Статті в журналах: "Differential quadrature-based elements"

1

Verma, Anjali, and Ram Jiwari. "Cosine expansion based differential quadrature algorithm for numerical simulation of two dimensional hyperbolic equations with variable coefficients." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 7 (September 7, 2015): 1574–89. http://dx.doi.org/10.1108/hff-08-2014-0240.

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Purpose – The purpose of this paper is to present the computational modeling of second-order two-dimensional nonlinear hyperbolic equations by using cosine expansion-based differential quadrature method (CDQM). Design/methodology/approach – The CDQM reduced the equations into a system of second-order differential equations. The obtained system is solved by RK4 method by converting into a system of first ordinary differential equations. Findings – The computed numerical results are compared with the results presented by other workers (Mohanty et al., 1996; Mohanty, 2004) and it is found that the present numerical technique gives better results than the others. Second, the proposed algorithm gives good accuracy by using very less grid point and less computation cost as comparison to other numerical methods such as finite difference methods, finite elements methods, etc. Originality/value – The author extends CDQM proposed in (Korkmaz and Dağ, 2009b) for two-dimensional nonlinear hyperbolic partial differential equations. This work is new for two-dimensional nonlinear hyperbolic partial differential equations.

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2

Kucharov, Olim, Fozil Turaev, Sergey Leonov, and Kholida Komilova. "Numerical study of nonlinear problems in the dynamics of thin-walled structural elements." E3S Web of Conferences 264 (2021): 05056. http://dx.doi.org/10.1051/e3sconf/202126405056.

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Mathematical model of the problem of vibration of thin-walled structural elements has been constructed based on Kirchhoff-Love theory. The problem is reduced, using the Bubnov-Galerkin method, to the solution of a set of nonlinear integro-differential Volterra type equations with weakly-singular kernels of relaxation. A numerical method based on the use of quadrature formulae being used for their solution. The influence of rheological parameters of the material on the values of critical velocity and amplitude-frequency characteristics of viscoelastic thin-walled structural elements is analyzed. It is shown that tacking account viscoelastic properties of the material of thin-walled structures lead to a decrease in the critical rate of gas flow.

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3

JANCHAI, Prawech. "Voltage-mode Second-order Filter and Quadrature Oscillator Based-on Differential Difference Current Conveyors and Only Grounded Elements." PRZEGLĄD ELEKTROTECHNICZNY 1, no. 9 (September 2, 2020): 64–69. http://dx.doi.org/10.15199/48.2020.09.13.

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4

Orlov, Victor, and Magomedyusuf Gasanov. "The Maximum Domain for an Analytical Approximate Solution to a Nonlinear Differential Equation in the Neighborhood of a Moving Singular Point." Axioms 12, no. 9 (August 30, 2023): 844. http://dx.doi.org/10.3390/axioms12090844.

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This paper presents the final stage in the study of the analytical approximate solution to a class of nonlinear differential equations unsolvable in quadrature in the general case in the neighborhood of a perturbed value of a moving singular point. An a priori error estimation is proven. The scope of application of the analytical approximate solution is extended; the formula for calculating this scope is obtained. The proof of the theorem is based on the application of elements of differential calculus. Theoretical results are supported by numerical calculations, which validate their reliability. The authors report a numerical comparison between the results, obtained in the paper, and the findings that were published earlier.

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5

Khudayarov, Bakhtiyar, Fozilzhon Turaev, and Olimzhon Kucharov. "Computer simulation of oscillatory processes of viscoelastic elements of thin-walled structures in a gas flow." E3S Web of Conferences 97 (2019): 06008. http://dx.doi.org/10.1051/e3sconf/20199706008.

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Results of numerical investigation of dynamic behavior of deformed wing aircraft in a gas flow are presented in the paper. Vibrations with respect to deflections are described by a system of integro-differential equations in partial derivatives. Using the Bubnov-Galerkin method, the problem is reduced to a system of ordinary integro-differential equations, where time is an independent variable. The solutions of integro-differential equations are determined by a numerical method based on the use of quadrature formulas. Computational algorithms and a package of applied programs have been created to solve problems on nonlinear flutter of viscoelastic elements of an aircraft. The reliability of the solution of the problem is confirmed by comparison with known numerical and analytical results. The effect of different boundary conditions on critical flutter velocity is studied. Critical velocity and critical flutter time of viscoelastic plates are determined. It is shown that the singularity parameter α affects not only the vibrations of viscoelastic systems, but also critical time and critical flutter velocity. It is stated that consideration of viscoelastic properties of plate material leads to 40 60% decrease in critical flutter velocity.

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6

Mirzaev, Sayibdjan, Majid Yusupov, Barna Rakhmankulova, Feruza Umarova, and Gulnaz Abdikayimova. "Vertical vibrations of traction engine with viscoelastic suspension." E3S Web of Conferences 365 (2023): 01022. http://dx.doi.org/10.1051/e3sconf/202336501022.

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The tasks of a traction engine with suspension elements and additional devices for converting movement (DCM) are considered. The object of protection, the estimated dynamic state, is solid with mass M and moment of inertia J relative to the center of gravity. To account the suspension material's rheological properties, the Boltzmann-Volterra principle is used. Mathematical models of the problem under consideration are obtained, which are described by the systems of integro-differential equations. A solution method based on quadrature formulas is developed, and a computer program is compiled based on its basis, the results of which are reflected in the graphs. The influence of DCM and rheological properties of the suspension material on the shape of the vertical vibrations of the object is investigated.

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7

Kiarasi, Faraz, Masoud Babaei, Kamran Asemi, Rossana Dimitri, and Francesco Tornabene. "Three-Dimensional Buckling Analysis of Functionally Graded Saturated Porous Rectangular Plates under Combined Loading Conditions." Applied Sciences 11, no. 21 (November 6, 2021): 10434. http://dx.doi.org/10.3390/app112110434.

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The present work studies the buckling behavior of functionally graded (FG) porous rectangular plates subjected to different loading conditions. Three different porosity distributions are assumed throughout the thickness, namely, a nonlinear symmetric, a nonlinear asymmetric and a uniform distribution. A novel approach is proposed here based on a combination of the generalized differential quadrature (GDQ) method and finite elements (FEs), labeled here as the FE-GDQ method, while assuming a Biot’s constitutive law in lieu of the classical elasticity relations. A parametric study is performed systematically to study the sensitivity of the buckling response of porous structures, to different input parameters, such as the aspect ratio, porosity and Skempton coefficients, along with different boundary conditions (BCs) and porosity distributions, with promising and useful conclusions for design purposes of many engineering structural porous members.

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8

Petrov, Andrey, Sergey Aizikovich, and Leonid A. Igumnov. "Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method." Key Engineering Materials 743 (July 2017): 158–61. http://dx.doi.org/10.4028/www.scientific.net/kem.743.158.

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Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

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9

Furqan, Muhammad, Faisal Ahmed, Reinhard Feger, Klaus Aufinger, Walter Hartner, and Andreas Stelzer. "A SiGe-based fully-integrated 122-GHz FMCW radar sensor in an eWLB package." International Journal of Microwave and Wireless Technologies 9, no. 6 (February 10, 2017): 1219–30. http://dx.doi.org/10.1017/s1759078717000095.

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High-performance SiGe HBTs and advancements in packaging processes have enabled system-in-package (SiP) designs for millimeter-wave applications. This paper presents a 122-GHz bistatic frequency modulated continuous wave (FMCW) radar SiP. The intended applications for the SiP are short-range distance and angular position measurements as well as communication links between cooperative radar stations. The chip is realized in a 130-nm SiGe BiCMOS technology and is based on a fully differential frequency-multiplier chain with in phase quadrature phase receiver and a binary phase shift keying modulator in the transmit chain. On-wafer measurement results show a maximum transmit output power of 2.7 dBm and a receiver gain of 11 dB. The chip consumes a DC power of 570 mW at a supply voltage of 3.3 V. The fabricated chip is integrated in an embedded wafer level ball grid array (eWLB) package. Transmit/receive rhombic antenna arrays with eight elements are designed in two eWLB packages with and without backside metal, with a measured peak gain of 11 dBi. The transceiver chip size is 1.8 mm × 2 mm, while the package size is 12 mm × 6 mm, respectively. FMCW measurements have been conducted with a sweep bandwidth of up to 17 GHz and a measured range resolution of 1.5 cm has been demonstrated. 2D positions of multiple targets have been computed using two coherently linked radar stations.

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10

Wang, Haijun, Weihua Jiang, Qing Hu, Jianjun Zhang, and Yanqing Jia. "Differential Evolution Algorithm-Aided Time-Varying Carrier Frequency Offset Estimation for OFDM Underwater Acoustic Communication." Journal of Marine Science and Engineering 10, no. 12 (November 28, 2022): 1826. http://dx.doi.org/10.3390/jmse10121826.

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Orthogonal frequency division multiplexing (OFDM) is the preferred scheme for high-speed communication in the field of underwater acoustic communication. However, it is very sensitive to the carrier frequency offset (CFO). This study used a time-varying CFO estimation method aided by the differential evolution (DE) algorithm to accurately estimate the CFO of an OFDM system. This method was based on the principle that the received OFDM signal with inter-carrier interference could be considered by a Multi Carrier-code division multiple access (MC-CDMA) system on the receiver side because MC-CDMA is a technology that combines OFDM and code division multiple access (CMDA). Because it is suitable for solving problems where there are dependencies between adjacent variables, the DE algorithm was used to capture the varying CFO values on the adjacent blocks. The spreading code of the MC-CDMA was obtained based on the estimated CFO values, which were elements in the DE solutions. Then the received signal was reconstructed. The Root-Mean-Square Error between the reconstructed and actual received signals was used as the cost function, and the CFO was estimated using the DE algorithm because of its powerful parallel search capability. The simulation results showed that the proposed method had a high estimation accuracy. Compared with other intelligent optimization algorithms such as the genetic algorithm and simulated annealing mutated-genetic algorithm, the time-varying CFO estimation performance of the DE algorithm was better because of its unique ability to solve problems with dependencies between adjacent variables. Specifically, under the condition of a high signal-to-noise ratio, the improvement of estimation accuracy reaches 36.13%, and the Bit Error Rate of demodulation is thus reduced by 75%, compared with the reference algorithms. In addition, the proposed method also has good applicability to modulation methods. For phase-shift keying and quadrature amplitude modulation, in particular, the proposed method not only achieved high-precision time-varying CFO estimation values, but also reduced the demodulation deterioration caused by noise.

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11

Farazin, Ashkan, Chunwei Zhang, and Azher M. Abed. "Vibrations of composite structures: Finite element and analytical investigation." Polymers and Polymer Composites 30 (January 2022): 096739112211129. http://dx.doi.org/10.1177/09673911221112956.

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In this examination, the free vibrations of complete composite shells with rectangular openings based on first-order shear deformation theory have been studied. The equations are generally written in such a way that they can be converted to any of Donnell, Love, or Sanders theories. To study the shell with the opening of the problem-solving space, it is elementalized in such a way that the boundary conditions and loading are uniform at the edges of each element. For each element, the governing equations, the boundary conditions of the edges, and the compatibility conditions at the common boundary of the adjacent elements are discretized by the generalized differential quadrature method in the longitudinal and peripheral directions, and by assembling them, a system of algebraic equations is formed. Finally, the natural frequency of the structure is calculated using the solution of the eigenvalue. To validate this method, the results are compared with the results of some articles as well as the results of Abaqus finite element software. After ensuring the efficiency of the present method, it has been used to study the effect of different parameters on the vibrational behavior of shells with and without apertures. These studies show that relatively small openings (c/L <0.3) have little effect on the natural frequency of the shell, regardless of the material and the porcelain layer of the shell. While reducing the ratio of length to radius or increasing the thickness of the shell is also effective in reducing the effects of opening. In addition, the effect of peripheral openings is far less than longitudinal openings.

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12

Algül, İlke, and Ahmet Sinan Oktem. "Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells." Applied Sciences 12, no. 24 (December 7, 2022): 12547. http://dx.doi.org/10.3390/app122412547.

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This study aimed to provide a static solution to the boundary value problem presented by symmetric (0°/90°/0°) and antisymmetric (0°/90°) cross-ply composite, moderately thick shallow shells and plates (a special case of the shells) subjected to mixed-type unsolved boundary conditions. The boundary-discontinuous double Fourier series (BDM) method, in which displacements are expressed in trigonometric functions, is employed in a well-established framework. The analytical solution obtained using the BDM is compared with the successful integration of the generalized differential quadrature (GDQ) method for the static analysis of composite shells with a roller skate-type boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to simply supported constraints. Comprehensive results are presented in order to show the effects of curvature on the deflections and stresses of moderately thick shallow shells made up of symmetric and antisymmetric cross-ply laminated composite materials. The validity of the proposed model is authenticated through the available HSDT-based literature review, and the convergence characteristics are demonstrated. The changing trends of displacements and stresses are explained in detail by investigating the effect of various parameters such as lamination, material properties, the effect of curvature, etc. Based on the results obtained using the proposed static solution, analytical BDM results were found to be in very close agreement with the numerical GDQ method, especially for symmetric lamination. However, the results obtained using the BDM and GDQ methods for antisymmetric lamination show differences, possibly due to the presence of a discontinuity in the derivatives originating from the bending–stretching matrix in antisymmetric lamination. Important numerical results presented include the sensitivity of the predicted response quantities of interest to material properties, lamination, and thickness effects, as well as their interactions. The results presented here may also serve as benchmark comparison points with numerical solutions such as finite elements, boundary elements, etc.

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13

Nie, G. J., and Zheng Zhong. "Second-Order Elasto-Plastic Analysis of Frames by Differential Quadrature Element Method." Key Engineering Materials 340-341 (June 2007): 1321–26. http://dx.doi.org/10.4028/www.scientific.net/kem.340-341.1321.

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A new differential quadrature element model is presented for the second-order elasto-plastic analysis of frames in this study. The new model is based on the differential quadrature method (DQM) and the finite-cut technique. Firstly the basic equilibrium differential equations of members, the compatibility conditions of joints and the equilibrium equations of joints for the second-order analysis of frames are established. The differential quadrature method is used to discretize the basic equations and then the stiffness equations of the whole structure can be derived. While the corresponding boundary conditions are considered, the mechanical behavior of frames can be obtained. The yielding development along the axis of the member can be taken into consideration by selecting several discrete points and simultaneously the yielding development across the section can be considered using the layered approach. The full historical second-order elasto-plastic analysis is achieved by the incremental iterative algorithm. According to the new model derived in this paper, the interrelated structural calculating program is worked out. The results of numerical examples demonstrate the validity of the differential quadrature element model (DQEM). The new model can be used in the second-order elasto-plastic analysis of arbitrary frames.

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14

Kővári, Zs, K. G. Strassmeier, K. Oláh, L. Kriskovics, K. Vida, T. A. Carroll, T. Granzer, et al. "Surface magnetic activity of the fast-rotating G5 giant IN Comae, central star of the faint planetary nebula LoTr 5." Astronomy & Astrophysics 624 (April 2019): A83. http://dx.doi.org/10.1051/0004-6361/201834810.

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Context. On the asymptotic giant branch, low to intermediate mass stars blow away their outer envelopes, forming planetary nebulae. Dynamic interaction between the planetary nebula and its central progenitor is poorly understood. The interaction is even more complex when the central object is a binary star with a magnetically active component, as is the case for the target in this paper. Aims. We aim to quantify the stellar surface activity of the cool binary component of IN Com and aim to explain its origin. In general, we need a better understanding of how central binary stars in planetary nebulae evolve and how this evolution could develop such magnetically active stars as IN Com. Methods. We present a time series of 13 consecutive Doppler images covering six months in 2017 that we used to measure the surface differential rotation with a cross-correlation method. Hitherto unpublished high-precision photometric data from 1989 to 2017 are presented. We applied Fourier-transformation-based frequency analysis to both photometry and spectra. Very high resolution (R ≈ 200 000) spectra were used to update IN Com’s astrophysical parameters by means of spectral synthesis. Results. Our time-series Doppler images show cool and warm spots coexisting with an average surface temperature contrast of −1000 K and +300 K with respect to the effective temperature. Approximately 8% of the stellar surface is covered with cool spots and ∼3% with warm spots. A consistent cool polar spot is seen in all images. The average lifetime of the cool spots is not much more than a few stellar rotations (one month), while the warm spots appear to live longer (three months) and are mostly confined to high latitudes. We found anti-solar surface differential rotation with a shear coefficient of α = −0.026 ± 0.005 suggesting an equatorial rotation period of 5.973 ± 0.008 d. We reconfirm the 5.9 day rotation period of the cool star from photometry, radial velocities, and Hα line-profile variations. A long-term V-brightness variation with a likely period of 7.2 yr is also found. It appears in phase with the orbital radial velocity of the binary system in the sense that it is brightest at highest velocity and faintest at lowest velocity, that is, at the two phases of quadrature. We redetermine [Ba/Fe], [Y/Fe], and [Sr/Fe] ratios and confirm the overabundance of these s-process elements in the atmosphere of IN Com.

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15

Doss, L. Jones Tarcius, and A. P. Nandini. "Discrete Mixed Petrov-Galerkin Finite Element Method for a Fourth-Order Two-Point Boundary Value Problem." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/962070.

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A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal ordera priorierror estimates are obtained without any restriction on the mesh.

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16

Bayat, Mohammad, and Mohammad Mohammadi Aghdam. "Micromechanical analysis of unidirectional composites using a least-squares-based differential quadrature element method." Journal of Mechanics of Materials and Structures 7, no. 2 (May 6, 2012): 119–35. http://dx.doi.org/10.2140/jomms.2012.7.119.

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17

Chang-New Chen. "Differential Quadrature, Generalized Methods, Related Discrete Element Analysis Methods And EDQ Based Time Integration Method." Recent Patents on Engineering 1, no. 2 (June 1, 2007): 163–76. http://dx.doi.org/10.2174/187221207780832147.

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18

Xing, Yufeng, Mingbo Qin, and Jing Guo. "A Time Finite Element Method Based on the Differential Quadrature Rule and Hamilton’s Variational Principle." Applied Sciences 7, no. 2 (February 4, 2017): 138. http://dx.doi.org/10.3390/app7020138.

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19

Zhao, Jingjun, Jingyu Xiao, and Yang Xu. "Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/857205.

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A finite element method (FEM) for multiterm fractional partial differential equations (MT-FPDEs) is studied for obtaining a numerical solution effectively. The weak formulation for MT-FPDEs and the existence and uniqueness of the weak solutions are obtained by the well-known Lax-Milgram theorem. The Diethelm fractional backward difference method (DFBDM), based on quadrature for the time discretization, and FEM for the spatial discretization have been applied to MT-FPDEs. The stability and convergence for numerical methods are discussed. The numerical examples are given to match well with the main conclusions.

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20

Arifeen, Shams Ul, Sirajul Haq, and Farhan Golkarmanesh. "Computational Study of Multiterm Time-Fractional Differential Equation Using Cubic B-Spline Finite Element Method." Complexity 2022 (November 23, 2022): 1–15. http://dx.doi.org/10.1155/2022/3160725.

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Due to the symmetry feature in nature, fractional differential equations precisely measure and describe biological and physical processes. Multiterm time-fractional order has been introduced to model complex processes in different physical phenomena. This article presents a numerical method based on the cubic B-spline finite element method for the solution of multiterm time-fractional differential equations. The temporal fractional part is defined in the Caputo sense while the B-spline finite element method is employed for space approximation. In addition, the four-point Gauss−Legendre quadrature is applied to evaluate the source term. The stability of the proposed scheme is discussed by the Von Neumann method, which verifies that the scheme is unconditionally stable. L 2 and L ∞ norms are used to check the efficiency and accuracy of the proposed scheme. Computed results are compared with the exact and available methods in the literature, which show the betterment of the proposed method.

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21

Shafiei, Navvab, Mohammad Kazemi, and Laleh Fatahi. "Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method." Mechanics of Advanced Materials and Structures 24, no. 3 (September 6, 2016): 240–52. http://dx.doi.org/10.1080/15376494.2015.1128025.

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22

Ansari, R., J. Torabi, and R. Hassani. "Vibration analysis of FG-CNTRC plates with an arbitrarily shaped cutout based on the variational differential quadrature finite element method." Materials Research Express 6, no. 12 (December 5, 2019): 125086. http://dx.doi.org/10.1088/2053-1591/ab5b57.

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Boutiba, Malika, Selma Baghli-Bendimerad, and Abbès Benaïssa. "Three Approximations of Numerical Solution's by Finite Element Method for Resolving Space-Time Partial Differential Equations Involving Fractional Derivative's Order." Mathematical Modelling of Engineering Problems 9, no. 5 (December 13, 2022): 1179–86. http://dx.doi.org/10.18280/mmep.090503.

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In this paper, we apply to a class of partial differential equation the finite element method when the problem is involving the Riemann-Liouville fractional derivative for time and space variables on a bounded domain with bounded conditions. The studied equation is obtained from the standard time diffusion equation by replacing the first order time derivative by  for 0<<1 and for the second standard order space derivative by  for 1<<2 respectively. The existence of the unique solution is proved by the Lax-Milgram Lemma. We present here three schemes to approximate numerically the time derivative and use the finite element method for the space derivative using the Hadamard finite part integral and the Diethlem's first degree compound quadrature formula, the second approach is based on the link between Riemann-Liouville and Caputo fractional derivative, when the third method was based on the approximation of the Riemann-Liouville by the Grunwald-Letnikov fractional derivative. For the approximation of the space fractional derivative, the finite element method is introduced for all the three approaches. Finally, to check the effectiveness of the three methods, a numerical example was given.

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24

Mohammadian, Mostafa, Seyed Mahmoud Hosseini, and Mohammad Hossein Abolbashari. "Lateral vibrations of embedded hetero-junction carbon nanotubes based on the nonlocal strain gradient theory: Analytical and differential quadrature element (DQE) methods." Physica E: Low-dimensional Systems and Nanostructures 105 (January 2019): 68–82. http://dx.doi.org/10.1016/j.physe.2018.08.022.

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Mohammadian, Mostafa, and Seyed Mahmoud Hosseini. "A size-dependent differential quadrature element model for vibration analysis of FG CNT reinforced composite microrods based on the higher order Love-Bishop rod model and the nonlocal strain gradient theory." Engineering Analysis with Boundary Elements 138 (May 2022): 235–52. http://dx.doi.org/10.1016/j.enganabound.2022.02.017.

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Trabelssi, M., and S. El-Borgi. "A novel formulation for the weak quadrature element method for solving vibration of strain gradient graded nonlinear nanobeams." Acta Mechanica, September 26, 2022. http://dx.doi.org/10.1007/s00707-022-03321-4.

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AbstractA novel formulation of the weak form quadrature element method, referred to as the locally adaptive weak quadrature element method, is proposed to develop elements for nonlinear graded strain gradient Timoshenko and Euler–Bernoulli nanobeams. The equations of motion are obtained based on Hamilton principle while accounting for the position of the physical neutral axis. The proposed elements use Gauss quadrature points to ensure full integration of the variational statement. The proposed formulation develops matrices based on the differential quadrature method which employs Lagrange-based polynomials. These matrices can be modified to accommodate any number of extra derivative degrees of freedom including third-order beams and higher-order strain gradient beams without requiring an entirely new formulation. The performance of the proposed method is evaluated based on the free vibration response of the linear and nonlinear strain gradient Timoshenko and Euler–Bernoulli nanobeams. Both linear and nonlinear frequencies are evaluated for a large number of configurations and boundary conditions. It is shown that the proposed formulation results in good accuracy and an improved convergence speed as compared to the locally adaptive quadrature element method and other weak quadrature element methods available in the literature.

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27

Trabelssi, M., and S. El-Borgi. "Vibration of nonlocal strain gradient functionally graded nonlinear nanobeams using a novel locally adaptive strong quadrature element method." Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, November 21, 2022, 239779142211294. http://dx.doi.org/10.1177/23977914221129426.

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The primary objective of this paper is to propose a novel method to derive Differential Quadrature Method matrices with several degrees of freedom at the boundaries that can be used to build Strong Quadrature Elements to solve fourth and higher-order equations of motion. The proposed method, referred to as Locally adaptive Strong Quadrature Element Method, is applied to higher-order equations of motion for nonlinear graded Timoshenko and Euler-Bernoulli nanobeams formulated using the Second Strain Gradient Theory or the Nonlocal Strain Gradient Theory. To limit the formulation complexity, the proposed approach is based on the regular formulation of the differential quadrature method combined with custom-built transfer matrices. Moreover, it does not require a different formulation for fourth and sixth-order equations and can be extended beyond sixth-order equations. Validation was carried out using examples from the literature as well as data obtained using the classical Locally adaptive Quadrature Element Method. Both linear and nonlinear frequencies were evaluated for a large number of configurations and boundary conditions. The proposed approach resulted in good accuracy and a convergence speed comparable to the conventional Locally adaptive Quadrature Element Method.

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Mohamed, Nurul Akmal, Nur Fadhilah Ibrahim, Mohd Rozni Md Yusof, Nurul Farihan Mohamed, and Nurul Huda Mohamed. "IMPLEMENTATIONS OF BOUNDARY–DOMAIN INTEGRO-DIFFERENTIAL EQUATION FOR DIRICHLET BVP WITH VARIABLE COEFFICIENT." Jurnal Teknologi 78, no. 6-5 (June 13, 2016). http://dx.doi.org/10.11113/jt.v78.9003.

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In this paper, we present the numerical results of the Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet problem for an elliptic type Partial Differential Equation (PDE) with a variable coefficient. The numerical constructions are based on discretizing the boundary of the problem region by utilizing continuous linear iso-parametric elements while the domain of the problem region is meshed by using iso-parametric quadrilateral bilinear domain elements. We also use a semi-analytic method to handle the integration that exhibits logarithmic singularity instead of using Gauss-Laguare quadrature formula. The numerical results that employed the semi-analytic method give better accuracy as compared to those when we use Gauss-Laguerre quadrature formula. The system of equations that obtained by the discretized BDIDE is solved by an iterative method (Neumann series expansion) as well as a direct method (LU decomposition method). From our numerical experiments on all test domains, the relative errors of the solutions when applying semi-analytic method are smaller than when we use Gauss-Laguerre quadrature formula for the integration with logarithmic singularity. Unlike Dirichlet Boundary Integral Equation (BIE), the spectral properties of the Dirichlet BDIDE is not known. The Neumann iterations will converge to the solution if and only if the spectral radius of matrix operator is less than 1. In our numerical experiment on all the test domains, the Neumann series does converge. It gives some conclusions for the spectral properties of the Dirichlet BDIDE even though more experiments on the general Dirichlet problems need to be carried out.

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29

Sun, Guangjun, Hongjing Li, Tong Wang, and Qiang Xu. "Out-of-Plane Free Vibration Analysis of Continuous Curved Girders with Combined Linetypes Using Differential Quadrature Element Method." International Journal of Structural Stability and Dynamics, January 24, 2022. http://dx.doi.org/10.1142/s0219455422500602.

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In this study, the out-of-plane free vibration equations of circular curved and clothoid transition curved girders with various boundary conditions were derived. The continuous curved girder with combined linetypes was discretized into separate curved girder elements. Based on the differential quadrature element method (DQEM), the Chebyshev–Gauss–Labatto unequal mesh division was applied to discretize the vibration equations of the continuous curved girder into those of the curved girder elements, along with the boundary conditions, internal geometric compatibility conditions, and force balance conditions. The out-of-plane natural frequency equations were derived considering the boundary conditions by the substitution method. The free vibration was solved for continuous curved girders with combined linetypes of circular and clothoid transition curves. The effectiveness of the DQEM was verified, and the effect of the number of grid points on the accuracy of the solution was evaluated. In addition, the effects of the bending/torsional stiffness ratio, warping coefficient, and boundary conditions on the natural frequencies of continuous curved girders with combined linetypes were investigated. The results showed that the DQEM can be efficiently used to solve the free vibration of multispan continuous curved girders with various linetype combinations. The frequencies of the continuous curved girder with combined linetypes decrease with the weakening of boundary constraints, but increase with increasing warping coefficient and bending/torsional stiffness ratio.

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Ansari, Reza, Ramtin Hassani, Emad Hasrati, and Hessam Rouhi. "Studying nonlinear vibrations of composite conical panels with arbitrary-shaped cutout reinforced with graphene platelets based on higher-order shear deformation theory." Journal of Vibration and Control, June 24, 2021, 107754632110248. http://dx.doi.org/10.1177/10775463211024847.

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

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Michel, Sixtine, Davide Torlo, Mario Ricchiuto, and Rémi Abgrall. "Spectral Analysis of High Order Continuous FEM for Hyperbolic PDEs on Triangular Meshes: Influence of Approximation, Stabilization, and Time-Stepping." Journal of Scientific Computing 94, no. 3 (January 21, 2023). http://dx.doi.org/10.1007/s10915-022-02087-0.

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AbstractIn this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and Bernstein), the stabilization techniques (streamline-upwind Petrov–Galerkin, continuous interior penalty, orthogonal subscale stabilization) and the time discretization (Runge–Kutta (RK), strong stability preserving RK and deferred correction). This is an extension of the one dimensional study by Michel et al. (J Sci Comput 89(2):31, 2021. https://doi.org/10.1007/s10915-021-01632-7), whose results do not hold in multi-dimensional frameworks. The study ranks these schemes based on efficiency (most of them are mass-matrix free), stability and dispersion error, providing the best CFL and stabilization coefficients. The challenges in two-dimensions are related to the Fourier analysis. Here, we perform it on two types of periodic triangular meshes varying the angle of the advection, and we combine all the results for a general stability analysis. Furthermore, we introduce additional high order viscosity to stabilize the discontinuities, in order to show how to use these methods for tests of practical interest. All the theoretical results are thoroughly validated numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that Cubature elements combined with SSPRK and OSS stabilization is the most promising combination.

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32

Jin, Bangti, and Zhi Zhou. "Recovery of a space-time-dependent diffusion coefficient in subdiffusion: stability, approximation and error analysis." IMA Journal of Numerical Analysis, September 26, 2022. http://dx.doi.org/10.1093/imanum/drac051.

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Abstract In this work we study an inverse problem of recovering a space-time-dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian–Caputo fractional derivative of order $\alpha \in (0,1)$ in time. The main technical challenges of both theoretical and numerical analyses lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results using a novel test function, which leads to a stability bound in $L^2(0,T;L^2(\varOmega ))$ under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for recovering the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal $L^p$ regularity and a nonstandard energy argument. Under the positivity condition, we obtain a standard $\ell ^2(L^2(\varOmega ))$ error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples.

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33

Tornabene, Francesco, Nicholas Fantuzzi, Francesco Ubertini, and Erasmo Viola. "Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey." Applied Mechanics Reviews 67, no. 2 (March 1, 2015). http://dx.doi.org/10.1115/1.4028859.

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A survey of several methods under the heading of strong formulation finite element method (SFEM) is presented. These approaches are distinguished from classical one, termed weak formulation finite element method (WFEM). The main advantage of the SFEM is that it uses differential quadrature method (DQM) for the discretization of the equations and the mapping technique for the coordinate transformation from the Cartesian to the computational domain. Moreover, the element connectivity is performed by using kinematic and static conditions, so that displacements and stresses are continuous across the element boundaries. Numerical investigations integrate this survey by giving details on the subject.

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Ford, Neville, Jingyu Xiao, and Yubin Yan. "A finite element method for time fractional partial differential equations." Fractional Calculus and Applied Analysis 14, no. 3 (January 1, 2011). http://dx.doi.org/10.2478/s13540-011-0028-2.

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AbstractIn this paper, we consider the finite element method for time fractional partial differential equations. The existence and uniqueness of the solutions are proved by using the Lax-Milgram Lemma. A time stepping method is introduced based on a quadrature formula approach. The fully discrete scheme is considered by using a finite element method and optimal convergence error estimates are obtained. The numerical examples at the end of the paper show that the experimental results are consistent with our theoretical results.

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Houalef, Ihab Eddine, Ismail Bensaid, Ahmed Saimi, and Abdelmadjid Cheikh. "Free Vibration Analysis of Functionally Graded Carbon Nanotube-Reinforced Higher Order Refined Composite Beams Using Differential Quadrature Finite Element Method." European Journal of Computational Mechanics, February 6, 2023. http://dx.doi.org/10.13052/ejcm2642-2085.3143.

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Present paper deals on the free vibration investigation of carbon nanotube-reinforced composite (CNTs) beams, based on refined third order shear deformation finite element beam theory. The particularity of this model is that, it can capture shear deformation effect without using of any shear correction factor by satisfying shear stress free at free edges. The carbon nanotubes are supposed to be immersed in a polymeric matrix with functionally graded pattern across the thickness direction of the beam, and their material properties are evaluated using the rule of mixture. The differential equations of motion and related boundary conditions are extracted using Lagrange’s principle and solved employing a robust numerical tool called, Differential Quadrature Finite Element Method (DQFEM) for the first time, with high convergence speed, fast calculus performance as well as a good numerical stability. The obtained results have been validated with those available in literature, in order to show the correctness of the present model. Afterwards, a deep parametric study is performed to examine the effects of various geometrical and material parameters on the vibration behavior of FG-CNTs beams.

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Saimi, Ahmed, Ismail Bensaid, and Ahmed Fellah. "Effect of crack presence on the dynamic and buckling responses of bidirectional functionally graded beams based on quasi-3D beam model and differential quadrature finite element method." Archive of Applied Mechanics, May 4, 2023. http://dx.doi.org/10.1007/s00419-023-02429-w.

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