Number Of Mass Points Research Articles

Peridynamics-coupled finite element method implication in concrete material crack prediction

This study presents a multi-bond element stiffness matrix to couple peridynamics with the finite element method, where mass points are denoted as elements with the degree of freedom to have a certain number of mass points. The proposed peridynamics-coupled finite element method (PDFEM) based on multi-bond element is demonstrated to be effective in the crack and fracture prediction of concrete materials. In addition, experimental test results and the compression and bidirectional load numerical simulations’ results are compared, and good agreement is observed. In the single compression simulation of concrete specimen, fracture crack occurs along with the compression load direction. In bidirectional load simulations of concrete specimen, different settings of load lead to changeable fracture cracks. Furthermore, the comparison with finite element method (FEM) also shows good agreement. The crack, mass point fracture percentage, and stress state of the concrete specimen are investigated under different load conditions and simulation times, and the results demonstrate a promising prospect of the proposed PDFEM in the field of fracture prediction of concrete materials.

A new multiple-arc model of the resonant Kuiper belt objects – Plutinos

ABSTRACT To incorporate the gravitational influence of Kuiper belt objects (KBOs) in planetary ephemerides, uniform-ring models are commonly employed. In this paper, for representing the KBO population residing in Neptune’s 2:3 mean motion resonance (MMR), known as the Plutinos, we introduce a three-arc model by considering their resonant characteristics. Each ‘arc’ refers to a segment of the uniform ring and comprises an appropriate number of point masses. Then the total perturbation of Plutinos is numerically measured by the change in the Sun–Neptune distance ($\Delta d_{\mathrm {SN}}$). We conduct a comprehensive investigation to take into account various azimuthal and radial distributions associated with the resonant amplitudes (A) and eccentricities (e) of Plutinos, respectively. The results show that over a 100-yr period: (1) at the smallest $e=0.05$, the Sun–Neptune distance change $\Delta d_{\mathrm {SN}}$ caused by Plutinos decreases significantly as A reduces. It can deviate from the value of $\Delta d_{\mathrm {SN}}$ obtained in the ring model by approximately 100 km; (2) as e increases in the medium range of 0.1–0.2, the difference in $\Delta d_{\mathrm {SN}}$ between the arc and ring models becomes increasingly significant; (3) at the largest $e\gtrsim 0.25$, $\Delta d_{\mathrm {SN}}$ can approach zero regardless of A, and the arc and ring models exhibit a substantial difference in $\Delta d_{\mathrm {SN}}$, reaching up to 170 km. Then the applicability of our three-arc model is further verified by comparing it to the perturbations induced by observed Plutinos on the positions of both Neptune and Saturn. Moreover, the concept of the multiple-arc model, designed for Plutinos, can be easily extended to other MMRs densely populated by small bodies.

Multiplane gravitational lenses with an abundance of images

We consider gravitational lensing of a background source by a finite system of point-masses. The problem of determining the maximum possible number of lensed images has been completely resolved in the single-plane setting (where the point masses all reside in a single lens plane), but this problem remains open in the multiplane setting. We construct examples of K-plane point-mass gravitational lens ensembles that produce ∏i=1K(5gi−5) images of a single background source, where gi is the number of point masses in the ith plane. This gives asymptotically (for large gi with K fixed) 5K times the minimal number of lensed images. Our construction uses Rhie’s single-plane examples and a structured parameter-rescaling algorithm to produce preliminary systems of equations with the desired number of solutions. Utilizing the stability principle from the differential topology, we then show that preliminary (nonphysical) examples can be perturbed to produce physically meaningful examples while preserving the number of solutions. We provide numerical simulations illustrating the result of our construction, including positions of lensed images and the structure of critical curves and caustics. We observe an interesting “caustic of multiplicity” phenomenon that occurs in the nonphysical case and has a noticeable effect on the caustic structure in the physically meaningful perturbative case.

Optimal Synthesis of Unconventional Links for Modular Reconfigurable Manipulators

Abstract Customization of manipulators having unconventional parameters and link shapes have gained attention to accomplish nonrepetitive tasks in a given cluttered environment. Adaptive modular and reconfigurable designs are being used to achieve customization and have provided time and cost-effective solutions. Major challenges are associated to provide the systematic approach on the design and realization of modular components considering connectivity and integration. This article focuses on the architectural synthesis of the modular links, optimized with respect to the dynamic torques while following a prescribed set of trajectories. The design methodology is proposed as an Architecture Prominent Sectioning−k strategy, which assumes a modular link as an equivalent system of k number of point masses, performing optimization to minimize the joint torques and map the resulting re-adjusted point masses into a possible architecture. The proposed strategy is general and can be applied to planar or spatial manipulators with n−DoF even with nonparallel and nonperpendicular jointed configurations. The design of optimal curved links is realized resulting from the optimized solution considering the dynamics of the modular configurations over primitive trajectories. The proposed modular library of unconventional curved link modules with joint modules have shown lesser requirement of the joint torques compared to the conventional straight links.

On the number of equilibria balancing Newtonian point masses with a central force

We consider the critical points (equilibria) of a planar potential generated by n Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular-restricted n-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by n + 1, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in n. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as “ring configurations,” consisting of n − 1 equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as 5n − 5 equilibria. We verify analytically that the ring configuration has at least 5n − 5 equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell’s problem from electrostatics and the image counting problem from gravitational lensing.

Application of the nonlinear optimisation in regional gravity field modelling using spherical radial base functions

The gravity field is a signature of the mass distribution and interior structure of the Earth, in addition to all its geodetic applications especially geoid determination and vertical datum unification. Determination of a regional gravity field model is an important subject and needs to be investigated and developed. Here, the spherical radial basis functions (SBFs) are applied in two scenarios for this purpose: interpolating the gravity anomalies and solving the fundamental equation of physical geodesy for geoid or disturbing potential determination, which has the possibility of being verified by the Global Navigation Satellite Systems (GNSS)/levelling data. Proper selections of the number of SBFs and optimal location of the applied SBFs are important factors to increase the accuracy of estimation. In this study, the gravity anomaly interpolation based on the SBFs is performed by Gauss-Newton optimisation with truncated singular value decomposition, and a Quasi-Newton method based on line search to solve the minimisation problems with a small number of iterations is developed. In order to solve the fundamental equation of physical geodesy by the SBFs, the truncated Newton optimisation is applied as the Hessian matrix of the objective function is not always positive definite. These two scenarios are applied on the terrestrial free-air gravity anomalies over the topographically rough area of Auvergne. The obtained accuracy for the interpolated gravity anomaly model is 1.7 mGal with the number of point-masses about 30% of the number of observations, and 1.5 mGal in the second scenario where the number of used kernels is also 30%. These accuracies are root mean square errors (RMSE) of the differences between predicted and observed gravity anomalies at check points. Moreover, utilising the optimal constructed model from the second scenario, the RMSE of 9 cm is achieved for the differences between the gravimetric height anomalies derived from the model and the geometric height anomalies from GNSS/levelling points.

Forward and Inverse Problems for a Finite Krein–Stieltjes String. Approximation of Constant Density by Point Masses

A dynamic inverse problem for a dynamical system that describes the propagation of waves in a Krein string is considered. The problem is reduced to an integral equation, and an important special case is considered where the string density is determined by a finite number of point masses distributed over the interval. An equation of Krein type, with the help of which the string density is restored, is derived. The approximation of constant density by point masses uniformly distributed over the interval and the effect of the appearance of a finite wave propagation velocity in the dynamical system are also studied.

On the Minimum Number of Distinct Eigenvalues in the Problem for a Tree Formed by Stieltjes Strings

We consider spectral problems related to small vibrations of a tree formed by Stieltjes strings. It is shown that the minimum number of distinct eigenvalues of this problem is equal to the maximum length (measured as the number of point masses) of paths in this tree.

The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points

This paper studies an $n$ -dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when $n=1$ the capacity-achieving input distribution is discrete with finitely many mass points, and when $n>1$ the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, not even a bound on the exact number of mass points/shells was available. This paper provides an alternative proof of the finiteness of the number mass points/shells of the capacity-achieving input distribution while producing the first firm bounds on the number of mass points and shells, paving an alternative way for approaching many such problems. The first main result of this paper is an order tight implicit bound which shows that the number of mass points in the capacity-achieving input distribution is within a factor of two from the number of zeros of the downward shifted capacity-achieving output probability density function. Next, this implicit bound is utilized to provide a first firm upper on the support size of optimal input distribution, an $O(\mathsf {A}^{2})$ upper bound where $\mathsf {A}$ denotes the constraint on the input amplitude. The second main result of this paper generalizes the first one to the case when $n>1$ , showing that, for each and every dimension $n\ge 1$ , the number of shells that the optimal input distribution contains is $O(\mathsf {A}^{2})$ . Finally, the third main result of this paper reconsiders the case $n=1$ with an additional average power constraint, demonstrating a similar $O(\mathsf {A}^{2})$ bound.

Tensegrity system dynamics with rigid bars and massive strings

This paper provides a single matrix-second-order nonlinear differential equation to simulate the dynamics of tensegrity systems with rigid bars and massive strings. The paper allows one to distribute the string mass into any specified number of point masses along the string while preserving the exact rigid bar dynamics. This formulation also allows modeling of skins and surfaces as a finite set of strings in the tensegrity dynamics. To reduce the complexity of the model, non-minimal coordinates (6 degrees of freedom for each bar instead of 5) were chosen. This is the key to give more accurate results in computer simulations since the mathematical structure of the model is simplified and exploited during numerical computations. A bar length correction algorithm is also provided for both class-1 and class- $k$ tensegrity systems to correct the erroneous change in bar length because of computational errors during numerical integration. We characterize the control variable as the force density in each string. This allows control laws to be developed independently of the material chosen for the structural elements. A nonlinear transformation back to the physical control variables involves the material properties.

On the capacity and energy efficiency of non‐coherent Rayleigh fading channels with additive Gaussian mixture noise

This paper studies the capacity and energy efficiency of non-coherent Rayleigh fading channels with Gaussian mixture noise where neither the transmitter nor the receiver has the knowledge of channel state information. The channel under consideration is suited for cellular networks having multi-tier heterogeneous architectures in which the channel conditions change rapidly. In the first part of the paper, we characterize the structure of a capacity-achieving input signal. Specifically, we establish an integrable upper bound on the integrand in the output entropy and demonstrate that there exists a unique optimal input. By formulating the Kuhn-Tucker condition and establishing a diverging lower bound on it, we show that the optimal input is discrete having a finite number of mass points. Using this result, we investigate the capacity and energy efficiency of the considered channel in the second part of the paper. In particular, we first develop a numerical method to evaluate the optimal input and compute the capacity. The energy efficiency, which is related to the capacity and optimal input in low-power regimes, is examined by calculating the minimum bit energy and wideband slope of the spectral-efficiency curve. We also analytically show the optimality of an on-off signal in this regime.

Linear mixed models with marginally symmetric nonparametric random effects

Linear mixed models (LMMs) are used as an important tool in the data analysis of repeated measures and longitudinal studies. The most common form of LMMs utilizes a normal distribution to model the random effects. Such assumptions can often lead to misspecification errors when the random effects are not normal. One approach to remedy the misspecification errors is to utilize a point-mass distribution to model the random effects; this is known as the nonparametric maximum likelihood-fitted (NPML) model. The NPML model is flexible but requires a large number of parameters to characterize the random-effects distribution. It is often natural to assume that the random-effects distribution be at least marginally symmetric. The marginally symmetric NPML (MSNPML) random-effects model is introduced, which assumes a marginally symmetric point-mass distribution for the random effects. Under the symmetry assumption, the MSNPML model utilizes half the number of parameters to characterize the same number of point masses as the NPML model; thus the model confers an advantage in economy and parsimony. An EM-type algorithm is presented for the maximum likelihood (ML) estimation of LMMs with MSNPML random effects; the algorithm is shown to monotonically increase the log-likelihood and is proven to be convergent to a stationary point of the log-likelihood function in the case of convergence. Furthermore, it is shown that the ML estimator is consistent and asymptotically normal under certain conditions, and the estimation of quantities such as the random-effects covariance matrix and individual a posteriori expectations is demonstrated. A simulation study is used to illustrate the gains in efficiency of the MSNPML model over the NPML model under the assumption of symmetry. A pair of real data applications are then used to demonstrate the manner in which the MSNPML model can be used to draw useful statistical inference.

Maximizing Probability Bounds Under Moment-Matching Restrictions

The problem of characterizing a distribution by its moments dates to work by Chebyshev in the mid-nineteenth century. There are clear (and close) connections with characteristic functions, moment spaces, quadrature, and other very classical mathematical pursuits. Lindsay and Basak posed the specific question of how far from normality could a distribution be if it matches k normal moments. They provided a bound on the maximal difference in cdfs, and implied that these bounds were attained. It will be shown here that in fact the bound is not attained if the number of even moments matched is odd. An explicit solution is developed as a symmetric distribution with a finite number of mass points when the number of even moments matched is even, and this bound for the even case is shown to hold as an explicit limit for the subsequent odd case. As Lindsay noted, the discrepancies can be sizable even for a moderate number of matched moments. Some comments on implications are proffered.

Capacity analysis for pulse amplitude modulated visible light communications with dimming control

This paper focuses on the capacity-approaching, nonuniform signaling for the pulse amplitude modulated (PAM) visible light communications under the non-negativity, peak power, and dimmable average power constraints. The input distribution is characterized by three parameters, i.e., the intensities, the probabilities, and the number of mass points in the PAM constellation. In the open literature, no analytical expression can be used to obtain the capacity-achieving input distribution. In this paper, a computationally simple but capacity-approaching input distribution is alternatively derived by determining the three aforementioned parameters. The resulting input distribution can serve as a useful tool not to approach the channel capacity but to guide the practical system design. Numerical results substantiate that the derived input distribution is a capacity-approaching distribution and can offer a better performance gain in comparison with the commonly employed uniform input distribution.

Gyroscopic Motion: Show Me the Forces!

Gyroscopes are frequently used in physics lecture demonstrations and in laboratory activities to teach students about rotational dynamics, namely, angular momentum and torque. Use of these powerful concepts makes it difficult for students to fully comprehend the mechanism that keeps the gyroscope from falling under the force of gravity. The analysis of gyroscopic motion presented here is in terms of linear forces and linear momentum, allowing students to understand gyroscopic motion using familiar concepts. Our simulation using VPython1 models the gyroscope as a number of point masses evenly spaced in a radial pattern. Students can alter the parameters of the gyroscope including starting position, mass, rotational velocity of the masses, and the force of gravity. All of this allows for a simulation intended to help students understand how such a complex system works.

Noise enhanced hypothesis-testing according to restricted Neyman–Pearson criterion

Noise enhanced hypothesis-testing is studied according to the restricted Neyman–Pearson (NP) criterion. First, a problem formulation is presented for obtaining the optimal probability distribution of additive noise in the restricted NP framework. Then, sufficient conditions for improvability and nonimprovability are derived in order to specify if additive noise can or cannot improve detection performance over scenarios in which no additive noise is employed. Also, for the special case of a finite number of possible parameter values under each hypothesis, it is shown that the optimal additive noise can be represented by a discrete random variable with a certain number of point masses. In addition, particular improvability conditions are derived for that special case. Finally, theoretical results are provided for a numerical example and improvements via additive noise are illustrated.

Using Hermite Bases in Studying Capacity-Achieving Distributions Over AWGN Channels

This paper studies classes of generic deterministic, discrete time, memoryless, and “nonlinear” additive white Gaussian noise (AWGN) channels. Subject to multiple types of constraints such as the even-moment and compact-support constraints or a mixture, the optimal input is proved to be discrete with finite number of mass points in the vast majority of the cases. Only under the even-moment constraint and for special cases that emulate the average power constrained linear channel, capacity is found to be achieved by an absolutely continuous input. The results are extended to channels where the distortion is generally piecewise nonlinear where the discrete nature of the optimal input is conserved. These results are reached through the development of methodology and tools that are based on standard decompositions in a Hilbert space with the Hermite polynomials as a basis, and it is showcased how these bases are natural candidates for general information-theoretic studies of the capacity of channels affected by AWGN. Intermediately, novel results regarding the output rate of decay of Gaussian channels are derived. Namely, the output probability distribution of any channel subjected to additive Gaussian noise decays necessarily “slower” than the Gaussian itself. Finally, numerical computations are provided for some sample cases, optimal inputs are determined, and capacity curves are drawn. These results put into question the accuracy of adopting the widely used expression 1(1+ SNR) for computing capacities of Gaussian deterministic channels.

Discrete Laguerre–Sobolev expansions: A Cohen type inequality

C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted L p spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre–Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B.Xh. Fejzullahu and F. Marcellán.

Even solutions of the Toda system with prescribed asymptotic behavior

Every solution of the Toda system, describing the behavior of a finite number of mass points on the line, each one interacting with its neighbors, is asymptotically linear at infinity. We show the existence and uniqueness of even solution with suitable prescribed asymptotic behavior, by analyzing a system of algebraic equations derived from the relation between the slopes and the intercepts of the asymptotic lines.

Extremely large displacement dynamic analysis of elastic–plastic plane frames

AbstractThis paper presents a numerical analysis of large displacement responses of elastic–plastic plane frames under static and dynamic loads, by applying the vector‐form intrinsic finite element (VFIFE or V‐5) method. The VFIFE method defines the structure into a number of mass points, and applies Newton's second law and the internal force equilibrium to describe the motions of each mass point. By tracing the motions of all the mass points, it can analyze the large geometrical and material nonlinear changes during the motion of the structure without using the geometrical stiffness matrix and iterative procedures. Three different numerical examples are presented to demonstrate both the capability and the accuracy of the VFIFE method in a nonlinear dynamic analysis of frame structures with extremely large displacement. Copyright © 2011 John Wiley & Sons, Ltd.