Planar Problem Research Articles

Fuzzy c-means and simulated annealing for planar location-routing problem

Location-routing problem (LRP) contains two Np-hard problems as, facility location (FL) and vehicle routing problem (VRP), in the same content. Since both problems directly affect the cost of distributions of the products and supply chain, the decision of location and routing is important for the success of companies. Therefore, many attempts are made to solve LRP problem in the literature. Researchers proposed exact and heuristic methods for LRP. However, exact methods cannot provide solutions for considerably large instances. In this paper, a new heuristic method is proposed for continuous or planar LRP. The proposed method contains fuzzy c-means for continuous location problem and simulated annealing algorithm for vehicle routing problem, respectively. The proposed method is applied to both capacitated and uncapacitated LRP instances that are widely used in the literature. Results of the proposed method are compared with successful researches that are made on this problem in terms of the total cost.

Assessment of interaction the new construction object at full load and the surrounding building in the dense urban development conditions

The work is devoted to research the impact of a new building at full load after completion of its construction has been investigated through an assessment of the interaction with the building envelope, the ground mass of the base and the foundation of the existing five-story building, in comparison with the assessment of the impact of a deep excavation in the process of engineering preparation for this construction, based on the developed methodology for studying the interaction of new construction objects and nearby buildings in complex engineering and geological conditions and dense development conditions.
 For numerical studies of the stability of the combined space, the proposed methodology has been used, which is built on the initial relations of the variational equations of equilibrium and the equation of the load surface in the six-dimensional stress space, which is a development of the theory of the limit stress state of the soil half-space based on the introduction of an extended yield criterion for a plane problem of the nonlinear theory of elasticity and plasticity, which provides for the determination of the value of the second critical load, at which solid sections of the ultimate stress state appear in the soil half-space.
 From the analysis of the movements of the nodes located on the vertical face of the «wall in the soil» structure (retaining wall of the pit), it can be concluded that the transverse displacements of the retaining wall compared to the calculations according to the first option have changed to the opposite, which is associated with the formation of a sliding prism under the passive pressure on the retaining wall from the side of the pile foundation of the new building. This indicates a change in the stress-deformation state of the soil base of the existing building, from the influence of the deep excavation during the construction process, which was evaluated according to the first version of calculations, to the impact of the new building after the end of construction, which was analyzed according to the second version of calculations. In this way, the change in the stress-deformation state of the main urban building is determined, which is a criterion for determining the impact of adjacent construction in the urban area on the condition of the ground bases and foundations of adjacent buildings, their preservation and ensuring safe operation.

The phase space structure of retrograde mean motion resonances with Neptune: the 4/5, 7/9, 5/8 and 8/13 cases

We compute planar and three-dimensional retrograde periodic orbits in the vicinity of the restricted three-body problem (RTBP) with the Sun and Neptune as primaries and we concentrate on the dynamics of higher-order exterior mean motion resonances with Neptune. By using the circular planar model as the basic model, families of retrograde symmetric periodic orbits are computed at the 4/5, 7/9, 5/8 and 8/13 resonances. We determine the bifurcation points from the planar circular to the planar elliptic problem and we find all the corresponding families. In order to obtain a global view of the families of periodic orbits, the eccentricity of the primaries takes values in the whole interval 0<e'<1. Then, we find all the possible vertical critical orbits (v.c.o) of the planar circular problem and we proceed to the three-dimensional circular restricted 3-body problem. In this model, retrograde periodic orbits are generated mainly from the retrograde v.c.o. Also, if we continue families of direct orbits for i>90^circ , then we can obtain families of 3D symmetric retrograde periodic orbits. The linear stability is examined too. Stable periodic orbits are associated with phase space domains of resonant motion where TNOs can be captured. In order to study the phase space structure of the above resonances, we construct dynamical stability maps for the whole inclination interval (0<i<180^circ ) by using the well-known “MEGNO Chaos Indicator”. Finally, we discuss about TNOs which are currently located at these resonances.

An ultrasonic wave‐based framework for imaging internal cracks in concrete

Nondestructive detection and sizing of internal cracks initiated in reinforced concrete have been critical problems. Geometric characterization of internal cracks and delaminations contributes to predicting their propagation path and failure pattern followed by structural prognostics. There have been limited studies on quantitative detection and characterization of vertical cracks in concrete using wave-based NDE techniques. This study focuses on solving the problem of imaging deep internal crack planes in the concrete medium by leveraging the ultrasonic shear horizontal (SH) waves from a transducer array. Post-processing the array full matrix capture (FMC) data with both total focusing method (TFM) and plane wave imaging (PWI) reveals their efficiency in mapping planar defects inclined between 0° and 60°. However, their performance on vertical/near-vertical defects is found to be inferior with imaging limited to the tip of the defect planes. Further, half-skip modes of wave dispersion have been adopted in addition to the directly scattered pulses for imaging the vertical and near-vertical cracks. We propose an imaging framework based on the outcomes of our investigation on the best suitable methodology to detect and map planar defects like cracks inclined in the range of 0–90°. Statistical quantification of defect inclinations from the reconstructed images is compared to the ground-truth orientations, and they are found to be 94%–99% accurate. Besides accuracy, the computational efficiency of the proposed techniques makes them desirable and reliable for quick on-site inspections on built infrastructure.

The high-order smooth interpolated reproducing kernel particle method for elastodynamics problems

Combining the interpolation reproducing kernel particle method (IRKPM) with the integral weak form of elastodynamics, we present a high-order smooth interpolated reproducing kernel particle method for an elastodynamics plane problem. The shape function of IRKPM not only has the interpolation property at any point but also has a high-order smoothness not lower than that of the kernel function. This new method overcomes the difficulties of most meshless methods in dealing with essential boundary conditions and ensures high numerical accuracy. For time domain integration, we use the classical Newmark average acceleration method. By numerical examples we demonstrate that the proposed method has the advantages of higher accuracy, smaller scale of solving problem, and direct application of boundary conditions.

Expansive behavior of high-strength self-stressing and self-compacting concrete: Experimental study and analytical model

This paper presents an experimental investigation on the expansive behavior of high-strength self-compacting concrete (HSCC) with various HCSA expansive agent (EA) contents ranging from 0 % to 12 %. The fresh properties, compressive strength and expansive ratios at free and restrained conditions are studied. The expansive process and expansive mechanism of self-stressing HSCC are analyzed with a combination of X-ray diffraction (XRD) and scanning electron microscopy (SEM) analysis. Moreover, the self-stress of expansive HSCC under the confinement of steel-tube is measured. The results show that the compressive strength of HSCC decreases with increasing EA contents, which should not be over 12 % due to the large expansions damage the concrete matrix seriously. The expansive ratios increase significantly with increasing EA contents, as well as increasing amplitudes. The free expansive ratios of concrete can reach 1.08 × 10−4, 3.03 × 10−4, 5.13 × 10−4 and 7.85 × 10−4 with the EA content ranging from 6 % to 12 % at 28 curing days, respectively. Moreover, the self-stresses of expansive HSCC with EA contents of 8–12 % are 3.2–3.7 MPa. Finally, under the framework of plane strain model in elasticity theory, a calculation analytical method is developed to predict the restrained expansive ratios of HSCC confined by steel tubes. The computational formula is also established to calculate the self-stress of expansive HSCC using the axisymmetric plane problems in elasticity. The model formula provides a good description of expansive responses of self-stressing HSCC.

Cortical Bone Model with a Microcrack under Tensile Loading

The fracture mechanics of cortical bone has received much attention in biomedical engineering. It is a fundamental question how the material constants and the geometric parameters of the cortical bone affect the fracture behavior of the cortical bone. In this work, the plane problem for cortical bone with a microcrack located in the interstitial tissue under tensile loading was considered. Using the solution for the continuously distributed edge dislocations as Green's functions, the problem was formulated as singular integral equations with Cauchy kernels. The numerical results suggest that a soft osteon promotes microcrack propagation, while a stiff osteon repels it, but the interaction effect between the microcrack and the osteon is limited near the osteon. This study not only sheds light on the fracture mechanics behavior of cortical bone but also offers inspiration for the design of bioinspired materials in biomedical engineering.

A Fully Implicit Log-Conformation Tensor Coupled Algorithm for the Solution of Incompressible Non-Isothermal Viscoelastic Flows.

In this work, a block-coupled algorithm is presented, which can compute laminar, incompressible, non-isothermal, viscoelastic flow problems based on the log-conformation tensor approach. The inter-equation coupling of the incompressible Cauchy linear momentum and mass conservation equations is obtained in a procedure based on the Rhie–Chow interpolation. The divergence of the log-conformation tensor term in the linear momentum equations is implicitly discretized in this work. In addition, the velocity field is considered implicitly in the log-conformation tensor constitutive equations by expanding the advection, rotation and the rate of deformation terms with a Taylor series expansion truncated at the second-order error term. Finally, the advection and diffusion terms in the energy equation are also implicitly discretized. The mass, linear momentum, log-conformation tensor constitutive model and energy-discretized linear equations are joined into a block-matrix following a monolithic framework. Validation of the newly developed algorithm is performed for the non-isothermal viscoelastic matrix-based Oldroyd-B fluid flow in the axisymmetric 4:1 planar sudden contraction benchmark problem.

An enhanced artificial hummingbird algorithm and its application in truss topology engineering optimization

In view of the shortcomings such as slow search speed, low optimization precision and premature convergence of artificial hummingbird algorithm, an enhanced artificial hummingbird algorithm based on golden sine factor named DGSAHA is proposed. Firstly, chaos mapping is used to generate the initial candidate solution to increase the diversity of the population, which lays the foundation for the global search. Then, perturb the individuals by means of the differential variation between individuals on the group, thereby enhancing the diversity of the population, preserving the excellent individuals, eliminating the inferior individuals, and guiding the search process to approach the global optimal solution, avoiding the phenomenon of premature convergence. Finally, the golden sine factor were introduced in the guided foraging stage is conducive to the full exploration of the global optimal solution, reducing the search space for individuals to approach the optimal solution. And, it facilitates the balance between “exploration” and “exploitation” of algorithm. Thereby, the accuracy and speed of the DGSAHA can be improved to a certain extent. 25 classic functions, the CEC2014 and CEC2019 benchmark functions were tested, and several representative meta-heuristic algorithms and its improved algorithm are compared for evaluate the validity of DGSAHA. Meanwhile, the dimensional scalability of the variable-dimensional test function is discussed. The results of non-parametric statistical analysis and performance index show that DGSAHA in this paper has better comprehensive optimization performance, significantly improves the search speed and convergence precision, and has strong ability to get rid of the local optimal solution. Finally, the performance of DGSAHA and the practicability of truss structure are answered by three engineering examples of plane and space truss topology optimization problem. This optimization problem considers not only the static constraints such as stress, displacement and buckling, but also the dynamic constraints of frequency and motion stability. In order to avoid singularity and unnecessary analysis, the stiffness, mass and load matrices are reconstructed in finite element analysis. A lighter truss structure than the existing solution is obtained. The validity, extensibility and practicability of the algorithm are further illustrated.

Improved convergence in planar nanophotonic topology optimization via the multigradient

Adjoint Topology Optimization is a physics-informed inverse-design strategy which has been widely applied in electromagnetics and nanophotonics to create highly performant meta-devices. By interfering the electric fields of both forward and reverse (i.e., adjoint) simulations within a device, it is possible to compute a permittivity gradient which will improve its performance. In the specific case of planar metasurface design, the permittivity gradient will vary depending on the z-height, which presents a problem for designing devices suitable for single-layer lithographic fabrication, since such products have uniform material properties in the z-direction. State-of-the-art applications of adjoint topology optimization address this issue by averaging together the gradients at multiple z-heights. This article proposes an alternative strategy for combining these same gradients based on the multigradient. This modification produces a notable speedup of as much as 2–3 times for a representative supercell design problem, indicating this new strategy may be preferable for planar nanophotonic design problems in general.

Covering convex polygons by two congruent disks

We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an O(nlog⁡n)-time algorithm for the two-center problem for a convex polygon, where n is the number of vertices of the polygon. This improves upon the previous best algorithm for the problem.

Special Issue on Experimental Solid Mechanics

In the present special volume, the German Association of Applied Mathematics and Mechanics (GAMM) Expert Committee “Experimental Solid Mechanics” was again given the opportunity to summarize the current scientific activities of individual participating working groups in Germany. Both optical measurement methods from surface information as well as radiation-based methods for detecting the internal states present in material bodies are receiving increasing interest. Nowadays, the required measurement systems can simply be purchased, or they can be developed in-house. In addition, there are still many scientific questions left open in the evaluation of the found measurement data. In the meantime, image correlation methods are often used to determine the surface deformation of components, the discrete data of which are now evaluated using proprietary software tools, or are coupled with infrared thermography systems, in particular to determine the dissipating energy of mechanically loaded components. In the first issue of this special volume, four contributions are compiled proposing (1) a shear evaluation tool using digital image correlation (DIC) for plane problems [5], the coupling of infrared thermography (IRT) and 3D DIC for both (2) identifying material parameters in metal plasticity [6] as well as (3) applicability studies of foams and auxetic materials [3], and, finally, (4) identifying the heat conductivity parameters in transversal isotropy using IRT [8]. The second issue treats distance measurements of laminate layers using microscopical images [4], and three further contributions on μ-CT measurements. First, new in-situ measurements in granular media using μ-X-ray computer tomography (CT) combined with ultrasonic wave propagation is investigated [7]. A further contribution treats the influence of the pores on the fatigue properties of particular additively manufactured parts [2], and, finally, a study on the micro-structural characterization and stochastic modeling of open-cell foam using μ-CT image analysis [1]. All articles contribute to contact-less measurement sensing and evaluation in the field of solid mechanics. We gratefully acknowledge the German Association of Applied Mathematics and Mechanics (GAMM) for scientific support of the expert committee in the last years.

Multi-input Unet model based on the integrated block and the aggregation connection for MRI brain tumor segmentation

With the growth of data information and the development of computer equipment, it is extremely time-consuming and laborious to rely on the traditional manual segmentation of brain medical images. To solve the above problems, this paper proposes a multi-input Unet model based on the integrated block and the aggregation connection to achieve efficient and accurate segmentation of tumor structure. Besides, this paper solves the low-resolution problem in sagittal and coronal planes, which can effectively improve memory efficiency. The proposed algorithm is innovative in three aspects. Firstly, by inputting the mask images which can effectively represent the tumor location characteristics, it can provide more information about the spatial relationship to alleviate the problems of fuzzy boundary and low contrast between the lesion region and healthy brain tissue. Then, the integrated block extracts the tumor local information in different receptive domains by a multi-scale convolution kernel. The aggregation connection realizes the implicit deep connection of context information, which combines the shallow and deep information of the brain with strong geometric spatial relationships. Meanwhile, to effectively alleviate the waste of memory resources caused by redundant and background information in medical images, the amount of calculation in model training is reduced by dimension reduction of the feature map. An ablation experiment is used to verify the innovation of the proposed algorithm on the BraTS dataset, which compares with the state-of-the-art brain tumor segmentation methods. The precision of the proposed multi-input Unet model for the whole tumor and core lesion is 0.92 and 0.90, respectively.

Exact solution of the plane problems for poroelastic rectangle and semi‐strip

AbstractThe exact solutions of the plane problems of poroelasticity for a rectangle and a semi‐strip are derived in the article. The original problems are reduced to one‐dimensional problems with the help of finite and semi‐infinite Fourier transforms for a rectangle and a semi‐strip, respectively. The one‐dimensional boundary value problems are formulated in vector form, and the apparatus of matrix differential calculation is used. Analytical formulas for displacements, stress, and pore pressure are derived for the rectangle and semi‐strip. The investigation of stress and pore pressure for different load types and different poroelastic materials is done for the rectangle and semi‐strip.

Analytical solution of an orthotropic elasticity for a frictional rigid flat punch on a half plane with an oblique edge crack using a mapping function

A solution of an orthotropic elastic plane problem is derived for a frictional rigid flat punch on a half plane with an oblique edge crack. A mapping function is used for arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped half plane problems can be solved by exchanging the mapping function. Therefore, once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, the stress distributions are shown for a frictional coefficient and an oblique edge crack angle. Four elastic constants are expressed by two independent parameters.

Equilibrium points and their linear stability in the planar equilateral restricted four-body problem: a review and new results

In this work, we revisit the planar restricted four-body problem to study the dynamics of an infinitesimal mass under the gravitational force produced by three heavy bodies with unequal masses, forming an equilateral triangle configuration. We unify known results about the existence and linear stability of the equilibrium points of this problem which have been obtained earlier, either as relative equilibria or a central configuration of the planar restricted \((3 + 1)\)-body problem. It is the first attempt in this direction. A systematic numerical investigation is performed to obtain the resonance curves in the mass space. We use these curves to answer the question about the existing boundary between the domains of linear stability and instability. The characterization of the total number of stable points found inside the stability domain is discussed.

"Periodic solutions for certain Hamiltonian systems in arbitrary dimension and global parametrization of some manifolds"

"t has been recently shown that the limit cycle situation is not valid for Hamiltonian systems in dimension two, under appropriate conditions. The applications concern global parametrizations of closed curves in the plane and optimal design problems. Here, we discuss a partial extension of this result, for certain Hamiltonian-type systems in higher dimension."

Artificial Controlling of the Collinear Liberation Points Using Lorentz Force in the Restricted Three-Body Problem

This work studies the possibility of generating artificial collinear liberation points for the planar circular restricted three-body problem using Lorentz force affecting a charged spacecraft due to the magnetic field of a planet. It is considered to be a magnetic dipole inclined by angle α with the spin axis of the planet. The acceleration components for Lorentz force are first derived in an inertial planet-center coordinate system. Then, they are transformed into the rotating coordinate system of the three-body system, with the planet naturally the smaller primary in a planet-Sun system. The equations for the liberation points are derived including the charge per unit mass as the controlling parameter. Finally, the values of the charge per unit mass required for controlling the collinear liberation point positions are derived. A numerical application for the Sun-Jupiter system is introduced and the relation between the position of the artificial liberation point and the charge per unit mass is presented graphically.

Оцінка пластичності поверхневого шару металу при обкочуванні деталей циліндричної форми кулькою

The article deals with the study of the peculiarities of the process of plastic deformation of the surface layer of cylindrical parts during rolling with a ball. The stress-strain state and the regularities of its change, depending on the process parameters in the surface layer of the part, were determined by the slip line method and the finite element method.The vast majority of parts of mechanisms and machines work in contact with the relevant parts, namely various friction pairs, shafts, rollers, gears or tools, product housings, strikers, and so on. Specific requirements for these parts are due to the peculiarities of the operation of the latter. As a rule, it is impossible to meet all the requirements when making parts from one material. This mainly applies to the requirements of combining the high strength characteristics of the material of the internal volume of the part and the significant hardness of the same material on its surface in contact with the corresponding part. In addition, the surface of the parts under certain conditions must have high wear resistance and low coefficient of friction (antifriction parts) or high coefficient of friction (friction parts), or self-changing surface layer, and sometimes the entire surface must have some porosity and sweat during operation .The slip line method can be attributed to the traditional methods for solving problems of the theory of plasticity. This method can be used to solve problems in which the equations of equilibrium and plasticity create a closed system of differential equations of hyperbolic type. Therefore, this method can be used for solving plane problems, and only in some cases for solving axisymmetric problems. The solutions that can be obtained if a certain area of slip lines is directly known are called straight lines. When information about the shape of the initial slip lines is insufficient, inverse and semi-inverse methods are used.A numerical study of the regularities of damage accumulation was carried out, calculations of the accumulated deformation, the stress state index, and the distributions of the used plastic resource over the volume of the plastic zone when rolling with a ball in the environment of a specialized software complex were carried out. The adequacy of the model is checked in comparison with the available experimental data.On the basis of the research carried out in the article, recommendations can be developed for the choice of processing modes that provide the most rational parameters of the state of the surface layer of parts such as bodies of revolution.

Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures

We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ of radius $R+2r$ and the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping over the moving balls $\mathbf B_1,\dots,\mathbf B_n$. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius $R$ tends to infinity. We obtain a corresponding planar problem consisting of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and the same radius $r$ that are rolling without slipping over a fixed plane $\Sigma_0$, and a moving plane $\Sigma$ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler-Jacobi theorem.