Special Issue "Analytical and Numerical Methods for Linear and Nonlinear Analysis of Structures at Macro, Micro and Nano Scale"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: 31 December 2021.

Special Issue Editors

Dr. Krzysztof Kamil Żur
E-Mail Website
Guest Editor
Faculty of Mechanical Engineering, Bialystok University of Technology, 15-351 Bialystok, Poland
Interests: applied mathematics; linear and non-linear mechanics of composite structures at macro, micro, and nano scale; non-local continuum mechanics; smart materials and structures; composite materials
Special Issues and Collections in MDPI journals
Dr. Jinseok Kim
E-Mail Website
Guest Editor
Department of Mechanical and Aerospace Engineering, Western Michigan University, Kalamazoo, MI 49008-5343, USA
Interests: computational solid mechanics; finite element method; plates and shell theories; non-local mechanics; composite materials; smart materials
Prof. Dr. J. N. Reddy
grade E-Mail Website
Guest Editor
J. Mike Walker ’66 Department of Mechanical Engineering, Texas A & M University, College Station, TX 77843-3123, USA
Interests: linear and nonlinear finite element analysis; variational methods; plate and shell theories; composite structures; numerical heat transfer; computational fluid dynamics; non-local theories; applied mathematics

Special Issue Information

Dear Colleagues,

The mathematical models of physical phenomena are based on the fundamental scientific laws of physics.  Mathematical models consist of a combination of algebraic and differential (sometimes even integral) equations. Mathematical models of structural elements (e.g., beams, plates, and shells) based on continuum assumption require a proper treatment of the kinematic, kinetic, and constitutive issues accounting for possible sources of non-local and non-classical continuum mechanics concepts and solving associated boundary value problems. The development of mathematical models and their solutions by analytical and numerical methods have been the focus of many researchers. In particular, the mechanical response of ultrasmall structures has received a great deal of attention because of their wide applications in high-tech devices, such as nanoelectromechanical and microelectromechanical systems.

This special issue is aimed at collecting high-quality papers on the latest developments, techniques, and approaches for the modeling and simulation of the mechanical behavior of structures at macro, micro and nano scales. Advanced accurate numerical and analytical methods to solve PDEs are of high interest. The vibrational response, buckling instability, wave propagation analysis, and static deformation of macro, micro and nano scales’ structural components are covered in this special issue.

Dr. Krzysztof Kamil Żur
Dr. Jinseok Kim
Prof. Dr. J. N. Reddy
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • linear and nonlinear boundary/initial value problems
  • continuum mechanics
  • elasticity theories
  • micro/nanostructured systems
  • macroscale structures
  • mechanical response
  • discretization methods
  • numerical analysis
  • analytical methods
  • multifield loads
  • partial differential equations
  • verification and validation
  • fast numerical algorithms

Published Papers (3 papers)

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Research

Article
k-Version of Finite Element Method for BVPs and IVPs
Mathematics 2021, 9(12), 1333; https://doi.org/10.3390/math9121333 - 09 Jun 2021
Viewed by 225
Abstract
The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) [...] Read more.
The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work. Full article
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Article
Direct Collocation with Reproducing Kernel Approximation for Two-Phase Coupling System in a Porous Enclosure
Mathematics 2021, 9(8), 897; https://doi.org/10.3390/math9080897 - 17 Apr 2021
Viewed by 300
Abstract
The direct strong-form collocation method with reproducing kernel approximation is introduced to efficiently and effectively solve the natural convection problem within a parallelogrammic enclosure. As the convection behavior in the fluid-saturated porous media involves phase coupling, the resulting system is highly nonlinear in [...] Read more.
The direct strong-form collocation method with reproducing kernel approximation is introduced to efficiently and effectively solve the natural convection problem within a parallelogrammic enclosure. As the convection behavior in the fluid-saturated porous media involves phase coupling, the resulting system is highly nonlinear in nature. To this end, the local approximation is adopted in conjunction with Newton–Raphson method. Nevertheless, to unveil the performance of the method in the nonlinear analysis, only single thermal natural convection is of major concern herein. A unit square is designated as the model problem to investigate the parameters in the system related to the convergence; several benchmark problems are used to verify the accuracy of the approximation, in which the stability of the method is demonstrated by considering various initial conditions, disturbance of discretization, inclination, aspect ratio, and reproducing kernel support size. It is shown that a larger support size can be flexible in approximating highly irregular discretized problems. The derivation of explicit operators with two-phase variables in solving the nonlinear system using the direct collocation is carried out in detail. Full article
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Article
Trigonometric Solution for the Bending Analysis of Magneto-Electro-Elastic Strain Gradient Nonlocal Nanoplates in Hygro-Thermal Environment
Mathematics 2021, 9(5), 567; https://doi.org/10.3390/math9050567 - 07 Mar 2021
Cited by 2 | Viewed by 458
Abstract
Nanoplates have been extensively utilized in the recent years for applications in nanoengineering as sensors and actuators. Due to their operative nanoscale, the mechanical behavior of such structures might also be influenced by inter-atomic material interactions. For these reasons, nonlocal models are usually [...] Read more.
Nanoplates have been extensively utilized in the recent years for applications in nanoengineering as sensors and actuators. Due to their operative nanoscale, the mechanical behavior of such structures might also be influenced by inter-atomic material interactions. For these reasons, nonlocal models are usually introduced for studying their mechanical behavior. Sensor technology of plate structures should be formulated with coupled mechanics where elastic, magnetic and electric fields interact among themselves. In addition, the effect of hygro-thermal environments are also considered since their presence might effect the nanoplate behavior. In this work a trigonometric approach is developed for investigating smart composite nanoplates using a strain gradient nonlocal procedure. Convergence of the present method is also reported in terms of displacements and electro-magnetic potentials. Results agree well with the literature and open novel applications in this field for further developments. Full article
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