-infinite strip and a finite rectangle, which can be used to calculate the residual stresses directly from the measured data; thereby, eliminating the need for an FE approach. The technique is then used to determine the residual stresses in a variable-polarity plasma-arc welded plate and the results show good agreement with independent neutron diffraction measurements.

Solving two-dimensional Cartesian unsteady heat conduction problems for small values of the time

This paper is intended to provide very accurate analytical solutions modeling transient heat conduction processes in 2D Cartesian finite bodies for small values of the time. Analysis of diffusion of thermal deviation effects indicates that, when the space and time coordinates satisfy a criterion developed in the paper, the simple transient 1D semi-infinite solutions may be “used” for generating extremely accurate values for temperature and heat flux at any point of a finite rectangle. Also, they may be “used” with excellent accuracy as short-time solutions when the time-partitioning method is applied (so avoiding the usually difficult integration of the short-cotime Green's functions). A complex 2D semi-infinite problem is solved explicitly and evaluated numerically as part of the analysis. The proposed criterion is based on an accuracy of one part in 10n (n = 1,2,...,10,...), where n = 2 is for engineering insight and visual comparison while n = 10 is for verification purposes of large numerical codes.

A Spring-Damping Regularization and a Novel Lie-Group Integration Method for Nonlinear Inverse Cauchy Problems

In this paper, the solutions of inverse Cauchy problems for quasilinear elliptic equations are resorted to an unusual mixed group-preserving scheme (MGPS). The bottom of a finite rectangle is imposed by overspecified boundary data, and we seek unknown data on the top side. The spring-damping regularization method (SDRM) is introduced by converting the governing equation into a new one, which includes a spring term and a damping term. The SDRM can further stabilize the inverse Cauchy problems, such that we can apply a direct numerical integration method to solve them by using the MGPS. Several numerical examples are examined to show that the SDRM+MGPS can overcome the ill-posed behavior of the inverse Cauchy problem. The present algorithm has good efficiency and stability against the disturbance from random noise, even with an intensity being large up to 10%, and the computational time is very saving.

Estimation of P-value of MAX test with double triangle diagram for 2 × 3 SNP case-control tables

Single nucle otide polymorphisms (SNPs) are the most popular markers in genetic epidemiology. Multiple tests have been applied to evaluate genetic effect of SNPs, such as Pearson's test with two degrees of freedom, three tests with one degree of freedom (chi(2) tests for dominant and recessive modes and Cockran-Armitage trend test for additive mode) as well as MAX3 test and MAX test, which are combination of four tests mentioned earlier. Because MAX test is a combination of Pearson's test of two degrees of freedom and two tests of one degree of freedom, the probability density function (pdf) of MAX statistics does not match pdf of chi(2) distribution of either one or two degrees of freedom. In order to calculate P-value of MAX test, we introduced a new diagram, Double Triangle Diagram, which was an extension of de Finetti diagram in population genetics which characterized all of the tests for 2 x 3 tables. In the diagram the contour lines of MAX statistics were consisted of elliptic curves and two tangent lines to the ellipses in the space. We normalized the ellipses into regular circles and expressed P-value of MAX test in an integral form. Although a part of the integral was not analytically solvable, it was calculable with arbitrary accuracy by dividing the area under pdf into finite rectangles. We confirmed that P-values from our method took uniform distribution from 0 to 1 in three example marginal count sets and concluded that our method was appropriate to give P-value of MAX test for 2 x 3 tables.

Algebraic aspects of the Hirzebruch signature operator and applications to transitive Lie algebroids

The index of the classical Hirzebruch signature operator on a manifold M is equal to the signature of the manifold. The examples of Lusztig ((10), 1972) and Gromov ((4), 1985) present the Hirzebruch signature operator for the cohomology (of a manifold) with coefficients in a flat symmetric or symplectic vector bundle. In (6), we gave a signature operator for the cohomology of transitive Lie algebroids. In this paper, firstly, we present a general approach to the signature operator, and the above four examples become special cases of a single general theorem. Secondly, due to the spectral sequence point of view on the signature of the cohomology algebra of certain filtered DG-algebras, it turns out that the Lusztig and Gromov examples are important in the study of the signature of a Lie algebroid. Namely, under some natural and simple regularity assumptions on the DG-algebra with a decreasing filtration for which the second term lives in a finite rectangle, the signature of the second term of the spectral sequence is equal to the signature of the DG algebra. Considering the Hirzebruch-Serre spectral sequence for a transitive Lie algebroid A over a compact oriented manifold for which the top group of the real cohomology of A is nontrivial, we see that the second term is just identical to the Lusztig or Gromov example (depending on the dimension). Thus, we have a second signature operator for Lie algebroids.

A REYNOLDS–UNIFORM NUMERICAL METHOD FOR PRANDTL'S BOUNDARY LAYER PROBLEM FOR FLOW PAST A PLATE WITH MASS TRANSFER

In this paper we consider Prandtl's boundary layer problem for incompressible laminar flow past a plate with transfer of fluid through the surface of the plate. When the Reynolds number is large the solution of this problem has a parabolic boundary layer. In a neighbourhood of the plate the solution of the problem has an additional singularity which is caused by the absence of the compartability conditions. To solve this problem outside nearest neighbourhood of the leading edge, we construct a direct numerical method for computing approximations to the solution of the problem using a piecewise uniform mesh appropriately fitted to the parabolic boundary layer. To validate this numerical method, the model Prandtl problem with self-similar solution was examined, for which a reference solution can be computed using the Blasius problem for a nonlinear ordinary differential equation. For the model problem, suction/blowing of the flow rate density is v0(x)=-vi2-1/2Re1/2x-1/2, where the Reynolds number Re can be arbitrarily large and vi is the intensity of the mass transfer with arbitrary values in the segment [-.3,.3]. We considered the Prandtl problem in a finite rectangle excluding the leading edge of the plate for various values of Re which can be arbitrary large and for some values of vi, when meshes with different number of mesh points were used. To find reference solutions for the the velocity components and their derivatives with required accuracy, we solved the Blasius problem using a semi–analytical numerical method. By extensive numerical experiments we showed that the direct numerical method constructed in this paper allows us to approximate both the solution and its derivatives Re–uniformly for different values of vi.

A Reynolds‐uniform numerical method for Prandtl's boundary layer problem for flow past a wedge

AbstractIn this paper, we deal with Prandtl's boundary layer problem for incompressible laminar flow past a wedge. When the Reynolds number is large the solution of this problem has a parabolic boundary layer. We construct a direct numerical method for computing approximations to the solution of this problem using a piecewise uniform fitted mesh technique appropriate to the parabolic boundary layer. We use the numerical method to approximate the self‐similar solution of Prandtl's problem in a finite rectangle excluding the leading edge of the wedge, which is the source of an additional singularity caused by incompatibility of the problem data. We verify that the constructed numerical method is robust, in the sense that the computed errors for the velocity components and their derivatives in the discrete maximum norm are Reynolds uniform. We construct and apply a special numerical method related to the Falkner–Skan technique to compute a reference solution for the error analysis of the velocity components and their derivatives. By means of extensive numerical experiments we show that the constructed direct numerical method is Reynolds uniform. Copyright © 2003 John Wiley & Sons, Ltd.

Two-dimensional packing problems: A survey

We consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste. In two-dimensional bin packing problems these units are finite rectangles, and the objective is to pack all the items into the minimum number of units, while in two-dimensional strip packing problems there is a single standardized unit of given width, and the objective is to pack all the items within the minimum height. We discuss mathematical models, and survey lower bounds, classical approximation algorithms, recent heuristic and metaheuristic methods and exact enumerative approaches. The relevant special cases where the items have to be packed into rows forming levels are also discussed in detail.

Sample function properties of two-parameter Markov processes

This paper is concerned with the sample function properties of two-parameter*-Markov processes. Some sufficient conditions in terms of transition functions are given for the sample functions to be bounded (more precisely, bounded on any finite rectangle), to be continuous or to possess open quadrantal limits.

Transient diffusion and migration to a disk electrode

A coordinate transformation is implemented that greatly facilitates the numerical solution of the equations describing transient ionic diffusion and migration to a disk electrode. The transformed coordinates are a modified form of the classical oblate spheroidal coordinates; singularities arising at the electrode edge are removed, and the infinite model space is collapsed down to a finite rectangle. The resulting savings in storage and execution time make it feasible to do numerical calculations that were previously difficult and computationally expensive