A regular variational boundary model for free vibrations of magneto-electro-elastic structures
TL;DR: In this article, a regular variational boundary element formulation for dynamic analysis of two-dimensional magneto-electro-elastic domains is presented, where the domain variables are approximated by using a superposition of weighted regular fundamental solutions of the static magnetoelectroelastic problem and the boundary variables are expressed in terms of nodal values.
Abstract: In this paper a regular variational boundary element formulation for dynamic analysis of two-dimensional magneto-electro-elastic domains is presented. The method is based on a hybrid variational principle expressed in terms of generalized magneto-electro-elastic variables. The domain variables are approximated by using a superposition of weighted regular fundamental solutions of the static magneto-electro-elastic problem, whereas the boundary variables are expressed in terms of nodal values. The variational principle coupled with the proposed discretization scheme leads to the calculation of frequency-independent and symmetric generalized stiffness and mass matrices. The generalized stiffness matrix is computed in terms of boundary integrals of regular kernels only. On the other hand, to achieve meaningful computational advantages, the domain integral defining the generalized mass matrix is reduced to the boundary through the use of the dual reciprocity method, although this implies the loss of symmetry. A purely boundary model is then obtained for the computation of the structural operators. The model can be directly used into standard assembly procedures for the analysis of non-homogeneous and layered structures. Additionally, the proposed approach presents some features that place it in the framework of the weak form meshless methods. Indeed, only a set of scattered points is actually needed for the variable interpolation, while a global background boundary mesh is only used for the integration of the influence coefficients. The results obtained show good agreement with those available in the literature proving the effectiveness of the proposed approach.
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TL;DR: In this article, the first order shear deformation theory considering the von Karman's nonlinear strains is used to obtain the equations of motion, whereas Maxwell equations for electrostatics and magnetostatics are used to model the electric and magnetic behavior.
100 citations
TL;DR: In this paper, the authors studied the nonlinear and linear free vibration of symmetrically laminated magneto-electro-elastic doubly-curved thin shell resting on an elastic foundation.
Abstract: Nonlinear and linear free vibration of symmetrically laminated magneto-electro-elastic doubly-curved thin shell resting on an elastic foundation is studied analytically The shell is considered to be simply-supported on all edges and the magneto-electro-elastic body is poled along the z direction and subjected to electric and magnetic potentials between the upper and lower surfaces To obtain the equations of motion, the Donnell shell theory in the presence of rotary inertia effect is used Moreover, Gauss' laws for electrostatics and magnetostatics are used to model the electric and magnetic behavior The nonlinear partial differential equations of motion are reduced to a single nonlinear ordinary differential equation by introducing a force function and using the single-term Galerkin method The resulting equation is solved analytically by Lindstedt-Poincare perturbation method After validation of the present study, several numerical studies are done to investigate the effects of foundation parameters, geometrical properties of the shell, and electric and magnetic potentials on the linear and nonlinear behavior of these smart shells
65 citations
TL;DR: In this article, a numerical model based on the dual reciprocity boundary element method (DRBEM) for studying the transient magneto-thermo-viscoelastic stresses in a nonhomogeneous anisotropic solid subjected to a heat source is presented.
Abstract: A numerical model based on the dual reciprocity boundary element method (DRBEM) for studying the transient magneto-thermo-viscoelastic stresses in a nonhomogeneous anisotropic solid subjected to a heat source is presented. The formulation is tested through its application to the problem of a solid placed in a constant primary magnetic field acting in the direction of the z-axis and rotating about this axis with a constant angular velocity. In the case of plane deformation, a numerical scheme for the implementation of the method is presented and the numerical computations are carried out for the temperature, displacement components and stress components. The validity of DRBEM is examined by considering a magneto-thermo-viscoelastic solid occupies a rectangular region and good agreement is obtained with existent results. The results obtained are presented graphically to show the effects of inhomogeneity and heat source on the temperature, displacement components and thermal stress components.
58 citations
TL;DR: The present study is in no ways exhaustive to the methods and results observed, but it may be considered as a guide to researchers and scholars about the behavior of MEE materials, wherein critical observations and analyses’ techniques are discussed.
Abstract: Magneto-electro-elastic (MEE) materials have been receiving a special attention from the research community owing to their specialized performance and coupled behavior under thermal, electric, magnetic and mechanical loads. The possibility of prospective energy conversion means, have additionally been added to the cause of researching about these materials. Therefore, the review presented here may be considered as a topical discussion on MEE materials and structures. Through this paper, all critical concepts revolving around MEE materials are discussed in separate sections ranging from the very definition of MEE materials, their material phenomenon, types and properties, to certain fundamental theories and micromechanical models, structural analyses of MEE structures and their nano-sized counterparts, effects of various external and internal parameters and prospective applications of these materials. The present study is in no ways exhaustive to the methods and results observed, but it may be considered as a guide to researchers and scholars about the behavior of MEE materials, wherein critical observations and analyses’ techniques are discussed.
56 citations
TL;DR: In this article, the transient magneto-thermo-visco-elastic stresses in a nonhomogeneous anisotropic solid placed in a constant primary magnetic field acting in the direction of the z-axis and rotating about it with a constant angular velocity were studied.
Abstract: The main objective of this paper is to study the transient magneto-thermo-visco-elastic stresses in a non-homogeneous anisotropic solid placed in a constant primary magnetic field acting in the direction of the z -axis and rotating about it with a constant angular velocity. The system of fundamental equations is solved by means of a dual-reciprocity boundary element method (DRBEM). In the case of plane deformation, a numerical scheme for the implementation of the method is presented and the numerical computations are carried out for the temperature, displacement components and thermal stress components. The validity of DRBEM is examined by considering a magneto-thermo-visco-elastic solid occupies a rectangular region and good agreement is obtained with the results obtained by other methods. The results obtained are presented graphically to show the effect of inhomogeneity on the displacement components and thermal stress components.
54 citations
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29 Jul 2002TL;DR: In this paper, Galerkin et al. defined mesh-free methods for shape function construction, including the use of mesh-less local Petrov-Galerkin methods.
Abstract: Preliminaries Physical Problems in Engineering Solid Mechanics: A Fundamental Engineering Problem Numerical Techniques: Practical Solution Tools Defining Meshfree Methods Need for Meshfree Methods The Ideas of Meshfree Methods Basic Techniques for Meshfree Methods Outline of the Book Some Notations and Default Conventions Remarks Meshfree Shape Function Construction Basic Issues for Shape Function Construction Smoothed Particle Hydrodynamics Approach Reproducing Kernel Particle Method Moving Least Squares Approximation Point Interpolation Method Radial PIM Radial PIM with Polynomial Reproduction Weighted Least Square (WLS) Approximation Polynomial PIM with Rotational Coordinate Transformation Comparison Study via Examples Compatibility Issues: An Analysis Other Methods Function Spaces for Meshfree Methods Function Spaces Useful Spaces in Weak Formulation G Spaces: Definition G1h Spaces: Basic Properties Error Estimation Concluding Remarks Strain Field Construction Why Construct Strain Field? Historical Notes How to Construct? Admissible Conditions for Constructed Strain Fields Strain Construction Techniques Concluding Remarks Weak and Weakened Weak Formulations Introduction to Strong and Weak Forms Weighted Residual Method A Weak Formulation: Galerkin A Weakened Weak Formulation: GS-Galerkin The Hu-Washizu Principle The Hellinger-Reissner Principle The Modified Hellinger-Reissner Principle Single-Field Hellinger-Reissner Principle The Principle of Minimum Complementary Energy The Principle of Minimum Potential Energy Hamilton's Principle Hamilton's Principle with Constraints Galerkin Weak Form Galerkin Weak Form with Constraints A Weakened Weak Formulation: SC-Galerkin Parameterized Mixed Weak Form Concluding Remarks Element Free Galerkin Method EFG Formulation with Lagrange Multipliers EFG with Penalty Method Summary Meshless Local Petrov-Galerkin Method MLPG Formulation MLPG for Dynamic Problems Concluding Remarks Point Interpolation Methods Node-Based Smoothed Point Interpolation Method (NS-PIM) NS-PIM Using Radial Basis Functions (NS-RPIM) Upper Bound Properties of NS-PIM and NS-RPIM Edge-Based Smoothed Point Interpolation Methods (ES-PIMs) A Combined ES/NS Point Interpolation Methods (ES/NS-PIM) Strain-Constructed Point Interpolation Method (SC-PIM) A Comparison Study Summary Meshfree Methods for Fluid Dynamics Problem Introduction Navier-Stokes Equations Smoothed Particle Hydrodynamics Method Gradient Smoothing Method (GSM) Adaptive Gradient Smoothing Method (A-GSM) A Discussion on GSM for Incompressible Flows Other Improvements on GSM Meshfree Methods for Beams PIM Shape Function for Thin Beams Strong Form Equations Weak Formulation: Galerkin Formulation A Weakened Weak Formulation: GS-Galerkin Three Models Formulation for NS-PIM for Thin Beams Formulation for Dynamic Problems Numerical Examples for Static Analysis Numerical Examples: Upper Bound Solution Numerical Examples for Free Vibration Analysis Concluding Remarks Meshfree Methods for Plates Mechanics for Plates EFG Method for Thin Plates EFG Method for Thin Composite Laminates EFG Method for Thick Plates ES-PIM for Plates Meshfree Methods for Shells EFG Method for Spatial Thin Shells EFG Method for Thick Shells ES-PIM for Thick Shells Summary Boundary Meshfree Methods RPIM Using Polynomial Basis RPIM Using Radial Function Basis Remarks Meshfree Methods Coupled with Other Methods Coupled EFG/BEM Coupled EFG and Hybrid BEM Remarks Meshfree Methods for Adaptive Analysis Triangular Mesh and Integration Cells Node Numbering: A Simple Approach Bucket Algorithm for Node Searching Relay Model for Domains with Irregular Boundaries Techniques for Adaptive Analysis Concluding Remarks MFree2D(c) Overview Techniques Used in MFree2D Preprocessing in MFree2D Postprocessing in MFree2D Index References appear at the end of each chapter.
1,768 citations
Book•
01 Jan 1991
TL;DR: In this article, the authors present the boundary element method for Laplace's Equation, which is used to solve the Torsion Problem with different approximations of functions.
Abstract: 1 Introduction.- 2 The Boundary Element Method for Equations ?2u = 0 and ?2u = b.- 2.1 Introduction.- 2.2 The Case of the Laplace Equation.- 2.2.1 Fundamental Relationships.- 2.2.2 Boundary Integral Equations.- 2.2.3 The Boundary Element Method for Laplace's Equation.- 2.2.4 Evaluation of Integrals.- 2.2.5 Linear Elements.- 2.2.6 Treatment of Corners.- 2.2.7 Quadratic and Higher-Order Elements.- 2.3 Formulation for the Poisson Equation.- 2.3.1 Basic Relationships.- 2.3.2 Cell Integration Approach.- 2.3.3 The Monte Carlo Method.- 2.3.4 The Use of Particular Solutions.- 2.3.5 The Galerkin Vector Approach.- 2.3.6 The Multiple Reciprocity Method.- 2.4 Computer Program 1.- 2.4.1 MAINP1.- 2.4.2 Subroutine INPUT1.- 2.4.3 Subroutine ASSEM2.- 2.4.4 Subroutine NECMOD.- 2.4.5 Subroutine SOLVER.- 2.4.6 Subroutine INTERM.- 2.4.7 Subroutine OUTPUT.- 2.4.8 Results of a Test Problem.- 2.5 References.- 3 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y).- 3.1 Equation Development.- 3.1.1 Preliminary Considerations.- 3.1.2 Mathematical Development of the DRM for the Poisson Equation.- 3.2 Different f Expansions.- 3.2.1 Case f = r.- 3.2.2 Case f = 1+ r.- 3.2.3 Case f = 1 at One Node and f = r at Remaining Nodes.- 3.3 Computer Implementation.- 3.3.1 Schematized Matrix Equations.- 3.3.2 Sign of the Components of r and its Derivatives.- 3.4 Computer Program 2.- 3.4.1 MAINP2.- 3.4.2 Subroutine INPUT2.- 3.4.3 Subroutine ALFAF2.- 3.4.4 Subroutine RHSVEC.- 3.4.5 Comparison of Results for a Torsion Problem using Different Approximating Functions.- 3.4.6 Data and Output for Program 2.- 3.5 Results for Different Functions b = b(x,y).- 3.5.1 The Case ?2u = ?x.- 3.5.2 The Case ?2u = ?x2.- 3.5.3 The Case ?2u = a2 ? x2.- 3.5.4 Results using Quadratic Elements.- 3.6 Problems with Different Domain Integrals on Different Regions.- 3.6.1 The Subregion Technique.- 3.6.2 Integration over Internal Region.- 3.7 References.- 4 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u).- 4.1 Introduction.- 4.2 The Convective Case.- 4.2.1 Results for the Case ?2u = ??u/?x.- 4.2.2 Results for the Case ?2u = ?(?u/?x+ ?u/?y).- 4.2.3 Internal Derivatives of the Problem Variables.- 4.3 The Helmholtz Equation.- 4.3.1 DRM Formulations.- 4.3.2 DRM Results for Vibrating Beam.- 4.3.3 Results for Non-Inversion DRM.- 4.4 Non-Linear Cases.- 4.4.1 Burger's Equation.- 4.4.2 Spontaneous Ignition: The Steady-State Case.- 4.4.3 Non-Linear Material Problems.- 4.5 Computer Program 3.- 4.5.1 MAINP3.- 4.5.2 Subroutine ALFAF3.- 4.5.3 Subroutine RHSMAT.- 4.5.4 Subroutine DERIVXY.- 4.5.5 Results of Test Problems.- 4.6 Three-Dimensional Analysis.- 4.6.1 Equations of the Type ?2u = b(x, y, z).- 4.6.2 Equations of the Type ?2u = b(x, y, z, u).- 4.7 References.- 5 The Dual Reciprocity Method for Equations of the Type ?2u = b(x, y, u, t).- 5.1 Introduction.- 5.2 The Diffusion Equation.- 5.3 Computer Program 4.- 5.3.1 MAINP4.- 5.3.2 Subroutine ASSEMB.- 5.3.3 Subroutine VECTIN.- 5.3.4 Subroutine BOUNDC.- 5.3.5 Results of a Test Problem.- 5.3.6 Data Input.- 5.3.7 Computer Output.- 5.3.8 Further Applications.- 5.3.9 Other Time-Stepping Schemes.- 5.4 Special f Expansions.- 5.4.1 Axisymmetric Diffusion.- 5.4.2 Infinite Regions.- 5.5 The Wave Equation.- 5.5.1 Infinite and Semi-Infinite Regions.- 5.6 The Transient Convection-Diffusion Equation.- 5.7 Non-Linear Problems.- 5.7.1 Non-Linear Materials.- 5.7.2 Non-Linear Boundary Conditions.- 5.7.3 Spontaneous Ignition: Transient Case.- 5.8 References.- 6 Other Fundamental Solutions.- 6.1 Introduction.- 6.2 Two-Dimensional Elasticity.- 6.2.1 Static Analysis.- 6.2.2 Treatment of Body Forces.- 6.2.3 Dynamic Analysis.- 6.3 Plate Bending.- 6.4 Three-Dimensional Elasticity.- 6.4.1 Computational Formulation.- 6.4.2 Gravitational Load.- 6.4.3 Centrifugal Load.- 6.4.4 Thermal Load.- 6.5 Transient Convection-Diffusion.- 6.6 References.- 7 Conclusions.- Appendix 1.- Appendix 2.- The Authors.
1,010 citations
TL;DR: In this paper, a review of the magnetoelectric (ME) effect in single phase and composite materials is presented, where the authors mainly emphasize their investigations of ME particulate composites and laminate composites, and summarize the important results.
Abstract: In the past few decades, extensive research has been conducted on the magnetoelectric (ME) effect in single phase and composite materials. Dielectric polarization of a material under a magnetic field or an induced magnetization under an electric field requires the simultaneous presence of long-range ordering of magnetic moments and electric dipoles. Single phase materials suffer from the drawback that the ME effect is considerably weak even at low temperatures, limiting their applicability in practical devices. Better alternatives are ME composites that have large magnitudes of the ME voltage coefficient. The composites exploit the product property of the materials. The ME effect can be realized using composites consisting of individual piezomagnetic and piezoelectric phases or individual magnetostrictive and piezoelectric phases. In the past few years, our group has done extensive research on ME materials for magnetic field sensing applications and current measurement probes for high-power electric transmission systems. In this review article, we mainly emphasize our investigations of ME particulate composites and laminate composites and summarize the important results. The data reported in the literature are also compared for clarity. Based on these results, we establish the fact that magnetoelectric laminate composites (MLCs) made from the giant magnetostrictive material, Terfenol-D, and relaxor-based piezocrystals are far superior to the other contenders. The large ME voltage coefficient in MLCs was obtained because of the high piezoelectric voltage coefficient of the piezocrystals and large elastic compliances. In addition, an optimized thickness ratio between the piezoelectric and magnetostrictive phases and the direction of the magnetostriction also influence the magnitude of the ME coefficient.
647 citations
21 Jan 2009
586 citations