Article
Mathematics
Tatiana Shaposhnikova, Alexander Podolskiy
Summary: This paper studies the homogenization of the optimal control problem for the Dirichlet cost functional with the Poisson state equation specified in a bounded domain Omega. It considers rapidly alternating boundary conditions on a part of the boundary and sets Robin type boundary condition with a large coefficient on subsets of this part. Neumann boundary condition is set on the remaining part of the boundary. The critical case, characterized by the presence of a strange term, is examined.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2022)
Article
Mechanics
V. Monchiet, G. Bonnet
Summary: In this paper, the FFT method is extended to handle the homogenization problem of composite conductors with uniform boundary conditions. The method applies a transformation to build a periodic problem from the solution with uniform boundary conditions. The extended domain obtained by mirror symmetry of the unit cell is used to solve the conductivity equation under an applied periodic polarization field. The effectiveness of the proposed method is validated by comparing the effective conductivity obtained with FFT to finite element solutions. The method can be applied to various microstructure geometries, including cells obtained through imaging devices.
EUROPEAN JOURNAL OF MECHANICS A-SOLIDS
(2024)
Article
Mathematics, Applied
Mifodijus Sapagovas, Jurij Novickij
Summary: This paper considers the stability of the alternating direction method for wave equation with integral boundary conditions in an energy norm. The proof of stability is based on the properties of eigenvalues and eigenvectors of the corresponding difference operators. The main properties of the alternating direction method are theoretically and numerically proven.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Genglin Li, Youshan Tao
Summary: In this study, an inhomogeneous Neumann boundary value problem for a chemotaxis-convection system modeling tumor-related angiogenesis in the early phase is considered. It is shown that for any given suitably regular initial data, the corresponding inhomogeneous initial-boundary value problem possesses a unique classical solution that is global-in-time and uniformly bounded through homogenization and a priori estimates.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Engineering, Mechanical
Brian Painter, Marco Amabili
Summary: In this study, the non-planar vibration response of a beam with initial geometric imperfections is investigated using a geometrically nonlinear beam model. The experimental data shows good agreement with the beam model response around the resonant frequency of the first bending mode.
NONLINEAR DYNAMICS
(2023)
Article
Engineering, Multidisciplinary
Hannes Grimm-Strele, Matthias Kabel
Summary: The FFT-based homogenization method is established as a fast, accurate, and robust tool for periodic homogenization in solid mechanics. The implementation of nonperiodic boundary conditions efficiently reduces runtime and memory requirements compared to the domain mirroring approach. The use of periodic boundary conditions for nonperiodic geometries yields vastly different results than with PMUBC, and the influence of the discretization method on the results is examined.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2021)
Article
Chemistry, Physical
Arkadiusz Denisiewicz, Mieczyslaw Kuczma, Krzysztof Kula, Tomasz Socha
Summary: This study focuses on the computational modeling and laboratory testing of high-performance concrete (HPC) to enhance its properties. Different boundary conditions were used to analyze the influence on HPC properties, and a comparison with experimental data was conducted. The research also considered the damage behavior and microstructure of HPC, using numerical simulations and comparing the results with experimental data to achieve good agreement.
Article
Mathematics, Applied
Akambadath Keerthiyil Nandakumaran, Kasinathan Sankar
Summary: This article studies the homogenization of heat equations in a domain with randomly oscillating boundary parts. A new homogenization technique is utilized to deal with the evolving domains, and a corrector result is established.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics
A. Mouze, V Munnier
Summary: The study explores the optimal boundary behavior of log-frequently hypercyclic functions with respect to the Taylor shift on H(D) using average L-p-norms, and also examines the growth of frequently or log-frequently hypercyclic functions for the differentiation operator on H(C). These results highlight the similarities and differences between the lower and upper bounds on the growth of frequently and log-frequently hypercyclic functions under different operators on different function spaces.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics
Olga Stikoniene, Mifodijus Sapagovas
Summary: In this paper, we investigate the convergence of the Peaceman-Rachford alternating direction implicit method for approximating the two-dimensional elliptic equations in a rectangular domain with nonlocal integral conditions. The main goal is to analyze the spectrum structure of the difference eigenvalue problem with nonlocal conditions. Convergence of the iterative method is proven when the system of eigenvectors is complete. The main results are generalized for approximating the differential problem with truncation error O(h4).
MATHEMATICAL MODELLING AND ANALYSIS
(2023)
Article
Mechanics
Marek Wojciechowski
Summary: This paper investigates generalized boundary conditions for mesoscopic statistical volume elements (SVEs) in computational homogenization methods. The method combines both classical uniform kinematic constraining and uniform static loading of the SVE, resulting in apparent homogenized parameters. The method is implemented within the finite element framework and can be used for random composites without assumptions on the SVE geometry.
COMPOSITE STRUCTURES
(2022)
Article
Mechanics
Shizhen Yin, Marek-Jerzy Pindera
Summary: This study incorporates homogeneous traction and displacement boundary conditions into a hybrid homogenization theory for unidirectional composites with random fiber distributions. The study investigates the convergence of homogenized moduli and local stress field statistics with representative volume element size. The results show that fiber randomness and boundary condition type have significant effects on the statistics of local stress fields, while the choice of boundary conditions influences the location of plasticity and failure initiation at the fiber/matrix interfaces.
EUROPEAN JOURNAL OF MECHANICS A-SOLIDS
(2022)
Article
Mathematics, Interdisciplinary Applications
Felix Ernesti, Jonas Lendvai, Matti Schneider
Summary: This study investigates the influence of boundary conditions on the effective crack energy evaluated on microstructure cells and provides a different approach based on fast marching algorithms which allows a liberal choice of boundary conditions.
COMPUTATIONAL MECHANICS
(2023)
Article
Engineering, Civil
Lei Liu, Jie Feng, Xuanyu Mu, Qingqi Pei, Dapeng Lan, Ming Xiao
Summary: Vehicular Edge Computing (VEC) has gained research interest for its potential in reducing response delay and alleviating bandwidth pressure. However, effectively aggregating and scheduling network resources in VEC to meet diverse tasks and demands remains a challenge.
IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
(2023)
Article
Materials Science, Multidisciplinary
A. G. Kolpakov, S. Rakin
Summary: This study presents a procedure for reducing a three-dimensional problem to several two-dimensional problems for plates with a unidirectional system of inhomogeneities, demonstrating the existence of boundary layers that result in wrinkling of the top and bottom surfaces of the plate and influencing its strength and interaction with surrounding media.
MECHANICS OF ADVANCED MATERIALS AND STRUCTURES
(2022)
Article
Mathematics, Applied
D. Borisov, P. Exner
Summary: The research introduces a new method for controlling gaps in two-dimensional periodic systems, using dispersion curves of specific waveguide systems to control the widths of the gaps and attain corresponding band functions at internal points of the Brillouin zone.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2021)
Article
Mathematics, Applied
D. Borisov, A. M. Golovina
Summary: This paper studies a general elliptic operator in an infinite multi-dimensional cylinder with several distant perturbations, showing that under certain conditions, the resonances of this type of operator are finite in number and the leading terms in the asymptotic expansion of these resonances are exponentially small. The authors conjecture that this scenario is unique when distant perturbations produce only a finite number of resonances near a real number lambda(0).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics
Denis Borisov
Summary: The paper explores a general second order self-adjoint elliptic operator on an arbitrary metric graph with a glued small graph. By introducing a special operator and assuming no embedded eigenvalues, it is proven that the spectrum of the perturbed operator converges to that of the limiting operator, along with the convergence of the spectral projectors. Additionally, it is shown that the eigenvalues and eigenfunctions of the perturbed operator converging to limiting discrete eigenvalues are analytic in epsilon.
Article
Nanoscience & Nanotechnology
Giuseppe Cardone, Tiziana Durante
Summary: This article studies a second order elliptic operator in a planar waveguide, where small holes are distributed along a curve and subject to classical boundary conditions on the holes. Under the weak assumptions on the perforation, all possible homogenized problems are described.
NANOSYSTEMS-PHYSICS CHEMISTRY MATHEMATICS
(2022)
Article
Physics, Multidisciplinary
Miloslav Znojil, Denis I. Borisov
Summary: This paper demonstrates the different applications of Arnold's one-dimensional polynomial potentials in classical catastrophe theory and quantum mechanics, particularly within specific dynamical regimes. By relaxing constraints and utilizing perturbative methods, the characteristics of these potentials are investigated.
Article
Mathematics, Applied
Srinivasan Aiyappan, Giuseppe Cardone, Carmen Perugia, Ravi Prakash
Summary: This paper studies the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary, while the Dirichlet condition is considered on the smooth separate part. By using the unfolding method and making natural hypotheses on the regularity of the domain, we prove the weak L-p-convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2022)
Article
Mathematics
D. I. Borisov, P. Exner
Summary: In this paper, a new type of approximation for a second-order elliptic operator with a point interaction in a planar domain is presented. The approximation is of a geometric nature and consists of operators with the same symbol and regular coefficients on the domain with a small hole. The boundary condition is imposed at the boundary of the hole with a coefficient depending on the linear size of the hole. It is shown that as the hole shrinks to a point and the parameter in the boundary condition is scaled appropriately, the approximating family converges in the norm-resolvent sense to the operator with the point interaction. The convergence is established with respect to several operator norms and the convergence rates are estimated.
BULLETIN OF MATHEMATICAL SCIENCES
(2023)
Article
Mathematics, Applied
D. I. Borisov, A. L. Piatnitski, E. A. Zhizhina
Summary: This paper focuses on the spectral properties of a bounded self-adjoint operator in L-2(R-d) that is the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. The essential and discrete spectra of this operator are studied. The essential spectrum of the sum is shown to be the union of the essential spectrum of the convolution operator and the image of the potential. Sufficient conditions for the existence of discrete spectrum are provided, and lower and upper bounds for the number of discrete eigenvalues are obtained. The spectral properties of the operators considered in this work are compared with those of classical Schrodinger operators.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
G. Cardone, A. Fouetio, S. Talla Lando, J. L. Woukeng
Summary: This work investigates the global dynamics of 2D stochastic tidal equations in a highly heterogeneous environment. Using the stochastic version of the sigma-convergence method and the Prokhorov and Skorokhod compactness theorems, the paper proves that the dynamics at the macroscopic level are of the same type as those at the microscopic level, but with non-oscillating parameters. A corrector-type result is also established.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics, Applied
D. I. Borisov, J. Kriz
Summary: In this paper, we consider a second order linear elliptic equation in a finely perforated domain. The shapes and distribution of cavities in the domain are arbitrary and non-periodic, and the boundary conditions can be either Dirichlet or nonlinear Robin conditions. We show that under certain conditions, the solution to our problem tends to zero as the perforation becomes finer.
ANALYSIS AND MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
D. I. Borisov
Summary: In this paper, a boundary value problem for a general second-order linear equation in a perforated domain is considered. The perforation is made by small cavities with a minimal distance between them also being small. Minimal natural geometric conditions are imposed on the shapes of the cavities, while no conditions are imposed on their distribution in the domain. A nonlinear Robin condition is imposed on the boundaries of the cavities. The main results of the paper demonstrate the convergence of the solution of the perturbed problem to that of the homogenized one, providing estimates for the convergence rates in W-2(1)- and L-2-norms uniformly in the L-2-norm of the right-hand side in the equation.
APPLICABLE ANALYSIS
(2023)
Article
Mathematics
Denis Ivanovich Borisov
Summary: We introduce the concept of point interaction for general non-self-adjoint elliptic operators in planar domains. By cutting out a small cavity around the point, we show that these operators can be geometrically approximated. A special Robin-type boundary condition with a nonlocal term is imposed on the boundary of the cavity. As the cavity shrinks to a point, the perturbed operator converges in the norm resolvent sense to a limiting operator with a point interaction containing an arbitrary complex-valued coupling constant. We establish convergence rates for several operator norms. The convergence of the spectrum is proven as a corollary of the norm resolvent convergence.
Article
Mathematics, Applied
S. Aiyappan, Giuseppe Cardone, Carmen Perugia
Summary: In this study, we examine the asymptotic behavior of a linear optimal control problem posed on a locally periodic rapidly oscillating domain. The problem involves an L2-cost functional constrained by a Poisson problem with a mixed boundary condition: a homogeneous Neumann condition on the oscillating part of the boundary and a homogeneous Dirichlet condition on the remaining part.
APPLICABLE ANALYSIS
(2023)
Article
Mathematics
D. I. Borisov
Summary: We study the existence of a limiting operator and the convergence of resolvent norm for a general second order matrix operator subject to a classical boundary condition in a multi-dimensional domain, perturbed by a first order differential operator depending on a small multi-dimensional parameter. Our main results show that the convergence of resolvent norm is equivalent to the convergence of coefficients in the perturbing operator in certain spaces of multipliers. Furthermore, we find that the convergence in these spaces is equivalent to the convergence of local mean values over small pieces of the domain. Our results are supported by examples of non-periodically oscillating perturbations.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
D. I. Borisov, A. L. Piatnitski, E. A. Zhizhina
Summary: This article considers a multiplication operator in L2(R) multiplied by a complex-valued potential, to which a convolution operator multiplied by a small parameter is added. The essential spectrum of such an operator is found in an explicit form, and it is shown that the entire spectrum is located in a thin neighborhood of the spectrum of the multiplication operator.
