Hasil untuk "Analytic mechanics"

S. Nemat-Nasser, H. Horii

M. Mamajonov

In this paper a boundary value problem for a fourth-order equation of parabolic-hyperbolic type within a pentagonal domain was investigated. In the equation under consideration, one characteristic aligns with the Ox axis while the other aligns with the Oy axis. Initially, the problem was examined within the lower triangle of the specified domain. Utilizing a differential equation solution construction method, a solution to the formulated problem was derived. Subsequently, within the rectangles of the domain, employing the continuation method, two relationships between the solution’s traces were established. Moreover, from the parabolic segment of the domain, two additional relationships between unknown traces will be derived. Solving this system of four equations enables determination of these traces, thereby resolving the problem.

Alexandre Epalle, Isabelle Ramière, Guillaume Latu et al.

Parallel implementation of numerical adaptive mesh refinement (AMR)strategies for solving 3D elastostatic contact mechanics problems is an essential step toward complex simulations that exceed current performance levels. This paper introduces a scalable, robust, and efficient algorithm to deal with 2D and 3D elastostatics contact problems between deformable bodies in a finite element framework. The proposed solution combines a treatment of the contact problem by a node-to-node pairing algorithm with a penalization technique and a non-conforming h-adaptive refinement of quadrilateral/hexahedral meshes based on an estimate-mark-refine approach in a parallel framework. One of the special features of our parallel strategy is that contact paired nodes are hosted by the same MPI tasks, which reduces the number of exchanges between processes for building the contact operator. The mesh partitioning introduced in this paper respects this rule and is based on an equidistribution of elements over processes, without any other constraints. In order to preserve the domain curvature while hierarchical mesh refinement, super-parametric elements are used. This functionality enables the contact zone to be well detected during the AMR process, even for an initial coarse mesh and low-order discretization schemes. The efficiency of our contact-AMR-HPC strategy is assessed on 2D and 3D Hertzian contact problems. Different AMR detection criteria are considered. Various convergence analyses are conducted. Parallel performances up to 1024 cores are illustrated. Furthermore, memory footprint and preconditionners performance are analyzed.

K. Brentner, F. Farassat

José F. Fernando

Let $X\subset{\mathbb R}^n$ be a (global) real analytic surface. Then every positive semidefinite meromorphic function on $X$ is a sum of $10$ squares of meromorphic functions on $X$. As a consequence, we provide a real Nullstellensatz for (global) real analytic surfaces.

Peidong Wu, E. Giessen

B. Voight

Xueling Cheng, Qilong Li, Hongyan Chen et al.

To accurately calculate the turbulent exchange coefficient, the contribution of multi-scale turbulent transportation needs to be considered, especially in the complex terrain of the coastal area. In September 2019, a comprehensive observation experiment on the offshore atmospheric boundary layer was carried out at the Yangmeikeng Ecological Monitoring Station and Sai Chung Gulf. Through scale decomposition, it is shown that the turbulent motion in the atmospheric boundary layer in the coastal area is affected by the underlying surface, such as that of the coastal land or the sea–land boundary. This is the main reason behind the phenomenon whereby different scales make different contributions to momentum flux. Different multi-scale characteristics of turbulent structures on the underlying surface affect the drag coefficient. Through wavelet transform and finite element method, the characteristics of the multi-scale flow structures produced by the complicated offshore terrain are analysed. It is found that large-scale flow structures enhance the pulsating intensity at the small scale, but the large-scale coherence characteristics are different from those at the small scale. In summary, in comparing these three sites, the flux exchange on the roof is greatest, followed by that on the tower. In the Gulf, the flux exchange is mainly dependent on small-scale structures, which are linked with the smallest values.

N.D. Markhabatov

In this paper, we study the rank characteristics for families of cubic theories, as well as new properties of cubic theories as pseudofiniteness and smooth approximability. It is proved that in the family of cubic theories, any theory is a theory of finite structure or is approximated by theories of finite structures. The property of pseudofiniteness or smoothly approximability allows one to investigate finite objects instead of complex infinite ones, or vice versa, to produce more complex ones from simple structures.

Matthias Aschenbrenner, Lou van den Dries

We show that maximal analytic Hardy fields are $η_1$ in the sense of Hausdorff. We also prove various embedding theorems about analytic Hardy fields. For example, the ordered differential field $\mathbb T$ of transseries is shown to be isomorphic to an analytic Hardy field.

S. Shaimardan, N.S. Tokmagambetov, Y. Aikyn

A large number of the most diverse problems related to almost all the most important branches of mathematical physics and designed to answer topical technical questions are associated with the use of Bessel functions. This paper introduces a h-difference equation analogue of the Bessel differential equation and investigates the properties of its solution, which is express using the Frobenius method by assuming a generalized power series. The authors find discrete analogue formulas for Bessel function and the h-Neumann function and these are solutions presented by a series with the h-fractional function th(α). Lastly they obtain the linear dependencies between h-functions Bessel on Ta .

A. Myrzagaliyeva

This work establishes necessary and sufficient conditions for the boundedness of one variable differential operator acting from a weighted Sobolev space W^l_p,v to a weighted Lebesgue space on the positive real half line. The coefficients of differential operators are often assumed to be pointwise multipliers of function spaces. The author introduces pointwise multipliers in weighted Sobolev spaces; obtains the description of the space of multipliers M(W_1 → W_2) for a pair of weighted Sobolev spaces (W_1, W_2) with weights of general type.

W. Schiehlen

C. M. Portela, J. Greer, D. Kochmann

Three-dimensional (3D), lattice-based micro- and nano-architected materials can possess desirable mechanical properties that are unattainable by homogeneous materials. Manufacturing these so-called structural metamaterials at the nano- and microscales typically results in non-slender architectures (e.g., struts with a high radius-to-length ratio r∕l), for which simple analytical and computational tools are inapplicable since they fail to capture the effects of nodes at strut junctions. We report a detailed analysis that quantifies the effect of nodes on the effective Young’s modulus (E∗) of lattice architectures with different unit cell geometries through (i) simple analytical constructions, (ii) reduced-order computational models, and (iii) experiments at the milli- and micrometer scales. The computational models of variable-node lattice architectures match the effective stiffness obtained from experiments and incur computational cost that are three orders-of-magnitude lower than alternative, conventional methods. We highlight a difference in the contribution of nodes to rigid versus non-rigid architectures and propose an extension to the classical stiffness scaling laws of the form E∗∝C_1(r∕l)α+C_2(r∕l)^β, which holds for slender and non-slender beam-based architectures, where constants C_1 and C_2 change with lattice geometry. We find the optimal scaling exponents for rigid architectures to be α=2 and β=4, and α=4 and β=6 for non-rigid architectures. These analytical, computational, and experimental results quantify the specific contribution of nodes to the effective stiffness of beam-based architectures and highlight the necessity of incorporating their effects into calculations of the structural stiffness. This work provides new, efficient tools that accurately capture the mechanics and physics of strut junctions in 3D, beam-based architected materials.

P. Goldstein, R. DeSalle

Pablo Ouro, Maxime Lazennec

Vertical axis turbine (VAT) arrays can achieve larger power generation per land area than their horizontal axis counterparts, due to the positive synergy from clustering VATs in close proximity. The VATs generate a three-dimensional wake that evolves unevenly over the vertical and transverse directions according to two governing length scales, namely the rotor's diameter and height. Theoretical wake models need to capture such a complex wake dynamics to enable reliable array design that maximises energy output. This paper presents two new theoretical VAT wake models based on super-Gaussian and Gaussian shape functions, which account for the three-dimensional velocity deficit distribution in the wake. The super-Gaussian model represents the initial elliptical shape with the superposition of vertical and lateral shape functions that progressively converge into an axisymmetric circular-shaped wake at a downstream distance that depends on the rotor's height-to-diameter aspect ratio. Our Gaussian model improves the initial wake width prediction taking into account the rectangular rotor's cross-section. Our models were well validated with large-eddy simulations (LES) of single VATs with varying aspect ratios and thrust coefficients operating in an atmospheric boundary layer. The super-Gaussian model attained a good agreement with LES in both near and far wake, whilst the Gaussian model represented well the far-wake region.

A.S. Berdyshev, A.R. Ryskan

In this work, the solvability of the problem with Neumann and Dirichlet boundary conditions for the Gellerstedt equation in four variables is investigated. The energy integral method is used to prove the uniqueness of the solution to the problem. In addition to it, formulas for differentiation, autotransformation, and decomposition of hypergeometric functions are applied. The solution is obtained explicitly and is expressed by Lauricella’s hypergeometric function.

A.R. Yeshkeyev, A.K. Issayeva, N.K. Shamatayeva

This article discusses the properties of atomic and prime models obtained with the some closure operator given on definable subsets of the semantic model some fixed Jonsson theory. The main result is to obtain the equivalence of the thus defined atomic and prime models, and this coincidence follows the assumption that there is some model with nice-defined properties.

O.A. Tarasova, A.V. Vasilyev, V.B. Vasilyev

We consider discrete analogue for simplest boundary value problem for elliptic pseudo-differential equation in a half-space with Dirichlet boundary condition in Sobolev-Slobodetskii spaces. Based on the theory of discrete boundary value problems for elliptic pseudo-differential equations we give a comparison between discrete and continuous solutions for certain model boundary value problem.

O.N. Stanzhytskyi, A.T. Assanova, M.A. Mukash

The averaging method, originally offered by Krylov and Bogolyubov for ordinary differential equations, is one of the most widespread and effective methods for the analysis of nonlinear dynamical systems. Further, the averaging method was developed and applied for investigating of various problems. Impulsive systems of differential equations supply as mathematical models of objects that, during their evolution, they are subjected to the action of short-term forces. Many researches have been devoted to non-fixed impulse problems. For these problems, the existence, stability, and other asymptotic properties of solutions were studied and boundary value problems for impulsive systems were considered. Questions of the existence of periodic and almost periodic solutions to impulsive systems also were examined. In this paper, the averaging method is used to study the existence of two-sided solutions bounding on the axis of impulse systems of differential equations with non-fixed times. It is shown that a one-sided, bounding, asymptotically stable solution to the averaged system generates a two-sided solution to the exact system. The closeness of the corresponding solutions of the exact and averaged systems both on finite and infinite time intervals is substantiated by the first and second theorems of N.N. Bogolyubov.