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A Strong Formulation Finite Element Method (SFEM) Based on RBF and GDQ Techniques for the Static and Dynamic Analyses of Laminated Plates of Arbitrary Shape
Abstract
This paper deals with the static and dynamic analysis of multi-layered plates with discontinuities. The two-dimensional First-order Shear Deformation Theory (FSDT) is used to derive the fundamental system of equations in terms of generalized displacements. The fundamental set, with its boundary conditions, is solved in its strong form. A new method termed Strong Formulation Finite Element Method (SFEM) is considered in the present paper to solve this kind of plates. This numerical methodology is the cohesion of derivative evaluation of partial differential systems of equations and a domain sub-division.
The numerical results in terms of natural frequencies and maximum deflections are compared to literature and to the same results obtained with a finite element code. The stability, accuracy and reliability of the present methodology is shown through several numerical applications.
... In particular, the present study considers a strong form pseudo-spectral technique implemented on arbitrarily shaped domains. In order words, the authors are decomposing the physical problem into quadrilateral domains and they are solving the mathematical problem upon each element with a strong form approach [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. As almost any numerical method the starting point is the functional approximation which might depend on the choice of basis functions and point collocation. ...
... The implementation of Eq. (13) is straightforward for partial differential systems of equations, nevertheless, only problems with regular domains can be solved. For this reason the authors extended the present pseudo-spectral approach to work with domains of arbitrary shape, such method has been termed Strong Formulation Finite Element Method (SFEM) [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. In order to achieve this, a mapping technique has been used which is able to transform a shape in Cartesian coordinates to a regular domain (of square shape). ...
... It has been assumed that no micro-couples can be applied to the solid but only in-plane pressures Q j , j ¼ 1; 2. The discrete system (14) cannot be solved directly without its discrete boundary conditions that can be easily carried out from Eq. (7). It is remarked that the implementation of the boundary conditions is the major drawback of the SFEM, while the implementation of the governing equation is quite straightforward [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. The solution is found by solving the standard linear problem ...
Cosserat theory of elasticity has been introduced for modelling micro-structured materials and structures. Micro-structured materials are able to re-distribute the stress resulting in lower stress peaks. Therefore, such effects are strongly underlined when composite structures have holes and discontinuities in which high stress gradients are generally observed in the classical theory of elasticity. However, general material configurations can be solved using numerical approaches, since exact solutions are only available for simple cases. The present paper deals with such problems using an advanced strong form pseudo-spectral method that uses domain decomposition to deal with geometric and material discontinuities.
... However, when the number of points in the whole domain increases the coefficient matrix of RBFs tends to be ill-conditioned and does not yield to accurate results. For this reason the idea of dividing the domain into sub-domains and the use of less points for each element to reduce the matrix ill-conditioning problem and simplify the initial problem has been proposed by Fantuzzi et al. [30] using strong formulation. Liu and colleagues recently proposed an edge-based smoothed finite element method (ES-FEM) that is based on weak-form formulations [21,31,32] instead of strong-formulations (RBF collocation) that are normally used. ...
... In this section, free vibration analysis of two kind of arbitrary shaped composite plates are performed. The RBFEM results of this work are compared with the GDQFEM [37] and SRBFEM [30] ...
... (a) (b) (c) In Table 6, the first ten natural frequencies of the bipolar plates are compared with the results of the GDQFEM [37], the SRBFEM [30] and the commercial codes ABAQUS [37]. The radial basis function of the SRBFEM is the same Wendland RBF as the present work. ...
A layerwise shear deformation theory for composite laminated plates is discretized using a radial basis function finite element method (RBFEM). The RBFEM is the radial basis function (RBF) method in weak-form and is a partially mesh free method. Therefore, elements of complex shapes can be easily constructed. Compact-support Wendland function is used in the RBFEM. A layerwise theory based on a linear expansion of Mindlin’s first-order shear deformation theory in thickness direction is employed for static and dynamic analysis. The combination of the RBFEM with the layerwise theory allows an accurate and very flexible prediction of the field variables. Laminated composite and sandwich plates were analyzed. The RBFEM solutions were compared with various models in literatures and showed very good agreements with exact and other high accurate results in literatures based on similar layerwise theories. The analysis of composite plates based on the layerwise theory indicates that the RBFEM is an effective method for high accuracy analysis of large-scale problems.
... The coefficients in Eqs. (26) and (27) are written as: ...
... To overcome the difficulties of DQ in computing the weighting coefficients, the generalized differential quadrature (GDQ) was developed by Shu [20], following the analysis of high-order polynomial approximation in a linear polynomial vector space. With the advent of the GDQ method in [20], new numerical applications have been proposed for structural mechanics application by Tornabene et al. [25] and investigation on static and dynamic behavior of multilayered structures via GDQ methods by Fantuzzi et al. [26]. Shafiei et al. [27] developed the governing equations and boundary conditions to the Hamilton's principle, and the governing equations are solved with the aid of the GDQM. ...
... Eqs. (26) and (27), the state coefficients in Eq. (63) for composite laminated Timoshenko and Reddy beams can be given in terms of displacement parameters as: ...
In this article, thermal effect on free vibration behavior of composite laminated microbeams based on the modified couple stress theory is presented. The proposed anisotropic model is developed by using a variational formulation. The governing equations and boundary conditions are obtained based on a modified couple stress theory and using the principle of minimum potential energy and considering different beam theories, i.e., Euler–Bernoulli, Timoshenko and Reddy beam theories. Unlike the classical beam theories, this model contains a material length scale parameter and can capture the size effect. Free vibration of a simply supported beam is solved by utilizing Fourier series. In addition, the fundamental frequency is achieved by using the generalized differential quadrature method for four types of cross-ply laminations with clamped–clamped, clamped–hinged and hinged–hinged boundary conditions for different beam theories. For investigating different parameters including temperature changes, material length scale parameter, beam thickness, some numerical results on different cross-ply laminated beams are presented. The fundamental frequency of different thin and thick beam theories is investigated by increasing slenderness ratio and thermal loads. The results prove that the modified couple stress theory increases the natural frequency under the thermal effects for free vibration of composite laminated microbeams.
... The authors employed the Generalized Differential Quadrature Finite Element Method [12] to study this problem. The highlights of the previous contribution were related to the use of a strong formulation finite element method which implemented the mapping technique (well-known feature of standard finite elements) and a strong formulation [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. However, this previous study had a limitation due to the use of Lagrangian shape functions for the mapping of the membrane elements. ...
... Therefore the present numerical approach is termed Strong Formulation Isogeometric Analysis (SFIGA). The SFIGA directly derives from the Strong Formulation Finite Element Method (SFEM) discussed in [35][36][37][38][39][40][41][42] and from the GDQFEM [31][32][33][34]. It is recalled that the SFEM is the general form of the GDQFEM because different basis function can be selected for the polynomial approximation, but the same Lagrangian shape functions are used in both techniques for the mapping technique. ...
... Detailed expressions of Eq. (6) were illustrated in the previous work [42]. For the sake of conciseness not all the expressions for performing the mapping technique will be reported, however, the a complete stepby-step procedure can be found in the works [31][32][33][34][35][36][37][38][39][40][41][42] where all the details about how the mapping technique is utilized within the strong form finite elements were shown. Only the blending function mapping is indicated in the following. ...
A strong form finite element technique, termed SFEM, has been presented recently. This approach resulted to be accurate and reliable for different engineering problems. The SFEM merges the high convergence rates of strong form pseudo-spectral methods and the versatility of domain decomposition techniques proper of the Finite Element Method (FEM). The governing differential equations and the compatibility conditions between two adjoining elements are transformed through the mapping technique. Due to its higher order nature given by the collocation of several points in each single element, classic 8 node elements are not often sufficient to map a geometry with the smallest amount of elements. Therefore, a new mapping approach based on blending functions is introduced in this paper for investigating membrane structures. In particular, isogeometric mapping based on Non-Uniform Rational Basis Spline (NURBS) will be considered. This kind of nonlinear mapping is generally associated with Isogeometric Analysis (IGA). Therefore, the present new approach is termed Strong Formulation Isogeometric Analysis (SFIGA). In order to prove the accuracy and stability of this technique several analytical and other results from the literature will be presented together with new applications.
... In the same period, Fantuzzi et al. [25,26] proposed another type of strong-form FEM (SFEM), in which a set of formulations computing the firstand second-order spatial partial derivatives are derived for 2D problems and are used to collocate the governing PDEs in solid mechanics. In SFEM, the continuity condition among elements is determined by the compatibility, and a mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the computational space. ...
... Various numerical examples [27][28][29][30][31] have proved that the above equation can give correct results. The important point is that Eq. 22 allows the final system of equations to have the same size as the conventional FEM, which is much smaller than those in FBM [8] and SFEM [25]. ...
In this article, the progress of frequently used advanced numerical methods is presented. According to the discretisation manner and manipulation dimensionality, these methods can be classified into four categories: volume-, surface-, line-, and point-operations–based methods. The volume-operation–based methods described in this article include the finite element method and element differential method; the surface-operation–based methods consist of the boundary element method and finite volume method; the line-operation–based methods cover the finite difference method and finite line method; and the point-operation–based methods mainly include the mesh free method and free element method. These methods have their own distinctive advantages in some specific disciplines. For example, the finite element method is the dominant method in solid mechanics, the finite volume method is extensively used in fluid mechanics, the boundary element method is more accurate and easier to use than other methods in fracture mechanics and infinite media, the mesh free method is more flexible for simulating varying and distorted geometries, and the newly developed free element and finite line methods are suitable for solving multi-physics coupling problems. This article provides a detailed conceptual description and typical applications of these promising methods, focusing on developments in recent years.
... [19,20]. Referring to these characteristics, some innovative techniques, like finite block method (FBM) [27,46], strong-form finite element method (SFEM) [13,14] and moving finite element method (MFEM) [37], have been published and successfully applied to solve the heat transfer and other engineering problems. ...
... where J ik is a component of the Jacobian matrix J . More details of these two equations can be found in some literature [13,14,19,20]. By using the formulation for the model discretized by free elements, a strong-form scheme called free element method (FREM) can be performed. ...
Our purpose is to establish a numerical method meeting the requirements of efficiency, stability, accuracy and easy-using. Faced with these demands, in this paper, a hybrid method combining the advantages of 2 weak-form meshless methods is proposed for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In the process of the calculation for an arbitrary model discretized by the hybrid method, two different weak-form methods are used for the nodes categorized as: (1) boundary nodes, (2) near boundary interior nodes and (3) interior nodes. A Galerkin free element method (GFREM) is proposed for the interior nodes of the Cartesian grid; and the local radial point interpolation method (LRPIM) is used for the other two types of nodes. The GFREM using the Cartesian grid can greatly improve the accuracy and efficiency of the hybrid scheme. Special decomposition technology to determine the irregular integral domain for LRPIM can increase the flexibility for complex models. In addition, the full implicit finite difference scheme and Newton–Raphson method are used for solving transient nonlinear problems. And non-crossing interface integral domain and support domain are employed for heterogeneous media. Compared with other methods, numerical results show better accuracy, stability and efficiency from the numerical experiments.
... Viola et al. [42] carried out free vibration analyses for composite plates with various shapes containing elliptic holes and slits using the generalized differential quadrature finite element method. Fantuzzi et al. [12] derived a FE model to study the dynamics of multi-layered plates with discontinuities. Capozucca and Bonci [8] carried out experimental vibration tests and FE analyses for simply supported pristine unidirectional CFRP laminated plates with and without discontinuities. ...
... With the substitution of Eqs. (10)- (12) into Eq. (9), the following equations of motion can be obtained: ...
... The implementation of the strong-form algorithm is relatively simple and does not require integration calculations. Novel strong-form algorithms such as the strong-form finite element method (SFEM) [26] and finite block method (FBM) [27,28] have been proposed and successfully applied to the mechanics, heat transfer, and other engineering problems. Then, Li et al. [29] solved the two-dimensional contact problems in functionally graded materials using FBM with point-point contact discretization. ...
In this paper, a new strong-form numerical method, the element differential method (EDM) is employed to solve two- and three-dimensional contact problems without friction. When using EDM, one can obtain the system of equations by directly differentiating the shape functions of Lagrange isoparametric elements for characterizing physical variables and geometry without the variational principle or any integration. Non-uniform contact discretization is used to enhance contact conditions, which avoids performing identical discretization along the contact surfaces of two contact objects. Two methods for imposing contact constraints are proposed. One method imposes Neumann boundary conditions on the contact surface, whereas the other directly applies the contact constraints as collocation equations for the nodes within the contact zone. The accuracy of the two methods is similar, but the multi-point constraints method does not increase the degrees of freedom of the system equations during the iteration process. The results of four numerical examples have verified the accuracy of the proposed method.
... The static behaviour of functionally graded shells under point and line loads was considered in [46]. A strong formulation finite element method for the statics and dynamics of laminated plates was proposed in [47]. The semi-analytical methods based on the classical laminate plate theory were formulated and used to consider some mechanical problems, e.g., for the dynamics and stability of functionally graded thin plates in [48]; for buckling of FML-FGM columns, made of such plates, with open cross-sections in [49]; similar columns with closed cross-sections in [50]; or for the imperfection sensitivity of the post-buckling of FML columns in [51]. ...
Dynamic problems of elastic non-periodically laminated solids are considered in this paper. It is assumed that these laminates have a functionally graded structure on the macrolevel along the x1-axis and non-periodic structure on the microlevel. However, along the other two directions, i.e., x2 and x3, their properties are constant. The effects of the size of a microstructure (the microstructure effect) on the behaviour of the composites can play a significant role. This effect can be described using the tolerance modelling method. This method allows us to derive model equations with slowly varying coefficients. Some of these terms can depend on the size of the microstructure. These governing equations of the tolerance model make it possible to determine formulas describing not only fundamental lower-order vibrations related to the macrostructure of these composite solids, but also higher-order vibrations related to the microstructure. Here, the application of the tolerance modelling procedure is shown to lead to equations of the tolerance model that can be used for non-periodically laminated solids. Then, these model equations are mainly used to analyse a simple example of vibrations for functionally graded composites with non-periodically laminated microstructure (FGL). Similar problems were investigated in the framework of the homogenised (macrostructural) model (Jędrysiak et al. 2006); the resulting equations neglect the microstructure effect.
... Civalek [12][13][14] proposed a discrete singular convolution method, where the geometric transformation technique of four-node and eight-node had been applied one after another. Tornabene [15][16][17] proposed the generalized DQFEM for the dynamics analysis of composite plates. In this method, the whole plate was subdivided into several subdomains of irregular shapes. ...
This paper reports a modeling and experimental study on the free vibration characteristics of plates with curved edges. The model is established by the first-order shear deformation theory (FSDT) and Chebyshev differential quadrature method (CDQM). The one-to-one coordinate transformation technique is introduced into the CDQM to map the plate with curved edges into a square plate. The admissible displacement functions of the square plate are expanded by two-dimensional Chebyshev polynomials and discretized by Gauss-Lobatto sampling points. The boundary conditions are applied to the plate according to the projection matrix method. Experimental studies of six aluminum plates with different shapes are carried out to investigate the vibration characteristics and verify the validity of the proposed CDQM. Furthermore, the results of the present CDQM are also compared with those of the finite element method (FEM) and existing numerical approaches to examine its efficiency and accuracy. The results show that the current CDQM can rapidly and accurately compute the vibration characteristics of plates with curved edges under different boundary conditions.
... In this method, the computational prob-lem is discretized into a number of zones and FREM is applied to set up the system of equations. ZFREM also takes the advantages of the finite block method (FBM) [39,40] and the strong-form finite element method (SFEM) [41,42] in division of zones. ...
Recently, the free element method(FREM), a novel strong-form method, is successfully used to solve thermal and mechanical problems. However, similar to some other strong-form methods, it is difficult to handle the geometric model with corner points. To overcome this weakness, the zonal free element method (ZFREM), which has strong adaptability to deal with complex geometry, is proposed by introducing the sub-domain mapping technique. Besides, the proposed method is firstly applied to solve piezoelectric problems. A unified formulation for piezoelectric problems in the case of electro-mechanical coupling field is established by the zonal free element method. Several two-dimensional (2D) and three-dimensional (3D) examples are carried out and the results are compared with available analytical solutions and finite element solutions to demonstrate validity and accuracy of the proposed method.
... Overall, low-order models preserving the main features of the underlying continuum formulations get rid of the complicatedness generally occurring in the analysis and interpretation of nonlinear phenomena when using richer models (e.g., finite elements), also possibly implemented within an effective unified perspective [13,37,38] . Thus, they allow easier analyses and deeper understanding of the basic, yet involved, effects of coupling on the finite amplitude vibrations of geometrically nonlinear structures. ...
An overview of extended research recently pursued on unified continuous/reduced-order modeling and nonlinear dynamics of thermomechanical composite plates of interest in aerospace, mechanical and civil engineering is presented. Reduced models exhibit the fundamental features of geometrical nonlinearity and thermomechanical coupling of the underlying continua. The role of multiphysics coupling and the main features of nonlinear response obtained with variably refined minimal models is highlighted. Besides transverse mechanical excitation and mechanically or thermally-induced buckling, a variety of active thermal excitations, of body or boundary nature, are considered. Features of thermal response obtained with variably refined thermal assumptions are compared, in view of detecting cheap, yet reliable, models to be used for systematic numerical investigations. The effects of two-way thermomechanical coupling on local and global nonlinear dynamics are addressed through bifurcation diagrams, phase portraits and planar cross sections of 4D basins of attraction, highlighting the important role played by the slow transient thermal dynamics solely detectable with coupled models in the steady outcome of the swifter mechanical response. Conditions allowing to utilize partially coupled models or even the uncoupled mechanical one with prescribed steady temperature are discussed
... A chaos problem for a rectangular functionally graded plate was investigated in Reference [37]. A strong formulation based on the GDQ technique to finite element method for multilayered plates was proposed in Reference [38], but a strong formulation of isogeometric analysis for composite laminated plates was shown in Reference [39]. A differential quadrature method and a layer-wise theory for composite plates were applied in Reference [40]. ...
In this paper, the problem of the stability of functionally graded thin plates with a microstructure is presented. To analyse this problem and take into consideration the effect of microstructure, tolerance modelling is used. The tolerance averaging technique allows us to replace the equation with non-continuous, tolerance-periodic, highly oscillating coefficients of the system of differential equations with slowly-varying coefficients, which describes also the effect of the microstructure. As an example, the buckling of a microstructured functionally graded plate band on a foundation is investigated. To obtain results, the tolerance model and the asymptotic model combined together with the Ritz method are used. It is shown that the tolerance model allows us to take into account the effect of microstructure on critical forces.
... Recently some innovative methods like the finite block method (FBM) [35] and strong-form finite element method (SFEM) [36] are proposed and used for fracture problems [37,38]. Same as the Differential Quadrature Method (DQM) [39], the feature of the FBM and SFEM is that the physical domain is divided into several blocks (SFEM mesh) and the PDEs are discretized directly in the strong-form for each block. ...
In this paper, a block-based Galerkin free element method is presented for the calculation of J-integral and mix-model stress intensity factors. Based on the free element collocation method, the Galerkin weak-form is constructed to achieve more accurate results. By absorbing the advantages of SFEM and FBM, the sub-domain mapping technique is used to simplify the geometry and to avoid the appearance of the deformity element. For fractural parameters, the Equivalent Domain Integral (EDI) method is used to predict the J-integral, and the interaction integral method is used to obtain the stress intensity factors. Some numerical examples are given to demonstrate the accuracy, convergence and stability of the proposed method for the computation of fractural parameters.
... Hyer and Lee [26] studied about the strength and buckling performance of variable stiffness composite plate using finite element method. Fantuzzi and Tornabene [27][28][29] studied the static and dynamic analysis of composite panels of arbitrary shapes with arbitrary cutouts. Chen et al. [30] studied the buckling response of variational stiffness composite laminated plate with pre-embedded delamination under axial compression using Rayleigh-Ritz method. ...
The present study deals with the investigation of buckling and postbuckling responses of functionally graded hybrid square plates which are symmetric about its mid plane. The cutouts of various sizes and shapes located at the center of the plate are considered. The plate is subjected to positive and negative in-plane shear loads. The boundary conditions used are simply supported on all the four edges of the plate. Three stacking sequences are considered to investigate the optimum stacking sequence at which the critical buckling and first failure loads are highest. Functionally graded hybrid plates (FH) subjected to negative in-plane shear load is more effective compared to plates subjected to positive in-plane shear load. Diamond shaped cutout with small sized perforation has the highest critical buckling and first ply failure loads amongst the FH plates with cutouts. It is observed that the direction of applied in-plane shear load, fiber stacking sequence; cutout shape and size substantially influence the strength and failure characteristics of functionally graded hybrid composite plates.
... The natural, simple and convenient methods to solve engineering problems governed by partial differential equations (PDEs) are the collocation-type ones [16][17][18], which discretize directly the partial derivatives of the PDEs in hand leading to a system of algebraic equations. Unfortunately, these types of strong-form methods usually have stability and reliability problems, that is, the computed results sometimes rely on the number and distribution of discretized points or their accuracy depends on some shape parameter [24][25][26][27]. To solve this issue, various techniques have been proposed for stabilizing the solution. ...
In this article, a completely new numerical method called the Local Least-Squares Element Differential Method (LSEDM), is proposed for solving general engineering problems governed by second order partial differential equations. The method is a type of strong-form finite element method. In this method, a set of differential formulations of the isoparametric elements with respect to global coordinates are employed to collocate the governing differential equations and Neumann boundary conditions of the considered problem to generate the system of equations for internal nodes and boundary nodes of the collocation element. For each outer boundary or element interface, one equation is generated using the Neumann boundary condition and thus a number of equations can be generated for each node associated with a number of element interfaces. The least-squares technique is used to cast these interface equations into one equation by optimizing the local physical variable at the least-squares formulation. Thus, the solution system has as many equations as the total number of nodes of the present heat conduction problem. The proposed LSEDM can ultimately guarantee the conservativeness of the heat flux across element surfaces and can effectively improve the solution stability of the element differential method in solving problems with hugely different material properties, which is a challenging issue in meshfree methods. Numerical examples on two- and three-dimensional heat conduction problems are given to demonstrate the stability and efficiency of the proposed method.
... It should be recalled that the use of composite materials in shell structures is becoming more and more popular due to the great advantages in terms of structural efficiency that can be achieved, as illustrated in [10][11][12][13][14][15][16]. This topic is investigated in many papers available in literature, especially as far as laminated composites [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] and functionally graded materials [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] are concerned. ...
A theoretical framework based on an Equivalent Single Layer (ESL) approach is proposed in this chapter to develop several Higher-order Shear Deformation Theories (HSDTs) in a unified and compact manner. In particular, the maximum order of kinematic expansion can be arbitrarily chosen in order to define more refined displacement fields. The Murakami’s function can be also included in the model to take into account the so-called zig-zag effect. The proposed theory is employed to describe the mechanical behavior of doubly-curved shell structures made of composite materials. In particular, the differential geometry is used to define accurately the curved surfaces at issue. The strong formulation of the governing equation is solved by means of a numerical approach based on the Generalized Differential Quadrature (GDQ) method. The accuracy of both the theoretical model and the numerical method is shown through some applications, in which the solutions are compared with the results obtained by means of a three-dimensional finite element model.
... It is important to specify that the discrete domain just defined denotes a rectangular domain from the computational point of view. At this point, a coordinate transformation, named mapping, can be introduced to define arbitrarily shaped curved surfaces, as the one depicted in Fig. 4 [90][91][92][93][94][95]. Analytically speaking, the mapping procedure in hand is given by the following expressions ...
The aim of this chapter is the development of an efficient and accurate higher-order formulation to solve the weak form of the governing equations that rule the mechanical behavior of doubly-curved shell structures made of composite materials, whose reference domain can be defined by arbitrary shapes. To this aim, a mapping procedure based on Non-Uniform Rational Basis Spline (NURBS) is introduced. It should be specified that the theoretical shell model is based on the Equivalent Single Layer (ESL) approach. In addition, the Generalized Integral Quadrature technique, that is a numerical tool which can guarantee high levels of accuracy with a low computational effort in the structural analysis of the considered shell elements, is introduced. The proposed technique is able to solve numerically the integrals by means of weighted sums of the values that a smooth function assumes in some discrete points placed within the reference domain.
... Recently, a new strong-form method combing advantages of finite element and mesh free methods, free element collocation method (FECM) is proposed by Gao et al. [12,13] for solving general boundary value problems like thermal and mechanical problems governed by the second order PDEs. The method of FECM is a type of collocating method based on the use of freely formed isoparametric elements as used in FEM as well as in the newly proposed strong-form methods FBM [14,15] and SFEM [16,17]. When establishing the system of equations by FECM node by node in the computational domain Ω which had been discretized into a series of distributed nodes, a local isoparametric element is formed by the collocation node and around nodes in order to disperse the variables. ...
In this paper, a new weak-form method (Galerkin free element method – GFrEM) is developed and implemented for solving general mechanical and fracture problems. This method combines the advantages of the finite element method and meshfree method in the aspects of setting up shape functions and generating computational meshes through node by node. The distinct feature of GFrEM is that only one collocation element is needed for each collocation node and the collocation element can be freely formed by the nodes surrounding the collocation node. In order to get the weak-form system equations, the weighted residual technique, with shape function as weight function, is used in the collocation elements. Numerical experiments are carried out to determine displacements, stresses and the stress-intensity factors in two-dimensional cracked structures, to verify the present method.
... Houmat [6,7] investigated nonlinear free vibration of laminated composite rectangular VSCL square and skew plates based on classical plate theory using FEM. Fantuzzi and Tornabene [8,9] presented their strong formulation finite element method (SFEM) for static and dynamic analysis of isotropic and composite panels of arbitrary shapes. The SFEM uses the differential quadrature method (DQM) in discretization of the equations. ...
The free vibration characteristics of the variable stiffness composite laminated plates containing embedded cutout of desired shapes is investigated. The circular, elliptical and quadrilateral perforation geometries are taken into account. In order to provide a strongly flexible calculation tool, an isogeometric analysis formulation based on the non-uniform rational B-splines (NURBS) along with the Nitsche technique is developed based on the first order shear deformation plate theory. The effects of change in the mechanical properties of tow steered laminates throughout the geometry due to fibers following prescribed curvilinear paths are taken into account in the integration procedures. The accuracy and reliability of the calculations is shown through some representative comparisons. The effects of change in the cutout geometry, location and orientation in conjunction with the curvilinear fiber placement are studied.
... Researchers have been focusing on mixed mode modeling of plates to overcome some of the accuracy problems of these theories especially when dealing with discontinuities, generic plate shapes or for solving problems of higher order dynamic vibrations [173][174][175]. Angioni et al [169] employed multiple plate theories simultaneously as an alternative to costly full scale 3D FEM solution. ...
The use of advance composite materials is increasing in various industrial applications such as renewable energy, transportation, medical devices, etc. As the demand for stability under high mechanical, thermal, electrical and combined loads is increasing, research is being focused on developing newer types of composites and developing analytical and numerical methods to study composite plates as well. The present work is aimed to provide a comprehensive review of research in the structural analysis of composite plates along-with research trends in the last 15 years. The article first presents the evolution of plate theories comparing their formulations, applicability and discusses some key papers, results and conclusions. Evolution of research from the equivalent shear deformation theories (ESL) such as first order theory and higher order theories based on various shape strain functions e.g., polynomial, trigonometric to layer-wise, zigzag and displacement potential theories is presented. The comparative analysis of various solution approaches is done based on a review of research work in the structural analysis of plates. This is followed by review of meshless analysis methods for composite materials highlighting problem domains where conventional finite element analysis (FEA) approach has limitations. This article also presents a discussion on the new methods of plate analysis such as region-by-region modeling, hierarchic modeling and mixed FE and neural network based modeling. An attempt has been done in this article to focus on research trends in the last 15 years.
... As far as the structural mechanics is concerned, the same approaches can be used to deal with irregular geometries which require a domain decomposition. Based on the formulation, the Strong Formulation Finite Element Method (SFEM) [50][51][52][53][54][55][56][57] and the Weak Formulation Finite Element Method (WFEM) [9,10,58] were developed by the authors in the last years. In the present paper, only regular domains are considered since its main aim is to investigate which combination of basis functions for the functional approximation and grid distribution for the discretization of the domain provides the best results in terms of accuracy and stability. ...
The aim of this work is to investigate and compare the accuracy and convergence behavior of two different numerical approaches based on Differential Quadrature (DQ) and Integral Quadrature (IQ) methods, respectively, when applied to the free vibration analysis of laminated plates and shells. The numerical methods at issue allow to solve the strong and the weak forms of the governing equations of these structural elements. A completely general approach is presented to evaluate numerically derivatives and integrals by using several basis functions (polynomial approximation) and grid distributions (discretization). The convergence analyses are performed for three different approaches: Strong Formulation (SF), Weak Formulation (WF) with continuity conditions, and Weak Formulation (WF) with continuity conditions. For each approach, a set of convergence graphs is shown, by varying both basis functions and discrete grids, in order to define the combinations that provide the best accuracy with reference to the exact solutions available in the literature.
... The mechanical analysis of laminated composite plates and shells is currently a recurring topic in the pertinent literature. In particular, the effect of the fiber orientation and the stacking sequence on the structural response has been hugely investigated in many papers to analyze the static [5][6][7][8][9][10][11][12][13][14][15][16][17] and dynamic behavior of such structures. ...
A mathematical scheme is proposed here to model a damaged mechanical configuration for laminated and sandwich structures. In particular, two kinds of functions defined in the reference domain of plates and shells are introduced to weaken their mechanical properties in terms of engineering constants: a two-dimensional Gaussian function and an ellipse shaped function. By varying the geometric parameters of these distributions, several damaged configurations are analyzed and investigated through a set of parametric studies. The effect of a progressive damage is studied in terms of displacement profiles and through-the-thickness variations of stress, strain, and displacement components. To this end, a posteriori recovery procedure based on the three-dimensional equilibrium equations for shell structures in orthogonal curvilinear coordinates is introduced. The theoretical framework for the two-dimensional shell model is based on a unified formulation able to study and compare several Higher-order Shear Deformation Theories (HSDTs), including the Murakami’s function for the so-called zig-zag effect. Thus, various higher-order models are used and compared to investigate also the differences which can arise from the choice of the order of the kinematic expansion. Their ability to deal with several damaged configurations is analyzed, too. The paper can be placed also in the field of numerical analysis, since the solution to the static problem at issue is achieved by means of the Generalized Differential Quadrature (GDQ) method, whose accuracy and stability are proven by a set of convergence analyses and by the comparison with the results obtained through a commercial finite element software.
... For this reason, the present problems are solved by using the so- displacements are set among the elements. Several papers have been published in this regard giving details on such implementation that involves also the corner points of the elements [19,20]. The Cartesian displacements are approximated as ...
The present study aims to show a novel numerical approach for investigating composite structures wherein inclusions and discontinuities are present. This numerical approach, termed Strong Formulation Finite Element Method (SFEM), implements a domain decomposition technique in which the governing partial differential system of equations is solved in a strong form. The provided numerical solutions are compared with the ones of the classic Finite Element Method (FEM). It is pointed out that the stress and strain components of the investigated model can be computed more accurately and with less degrees of freedom with respect to standard weak form procedures. The SFEM lies within the general framework of the so-called pseudo-spectral or collocation methods. The Differential Quadrature (DQ) method is one specific application of the previously cited ones and it is applied for discretizing all the partial differential equations that govern the physical problem. The main drawback of the DQ method is that it cannot be applied to irregular domains. In converting the differential problem into a system of algebraic equations, the derivative calculation is direct so that the problem can be solved in its strong form. However, such problem can be overcome by introducing a mapping transformation to convert the equations in the physical coordinate system into a computational space. It is important to note that the assemblage among the elements is given by compatibility conditions, which enforce the connection with displacements and stresses along the boundary edges. Several computational aspects and numerical applications will be presented for the aforementioned problems related to composite materials with discontinuities and inclusions.
... A two-dimensional analytical solution of a multilayered plate with a periodic structure along one in-plane direction is obtained by Wen-Ming He et al. [10], with using the two-scale asymptotic expansion method to develop a homogenized model. The finite element method is applied to static and dynamic analyses of laminated plates by Fantuzzi et al. [11]. An application of the spectral element method to consider vibration band gap properties of periodic plates is presented by Zhi-Jing Wu et al. [12]. ...
Thin periodic plates with uncertain properties in a periodicity cell are investigated. To describe dynamics of these plates the non-asymptotic tolerance modelling method, cf. Woźniak and Wierzbicki (Averaging techniques in thermomechanics of composite solids. Wydawnictwo Politechniki Częstochowskiej, Częstochowa, 15), Woźniak et al. (eds.) (Thermomechanics of microheterogeneous solids and structures. Tolerance averaging approach. Wydawnictwo Politechniki Łódzkiej, Łódź, 16), for those plates is applied. The governing equations of tolerance models based on this method take into account the effect of period lengths on the overall behaviour of the plate. Hence, the additional effects of the periodicity can be analysed, as higher order vibrations. Moreover, properties of the plate in the periodicity cell are determined uncertainly. To analyse an influence of random variables of the properties with a fixed probability distribution on vibrations of the plate the Monte Carlo analysis is applied.
... Analogously, the same scheme coupled with higher-order shear deformation theories was employed to achieve the numerical solution to similar problems [39][40][41][42][43][44][45][46][47][48][49][50][51]. Finally, it should be noted that the GDQ method can be used to solve the strong formulation when the reference domain is characterized by arbitrary geometries [52][53][54][55][56][57][58][59][60][61][62][63][64][65]. In general, the strong formulation needs a higher order of derivation in comparison with the corresponding weak form, which is introduced to weaken (or reduce) the order of differentiability [3]. ...
The main aim of the paper is to present a new numerical method to solve the weak formulation of the governing equations for the free vibrations of laminated composite shell structures with variable radii of curvature. For this purpose, the integral form of the stiffness matrix is computed numerically by means of the Generalized Integral Quadrature (GIQ) method. A two-dimensional structural model is introduced to analyze the mechanical behavior of doubly-curved shells. The displacement field is described according to the basic aspects of the general Higher-order Shear Deformation Theories (HSDTs), which allow to define several kinematic models as a function of the free parameter that stands for the order of expansion. Since an Equivalent Single Layer (ESL) approach is considered, the generalized displacements evaluated on the shell middle surface represent the unknown variables of the problem, which are approximated by using the Lagrange interpolating polynomials. The mechanical behavior of the structures is modeled through only one element that includes the double curvature in its formulation, which is transformed into a distorted domain by means of a mapping procedure based on the use of NURBS (Non-Uniform Rational B-Splines) curves, following the fundamentals of the well-known Isogeometric Analysis (IGA). For these reasons, the presented methodology is named Weak Formulation Isogeometric Analysis (WFIGA) in order to distinguish it from the corresponding approach based on the strong form of the governing equations (Strong Formulation Isogeometric Analysis or SFIGA), previously introduced by the authors. Several numerical applications are performed to test the current method. The results are validated for different boundary conditions and various lamination schemes through the comparison with the solutions available in the literature or obtained by a finite element commercial software.
... In addition, this study was limited to two-dimensional thin anisotropic plates. Fantuzzi et al. [24] used a strong formulation finite element method (SFEM) based on the radial basis function (RBF) and generalized differential quadrature (GDQ) techniques to conduct the dynamic analyses of multilayered plates of arbitrary shapes. The numerical results in terms of natural frequencies were compared to the literature and finite element results. ...
The free vibration behavior of quasi-isotropic carbon fiber laminated composite plates containing circular holes with free-clamped boundary conditions are numerically, analytically, and experimentally investigated. Finite element models based on classical plate theory (Kirchhoff) and the shear deformable theory (Mindlin) within the framework of equivalent single-layer and layer-wise concepts as well as the three-dimensional theory of elasticity are developed. These models are created using the finite element software, Abaqus, to determine the natural frequencies and the corresponding mode shapes. In addition, an analytical model based on Kirchhoff plate theory is developed. Using this approach, an equivalent bending-torsion beam model for cantilever laminated plates is extracted taking into account the reduction in local stiffness and mass induced by the center hole. Experimental vibration analyses are carried out using an optically-based vibration measurement tool to extract the frequency response functions and to measure the natural frequencies. Numerical and analytical natural frequency values are then compared with those obtained through experimental vibrational tests, and the accuracy of each finite element (FE) and analytical model type is assessed. It is shown that the natural frequencies obtained using the analytical and FE models are within 8% of the experimentally determined values.
... Therefore, many researchers used the mentioned value even for the laminated composite plates (e.g. see [44][45][46] ). For generality of the numerical results and easiness in comparison with their counterparts in the literature, the aforementioned value is used for the shear correction factor in conjunction with the FSDT throughout this study. ...
... Tornabene et al. [24] applied the radial basis function method to doubly-curved laminated composite shells. This work followed by Fantuzzi et al. [25,26], where they researches deals with Radial Basis Function method for laminated composite arbitrarily shaped plates. Recently few researcher shows significant interest in applying mesh free methods in their research [27][28][29][30]. ...
Two type of numerical approach namely, Radial
Basis Function and Spline approximation, used to analyse
the free vibration of anti-symmetric angle-ply laminated
plates under clamped boundary conditions. The equations
of motion are derived using YNS theory under first order
shear deformation. By assuming the solution in separable
form, coupled differential equations obtained in
term of mid-plane displacement and rotational functions.
The coupled differential is then approximated using Spline
function and radial basis function to obtain the generalize
eigenvalue problem and parametric studies are made to investigate
the effect of aspect ratio, length-to-thickness ratio,
number of layers, fibre orientation and material properties
with respect to the frequency parameter. Some results
are compared with the existing literature and other
new results are given in tables and graphs.
... The present numerical application presents an elliptic plate with two cut-outs [90]. One hole has a rectangular shape, and the other one has a squared shape. ...
In the present paper strong form finite elements are employed for the free vibration study of laminated arbitrarily shaped plates. In particular, the stability and accuracy of three different Fourier expansion-based differential quadrature techniques are shown. These techniques are used to solve the partial differential system of equations inside each computational element. The three approaches are called Harmonic Differential Quadrature (HDQ), Fourier Differential Quadrature (FDQ) and Improved Fourier expansion-based Differential Quadrature (IFDQ) methods. IFDQ method implements auxiliary functions in order to approximate functional derivatives up to the fourth order, with respect to FDQ method that has a basis made of sines and cosines. All the present applications are related to literature comparisons and the presentation of new results for further investigation within the same topic. A study of such kind has never been proposed in the liteature and it could be useful as a reference for future investigation in this matter.
The main aim of this book is to analyze the mathematical
fundamentals and the main features of the Generalized
Differential Quadrature (GDQ) and Generalized Integral
Quadrature (GIQ) techniques. Furthermore, another interesting
aim of the present book is to shown that from the two numerical
techniques mentioned above it is possible to derive two different
approaches such as the Strong and Weak Finite Element
Methods (SFEM and WFEM), that will be used to solve various
structural problems and arbitrarily shaped structures.
A general approach to the Differential Quadrature is
proposed. The weighting coefficients for different basis
functions and grid distributions are determined. Furthermore,
the expressions of the principal approximating polynomials
and grid distributions, available in the literature, are shown.
Besides the classic orthogonal polynomials, a new class of basis
functions, which depend on the radial distance between the
discretization points, is presented. They are known as Radial
Basis Functions (or RBFs). The general expressions for the
derivative evaluation can be utilized in the local form to reduce
the computational cost. From this concept the Local Generalized
Differential Quadrature (LGDQ) method is derived.
The Generalized Integral Quadrature (GIQ) technique can be
used employing several basis functions, without any restriction
on the point distributions for the given definition domain.
To better underline these concepts some classical numerical
integration schemes are reported, such as the trapezoidal rule or
the Simpson method. An alternative approach based on Taylor
series is also illustrated to approximate integrals. This technique
is named as Generalized Taylor-based Integral Quadrature (GTIQ)
method.
The major structural theories for the analysis of the
mechanical behavior of various structures are presented in depth
in the book. In particular, the strong and weak formulations
of the corresponding governing equations are discussed and
illustrated. Generally speaking, two formulations of the same
system of governing equations can be developed, which are
respectively the strong and weak (or variational) formulations.
Once the governing equations that rule a generic structural
problem are obtained, together with the corresponding boundary
conditions, a differential system is written. In particular, the
Strong Formulation (SF) of the governing equations is obtained.
The differentiability requirement, instead, is reduced through
a weighted integral statement if the corresponding Weak
Formulation (WF) of the governing equations is developed. Thus,
an equivalent integral formulation is derived, starting directly
from the previous one. In particular, the formulation in hand is
obtained by introducing a Lagrangian approximation of the
degrees of freedom of the problem.
The need of studying arbitrarily shaped domains or
characterized by mechanical and geometrical discontinuities
leads to the development of new numerical approaches that
divide the structure in finite elements. Then, the strong form or
the weak form of the fundamental equations are solved inside
each element. The fundamental aspects of this technique,
which the author defined respectively Strong Formulation Finite
Element Method (SFEM) and Weak Formulation Finite Element
Method (WFEM), are presented in the book.
In this article, a novel weak‐form zonal Petrov–Galerkin free element method is proposed for two‐ and three‐dimensional linear mechanical problems. By absorbing the advantages of finite block method and strong‐form finite element method, the block mapping technique is used in the free element method. Combining the characteristics of the meshless local Petrov–Galerkin method, the local Petrov–Galerkin formulation based on the zonal free element method is formed at last. Besides, the local integral domain selected in the local collocation element is circular or spherical to simplify programming. The transformation of the local integral domain between the physical and normalized spaces is given for two‐ and three‐dimensional problems. The comparison of accuracy and convergence between the new proposed Petrov–Galerkin method and the conventional methods is carried out. Some challenging examples including fracture mechanics problems and a complex 3D problem are given to validate the convergence and accuracy of the proposed method.
In the study, a method is proposed and developed for solid analysis that is a hybrid of the radial basis function and the finite element method (RBF-FEM). Based on the finite nodes, the method employs the radial basis functions to produce the shape functions with simplicity, especially when increasing the element node number. The formulation and applications of the method in the analysis of solids are examined. Several numerical examples are carried out to analyze the convergence, accuracy, and computational time cost. Its comeouts are then compared with that by the finite element method. The study also shows the factors that affect the accuracy of the method, such as the radial basis function types, radial basis function shape parameter, node number, and node position in an element. It is shown that the method quickly produces better results with large-node-number elements (seven and eight nodes) in comparison to the conventional numerical method. Discussion on the method and the extensibility of the approach are addressed in the conclusion.
Smart materials, whether natural or synthetic, often work in the thermo-electro-mechanical coupled field and are sensitive to stimulation from the external environment. This paper is aimed to simulate the static and transient thermo-electro-mechanical responses of the piezoelectric structures by the weak-form meshless method named thermal piezoelectric zonal Galerkin free element method (TP-ZGFREM). In this method, the computational domain is first divided into a group of zones and then each zone is discretized by nodes, which makes the method have a strong ability to complicated geometry. Then, the stability and accuracy of the results are ensured by using the Galerkin method in the process of establishing the equation for each node. Besides, the influence of temperature on the responses of piezoelectric structures is discussed in detail. The piezoelectric tuning fork with anisotropic material properties and the corrugated sandwich structure with piezoelectric bodies in practical engineering applications are given in the numerical examples, and the results that are a good approximation to the finite element method verify the maturity of the proposed method, which can provide a reference value for the practical design of smart structure subjected to thermal loads.
A novel strong-form numerical algorithm, piezoelectric vibration element differential method (PVEDM), is proposed for simulating the static deflection and forced vibration of the structure integrated with piezoelectric layers, with the host structure being homogeneous or functionally graded materials. A unified manner for the steady-state and dynamic responses of piezoelectric structures is set up by the proposed method, which draws on the merits of the finite element method and collocation method. In the whole process of assembling the system of equations, variational principle and integration are not required. Furthermore, the influence of boundary conditions on static deflection, and static shape control are investigated. Three examples of static and dynamic responses from one-layer structure, bimorph structure to the structure bonded with piezoelectric layers are given in turn. By comparing with analytical solution or ABAQUS, precise results are achieved, which verifies the the accuracy of the method.
In this paper, the fracture mechanics analysis in functionally graded materials and structures (FGMs) is presented. The elemental differential method, is extended to simulate the fracture behaviors of the functionally graded materials, in which the system of equations is established directly based on the equilibrium equations. The first and second order differentiations of the shape functions are utilized to interpolate the geometrical and physical variables within the isoparametric elements. A novel collocation strategy is introduced to construct the system of equations by the governing equations and the traction equilibrium equations according to the nodal distributions in the mesh grids of the structures. Furthermore, a mixed collocation element differential method is further proposed to handle the singular points in the computation domains such as the crack tips and structural corners. The weak-form formulations, such as the weighted residuals approach, are utilized to establish the system of equations for nodes within the domain of elements. Thus, the strong-weak form method can combine the superiorities of the standard finite element methods and the strong-form methods for the aspects of easily constructing shape functions and directly generating system of equations. Numerical examples about the stress intensity factors of the static and dynamic problems in functionally graded materials are presented to validate the proposed methods.
A Generalized Differential Quadrature (GDQ) as an accurate numerical technique based on non-uniform grid point distribution, Chebyshev-Gauss-Lobatto (CGL) and Roots of the Legendre Polynomial (RLP) is investigated for active vibration suppression of flexible spacecraft appendages embedded with piezoelectric (PZT) patches. The flexibility of the system is modeled as a sandwich panel with honeycomb core via high-order theories to monitor extra vibrations of the system for high accuracy missions. The coupled governing partial differential equations of the motion and the corresponding boundary conditions were derived through Hamilton's principle. The spacecraft is maneuvered by constant and harmonic torques with different excitation frequency to analyze the vibration sensitivity of the system. The Strain Rate Feedback (SRF) control law is utilized to apply the effects of PZTs action on vibration suppression of flexible appendages. The numerical study of the system characterized by coupled rigid-flexible (high-order) dynamic provides a powerful general tool for analysis of maneuvering spacecraft with smart sandwich appendages and demonstrates the importance of the proposed formulation for the prediction of higher mode vibration response of flexible parts.
Element differential method (EDM), as a newly proposed numerical method, has been applied to solve many engineering problems because it has higher computational efficiency and it is more stable than other strong‐form methods. However, due to the utilization of strong‐form equations for all nodes, EDM become not so accurate when solving problems with abruptly changed boundary conditions. To overcome this weakness, in this paper, the weak‐form formulations are introduced to replace the original formulations of element internal nodes in EDM, which produce a new strong‐weak‐form method, named as weak‐form element differential method (WEDM). WEDM has advantages in both the computational accuracy and the numerical stability when dealing with the abruptly changed boundary conditions. Moreover, it can even achieve higher accuracy than finite element method (FEM) in some cases. In this paper, the computational accuracy of EDM, FEM and WEDM are compared and analyzed. Meanwhile, several examples are performed to verify the robustness and efficiency of the proposed WEDM.
In this paper, a family of global elements (GEs) are constructed for modeling geometries and representing physical variables, based on a set of complete basis functions formulated in terms of normalized global coordinates. The main benefits of using GEs are that the elemental nodes can be distributed and numbered in an arbitrary manner and the global spatial partial derivatives of geometries and physical variables appearing in the governing equations of engineering problems can be directly derived. Based on the constructed GEs and their spatial derivatives of global shape functions, a simple and robust new numerical method, called as the Global-Element-based FRee Element Method (GEFREM), is proposed for solving general two-dimensional heat conduction problems. GEFREM inherits the advantages of the finite element method, mesh free method and free element method. A detailed description of GEFREM for solving general non-linear and inhomogeneous heat conduction problems will be presented in the paper and a number examples are given to verify the correctness and demonstrate the potential of the proposed method.
In this paper, a new numerical method, named as the Free Element Collocation Method (FECM), is proposed for solving general engineering problems governed by the second order partial differential equations (PDEs). The method belongs to the group of the collocation method, but the spatial partial derivatives of physical quantities are computed based on the isoparametric elements as used in FEM. The key point of the method is that the isoparametric elements used can be freely formed by the nodes around the collocation node. To achieve a narrow bandwidth of the final system of equations, elements with a central node are recommended. For this purpose, a new 21-node quadratic element for 3D problems is constructed for the first time. Attributed to the use of isoparametric elements, FECM can result in more stable results than the traditional collocation method. In addition, the elements can be freely formed by local nodes, FECM has the advantage of mesh-free methods to fit complicated geometries of engineering problems. A number of numerical examples of 2D and 3D thermal and mechanical problems are given to demonstrate the correctness and efficiency of the proposed method.
Many rectangular plate elements developed in the history of finite element method (FEM) have displayed excellent numerical properties, yet their applications have been limited due to inability to conform to the arbitrary geometry of plates and shells. Numerical manifold method (NMM), considered to be a generalization of FEM, can easily solve this issue by viewing a mesh made up of rectangular elements as mathematical cover. In this study, ACM element (Adini and Clough element from A. Adini, R.W. Clough, Analysis of plate bending by the finite element method, University of California, 1960), a typical rectangular plate element is first integrated in the framework of NMM. Then, vibration analysis of arbitrary shaped thin plates is conducted employing the tailored NMM. Using the definition of integral of scalar functions on manifolds, we developed a mathematically rigorous mass lumping scheme for creating a symmetric and positive definite lumped mass matrix that is easy to inverse. A series of numerical experiments have been studied and analyzed, including free and forced vibration of thin plates with various shapes, validating the proposed mass lumping scheme can supersede the consistent mass formulation in those cases.
The numerical simulation of sloshing waves for Laplace equation with nonlinear free surface boundary condition in a two-dimensional (2D) rectangular tank is performed using the Differential Quadrature Method (DQM). Application of the DQM to the Laplace equation and the nonlinear free surface boundary condition gives two sets of Ordinary Differential Equations (ODEs) in time. These two sets of ODEs are coupled with each other and can be expressed as a system of nonlinear ODEs which can be further discretized in time using various time integration schemes. The resultant system of nonlinear algebraic equations can then be solved using various iterative methods. In this study, the backward difference time integration scheme (of order six) in conjunction with the Newton-Raphson method is used to solve the resultant system of nonlinear ODEs. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison study with the available solutions in the literature is performed. Numerical results reveal that the DQM can be used as an effective tool for handling nonlinear sloshing problems.
In this study, a new formulation of finite element method (FEM) has been extracted to analyze 2D viscoelastic problems. As there has not been enough accuracy and not sufficient literature in classical finite element modeling of viscoelastic problems, using a new set of shape functions founded on radial basis functions (RBFs) is recommended. Applying these new, RBF-based shape functions instead of the classical Lagrangian ones, results in subtler answers and conducts a reconsideration over the usual numerical method. Hankel functions are chosen, enriched and summed up with polynomial terms. Therefore, they satisfy not only polynomial terms, but also the first- and second-order Bessel functions simultaneously; which, in the case of classic shape functions, happens only for the polynomial function field. This method illustrates an approach with faster convergence rate and better robustness in different manners. Hence, it is less time-consuming and economical. Finally, various numerical examples are provided for the comparison of analytical solution, classic FEM and Hankel-based FEM, which show the much better agreement of the proposed method with analytical solution in comparison to classic FEM. Also, the number of nodes and degrees of freedom are reduced noticeably while maintaining accuracy in the interpolation of the adopted procedure.
In this paper, finite element method (FEM) is reformulated using new shape functions to approximate the state variables (i.e., displacement field and its derivatives) and inhomogeneous term (i.e., inertia term) of Navier's differential equation. These shape functions and corresponding elements are called spherical Hankel hereafter. It is possible for these elements to satisfy the polynomial and the first and second kind of Bessel function fields simultaneously, while the classic Lagrange elements can only satisfy polynomial ones. These shape functions are so robust that with least degrees of freedom, much better results can be achieved in comparison with classic Lagrange ones. It is interesting that no Runge phenomenon exists in the interpolation of proposed shape functions when going to higher degrees of freedom, while it may occur in classic Lagrange ones. Moreover, the spherical Hankel shape functions have a significant robustness in the approximation of folded surfaces. Five numerical examples related to the usage of suggested shape functions in finite element method in solving problems are studied and their results are compared with those obtained from classic Lagrange shape functions and analytical solutions (if available) to show the efficiency and accuracy of the present method.
The linear and nonlinear flexure analysis of laminated plates with twenty theories with the five variable higher order shear deformation theory (HSDT) is investigated using multiquadratic radial basis function based meshfree method. The mathematical formulation of the actual physical problem of the plate subjected to transverse loading is presented utilizing von Karman nonlinear kinematics. These non-linear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. The different results for deflection and stresses are obtained for thin to a thick plate and compared with some published results. It is observed that some of the theories taken here are well suited for analysis of thin as well as a thick plate while some theories are suited only for thin plates.
In this article, free flexural vibration and supersonic flutter analyses are studied for cantilevered trapezoidal plates composed of two homogeneous isotropic face sheets and an orthotropic honeycomb core. The plate is modeled based on the first-order shear deformation theory, and aerodynamic pressure of external flow with desired flow angle is estimated via the piston theory. For this goal, first applying the Hamilton's principle, the set of governing equations and boundary conditions are derived. Then, using a transformation of coordinates, the governing equations and boundary conditions are converted from the original coordinates into new computational ones. Finally, the differential quadrature method is employed and natural frequencies, corresponding mode shapes, and critical speed are numerically achieved. Accuracy of the proposed solution is confirmed by the finite element simulations and published experimental results. After the validation, effect of various parameters on the vibration and flutter characteristics of the plate are investigated. It is concluded that geometry of hexagonal cells in the honeycomb core has a weak effect on the natural frequencies and critical speed of the sandwich plate, whereas thickness of the honeycomb core has main influence on the natural frequencies and the critical speed. Besides, it is shown that the honeycomb core thickness has optimum values that lead to the most growth in the natural frequencies or critical speed. These optimum magnitudes can be taken into account by designers to increase the natural frequencies or expand flutter boundaries and make aircrafts safer in supersonic flights. It is also concluded that geometrical parameters of the hexagonal cells and thickness of the honeycomb core have no significant effect on the value of the critical flow angle.
As a useful tool for designing wings and tail fins of aircrafts, this paper presents an optimization for flutter characteristics of cantilevered functionally graded sandwich plates. The plate is composed of an isotropic homogeneous core and two functionally graded face sheets. The plate is modeled based on the first-order shear deformation theory. The aerodynamic pressure is estimated using supersonic piston theory and using Hamilton's principle, the set of governing equations and boundary conditions are then derived. Applying a transformation of coordinates, governing equations and boundary conditions are converted and solved numerically by differential quadrature method. Natural frequencies, damping ratio, corresponding mode shapes, critical aerodynamic pressure, and flutter frequency are calculated. In order to achieve an optimum design, particle swarm optimization is employed to find the best values of aspect ratio, thickness of the plate, thickness of the core, power law index, and angles of the plate which increase critical aerodynamic pressure. Some constrains on the angles of the plate and its mass and area (lift force) are also considered.
In the framework of a unified 2D continuum formulation of the fully coupled thermomechanical laminated plate with von Karman nonlinearities, a consistent model with third order shear deformability and cubic temperature distribution along the thickness is proposed. Focusing on symmetric cross-ply laminates, an effective minimal dimension reduction is then pursued by expressing both in-plane displacement components and shear angles in terms of transverse displacement and thermal variables via kinematic condensations. The ensuing compact reduced order model with different coupling features allows to account for nonlinear mechanical and thermal phenomena, along with their interactions, under a variety of thermomechanical assumptions, boundary conditions and excitations. Upon proper validation in linear free dynamics and critical buckling, numerical investigation of the nonlinear dynamic response under different thermal conditions is accomplished, focusing on the variable features of its transient and post-buckled response, where the thermomechanical interaction plays a meaningful role.
In this paper, based on the first-order shear deformation theory for modeling the structure and the supersonic Piston theory to estimate the aerodynamic pressure, the set of governing equations and boundary conditions for flutter analysis of a trapezoidal thick plate with variable thickness are derived. Using a transformation of coordinates, governing equations and boundary conditions are converted from the original coordinates into a new computational one. Using differential quadrature method, natural frequencies, damping ratio, and corresponding mode shapes are derived, and critical aerodynamic pressure and flutter frequency are determined. Critical aerodynamic pressure of the plate is considered as an objective function to increase and using particle swarm optimization, optimum values of aspect ratio, thickness, variation of thickness, and angles of the plate are found. Meanwhile, some constrains on the volume (weight) and area (lift force) of the plate are considered. This constrained optimization can be considered as a useful tool for design wing and tail fin of aircrafts.
This manuscript comes from the experience gained over thirteen years (2003-2016) of study and research on Laminated Composite Doubly-Curved Shell Structures. The present book came alive when Professor Viola gave to Tornabene the book by Kraus (Thin Elastic Shells, 1967) and the book by Markuš (The Mechanics of Vibrations of Cylindrical Shells, 1988). After that life episode, Tornabene started to study the interesting world of shell structures, concluding his studies at the University of Bologna in 2003 with the Master Thesis (in Italian) entitled: Dynamic Behavior of Cylindrical Shells: Formulation and Solution. After that, he finished in 2007 his PhD in Structural Mechanics at the same University with the PhD Thesis (in Italian) entitled: Modelling and Solution of Shell Structures Made of Anisotropic Material. During these years Tornabene has met Dr. Nicholas Fantuzzi in occasion of his three level degrees at the University of Bologna. In fact, Tornabene was the co-advisor for the three theses that Dr. Fantuzzi discussed in his student carrier. Finally, Tornabene became Assistant Professor at the University of Bologna in 2012 and he published the book (in Italian) entitled: Mechanics of Shell Structures Made of Composite Materials. The Generalized Differential Quadrature Method.
The book by Tornabene represents the first manuscript in Italian language that treats the theoretical aspects about laminated composite shell structures using the differential geometry and that exposes the recovery procedure that allows to evaluate the stresses and the strains through the thickness of a doubly-curved shell structure. It is also the first book that presents to the Italian audience the Differential Quadrature Method and uses this methodology to solve the governing equations of laminated composite doubly-curved shell structures. Furthermore, the three fundamental aspects that characterize the book by Tornabene are two theoretical and one numerical. The first one is the theory considered for studying shell structures: the First-Order Shear Deformation Theory (FSDT). The second one is the use of the Differential Geometry as a powerful tool for describing the shell reference surface. In fact, a huge number of reference surfaces, useful to analyze various shell structures, has been collected by Tornabene in his book. Finally, the third aspect is the numerical technique, called Generalized Differential Quadrature Method. This method allows to approximate the derivatives of geometrical quantities and to solve the system of differential shell equations.
After the previous historical events, it is possible to introduce the present book that represents the translation and the generalization of the book by Tornabene for the worldwide audience. In particular, the present manuscript was written as an attempt to show, in an easy way, the theoretical aspects of doubly-curved composite shell structures. Furthermore, it represents a shortened version of the book entitled: Mechanics of Laminated Composite Doubly-Curved Shell Structures by the same authors, wherein also the numerical part has been presented. The present volume is aimed at Master degree and PhD students in structural and applied mechanics, as well as experts in these fields.
The title, Theory of Laminated Composite Doubly-Curved Shell Structures, illustrates the themes followed in the present volume. The main aim of this book is to analyze the static and dynamic behavior of moderately thick doubly-curved shells made of composite materials.
In fact, this book presents a general approach for studying doubly-curved laminated composite shell structures solved using a numerical methodology based on the strong formulation. The main reason for presenting this book to the engineering community is to review and extend the literature, about shell theories, that appeared in the last seventy years.
The present volume is divided into six chapters, in which static and dynamic analyses of several structural elements are provided in detail. Furthermore, the results of the adopted numerical technique are presented for several problems such as different loading and boundary conditions.
Starting from the Differential Geometry, fundamental tool for the analysis of the structures at issue, the first chapter presents the Theory of Composite Laminated Shell Structures. In the theoretical discussion the displacement field associated to the Reissner-Mindlin theory, also known as “First-order Shear Deformation Theory” (FSDT), is considered. Once the kinematic equations and the constitutive equations are introduced, the indefinite equilibrium equations and the natural boundary conditions are deducted through the Hamilton principle. The equations of doubly-curved shells are worked out and summarized in the scheme of physical theories and specialized to structures of revolution.
As far as the constitutive equations are concerned, particular attention is given to composite materials due to the increasing development in several structural engineering areas. The scientific interest in these materials, that have the high makings of application, suggested the static and dynamic analysis of composite shell structures. A new class of composite materials, recently introduced in literature, is also taken into account. As it is well-known, laminated composite materials are affected by inevitable problems of delamination due to the presence of interfaces where different materials are in contact. On the contrary, “Functionally Graded Materials” (FGMs) are characterized by a continuous variation of the mechanical properties, such as the elastic modulus, material density and Poisson ratio, along a particular direction. This feature is obtained by varying gradually, along a preferential direction, the volume fraction of the constituent materials with appropriate industrial processes. Therefore, FGMs are non-homogeneous materials, typically composed of metal and ceramic.
Starting from the analysis of shells of translation and doubly-curved shells of revolution, in the second chapter the fundamental equations of Main Shell Structures are presented. In this chapter it is shown how to carry out, through simple geometric relations, the governing equations of the elastic problem of conical and cylindrical shells, circular and rectangular plates and translational shells with a generic profile from the equations of doubly-curved shells of revolution.
In the third chapter the 3D Elasticity Equations in Orthogonal Curvilinear Coordinates are presented. They are the basis for a correct recovery of the stress and strain states through the shell thickness. The recovery procedure is necessary because certain effects, due to the transition from a three-dimensional theory to a two-dimensional one, are neglected and this is done to reduce the computational cost of the structural analysis. This simplification of the three-dimensional theory to an engineering theory is due to the introduction of suitable assumptions that limit the applicability of these theories within an appropriate validity range. The three-dimensional equations in curvilinear orthogonal coordinates are worked out through the Hamilton principle.
In the fourth chapter the Theory of Composite Thin Shells is derived in a simple and intuitive manner from the theory of moderately thick shells developed in the first chapter. In particular, the Kirchhoff-Love Theory and the Membrane Theory for composite shells are shown.
The fifth chapter exposes the Theory of Composite Arches and Beams. In particular, the equations of the Timoshenko Theory and the Euler-Bernoulli Theory, with and without curvature, are directly deducted from the equations of singly-curved shells of translation and of plates.
The sixth chapter presents the so-called General Shell Theory in which the curvature effect is embedded into the FSDT kinematic model. This effect is reflected into the stress resultants and strain characteristics of the model. Due to these considerations the stress resultants directly depend on the geometry of the structure in terms of curvature coefficients and the hypothesis of the symmetry of the in-plane shearing force resultants and the torsional or twisting moments is not valid. Furthermore, several numerical applications are presented in the chapter at hand for the sake of completeness.
This book is intended to be a reference for experts in structural, applied and computational mechanics. It can be also used as a text book, or a reference book, for a graduate or PhD courses on plates and shells, composite materials, vibration of continuous systems and stress recovery of the previous structures. Finally, the present book also has the same audience of the book by Professor Harry Kraus (1967). Thus, using his words: “The” present “book is aimed primarily at graduate students at the intermediate level in engineering mechanics, aerospace engineering, mechanical engineering and civil engineering, whose field of specialization is solid mechanics. Stress analysts in industry will find the” present “book a useful introduction that will equip them to read further in the literature of solutions to technically important shell problems, while research specialists will find it useful as an introduction to current theoretical work. This volume is not intended to be an exhaustive treatise on the theory of thin” and thick “elastic shells but, rather, a broad introduction from which each reader can follow his own interests further”. In addition, it is opinion of the authors that the present volume represents the continuation and the generalization of the work begun by Kraus in 1967.
As a first endeavor, the buckling analysis of functionally graded (FG) arbitrary straight-sided quadrilateral plates rested
on two-parameter elastic foundation under in-plane loads is presented. The formulation is based on the first order shear deformation
theory (FSDT). The material properties are assumed to be graded in the thickness direction. The solution procedure is composed
of transforming the governing equations from physical domain to computational domain and then discretization of the spatial
derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. After studying
the convergence of the method, its accuracy is demonstrated by comparing the obtained solutions with the existing results
in literature for isotropic skew and FG rectangular plates. Then, the effects of thickness-to-length ratio, elastic foundation
parameters, volume fraction index, geometrical shape and the boundary conditions on the critical buckling load parameter of
the FG plates are studied.
Three-dimensional free vibration analysis of functionally graded piezoelectric (FGPM) annular plates resting on Pasternak foundations with different boundary conditions is presented. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the influences of the Winkler and shearing layer elastic coefficients of the foundations on the non-dimensional natural frequencies of functionally graded piezoelectric annular plates. The analytical solution in the thickness direction can be acquired using the state-space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. Numerical results are given to demonstrate the convergency and accuracy of the present method. The influences of the material property graded index, circumferential wave number and thickness of the annular plate on the dynamic behavior are also investigated. Since three-dimensional free vibration analysis of FGPM annular plates on elastic foundations has not been implemented before, the new results can be used as benchmark solutions for future researches.
We present a natural element method to treat higher-order spatial derivatives in the Cahn–Hilliard equa-tion. The Cahn–Hilliard equation is a fourth-order nonlinear partial differential equation that allows to model phase sep-aration in binary mixtures. Standard classical C 0 -continuous finite element solutions are not suitable because primal varia-tional formulations of fourth-order operators are well-defined and integrable only if the finite element basis functions are piecewise smooth and globally C 1 -continuous. To ensure C 1 -continuity, we develop a natural-element-based spatial discretization scheme. The C 1 -continuous natural element shape functions are achieved by a transformation of the classical Farin interpolant, which is basically obtained by embedding Sibsons natural element coordinates in a Bernstein–Bézier surface representation of a cubic simplex. For the temporal discretization, we apply the (second-order accurate) trapezoidal time integration scheme supplemented with an adaptively adjustable time step size. Numerical examples are presented to demonstrate the efficiency of the computational algorithm in two dimensions. Both periodic Dirichlet and homogeneous Neumann boundary conditions are applied. Also constant and degenerate mobilities are con- sidered. We demonstrate that the use of C 1 -continuous natural element shape functions enables the computation of topolog-ically correct solutions on arbitrarily shaped domains.
Over the years the Differential Quadrature (DQ) method has distinguished because of its high accuracy, straightforward implementation and general ap- plication to a variety of problems. There has been an increase in this topic by several researchers who experienced significant development in the last years.
DQ is essentially a generalization of the popular Gaussian Quadrature (GQ) used for numerical integration functions. GQ approximates a finite in- tegral as a weighted sum of integrand values at selected points in a problem domain whereas DQ approximate the derivatives of a smooth function at a point as a weighted sum of function values at selected nodes. A direct appli- cation of this elegant methodology is to solve ordinary and partial differential equations. Furthermore in recent years the DQ formulation has been gener- alized in the weighting coefficients computations to let the approach to be more flexible and accurate. As a result it has been indicated as Generalized Differential Quadrature (GDQ) method.
However the applicability of GDQ in its original form is still limited. It has been proven to fail for problems with strong material discontinuities as well as problems involving singularities and irregularities. On the other hand the very well-known Finite Element (FE) method could overcome these issues because it subdivides the computational domain into a certain number of elements in which the solution is calculated. Recently, some researchers have been studying a numerical technique which could use the advantages of the GDQ method and the advantages of FE method. This methodology has got different names among each research group, it will be indicated here as Generalized Differential Quadrature Finite Element Method (GDQFEM).
The purpose of this PhD Thesis is to introduce the limitations of the di- rect GDQ method and more importantly the implementation technique of the GDQFEM. Moreover, in order to show the accuracy, stability and flexibility of the current methodology some numerical examples are shown. The examples are related to the mechanics of civil and mechanical engineering structures such as membranes, state plane structures and flat plates. The static and dynamic behaviour of these structures are proposed in the following chapters. Numerical comparisons with literature and FE analyses are reported and very good agreement is observed in all the computations.
Il presente volume scaturisce dall’esperienza maturata nel corso di circa nove anni di studio e di ricerca sulle strutture a guscio e sul metodo di Quadratura Differenziale. Comprendono il periodo della tesi di laurea in “Scienza delle Costruzioni”, i tre anni del Dottorato di Ricerca in “Meccanica delle Strutture”, e alcuni anni di Assegni di Ricerca svolti dall’autore presso l’Alma Mater Studiorum - Università di Bologna. Il titolo, Meccanica delle Strutture a Guscio in Materiale Composito, illustra il tema trattato e la prospettiva seguita nella scrittura del volume. Il presente elaborato si pone come obiettivo quello di analizzare il comportamento statico e dinamico dei gusci moderatamente spessi in materiale composito attraverso l’applicazione del Metodo Generalizzato di Quadratura Differenziale (GDQ Method). Una particolare attenzione viene riservata oltre che ai compositi fibrosi e laminati anche ai “functionally graded materials” (FGMs). Essi risultano materiali non omogenei, caratterizzati da una variazione continua delle proprietà meccaniche lungo una particolare direzione. La soluzione numerica GDQ viene confrontata con i risultati presenti in letteratura e con quelli forniti e ricavati mediante l’utilizzo di diversi programmi di calcolo strutturale basati sul metodo agli elementi finiti (FEM).
In this paper, an advanced version of the classic GDQ method, called the Generalized Differential Quadrature Finite Element Method (GDQFEM) is for-mulated to solve plate elastic problems with inclusions. The GDQFEM is com-pared with Cell Method (CM) and Finite Element Method (FEM). In particular, stress and strain results at fiber/matrix interface of dissimilar materials are pro-vided. The GDQFEM is based on the classic Generalized Differential Quadrature (GDQ) technique that is applied upon each sub-domain, or element, into which the problem domain is divided. When the physical domain is not regular, the map-ping technique is used to transform the fundamental system of equations and all the compatibility conditions. A differential problem defined on the regular master ele-ment in the computational domain is turned into an algebraic system. With respect to the very well-known Finite Element Method (FEM), the GDQFEM is based on a different approach: the direct derivative calculation is performed by using the GDQ rule. The imposition of the compatibility conditions between two boundaries are also used in the CM for solving contact problems. Since the GDQFEM is a higher-order tool connected with the resolution of the strong formulation of the system of equations, the compatibility conditions must be applied at each disconnection in order to capture the discontinuity between two boundaries, without losing accu-racy. A comparison between GDQFEM, CM and FEM is presented and very good agreement is observed.
This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method. The analyses are worked out considering the First-order Shear Deformation Theory (FSDT) for the above mentioned moderately thick structural elements. The effect of the shell curvatures is included from the beginning of the theory formulation in the kinematic model. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution struc-tures. The discretization of the system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. Comparisons between the present for-mulation and the Reissner-Mindlin theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs. Very good agreement is observed. Finally, new results are presented in order to invest-tigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the behav-ior of laminated shells of revolution.
We combine a layer-wise formulation and a generalized differential quadrature technique for predicting the static deformations and free vibration behaviour of sandwich plates. Through numerical experiments, the capability and efficiency of this strong-form technique for static and vibration problems are demonstrated, and the numerical accuracy and convergence are thoughtfully examined.
This study deals with a mixed static and dynamic optimization of four-parameter functionally graded material (FGM) doubly curved shells and panels. The two constituent functionally graded shell consists of ceramic and metal, and the volume fraction profile of each lamina varies through the thickness of the shell according to a generalized power-law distribution. The Generalized Differential Quadrature (GDQ) method is applied to determine the static and dynamic responses for various FGM shell and panel structures. The mechanical model is based on the so-called First-order Shear Deformation Theory (FSDT). Three different optimization schemes and methodologies are implemented. The Particle Swarm Optimization, Monte Carlo and Genetic Algorithm approaches have been applied to define the optimum volume fraction profile for optimizing the first natural frequency and the maximum static deflection of the considered shell structure. The optimization aim is in fact to reach the frequency and the static deflection targets defined by the designer of the structure: the complete four-dimensional search space is considered for the optimization process. The optimized material profile obtained with the three methodologies is presented as a result of the optimization problem solved for each shell or panel structure.
A simple and accurate mixed modal-differential quadrature formulation is proposed to study the dynamic behavior of beams in contact with fluid. Both free and forced vibration problems are considered. The proposed mixed methodology uses the modal technique for the structural domain while it applies the differential quadrature method (DQM) to the fluid domain. Thus, the governing partial differential equations of the beam and fluid are reduced to a set of ordinary differential equations in time. In the case of forced vibration, the Newmark time integration scheme is employed to solve the resulting system of ordinary differential equations. The proposed formulation, in general, combines the simplicity of the modal method and high accuracy and efficiency of the DQM. Its application is shown by solving some beam-fluid interaction problems. Comparisons with analytical solutions show that the present method is very accurate and reliable. To demonstrate its efficiency, the test problems are also solved using the finite element method (FEM). It is found that the proposed method can produce better accuracy than the FEM using less computational time. The technique presented in this investigation is general and can be used to solve various fluid-structure interaction problems.
A simple and accurate mixed finite element-differential quadrature formulation is proposed to study the free vibration of rectangular and skew Mindlin plates with general boundary conditions. In this technique, the original plate problem is reduced to two simple bar (or beam) problems. One bar problem is discretized by the finite element method (FEM) while the other by the differential quadrature method (DQM). The mixed method, in general, combines the geometry flexibility of the FEM and high accuracy and efficiency of the DQM and its implementation is more easier and simpler than the case where the FEM or DQM is fully applied to the problem. Moreover, the proposed formulation is free of the shear locking phenomenon that may be encountered in the conventional shear deformable finite elements. A simple scheme is also presented to exactly implement the mixed natural boundary conditions of the plate problem. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of rectangular and skew Mindlin plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of rectangular and skew Mindlin plates with general boundary conditions.
The aim of the contribution is to formulate a macroscopic mathematical model describing the dynamic behaviour of a certain composite thin plates. The plates are made of two-phase stratified composites with a smooth and a slow gradation of macroscopic properties along the stratification. The formulation of mathematical model of these plates is based on a tolerance averaging approach (Woźniak, Michalak, Jędrysiak in Thermomechanics of microheterogeneous solids and structures, 2008). The presented general results are illustrated by analysis of the natural frequencies for two cases of plates: a plate band and an annular plate. The spatial volume fractions of the two different isotropic homogeneous components are optimized so as to maximize or minimize the first natural frequency of the plate under consideration.
The paper establishes a relation between exact sequences, parametric finite elements, and perfectly matched layer (PML) techniques. We illuminate the analogy between the Piola-like maps used to define parametric H
1-, H(curl)-, H(div)-, and L
2-conforming elements, and the corresponding PML complex coordinates stretching for the same energy spaces. We deliver a method for obtaining PML-stretched bilinear forms (constituting the new weak formulation for the original problem with PML absorbing boundary layers) directly from their classical counterparts.
Laminated composite plates and shells are made from a variety of materials. They have quite different mechanical features compared with those made of single material, such as uncertain principal direction of the material, discontinuity of material between layers, highly geometrical and material nonlinearity, etc. Their failure modes include matrix cracking, debonding, delamination, crack deflection, multi delamination and delamination propagation, which are much more complex than those of single material. Based on different considerations, various methods have been proposed by scholars from different countries to study the failure of laminated composite plates and shells. This paper summarizes the fundamental theory of linear mechanics and reviews the development of nonlinear theories for laminated plates and shells. In particularly, theoretical systems and basic formulas are expatiated for the classical nonlinear theory of large deformation, the first order shear deformation theory, the high order shear deformation theory, the zig-zag theory, and the layer-wise theory. The relevance and differences among these theories are stated. Current research progress in the field of nonlinear mechanics for laminated composite plates and shells are overviewed and the latest achievements are introduced in research hotspots regarding the failure mechanism and optimization design of typical laminated composite plates and shells, the failure mechanism of laminated composite plates and shells in complex environments, material nonlinearity of composite plates and shells, failure mechanism of fiber reinforced delaminated composite plates and shells, and so on. Based on the review, prospects for future research in the area of nonlinear mechanics of laminated composite plates and shells are proposed. © 2017, Editorial Office of Chinese Journal of Theoretical and Applied Mechanics. All right reserved.
Composite materials consist of two or more materials which together produce desirable properties that may not be achieved with any of the constituents alone. Fiber-reinforced composite materials, for example, consist of high strength and high modulus fibers in a matrix material. Reinforced steel bars embedded in concrete provide an example of fiber-reinforced composites. In these composites, fibers are the principal loadcarrying members, and the matrix material keeps the fibers together, acts as a load-transfer medium between fibers, and protects fibers from being exposed to the environment (e.g., moisture, humidity, etc.).
In the analysis of sandwich laminates, where core and skin materials are so different, layerwise formulations should be adopted. In this paper the static and free vibration analysis of sandwich plates by the use of collocation with radial basis functions and using a new layerwise theory with independent rotations in each layer and thickness stretching was performed. With this formulation, transverse normal and shear deformations and stresses are accurately computed. The equations of motion were automatically implemented via a Unified Formulation and interpolated with radial basis functions. Finally composite laminated plate and sandwich plate examples were tested and discussed.
In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a powerful tool which work well in very general circumstances and so are becoming of widespread use as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.
Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper.
Spectral methods involve seeking the solution to a differential equation in terms of a series of known, smooth functions. They have recently emerged as a viable alternative to finite difference and finite element methods for the numerical solution of partial differential equations. The key recent advance was the development of transform methods for the efficient implementation of spectral equations. Spectral methods have proved particularly useful in numerical fluid dynamics where large spectral hydrodynamics codes are now regularly used to study turbulence and transition, numerical weather prediction, and ocean dynamics. In this monograph, we discuss the formulation and analysis of spectral methods. It turns out that several features of this analysis involve interesting extensions of the classical numerical analysis of initial value problems.
This monograph is based on part of a series of lectures presented by one of us (S.A.O.) at the NSF—CBMS Regional Conference held at Old Dominion University from August 2–6, 1976. This conference was supported by the National Science Foundation.
We should like to thank our colleagues M. Deville, M. Dubiner, M. Gunzburger, B. Gustaffson, D. Haidvogel, M. Israeli, and J. Ortega for helpful discussions. We are grateful to E. Cohen, A. Patera, and K. Pitman for their assistance in preparing graphs and tables. Some calculations were performed at the Computing Facility of the National Center for Atmospheric Research which is supported by the National Science Foundation. One of us (D.G.) would like to acknowledge support by the National Aeronautics and Space Administration while in residence at ICASE, NASA Langley Research Center, Hampton, Virginia. Both authors would like to acknowledge support by the Fluid Dynamics Branch of the Office of Naval Research and the Atmospheric Sciences Section of the National Science Foundation.
Hampton, Virginia, Cambridge, Massachusetts
September 1977
This paper studies the differential quadrature finite element method (DQFEM) systematically, as a combination of differential quadrature method (DQM) and standard finite element method (FEM), and formulates one- to three-dimensional (1-D to 3-D) element matrices of DQFEM. It is shown that the mass matrices of C0 finite element in DQFEM are diagonal, which can reduce the computational cost for dynamic problems. The Lagrange polynomials are used as the trial functions for both C0 and C1 differential quadrature finite elements (DQFE) with regular and/or irregular shapes, this unifies the selection of trial functions of FEM. The DQFE matrices are simply computed by algebraic operations of the given weighting coefficient matrices of the differential quadrature (DQ) rules and Gauss-Lobatto quadrature rules, which greatly simplifies the constructions of higher order finite elements. The inter-element compatibility requirements for problems with C1 continuity are implemented through modifying the nodal parameters using DQ rules. The reformulated DQ rules for curvilinear quadrilateral domain and its implementation are also presented due to the requirements of application. Numerical comparison studies of 2-D and 3-D static and dynamic problems demonstrate the high accuracy and rapid convergence of the DQFEM.
This article presents the Ritz method for the vibration analysis of sandwich plates having an orthotropic core and laminated facings. The planform of the plate may take on any arbitrary shape. On the basis of the Mindlin plate theory and the Ritz method, the governing eigenvalue equation for determining the natural frequencies was derived. The Ritz method was automated and made computationally effective for general-shaped plates with any boundary conditions by (1) adopting the product of polynomial functions and boundary equations that were raised to appropriate powers and (2) applying Green's theorem to transform the integration over the general-shaped domain into a closed line integration. The Ritz formulation and software were verified by the close agreement with vibration frequencies obtained by previous researchers for a wide range of subset plate problems involving isotropic, laminated, and sandwich plates of various shapes. Moreover, sample natural frequencies of sandwich plates with laminated facings are presented for some quadrilateral plate shapes. These frequencies should be useful as reference results to researchers who are developing new methods or software for vibration analysis of sandwich plates.
The results of a series of numerical experiments are presented to verify some of the important developments made in the first part of this paper. Firstly, the static solution of an algebraic system obtained through Strong Formulation Finite Element Method (SFEM) is presented. Secondly, the stress and strain recovery procedure is descripted for the present technique. It will be clear that the present approach is suitable for any strong formulation finite element methodology, due to the presented general approach based on the unknown displacements and on the elasticity equations. Thirdly, the numerical solutions for some classical and other numerical results found in literature are exposed. Finally, an arbitrarily shaped composite plate is solved and good agreement is observed for all the presented cases.
This paper provides a new technique for solving the static analysis of arbitrarily shaped composite plates by using Strong Formulation Finite Element Method (SFEM). Several papers in literature by the authors have presented the proposed technique as an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The present methodology joins the high accuracy of the strong formulation with the versatility of the well-known Finite Element Method (FEM). The continuity conditions among the elements is carried out by the compatibility or continuity conditions. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is easy and straightforward. The main novelty of this paper is the application of the stress and strain recovery once the displacement parameters are evaluated. Computer investigations concerning a large number of composite plates have been carried out. SFEM results are compared with those presented in literature and a perfect agreement is observed.
The original direct differential quadrature (DQ) method has been known to fail for problems with strong nonlinearity and material discontinuity as well as for problems involving singularity, irregularity, and multiple scales. But now researchers in applied mathematics, computational mechanics, and engineering have developed a range of innovative DQ-based methods to overcome these shortcomings. Advanced Differential Quadrature Methods explores new DQ methods and uses these methods to solve problems beyond the capabilities of the direct DQ method.
After a basic introduction to the direct DQ method, the book presents a number of DQ methods, including complex DQ, triangular DQ, multi-scale DQ, variable order DQ, multi-domain DQ, and localized DQ. It also provides a mathematical compendium that summarizes Gauss elimination, the Runge–Kutta method, complex analysis, and more. The final chapter contains three codes written in the FORTRAN language, enabling readers to quickly acquire hands-on experience with DQ methods.
Focusing on leading-edge DQ methods, this book helps readers understand the majority of journal papers on the subject. In addition to gaining insight into the dynamic changes that have recently occurred in the field, readers will quickly master the use of DQ methods to solve complex problems.
Collocation is based on the discretization of the strong form of the underlying partial differential equations, which requires basis functions of sufficient order and smoothness. Consequently, the use of isogeometric analysis (IGA) for collocation suggests itself, since splines can be readily adjusted to any order in polynomial degree and continuity required by the differential operators. In addition, they can be generated for domains of arbitrary geometric and topological complexity, directly linked to and fully supported by CAD technology. The major advantage of isogeometric collocation over Galerkin type IGA is the minimization of the computational effort for numerical quadrature. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
This paper provides a new technique for solving free vibration problems of composite arbitrarily shaped membranes by using Generalized Differential Quadrature Finite Element Method (GDQFEM). The proposed technique, also known as Multi-Domain Differential Quadrature (MDQ), is an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The multi-domain method can be directly applied to regular sub-domains of rectangular shape, as well as to elements of general shape when a coordinate transformation is considered. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the parent space, called computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is simple and straightforward. Computer investigations concerning a large number of membrane geometries have been carried out. GDQFEM results are compared with those presented in literature and a perfect agreement is observed. Membranes of complex geometry with a material inhomogeneity are also carefully examined. Numerical results referring to some new unpublished geometric shapes are reported to let comparisons with further research on this subject.
This work focuses on the static analysis of functionally graded (FGM) and laminated doubly-curved shells and panels resting on nonlinear and linear elastic foundations using the Generalized Differential Quadrature (GDQ) method. The First-order Shear Deformation Theory (FSDT) for the aforementioned moderately thick structural elements is considered. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Several types of shell structures such as doubly-curved shells (elliptic and hyperbolic hyperboloids), singly-curved (spherical, cylindrical and conical shells), and degenerate panels (rectangular plates) are considered in this paper. The main contribution of this paper is the application of the differential geometry within GDQ method to solve doubly-curved FGM shells resting on nonlinear elastic foundations. The linear Winkler-Pasternak elastic foundation has been considered as a special case of the nonlinear elastic foundation proposed herein. The discretization of the differential system by means of the GDQ technique leads to a standard nonlinear problem, and the Newton-Raphson scheme is used to obtain the solution. Two different four-parameter power-law distributions are considered for the ceramic volume fraction of each lamina. In order to show the accuracy of this methodology, numerical comparisons between the present formulation and finite element solutions are presented. Very good agreement is observed. Finally, new results are presented to show effects of various parameters of the nonlinear elastic foundation on the behavior of functionally graded and laminated doubly-curved shells and panels.
The aim of this work is to study the static behavior of 2D soft core plane state structures. Deflections and inter-laminar stresses caused by forces can have serious consequences for strength and safety of these structures. Therefore, an accurate identification of the variables in hand is of considerable importance for their technical design. It is well-known that for complex plane structures there is no analytical solution, only numerical procedures can be used to solve them. In this study two numerical techniques will be taken mainly into account: the Gen-eralized Differential Quadrature Finite Element Method (GDQFEM) and the Cell Method (CM). The former numerical technique is based on the classic Generalized Differential Quadrature (GDQ) rule and operates differently from the classic Finite Element Method (FEM). The principal novelty of this paper regards the compari-son, by means of several numerical applications about soft-core structures, among GDQFEM, CM and FEM. Such a comparison appears for the first time in the liter-ature and in this paper.
In the present paper the Generalized Differential Quadrature Finite El-ement Method (GDQFEM) is applied to deal with the static analysis of plane state structures with generic through the thickness material discontinuities and holes of various shapes. The GDQFEM numerical technique is an extension of the Gener-alized Differential Quadrature (GDQ) method and is based on the idea of conven-tional integral quadrature. In particular, the GDQFEM results in terms of stresses and displacements for classical and advanced plane stress problems with discon-tinuities are compared to the ones by the Cell Method (CM) and Finite Element Method (FEM). The multi-domain technique is implemented in a MATLAB code for solving irregular domains with holes and defects. In order to demonstrate the accuracy of the proposed methodology, several numerical examples of stress and displacement distributions are graphically shown and discussed.
This paper investigates the static analysis of doubly-curved laminated composite shells and panels. A theoretical
formulation of 2D Higher-order Shear Deformation Theory (HSDT) is developed. The middle surface
of shells and panels is described by means of the differential geometry tool. The adopted HSDT is
based on a generalized nine-parameter kinematic hypothesis suitable to represent, in a unified form,
most of the displacement fields already presented in literature. A three-dimensional stress recovery procedure
based on the equilibrium equations will be shown. Strains and stresses are corrected after the
recovery to satisfy the top and bottom boundary conditions of the laminated composite shell or panel.
The numerical problems connected with the static analysis of doubly-curved shells and panels are solved
using the Generalized Differential Quadrature (GDQ) technique. All displacements, strains and stresses
are worked out and plotted through the thickness of the following six types of laminated shell structures:
rectangular and annular plates, cylindrical and spherical panels as well as a catenoidal shell and an elliptic
paraboloid. Several lamination schemes, loadings and boundary conditions are considered. The GDQ
results are compared with those obtained in literature with semi-analytical methods and the ones computed
by using the finite element method.
The aim of this study is to clarify the discrepancy regarding the critical flow speed of straight pipes conveying fluids that appears to be present in the literature by using the Generalized Differential Quadrature method. It is well known that for a given “mass of the fluid” to the “mass of the pipe” ratio, straight pipes conveying fluid are unstable by a flutter mode via Hopf bifurcation for a certain value of the fluid speed, i.e. the critical flow speed. However, there seems to be lack of consensus if for a given mass ratio the system might lose stability for different values of the critical flow speed or only for a single speed value. In this paper an attempt to answer to this question is given by solving the governing equation following first the practical aspect related to the engineering problem and than by simply considering the mathematics of the problem. The
Generalized differential quadrature method is used as a numerical technique to resolve this problem. The differential governing equation is transformed into a discrete system of algebraic equations. The stability of the system is thus reduced to an eigenvalue problem. The relationship between the eigenvalue branches and the corresponding unstable flutter modes are shown via Argand diagram. The transfer of flutter-type instability from one eigenvalue branch to another is thoroughly investigated and discussed. The critical mass ratios, at which the transfer of the eigenvalue branches related to flutter take place, are determined.
This paper is focused on the Generalized Differential Quadrature (GDQ) Method to study the free vibration of conical shell structures. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion are expressed as functions of five kinematic parameters. Numerical solutions are obtained.
This paper investigates the dynamic behavior of moderately thick composite plates of arbitrary shape using the Generalized Differential Quadrature Finite Element Method (GDQFEM), when geometric dis-continuities through the thickness are present. In this study a five degrees of freedom structural model, which is also known as the First-order Shear Deformation Theory (FSDT), has been used. GDQFEM is an advanced version of the Generalized Differential Quadrature (GDQ) method which can discretize any derivative of a partial differential system of equations. When the structure under consideration shows an irregular shape, the GDQ method cannot be directly applied. On the contrary, GDQFEM can always be used by subdividing the whole domain into several sub-domains of irregular shape. Each irregular ele-ment is mapped on a parent regular domain where the standard GDQ procedure is carried out. The con-nections among all the GDQFEM elements are only enforced by inter-element compatibility conditions. The equations of motion are written in terms of displacements and solved starting from their strong for-mulation. The validity of the proposed numerical method is checked up by using Finite Element (FE) results. Comparisons in terms of natural frequencies and mode shapes for all the reported applications have been performed.
The accuracy of the multiquadratic radial basis functions collocation method in performing elasto-static analyses of multilayer composite plates is strongly affected by the shape parameter, a value that dominates the shape of the radial basis functions. The selection of an optimal value of the shape parameter is still an open problem and several approaches have been proposed in the open literature. In this paper, a novel algorithm based on the Principle of Minimum of the Total Potential Energy is presented. The effectiveness of this algorithm is assessed by static analysis of a laminated composite plate simply supported on all edges. Comparison with other algorithms for the selection of the shape parameter is made. The radial basis functions collocation method coupled with the present algorithm results very accurate in predicting maximum deflections and stresses in the range of the span-to-thickness ratio values considered.
Although global collocation with radial basis functions proved to be a very accurate means of solving interpolation and partial differential equations problems, ill-conditioned matrices are produced, making the choice of the shape parameter a crucial issue. The use of local numerical schemes, such as finite differences produces better conditioned matrices. For scattered points, a combination of finite differences and radial basis functions avoids the limitation of finite differences to be used on special grids. In this paper, we use a higher-order shear and normal deformation plate theory and a radial basis function—finite difference technique for predicting the static behavior of thick plates. Through numerical experiments on square and L-shaped plates, the accuracy and efficiency of this collocation technique is demonstrated, and the numerical accuracy and convergence are thoughtfully examined. This technique shows great potential to solve large engineering problems without the issue of ill-conditioning.
We conduct discrete spectrum analyses for a selection of mixed discretization schemes for the Stokes eigenproblem. In particular, we consider the MINI element, the Crouzeix–Raviart element, the Marker-and-Cell scheme, the Taylor–Hood element, the
${\mathbf{Q}_{k}/P_{k-1}}$
element, the divergence-conforming discontinuous Galerkin method, and divergence-conforming B-splines. For each of these schemes, we compare the spectrum for the continuous Stokes problem with the spectrum for the discrete Stokes problem, and we discuss the relationship of eigenvalue errors with solution errors associated with unsteady viscous flow problems.
Differential Quadrature (DQ) is a high-order numerical scheme, yielding very accurate results by use of very small number of nodal points. But it requires the functions to be determined highly differentiable. In the presence of material discontinuity in an elastic medium, direct application of DQ would yield poor results, and this issue has been addressed through a numerical example in this paper. After that, a multi-domain DQ approach has been proposed to solve the discontinuity difficulty. The approach is characterized by being first-order accurate at the interfaces of two different materials, but high-order accurate elsewhere. Numerical examples are given to demonstrate the effectiveness of the method.