Variational Differential Quadrature Research Articles

Improving the efficiency and stability of perovskite solar cell: Application of innovative machine learning algorithm

Solar cells are modern inventions that use the photovoltaic effect to directly convert light energy into electricity, generating electrical charges that are free to move through semiconductors. The semiconductor is typically utilized as the raw material for solar cells. In order to convert energy, electron–hole pairs that are responsible for producing light (photon) energy must be absorbed in a semiconductor, followed by charge carrier separation. Enhancing solar cells’ stability is crucial in engineering because they are used in a variety of environments. Moreover, graphene nanoplatelets (GPLs) have a great deal of potential to enhance ceramic–GNP composites’ mechanical, tribological, electrical, thermal, and biological characteristics, all at once. Machine learning algorithms (MLA) are often used to forecast how various systems would behave. In an MLA network, hyperparameters like the number of hidden layers and learning rate are often selected manually as required. MLA is used in this work to examine spinning cylindrical constructions’ stability at the microscale. In this context, the particle swarm optimization (PSO) is used to optimize the weights and biases of the network. The number of perceptions in the two hidden layers is optimized in a second parallel process using a genetic algorithm. The modified torque–stress theory (MCST) equation’s numerical solution for the dynamic behavior of GPL-reinforced perovskite solar cells was used to train the MLA. It is proposed to guide the spatial discretization of governing equations using the variational differential quadrature (VDQ) method as a direct discretization of the energy functional in the space domain. Lastly, the findings demonstrate that the stability of the current cantilevered solar cell reinforced by GPLs is significantly influenced by curvature, length scale, the shape of the solar cell, and mode number factors.

Data-driven solution for supersonic aerodynamics analysis of Perovskite solar cells considering the velocity of sound and Mach number

Due to the use of Perovskite solar cells (PSC) in aerospace applications, so, in this work for the first time, aerodynamic flutter and vibrational analysis of PSC are presented. For modeling the PSCs aerodynamic motion in the mathematical domain, the variational differential quadrature (VDQ) technique is used as a numerical operator for the discretization of various domains. Also, due to improving the stability of the PSC, carbon nanotubes with various distribution patterns as the reinforcement of the metal layer are presented. In this work for modeling the aerodynamics behavior, we use Krumhaar’s modified supersonic piston theory incorporating curvature influence, to simulate the aerodynamic stiffness and damping matrices. Also, for modeling the size effect, a modified couple stress theory with only one length scale factor is suggested. Data-driven solution with the aid of deep neural network with independent variables, input, hidden, and output layers, activation function, neural network’s intrinsic parameters, loss function, and hidden neurons is presented to solve the current applicable structure. The results show that the stability of the system is substantially impacted by the value of the yaw angle, and the link between Mach number and stability is heavily reliant on the value of the yaw angle. Another important outcome is that designing a spherical PSC and increasing its stability should have special attention to the a/b parameter. Also, future engineering research should have special attention that spherical solar cell has the highest natural frequency than other types of solar cells. Other important outcomes of the innovative PSC are presented in the results section for future handbooks on solar cells as sustainable energy.

Free vibration analysis of pre-stretched hyperelastic micromorphic continua with arbitrary shapes

A novel numerical strategy is developed in this article to study the free vibrations of hyperelastic micromorphic continua under bending load. In the proposed approach, the variational differential quadrature (VDQ) method and the idea of position transformation are used. The 3D micromorphic hyperelasticity is first formulated via vector-matrix relations which can be readily utilized in the coding process of numerical methods. The present numerical approach is able to address problems with irregular domains. To this end, the domain of elements is transformed into a regular one by the technique of mapping of position field based on the finite element shape functions. Being locking-free, simple implementation, computational efficiency and fast convergence rate are other features of the present variational approach. Three numerical examples including rectangular, sector and circular plates under bending load are considered whose free vibration behavior is analyzed. It is shown that the method is capable of predicting the relation between natural frequencies of micromorphic hyperelastic continua with load factor in an efficient way. The effects of internal length scale, scale-transition parameter and mode transition on the results are investigated.

Three-Dimensional Free Vibration Analysis of Hyperelastic Structures Under Bending Load Using the VDQ-Transformed Method

Using the ideas of variational differential quadrature (VDQ) technique and position transformation, an efficient numerical approach is developed herein in order to address the free vibration problem of compressible and nearly-incompressible solid bodies with arbitrary deformed shape within the framework of 3D hyperelasticity. The 3D hyperelasticity is first formulated by vector-matrix relations with the purpose of applying in coding process. An energy principle together with the Neo-Hookean strain energy function is also employed in the derivation of governing equations. The proposed numerical method is capable of addressing problems with irregular domains. Simple application, being free from the locking problem, and fast convergence rate are the key features of the approach. Hyperelastic rectangular/ sector plates and cylindrical panel subjected to bending load are selected as test problems whose free vibrations are analyzed. The developed numerical method is found to be capable of yielding accurate solutions to the considered problems. Moreover, the effects of mode transition and geometrical properties are investigated in the numerical examples.

Large deformation analysis of shell-type structures using the VDQ-transformed scheme: A two-point formulation based on 3D elasticity

Shell-type structures are frequently used in aerospace, marine and civil engineering. In this paper, a numerical approach called as VDQ-transformed is introduced to analyze the large deformations of hyperelastic shell-type structures based on the Saint Venant–Kirchhoff constitutive model in the context of three-dimensional (3D) nonlinear elasticity. According to the Euler–Lagrange description, the kinetic first Piola-Kirchhoff tensor and kinematic deformation gradient tensor are considered for the stress and strain measures in the formulation. By replacing the tensor form of formulations with matricized ones, the governing equations are written in a novel vector-matrix format which can be readily exploited for programming in numerical approaches. Moreover, discretizing is carried out by the variational differential quadrature (VDQ) method as a point-wise numerical method. To apply the VDQ technique, a transformation is used to map the irregular domain into the regular one. Some well-known benchmark problems for the large deformations of shells are solved to assess the approach. Compact matricized formulation, simple implementation, being locking-free, computational efficiency and fast convergence rate can be mentioned as the main features of the introduced numerical approach.

On finite deformation of hyperelastic shell-type structures: Cartesian formulation-based VDQ approach

In this article, a numerical approach is presented for the large deformation analysis of shell-type structures made of Neo-Hookean and Kirchhoff–St Venant materials within the framework of the seven-parameter shell theory. Work conjugate pair of the second Piola–Kirchhoff stress and Green–Lagrange strain tensors are taken for the macroscopic stress and strain measures in this total Lagrangian formulation. By defining displacement vector, deformation gradient and stress tensor in the Cartesian coordinate system, and using the chain rule for taking derivative of tensors, the complications of using the curvilinear coordinate system are bypassed. The variational differential quadrature (VDQ) technique as an effective numerical solution method is applied to obtain the weak form of governing equations. Being locking-free, simple implementation, computational efficiency and fast convergence rate are the main features of the proposed numerical approach. A number of well-known benchmark problems are solved in order to reveal the accuracy and efficiency of the method. It is shown that this approach is able to predict the large deformations of hyperelastic shell-type structures in an efficient way.

A variational differential quadrature solution to finite deformation problems of hyperelastic shell-type structures: a two-point formulation in Cartesian coordinates

A variational differential quadrature solution to finite deformation problems of hyperelastic shell-type structures: a two-point formulation in Cartesian coordinates

Dynamic Response of Rapidly Heated Rectangular Plates Made of Porous Functionally Graded Material

In this paper, the nonlinear thermally induced vibration in rectangular plates made of porous functionally graded materials (FGMs) due to thermal shock is studied based on a new numerical approach. The shear deformation influences are taken into consideration via Mindlin’s plate theory. The properties of the materials are also assumed to be temperature- and position-dependent. Moreover, the influence of elastic foundation is captured by the Winkler–Pasternak model. For deriving the governing equations, Hamilton’s principle, transient 1D Fourier-type heat conduction equation and von Kármán hypothesis are utilized. The relations of paper are written in a new matricized format for computational aims. The generalized differential quadarature (GDQ) method and Newmark’s direct integration scheme are used for solving the heat equation. Furthermore, solving motion equations is done based on the variational differential quadrature (VDQ) formulation. The effects of important parameters including material porosity and elastic foundation on the thermal shock response of porous FG plates are investigated in the numerical results.

Multi‐patch variational differential quadrature method for shear‐deformable strain gradient plates

AbstractThe integration of generalized differential quadrature techniques and finite element (FE) methods has been developed during the past decade for engineering problems within classical continuum theories. Hence, the main objective of the present study is to propose a novel numerical strategy called the multi‐patch variational differential quadrature (VDQ) method to model the structural behavior of plate structures obeying the shear deformation plate theory within the strain gradient elasticity theory. The idea is to divide the two‐dimensional solution domain of the plate model into sub‐domains, called patches, and then to apply the VDQ method along with the FE mapping technique for each patch. The formulation is presented in a weak form and due to the ‐continuity requirements the corresponding compatibility conditions are applied through the patch interfaces. The Lagrange multiplier technique and the penalty method are implemented to apply the higher‐order compatibility conditions and boundary conditions, respectively. To show the efficiency of the proposed method, numerical results are provided for plate structures with both regular and irregular solution domains. The provided numerical examples demonstrate the applicability and accuracy of the method in predicting the bending and vibration behavior of plate structures following the higher‐order plate model.

On the buckling and bending analysis of FG straight-sided quadrilateral nanoplates using a continuum mechanics-based surface elastic model

In this research, an elastic model based on the continuum mechanics is developed to study the static behaviors of functionally graded (FG) arbitrary straight-sided quadrilateral nanoplates. The model is constructed in the framework of Gurtin-Murdoch’s surface and Mindlin’s plate theories to account for the surface energy and shear deformation effects, simultaneously. The variational differential quadrature (VDQ) method is used along with a mapping technique to do the discretization process in a variational framework by means of differential and integral operators. Consequently, a weak form of governing equations is obtained from the energy quadratic representation of the problem. The solution method is of a distinguished feature as it involves just the first-order derivative of the field components in the mapping and discretization. After assuring the effectiveness of presented model by doing comparative studies, the critical buckling load and static deflection of the FG nanoplates with different shapes in geometry are investigated considering the surface effects. It is found that the surface energies effect on the static behavior of the rectangular nanoplates is more significant as compared to the non-rectangular nanoplates.

Studying nonlinear vibrations of composite conical panels with arbitrary-shaped cutout reinforced with graphene platelets based on higher-order shear deformation theory

In this article, the vibrational behavior of conical panels in the nonlinear regime made of functionally graded graphene platelet–reinforced composite having a hole with various shapes is investigated in the context of higher-order shear deformation theory. To achieve this aim, a numerical approach is used based on the variational differential quadrature and finite element methods. The geometrical nonlinearity is captured using the von Karman hypothesis. Also, the modified Halpin–Tsai model and rule of mixture are applied to calculate the material properties of graphene platelet–reinforced composite for various functionally graded distribution patterns of graphene platelets. The governing equations are derived by a variational approach and represented in matrix-vector form for the computational purposes. Moreover, attributable to using higher-order shear deformation theory, a mixed formulation approach is presented to consider the continuity of first-order derivatives on the common boundaries of elements. In the numerical results, the nonlinear free vibration behaviors of functionally graded graphene platelet–reinforced composite conical panels including square/circular/elliptical hole and with crack are studied. The effects of boundary conditions, graphene platelet reinforcement, and other important parameters on the vibrational response of panels are comprehensively analyzed.

Nonlinear Primary Resonant Characteristics of Higher-Order Shear Deformable FG-CNTRC Circular Cylindrical Panels

In this work, the nonlinear primary resonance of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) circular cylindrical panels is numerically studied. The FG-CNTRC cylindrical panels under the radial harmonic loading are modeled on the basis of the hyperbolic shear deformation shell theory (HSDST). The von Kármán hypothesis is employed to incorporate geometric nonlinearity into mathematical modeling. After representing the kinetic and strain energies and the external work in matrix forms (in terms of displacement vector), the variational differential quadrature (VDQ) method is utilized to obtain the discretized form of the energy functional on the space domain. Then, the nonlinear governing equations are achieved via Hamilton’s principle. In the next step, a multistep numerical solution approach is employed to illustrate the influences of geometrical parameters, subtended angle, CNTs distribution scheme and volume fraction on the primary resonant characteristics of FG-CNTRC cylindrical panels. The results are provided for the panels with clamped and simply-supported boundary conditions (BCs).

Free vibration analysis of postbuckled arbitrary-shaped FG-GPL-reinforced porous nanocomposite plates

Based upon the third-order shear deformation theory (TSDT) and a variational mixed formation, the postbuckling response and free vibration around buckled configurations of variously-shaped plates made of functionally graded graphene platelet (FG-GPL)-reinforced nanocomposite are numerically investigated considering the effect of porosity. The proposed numerical strategy is formulated according to the ideas of variational differential quadrature (VDQ) and finite element method (FEM), and can be employed for plates with different shapes (e.g. rectangular, skew or quadrilateral and annular) including arbitrary-shaped hole. The material properties of nanocomposite are approximated based upon the Halpin–Tsai model together with the closed-cell Gaussian Random field scheme for various distribution patterns of porosity and GPLs along the thickness direction. The governing equations are obtained according to Hamilton’s principle by novel vector-matrix relations which can be readily used in numerical methods. One of the main novelties of developed numerical approach is proposing an efficient technique according to the mixed formulation to accommodate the continuity of first-order derivatives on the common boundaries of elements for the used TSDT model. A number of numerical examples are given to investigate the influences of porosity coefficient/distribution pattern, GPL weight fraction/dispersion pattern, cutout and edge conditions on the free vibrations of postbuckled FG-GPL-reinforced porous nanocomposite plates.

Thermally nonlinear generalized coupled thermo-viscoelasticity of disks: a numerical variational approach

A novel numerical approach is proposed herein to study thermomechanical wave propagation in annular disks made of viscoelastic materials under inner thermal shock based on the Lord–Shulman (L–S) theory and the Kelvin–Voigt model. It is assumed that the temperature change is considerable in comparison with the reference temperature and the original nonlinear form of energy equation is considered accordingly. In the polar coordinate system, the coupled governing equations are obtained in a weak form which is then solved using the variational differential quadrature (VDQ) technique. The influences of viscoelastic character and thermal shock on the propagation of temperature and radial displacement of disk are investigated. In addition, the predictions of thermally linear and nonlinear models are compared.

Thermal Postbuckling of Temperature-Dependent Functionally Graded Nanocomposite Annular Sector Plates Reinforced by Carbon Nanotubes

In this study, the thermal buckling and postbuckling of functionally graded (FG) nanocomposite annular sector plates reinforced by carbon nanotubes (CNTs) are numerically analyzed. The effective material properties of FG nanocomposite are temperature-dependent (TD) and evaluated via the modified micromechanical method and rule of mixture. Based on the higher-order shear deformation theory (HSDT) and using the principle of virtual work and variational differential quadrature (VDQ) approach, the unified weak form of discretized nonlinear governing equilibrium equations is derived. Then, by using the linear part of equations and solving the derived eigenvalue problem, the critical temperature rise and associated mode shapes are obtained, which are used as the initial guess in solving the nonlinear thermal postbuckling problem. The pseudo-arc-length method and an iterative solver are employed to obtain the nonlinear thermal postbuckling equilibrium path of the FG nanocomposite annular sector plates. The influences of geometrical parameters, boundary conditions (BCs), CNT volume fraction, and CNT distribution pattern on the critical temperature rise and thermal postbuckling behavior of the FG nanocomposite annular sector plates are evaluated and discussed. Also, comparisons are made between the results considering the TD and temperature-independent (TID) properties. It is demonstrated that for higher values of sector angle, the effect of sector angle on the critical temperature rise and thermal postbuckling path is negligible. Moreover, by increasing the sector angle, the effect of BCs of straight edges vanishes, and the critical temperature rise and thermal postbuckling curves of for BCs of CSCS and SCSC approach those for CCCC and SSSS ones.

Nonlinear bending analysis of arbitrary-shaped porous nanocomposite plates using a novel numerical approach

In this paper, an efficient numerical strategy is used to study the geometrically nonlinear static bending of functionally graded graphene platelet-reinforced composite (FG-GPLRC) porous plates with arbitrary shape. Porous nanocomposite plates including cutout with various shapes can be modeled by the present approach. Four types of porous distribution scheme and four GPL dispersion patterns are selected, and the material properties are calculated based on the closed-cell Gaussian random field scheme, the Halpin–Tsai micromechanical model together with the rule of mixture. First, the variational statement of governing equations based on the virtual work principle and higher-order shear deformation theory (HSDT) is derived and presented in vector–matrix form for computational aims. Then, using the ideas of variational differential quadrature and finite element methods (VDQ and FEM), a numerical approach called as VDQ-FEM is used to address the considered problem. In VDQ-FEM, the domain of problem is first transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to obtain the set of Studying effects of porosity and GPL distributions and porosity coefficient matricized governing equations which is finally solved by means of the pseudo arc-length continuation algorithm. One of the main novelties of the present work in implementing VDQ-FEM is proposing an efficient way based on mixed-formulation to guarantee the continuity condition of first-order derivatives in entire domain for the used HSDT model. A detailed parametric study is conducted to investigate the nonlinear bending of FG-GPLRC porous plates with different shapes. In the numerical results, the effects of porosity coefficient, porosity distribution pattern, GPL distribution pattern and boundary conditions on the nonlinear bending response of plates are analyzed.

Thermal postbuckling analysis of FG-CNTRC plates with various shapes and temperature-dependent properties using the VDQ-FEM technique

Using the ideas of variational differential quadrature and finite element methods (VDQ and FEM), a novel numerical approach is developed in this paper to investigate the postbuckling behavior of plates with arbitrary shape under the action of thermal load within the framework of higher-order shear deformation theory (HSDT). It is considered that the plates are made of functionally graded carbon nanotube reinforced composite (FG-CNTRC). By the proposed approach, the thermal postbuckling behavior of plates with arbitrary-shaped cutout can be modeled. The rule of mixture is utilized to obtain the effective material properties of nanocomposite which are considered to be temperature-dependent. To implement the proposed method that can be named as VDQFEM, the variational temperature-dependent formulation of problem is first developed. This formulation is presented using novel vector-matrix relations with the aim of utilizing in programming in an efficient way. In the context of VDQFEM, the space domain of plate is first transformed into a number of finite elements. In the next step, the VDQ discretization technique is implemented within each element. Then, the assemblage procedure is performed to derive the set of matricized governing equations which is finally solved by means of the pseudo arc-length continuation algorithm. One of the main novelties of present approach is proposing an efficient way based on mixed-formulation to accommodate the continuity of first-order derivatives on the common boundaries of elements for the used higher-order shear deformable plate model. Quadrilateral plate, skew plate, annular plate, square plate with rectangular hole and rectangular plate with circular hole under different boundary conditions including CCCC, SSSS, CSCS and CCSS (S: simply-supported, C: clamped) are considered in the numerical example. First, validation studies are presented in terms of critical buckling temperature of CNTRC rectangular plates with various CNT distribution patterns. It is shown that the proposed method has the advantages of standard VDQ technique including fast convergence rate, being locking-free and simple implementation. Moreover, it is capable of considering polygon and concave domains. Also, the effects of geometrical properties, volume fraction and distribution pattern of CNTs and temperature-dependency of material properties on the thermal postbuckling of plates are studied.

Three-dimensional nonlinear primary resonance of functionally graded rectangular small-scale plates based on strain gradeint elasticity theory

Abstract In this paper, a size-dependent three-dimensional (3D) nonlinear weak formulation is provided to examine the nonlinear primary resonance problem for functionally graded rectangular small-scale plates. The small-scale factors are taken into formulation by choosing the Mindlin's strain gradeint theory (SGT). According to the variational differential quadrature (VDQ) method, first, the displacement field, nonlinear strain-displacement and constitutive relations as well as the potential and kinetic energies are expressed as the vector and matrix forms. Then, by applying the discretized form of differential operators obtained via the generalized differential quadrature (GDQ) method, the discretized form of aforementioned relations is achieved. Finally, Hamilton's principle is employed to access the weak form of 3D nonlinear governing equations of thick rectangular small-scale plates. The achieved formulation is solved via a multi-step numerical technique to address the size-dependent nonlinear primary resonance of considered system under the harmonic lateral force. In addition to reducing the run time, computational effort and CPU usage, the feature of proposed weak form formulation is that one can employ it in other solution approaches such as finite element method. Also, the use of this formulation provides the possibility of recovering models on the basis of other types of size-dependent theories such as modified strain gradient and modified couple stress theories (MSGT and MCST). In the numerical results, the effects of boundary conditions, small-scale parameter, material index and geometry are examined.

An integral nonlocal model for the free vibration analysis of Mindlin nanoplates using the VDQ method

Eringen’s nonlocal theory (ENT) provides an efficient tool to consider the size dependency for the structural analysis of small-scale structures. Recently, it has been indicated that in some cases, inconsistent results may be obtained using the Eringen differential model (EDM) and Eringen integral model (EIM) should be employed. In this regard, the vibration of nanoplates resting on the elastic medium is investigated on the basis of the EIM using variational differential quadrature (VDQ) method. On the basis of the first-order shear deformation theory (FSDT), the EIM and EDM formulations are given. An effective numerical technique is applied in the framework of the energy method to find the natural frequencies. Comprehensive results are reported to compare the EIM and EDM. The results show that using the proposed integral model, the paradox related to cantilever nanoplates is resolved.

Vibration analysis of FG-CNTRC plates with an arbitrarily shaped cutout based on the variational differential quadrature finite element method

The vibration analysis of functionally graded carbon nanotube-reinforced composite plates with the arbitrarily shaped cutout is presented using a novel numerical approach called variational differential quadrature finite element method (VDQFEM). The governing equations are expressed in matrix form based on Mindlin’s plate theory. In the proposed numerical approach, the space domain of the plate is first transformed into a number of sub-domains known as finite elements. Then, the variational differential quadrature (VDQ) discretization method is used within each element to obtain the mass and stiffness matrices. In order to use the VDQ method, the irregular domain of the element is transformed into a regular one employing the mapping technique. Finally, the assemblage procedure is performed to present total mass and stiffness matrices. The introduced numerical approach can be effectively used for structural analysis of arbitrarily shaped plates. A wide range of comparative and convergence studies are outlined to show the performance of the method. It is observed that the numerical results are rapidly converged and the proposed solution strategy can be successfully applied to examine the vibration of FG-CNTRC plates with different cutouts.