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Applications of Generalized Differential and Integral Quadrature Methods
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Applications of Generalized Differential and Integral Quadrature Methods

A special issue of Applied Sciences (ISSN 2076-3417).

Deadline for manuscript submissions: closed (31 March 2017) | Viewed by 34546

Special Issue Editor

Dr. Francesco Tornabene

grade E-Mail Website
Guest Editor
Department of Innovation Engineering, University of Salento, 73100 Lecce, Italy
Interests: theory of shells, plates, arches, and beams; generalized differential quadrature; FEM; SFEM; WFEM; IGA; advanced composite materials; functionally graded materials; nanomaterials and nanotechnology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The Generalized Differential Quadrature (GDQ) method has been widely used in scientific and engineering computation. Its first applications were related to fluid mechanics. Nevertheless, it has been employed soon in structural engineering for solving different problems, such as thermal, acoustics, vibrations, flows in porous media and so on. The main advantage of the GDQ method is that it can be easily applied to any field governed by partial differential equations, because the approximation and implementation of partial differential equations is easy and straightforward. Moreover, it is well known to be fast and reliable if compared to the classic Finite Element Method. The GDQ method was considered also in time-domain problems and for approximating integrals. Thus, the so-called Generalized Integral Quadrature (GIQ) method was introduced. Thanks to their great versatility, the GDQ and GIQ methods were implemented and used by a lot of researchers for investigating several fields of avant-garde nature. Nowadays, the GDQ method is particularly used in structural mechanics for investigating the static and dynamic behavior of systems made of composite and advanced materials, which are generally complex to study numerically, but they become easy to implement and analyze through this numerical technique. These applications are related, but not limited to, SMART composite structures, carbon-nano tubes reinforced composites, carbon-reinforced polymer composites, functionally graded materials, and numerical optimization of composite structures.

The Special Issue of Applied Sciences “Applications of Generalized Differential and Integral Quadrature Methods” aims to cover recent advances in the development of engineering, mathematical and physical applications of Generalized Differential Quadrature methods. Moreover, this special issue is aimed at new researchers in this field, who wants to learn the main novelties and applications related to these numerical techniques.

Prof. Francesco Tornabene
Guest Editor

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Keywords

  • Differential quadrature methods
  • Differential quadrature element methods
  • Numerical analysis
  • Engineering applications
  • Structural mechanics
  • Fluid dynamics
  • Thermal problems
  • Differential equations
  • Strong and weak formulation
  • Applied mechanics
  • Vibrations and acoustics

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Published Papers (6 papers)

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Research

804 KiB  
Article
An Improved Differential Quadrature Time Element Method
by Mingbo Qin, Yufeng Xing and Jing Guo
Appl. Sci. 2017, 7(5), 471; https://doi.org/10.3390/app7050471 - 3 May 2017
Cited by 7 | Viewed by 4525
Abstract
A Differential Quadrature Time Element Method (DQTEM) was proposed by the author and co-worker, its drawback is the need of larger storage capacity since the dimension of the coefficients matrix for solution is the product of both spatial degrees of freedom and temporal [...] Read more.
A Differential Quadrature Time Element Method (DQTEM) was proposed by the author and co-worker, its drawback is the need of larger storage capacity since the dimension of the coefficients matrix for solution is the product of both spatial degrees of freedom and temporal degrees of freedom. To solve this problem, an improved DQTEM is developed in this work, in which the differential quadrature method is used to discretize both spatial and time domains, sequentially, and the dimension of the coefficients matrix is greatly reduced without losing solution accuracy. Theoretical studies demonstrate the improved DQTEM features superiorities including higher-order accuracy, adequate stability and symplectic characteristics. The improvement of DQTEM is validated by extensive comparisons of the present DQTEM with the original DQTEM. Full article
(This article belongs to the Special Issue Applications of Generalized Differential and Integral Quadrature Methods)
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1759 KiB  
Article
Construction of Compact Finite Difference Schemes by Classic Differential Quadrature
by Fangzong Wang, Mingshuai Pan and Yong Wang
Appl. Sci. 2017, 7(3), 284; https://doi.org/10.3390/app7030284 - 14 Mar 2017
Viewed by 4424
Abstract
Using classic differential quadrature formulae and uniform grids, this paper systematically constructs a variety of high-order finite difference schemes, and some of these schemes are consistent with the so-called boundary value methods. The derived difference schemes enjoy the same stability and accuracy properties [...] Read more.
Using classic differential quadrature formulae and uniform grids, this paper systematically constructs a variety of high-order finite difference schemes, and some of these schemes are consistent with the so-called boundary value methods. The derived difference schemes enjoy the same stability and accuracy properties with correspondent differential quadrature methods but have a simpler form of calculation; thus, they can be seen as a compact format of classic differential quadrature methods. Through systematic Fourier stability analysis, the characteristics such as the dissipation, dispersion and resolution of the different schemes were studied and compared. Full article
(This article belongs to the Special Issue Applications of Generalized Differential and Integral Quadrature Methods)
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3713 KiB  
Article
A Time Finite Element Method Based on the Differential Quadrature Rule and Hamilton’s Variational Principle
by Yufeng Xing, Mingbo Qin and Jing Guo
Appl. Sci. 2017, 7(2), 138; https://doi.org/10.3390/app7020138 - 4 Feb 2017
Cited by 13 | Viewed by 4282
Abstract
An accurate and efficient Differential Quadrature Time Finite Element Method (DQTFEM) was proposed in this paper to solve structural dynamic ordinary differential equations. This DQTFEM was developed based on the differential quadrature rule, the Gauss–Lobatto quadrature rule, and the Hamilton variational principle. The [...] Read more.
An accurate and efficient Differential Quadrature Time Finite Element Method (DQTFEM) was proposed in this paper to solve structural dynamic ordinary differential equations. This DQTFEM was developed based on the differential quadrature rule, the Gauss–Lobatto quadrature rule, and the Hamilton variational principle. The proposed DQTFEM has significant benefits including the high accuracy of differential quadrature method and the generality of standard finite element formulation, and it is also a highly accurate symplectic method. Theoretical studies demonstrate the DQTFEM has higher-order accuracy, adequate stability, and symplectic characteristics. Moreover, the initial conditions in DQTFEM can be readily imposed by a method similar to the standard finite element method. Numerical comparisons for accuracy and efficiency among the explicit Runge–Kutta method, the Newmark method, and the proposed DQTFEM show that the results from DQTFEM, even with a small number of sampling points, agree better with the exact solutions and validate the theoretical conclusions. Full article
(This article belongs to the Special Issue Applications of Generalized Differential and Integral Quadrature Methods)
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4036 KiB  
Article
A Numerical Investigation on the Natural Frequencies of FGM Sandwich Shells with Variable Thickness by the Local Generalized Differential Quadrature Method
by Francesco Tornabene, Nicholas Fantuzzi, Michele Bacciocchi, Erasmo Viola and Junuthula N. Reddy
Appl. Sci. 2017, 7(2), 131; https://doi.org/10.3390/app7020131 - 27 Jan 2017
Cited by 113 | Viewed by 8886
Abstract
The main aim of the present paper is to solve numerically the free vibration problem of sandwich shell structures with variable thickness and made of Functionally Graded Materials (FGMs). Several Higher-order Shear Deformation Theories (HSDTs), defined by a unified formulation, are employed in [...] Read more.
The main aim of the present paper is to solve numerically the free vibration problem of sandwich shell structures with variable thickness and made of Functionally Graded Materials (FGMs). Several Higher-order Shear Deformation Theories (HSDTs), defined by a unified formulation, are employed in the study. The FGM structures are characterized by variable mechanical properties due to the through-the-thickness variation of the volume fraction distribution of the two constituents and the arbitrary thickness profile. A four-parameter power law expression is introduced to describe the FGMs, whereas general relations are used to define the thickness variation, which can affect both the principal coordinates of the shell reference domain. A local scheme of the Generalized Differential Quadrature (GDQ) method is employed as numerical tool. The natural frequencies are obtained varying the exponent of the volume fraction distributions using higher-order theories based on a unified formulation. The structural models considered are two-dimensional and require less degrees of freedom when compared to the corresponding three-dimensional finite element (FE) models, which require a huge number of elements to describe the same geometries accurately. A comparison of the present results with the FE solutions is carried out for the isotropic cases only, whereas the numerical results available in the literature are used to prove the validity as well as accuracy of the current approach in dealing with FGM structures characterized by a variable thickness profile. Full article
(This article belongs to the Special Issue Applications of Generalized Differential and Integral Quadrature Methods)
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4455 KiB  
Article
An Equivalent Layer-Wise Approach for the Free Vibration Analysis of Thick and Thin Laminated and Sandwich Shells
by Francesco Tornabene, Nicholas Fantuzzi, Michele Bacciocchi and Junuthula N. Reddy
Appl. Sci. 2017, 7(1), 17; https://doi.org/10.3390/app7010017 - 22 Dec 2016
Cited by 74 | Viewed by 6235
Abstract
The main purpose of the paper is to present an innovative higher-order structural theory to accurately evaluate the natural frequencies of laminated composite shells. A new kinematic model is developed starting from the theoretical framework given by a unified formulation. The kinematic expansion [...] Read more.
The main purpose of the paper is to present an innovative higher-order structural theory to accurately evaluate the natural frequencies of laminated composite shells. A new kinematic model is developed starting from the theoretical framework given by a unified formulation. The kinematic expansion is taken as a free parameter, and the three-dimensional displacement field is described by using alternatively the Legendre or Lagrange polynomials, following the key points of the most typical Layer-wise (LW) approaches. The structure is considered as a unique body and all the geometric and mechanical properties are evaluated on the shell middle surface, following the idea of the well-known Equivalent Single Layer (ESL) models. For this purpose, the name Equivalent Layer-Wise (ELW) is introduced to define the present approach. The governing equations are solved numerically by means of the Generalized Differential Quadrature (GDQ) method and the solutions are compared with the results available in the literature or obtained through a commercial finite element program. Due to the generality of the current method, several boundary conditions and various mechanical and geometric configurations are considered. Finally, it should be underlined that different doubly-curved surfaces may be considered following the mathematical framework given by the differential geometry. Full article
(This article belongs to the Special Issue Applications of Generalized Differential and Integral Quadrature Methods)
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7703 KiB  
Article
Harmonic Differential Quadrature Analysis of Soft-Core Sandwich Panels under Locally Distributed Loads
by Xinwei Wang and Zhangxian Yuan
Appl. Sci. 2016, 6(11), 361; https://doi.org/10.3390/app6110361 - 18 Nov 2016
Cited by 5 | Viewed by 4812
Abstract
Sandwich structures are widely used in practice and thus various engineering theories adopting simplifying assumptions are available. However, most engineering theories of beams, plates and shells cannot recover all stresses accurately through their constitutive equations. Therefore, the soft-core is directly modeled by two-dimensional [...] Read more.
Sandwich structures are widely used in practice and thus various engineering theories adopting simplifying assumptions are available. However, most engineering theories of beams, plates and shells cannot recover all stresses accurately through their constitutive equations. Therefore, the soft-core is directly modeled by two-dimensional (2D) elasticity theory without any pre-assumption on the displacement field. The top and bottom faces act like the elastic supports on the top and bottom edges of the core. The differential equations of the 2D core are then solved by the harmonic differential quadrature method (HDQM). To circumvent the difficulties in dealing with the locally distributed load by point discrete methods such as the HDQM, a general and rigorous way is proposed to treat the locally distributed load. Detailed formulations are provided. The static behavior of sandwich panels under different locally distributed loads is investigated. For verification, results are compared with data obtained by ABAQUS with very fine meshes. A high degree of accuracy on both displacement and stress has been observed. Full article
(This article belongs to the Special Issue Applications of Generalized Differential and Integral Quadrature Methods)
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