The smooth topology optimization for bi-dimensional functionally graded structures using level set-based radial basis functions

Wonsik Jung,Thanh T. Banh,Nam G. Luu,Dongkyu Lee 국제구조공학회 2023 Steel and Composite Structures, An International J Vol.47 No.5

This paper proposes an efficient approach for the structural topology optimization of bi-directional functionally graded structures by incorporating popular radial basis functions (RBFs) into an implicit level set (ILS) method. Compared to traditional element density-based methods, a level set (LS) description of material boundaries produces a smoother boundary description of the design. The paper develops RBF implicit modeling with multiquadric (MQ) splines, thin-plate spline (TPS), exponential spline (ES), and Gaussians (GS) to define the ILS function with high accuracy and smoothness. The optimization problem is formulated by considering RBF-based nodal densities as design variables and minimizing the compliance objective function. A LS-RBF optimization method is proposed to transform a Hamilton-Jacobi partial differential equation (PDE) into a system of coupled non-linear ordinary differential equations (ODEs) over the entire design domain using a collocation formulation of the method of lines design variables. The paper presents detailed mathematical expressions for BiDFG beams topology optimization with two different material models: continuum functionally graded (CFG) and mechanical functionally graded (MFG). Several numerical examples are presented to verify the method's efficiency, reliability, and success in accuracy, convergence speed, and insensitivity to initial designs in the topology optimization of two-dimensional (2D) structures. Overall, the paper presents a novel and efficient approach to topology optimization that can handle bi-directional functionally graded structures with complex geometries.

Dynamic analysis of porous functionally graded layered deep beams with viscoelastic core

Amr Assie,Şeref D. Akbaş,Abdallah M. Kabeel,Alaa A. Abdelrahman,Mohamed A Eltaher 국제구조공학회 2022 Steel and Composite Structures, An International J Vol.43 No.1

In this study, the dynamic behavior of functionally graded layered deep beams with viscoelastic core is investigated including the porosity effect. The material properties of functionally graded layers are assumed to vary continuously through thickness direction according to the power-law function. To investigate porosity effect in functionally graded layers, three different distribution models are considered. The viscoelastically cored deep beam is exposed to harmonic sinusoidal load. The composite beam is modeled based on plane stress assumption. The dynamic equations of motion of the composite beam are derived based on the Hamilton principle. Within the framework of the finite element method (FEM), 2D twelve –node plane element is exploited to discretize the space domain. The discretized finite element model is solved using the Newmark average acceleration technique. The validity of the developed procedure is demonstrated by comparing the obtained results and good agreement is detected. Parametric studies are conducted to demonstrate the applicability of the developed methodology to study and analyze the dynamic response of viscoelastically cored porous functionally graded deep beams. Effects of viscoelastic parameter, porosity parameter, graduation index on the dynamic behavior of porous functionally graded deep beams with viscoelastic core are investigated and discussed. Material damping and porosity have a significant effect on the forced vibration response under harmonic excitation force. Increasing the material viscosity parameters results in decreasing the vibrational amplitudes and increasing the vibration time period due to increasing damping effect. Obtained results are supportive for the design and manufacturing of such type of composite beam structures.

Wrinkling of a homogeneous thin solid film deposited on a functionally graded substrate

Masoud Noroozi 국제구조공학회 2020 Structural Engineering and Mechanics, An Int'l Jou Vol.74 No.2

Thin films easily wrinkle under compressive loading due to their small bending stiffness resulting from their tiny thickness. For a thin film deposited on a functionally graded substrate with non–uniform stiffness exponentially changes along the length span in this paper, the uniaxial wrinkling problem is solved analytically in terms of hyper–Bessel functions. For infinite, semi–infinite and finite length systems the wrinkling load and wrinkling wavenumber are determined and compared with those in literature. In comparison with a homogeneous substrate–bounded film in which the wrinkling pattern is uniform along the length span, for a functionally graded substrate–film system the wrinkles accumulate around the softer location of the functionally graded substrate. Therefore, the effective length of the film influenced by the wrinkles decreases, the amplitude of the wrinkles on softer regions of the functionally graded substrate grows and the wrinkling load of the functionally graded substrates with higher softening rate decreases more. The results of the current research are expected to be useful in science and technology of thin films and wrinkling of the structures especially living tissues.

Graded Lie superalgebras and super-replicable functions

Kang, Seok-Jin,Kim, Chang Heon,Koo, Ja Kyung,Oh, Young-Tak Elsevier 2005 Journal of algebra Vol.285 No.2

<P><B>Abstract</B></P><P>In this paper, we investigate the properties of super-replicable functions and their connections with graded Lie superalgebras. The Euler–Poincaré principle for the homology of graded Lie superalgebras yields a certain product identity called the generalized denominator identity. Applying the formal directional derivatives and the Laplacian, we derive a recursive supertrace formula for the graded Lie superalgebras with gradation-preserving endomorphisms. On the other hand, the super-replicable functions are characterized by certain product identities that have the same form as the generalized denominator identities for some graded Lie superalgebras. We derive many interesting relations among the Fourier coefficients of super-replicable functions and their super-replicates, and compute the supertraces of Monstrous Lie superalgebras associated with super-replicable functions. Finally, we study the properties of the hauptmoduln <SUB>J1,N</SUB> of <SUB>Γ1</SUB>(N) for N=5,8,10,12, which are super-replicable functions, and determine the Fourier coefficients of their super-replicates J1,N(m) (m⩾1).</P>

Hygro-thermal post-buckling analysis of a functionally graded beam

Akbas, Seref D. Techno-Press 2019 Coupled systems mechanics Vol.8 No.5

This paper presents post-buckling analysis of a functionally graded beam under hygro-thermal effect. The material properties of the beam change though height axis with a power-law function. In the nonlinear kinematics of the post-buckling problem, the total Lagrangian approach is used. In the solution of the problem, the finite element method is used within plane solid continua. In the nonlinear solution, the Newton-Raphson method is used with incremental displacements. Comparison studies are performed. In the numerical results, the effects of the material distribution, the geometry parameters, the temperature and the moisture changes on the post-buckling responses of the functionally graded beam are presented and discussed.

Geometrically nonlinear dynamic analysis of FG graphene platelets-reinforced nanocomposite cylinder: MLPG method based on a modified nonlinear micromechanical model

Mohammad Hossein Ghadiri Rad,Farzad Shahabian,Seyed Mahmoud Hosseini 국제구조공학회 2020 Steel and Composite Structures, An International J Vol.35 No.1

The present paper outlined a procedure for geometrically nonlinear dynamic analysis of functionally graded graphene platelets-reinforced (GPLR-FG) nanocomposite cylinder subjected to mechanical shock loading. The governing equation of motion for large deformation problems is derived using meshless local Petrov-Galerkin (MLPG) method based on total lagrangian approach. In the MLPG method, the radial point interpolation technique is employed to construct the shape functions. A micromechanical model based on the Halpin-Tsai model and rule of mixture is used for formulation the nonlinear functionally graded distribution of GPLs in polymer matrix of composites. Energy dissipation in analyses of the structure responding to dynamic loads is considered using the Rayleigh damping. The Newmark-Newton/Raphson method which is an incremental-iterative approach is implemented to solve the nonlinear dynamic equations. The results of the proposed method for homogenous material are compared with the finite element ones. A very good agreement is achieved between the MLPG and FEM with very fine meshing. In addition, the results have demonstrated that the MLPG method is more effective method compared with the FEM for very large deformation problems due to avoiding mesh distortion issues. Finally, the effect of GPLs distribution on strength, stiffness and dynamic characteristics of the cylinder are discussed in details. The obtained results show that the distribution of GPLs changed the mechanical properties, so a classification of different types and volume fraction exponent is established. Indeed by comparing the obtained results, the best compromise of nanocomposite cylinder is determined in terms of mechanical and dynamic properties for different load patterns. All these applications have shown that the present MLPG method is very effective for geometrically nonlinear analyses of GPLR-FG nanocomposite cylinder because of vanishing mesh distortion issue in large deformation problems. In addition, since in proposed method the distributed nodes are used for discretization the problem domain (rather than the meshing), modeling the functionally graded media yields to more accurate results.

A simple analytical approach for thermal buckling of thick functionally graded sandwich plates

El-Haina, Fouzia,Bakora, Ahmed,Bousahla, Abdelmoumen Anis,Tounsi, Abdelouahed,Mahmoud, S.R. Techno-Press 2017 Structural Engineering and Mechanics, An Int'l Jou Vol.63 No.5

This study aimed to presents a simple analytical approach to investigate the thermal buckling behavior of thick functionally graded sandwich by employing both the sinusoidal shear deformation theory and stress function. The material properties of the sandwich plate faces are continuously varied within the plate thickness according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. The thermal loads are considered as uniform, linear and non-linear temperature rises across the thickness direction. Numerical examples are presented to prove the effect of power law index, loading type and functionally graded layers thickness on the thermal buckling response of thick functionally graded sandwich.

A simple analytical approach for thermal buckling of thick functionally graded sandwich plates

Fouzia El-Haina,Ahmed Bakora,Abdelmoumen Anis Bousahla,Abdelouahed Tounsi,S. R. Mahmoud 국제구조공학회 2017 Structural Engineering and Mechanics, An Int'l Jou Vol.63 No.5

This study aimed to presents a simple analytical approach to investigate the thermal buckling behavior of thick functionally graded sandwich by employing both the sinusoidal shear deformation theory and stress function. The material properties of the sandwich plate faces are continuously varied within the plate thickness according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. The thermal loads are considered as uniform, linear and non-linear temperature rises across the thickness direction. Numerical examples are presented to prove the effect of power law index, loading type and functionally graded layers thickness on the thermal buckling response of thick functionally graded sandwich.

A General Dry Density Law for Sands

Emoke Imre,Janos Lorincz,Q. Phong Trang,Stephen Fityus,Jozsef Pusztai,Gabor Telekes,Tom Schanz 대한토목학회 2009 KSCE Journal of Civil Engineering Vol.13 No.4

The direct interpolation of a transfer function needs exponentially many data in terms of the number of the fractions in the grading curve. The suggested transfer function construction method - based on a double approximation technique, the grading entropy concept and at most quadratic many data in terms of the fraction number – is tested on the example of the dry density of sands here using some previously measured data. In the first approximation step a “preliminary transfer function” is interpolated in the nonnormalized grading entropy diagram on the basis of some “optimal” soil data. In the second approximation step the preliminary transfer function is extended to the space of the possible grading curves with the constant function. The so determined transfer function is tested against an independent “non-optimal” data set, measured on some soil series with basically continuous (i.e., not gap-graded) grading curves. The aim of this paper is to present the main results of the study supporting the goodness of the method and the predictability of the dry density transfer function. The direct interpolation of a transfer function needs exponentially many data in terms of the number of the fractions in the grading curve. The suggested transfer function construction method - based on a double approximation technique, the grading entropy concept and at most quadratic many data in terms of the fraction number – is tested on the example of the dry density of sands here using some previously measured data. In the first approximation step a “preliminary transfer function” is interpolated in the nonnormalized grading entropy diagram on the basis of some “optimal” soil data. In the second approximation step the preliminary transfer function is extended to the space of the possible grading curves with the constant function. The so determined transfer function is tested against an independent “non-optimal” data set, measured on some soil series with basically continuous (i.e., not gap-graded) grading curves. The aim of this paper is to present the main results of the study supporting the goodness of the method and the predictability of the dry density transfer function.

Exact solution of a thick walled functionally graded piezoelectric cylinder under mechanical, thermal and electrical loads in the magnetic field

Arefi, M.,Rahimi, G.H.,Khoshgoftar, M.J. Techno-Press 2012 Smart Structures and Systems, An International Jou Vol.9 No.5

The present paper deals with the analytical solution of a functionally graded piezoelectric (FGP) cylinder in the magnetic field under mechanical, thermal and electrical loads. All mechanical, thermal and electrical properties except Poisson ratio can be varied continuously and gradually along the thickness direction of the cylinder based on a power function. The cylinder is assumed to be axisymmetric. Steady state heat transfer equation is solved by considering the appropriate boundary conditions. Using Maxwell electro dynamic equation and assumed magnetic field along the axis of the cylinder, Lorentz's force due to magnetic field is evaluated for non homogenous state. This force can be employed as a body force in the equilibrium equation. Equilibrium and Maxwell equations are two fundamental equations for analysis of the problem. Comprehensive solution of Maxwell equation is considered in the present paper for general states of non homogeneity. Solution of governing equations may be obtained using solution of the characteristic equation of the system. Achieved results indicate that with increasing the non homogenous index, different mechanical and electrical components present different behaviors along the thickness direction. FGP can control the distribution of the mechanical and electrical components in various structures with good precision. For intelligent properties of functionally graded piezoelectric materials, these materials can be used as an actuator, sensor or a component of piezo motor in electromechanical systems.