Composite Rectangle Research Articles

Axial compressive behavior of concrete-encased CFST stub columns with open composite stirrups

The existing method to improve the coordination performance of the inner and outer parts of concrete-encased concrete-filled steel tube (CFST) composite columns by increasing the volume-stirrup ratio causes difficulties in construction due to over-dense stirrups. Thus, this paper proposes an open polygonal composite stirrup with high strength and high ductility CRB600H reinforced rebar, and seventeen specimens were constructed, and their axial compressive performance was tested. The main parameters considered were the volume-stirrup ratio, the steel tube size, the stirrup type and the stirrup strength. The test results indicated: For the specimens restrained by open octagonal composite stirrups, compared with the specimen of 0.5% volume-stirrup ratio, the compressive bearing capacity increased by 14.6%, 15.7% and 21.5% for volume-stirrup ratio of 0.73%, 1.07% and 1.61%, respectively. For the specimens restrained by open composite rectangle stirrups, compared with the specimen of 0.79% volume-stirrup ratio, the compressive bearing capacity increased by 7.5%, 6.1%, and -1.4% for volume-stirrup ratio of 1.12%, 1.58% and 2.24%, respectively. The restraint ability and the bearing capacity of the octagonal composite stirrup are better than other stirrup types. The specimens equipped with open polygonal composite stirrup not only had a higher ductility than those with the traditional closed-loop stirrup, but they also had a higher axial bearing capacity than those with an HPB300 strength grades stirrup. Therefore, the open composite stirrup can be used in practical engineering. A new calculation method was proposed based on the stress-strain models for confined concrete under different restrain conditions, and the predicted value was close to the experimental value.

Numerical discretization and fast approximation of a variably distributed-order fractional wave equation

We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size σ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom N is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting σ = O(N−1) leads to O(N3) operations of evaluating the temporal discretization coefficients. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only O(logN) terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of generating coefficients from O(N3) to O(N2 logN). Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.

Extrapolation Algorithm for Computing Multiple Cauchy Principal Value Integrals

In this paper, the computation of multiple (including two dimensional and three dimensional) Cauchy principal integral with generalized composite rectangle rule was discussed, with the density function approximated by the middle rectangle rule, while the singular kernel was analytically calculated. Based on the expansion of the density function, the asymptotic expansion formulae of error functional are obtained. A series is constructed to approach the singular point, then the extrapolation algorithm is presented, and the convergence rate is proved. At last, some numerical examples are presented to validate the theoretical analysis.

An iterative merging algorithm for soft rectangle packing and its extension for application of fixed-outline floorplanning of soft modules

We address an important variant of the rectangle packing problem, the soft rectangle packing problem, and explore its problem extension for the fixed-outline floorplanning with soft modules. For the soft rectangle packing problem with zero deadspace, we present an iterative merging packing algorithm that merges all the rectangles into a final composite rectangle in a bottom-up order by iteratively merging two rectangles with the least areas into a composite rectangle, and then shapes and places each pair of sibling rectangles based on the dimensions and position of their composite rectangle in an up-bottom order. We prove that the proposed algorithm can guarantee feasible layout under some conditions, which are weaker as compared with a well-known zero-dead-space packing algorithm. We then provide a deadspace distribution strategy, which can systematically assign deadspace to modules, to extend the iterative merging packing algorithm to deal with soft packing problem with deadspace. For the fixed-outline floorplanning with soft modules problem, we propose an iterative merging packing based hierarchical partitioning algorithm, which adopts a general hierarchical partitioning framework as proposed in the popular PATOMA floorplanner. The framework uses a recursive bipartitioning method to partition the original problem into a set of subproblems, where each subproblem is a soft rectangle packing problem and how to solve the subproblem plays a key role in the final efficiency of the floorplanner. Different from the PATOMA that adopts the zero-dead-space packing algorithm, we adopt our proposed iterative merging packing algorithm for the subproblems. Experiments on the IBM-HB benchmarks show that the proposed packing algorithm is more effective than the zero-dead-space packing algorithm, and experiments on the GSRC benchmarks show that our floorplanning algorithm outperforms three state-of-the-art floorplanners PATOMA, DeFer and UFO, reducing wirelength by 0.2%, 4.0% and 2.3%, respectively.

Error expansion of piecewise constant interpolation rule for certain two-dimensional Cauchy principal value integrals

The classical composite rectangle (midpoint) rule for the computation of two dimensional singular integrals is discussed, with the error functional of rectangle rule for computing two dimensional singular integrals, the local coordinate of certain point and the convergence results O(h2) are obtained. When the local coordinate is coincided with certain priori known coordinates, we get the convergence rate the same as the Riemann integral at certain point. At last, numerical examples are presented to illustrate our theoretical analysis which agree with it very well.

The Rectangle Rule for Computing Cauchy Principal Value Integral on Circle

The classical composite rectangle (constant) rule for the computation of Cauchy principle value integral with the singular kernel is discussed. We show that the superconvergence rate of the composite midpoint rule occurs at certain local coodinate of each subinterval and obtain the corresponding superconvergence error estimate. Then collation methods are presented to solve certain kind of Hilbert singular integral equation. At last, some numerical examples are provided to validate the theoretical analysis.

The extrapolation methods for computing Cauchy principal value integral on intervals

The generalised composite rectangle (constant) rule for the computation of Cauchy principle value integral with kernel 1/(x – s) is discussed, and the asymptotic expansion of error function is obtained. Based on the asymptotic expansion, a series is constructed to approach the singular point. An extrapolation algorithm is presented and the convergence rate is proved. At last, some examples are also illustrated to confirm the theoretical results and show the efficiency of the algorithms.

SEP Performance of Cross QAM Signaling with MRC over Fading Channels and its Arbitrarily Tight Approximation

The exact average symbol error probability (SEP) of the cross quadrature amplitude modulation signal in a single-input multiple-output system over independent but not necessarily identical fading channels is derived. The maximal-ratio combining (MRC) is considered as the diversity technique, and the average SEP is obtained by using the moment generating function (MGF) method. The obtained closed-form SEP expression is presented in terms of a finite sum of single integrals with finite limits and an integrand composed of a finite product of elementary functions. In addition, the arbitrarily tight approximations with the form of a sum of constant coefficient exponential functions for Gaussian Q-function and the generalization of its Craig's form are proposed by applying the composite rectangle and Simpson numerical integration rules, respectively. The proposed approximations are simple and accurate enough even with only a few terms of exponential functions, and they are particularly suitable for applications of averaging Q-function and the generalized Q-function over the fading distributions. As a result, the closed-form approximations of the SEP over the AWGN channel and fading multichannels are expressed as a finite sum of exponential functions and a finite sum of MGFs, respectively, such that it is convenient and rapid to evaluate the SEP performances. Both the simulation results and the approximations show excellent agreement with the exact analytical expressions.

The Superconvergence of Certain Two-Dimensional Cauchy Principal Value Integrals

Summary The composite rectangle (midpoint) rule for the computation of multi-dimensional singular integrals is discussed, and the superconvergence results is obtained. When the local coordinate is coincided with certain priori known coordinates, we get the convergence rate one order higher than the global one. At last, numerical examples are presented to illustrate our theoretical analysis which agree with it very well.

The Superconvergence of Certain Two-Dimensional Hilbert Singular Integrals

The composite rectangle (midpoint) rule for the computation of multi-dimensional singular integrals is discussed and the superconvergence results is obtained. When the local coordinate is coincided with certain priori known coordinates, we get the convergence rate one order higher than the global one. At last, numerical examples are presented to illustrate our theoretical analysis which agree with it very well.

The Rectangle Rule for Computing Hilbert Integral In Boundary Element Methods

The generalized composite rectangle (constant) rule for the computation of Cauchy principal value integral with the singular kernel cot[(x−s)/2] is discussed. Our study is based on the investigation of the point-wise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at certain local coordinate of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.

A rectangle rule for the computation of hypersingular integral

The composite rectangle rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel 1/(x − s)2 is discussed. For the case of singular point coinciding with the mesh point, a new quadrature rule which is based on the classical finite-part definition is presented. A kind of the hypersingular integral equation is solved by collocation methods and the maximal error estimation is given. Some numerical results are also illustrated to confirm the theoretical results and show the efficiency of the algorithms.

Saint-Venant torsion of orthotropic bars with inhomogeneous rectangular cross section

This work presents an exact solution of the Saint-Venant torsion of a straight bar with an orthotropic and inhomogeneous rectangular cross section whose material properties obey the exponential law in one direction. An approximate solution is also obtained for the material properties being arbitrarily distributed in one direction by using a layerwise model, in which the inhomogeneous rectangle is simulated by a composite rectangle composed of multiple rigidly connected homogeneous rectangular regions. The warping function, stresses and torsional rigidity are analytically expressed in terms of Fourier series for both solutions, in which the hyperbolic functions are not directly employed to avoid numerical difficulties. Some numerical examples are finally presented to verify the proposed method and the parametrical study is also performed.

A new figure fracturing algorithm for variable-shaped EB exposure-data generation

This paper discusses a new figure fracturing algorithm to be used in input data generation for the variable-shaped EB (electron beam) exposure system. In the input data generation of this kind of system, processing is required in which the individual polygons composing the mask pattern are represented as a set of trapezoids (figure fracturing). The figure fracturing considered in this paper differs from those in the conventional methods, in that the generation of a narrow trapezoid as well as the partitioning of the critical part is suppressed, in order to improve the accuracy of LSI mask fabrication. In this paper, the problem is solved as the fracturing of the composite rectangle into a set of rectangles by removing slant edges from the polygon. The relation graph is introduced to represent the relation among the partitioning line candidates. A graph-based algorithm is presented that efficiently selects the adequate partition lines from all possible candidates. The proposed method is applied to the mask fabrication for 64 M and 256 M DRAM, and a satisfactory fractured result is obtained in a practical execution time. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(8): 87–102, 2000

Solution of the mixed boundary value problem of heat conduction in a composite rectangle

The author offers a particular solution of the two-dimensional problem of steady heat conduction for a rectangular region composed of two different materials with mixed boundary conditions.