The Experts below are selected from a list of 300 Experts worldwide ranked by ideXlab platform
A Idesman - One of the best experts on this subject based on the ideXlab platform.
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the use of the local truncation error to improve arbitrary order finite elements for the linear wave and heat equations
Computer Methods in Applied Mechanics and Engineering, 2018Co-Authors: A IdesmanAbstract:Abstract The local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high-order finite elements in the 1-D, 2-D and 3-D cases on uniform and non-uniform meshes. Several applications of the local truncation error are considered in the paper. It is proven that for the 1-D wave equation with a piece-wise constant wave velocity, the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity, the Lumped Mass Matrix and the central-difference method with the time increments equal to the stability limit are used. It is shown in the 1-D and multidimensional cases that the optimal Lumped Mass Matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy. The minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high-order elements with improved accuracy; i.e., accuracy is improved from order 2 p (the conventional elements) to order 4 p (the new elements) where p is the order of the polynomial approximations. New high-order boundary conditions are developed in order to keep a high-order accuracy of the developed technique. The new elements can be equally applied to linear wave propagation and heat transfer problems. It is also shown that non-uniform meshes may lead to inaccurate results due to the increase in the local truncation error. The difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach. The presented numerical examples are in good agreement with the theoretical results. The approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches.
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Optimal reduction of numerical dispersion for wave propagation problems. Part 1: Application to 1-D isogeometric elements
Computer Methods in Applied Mechanics and Engineering, 2017Co-Authors: A IdesmanAbstract:A numerical technique with the optimal coefficients of the stencil equation has been suggested. Based on this approach, new high-order isogeometric elements with the reduced dispersion error have been developed for wave propagation problems in the 1-D case. By the modification of the Mass and stiffness matrices, the order of the dispersion error is improved from order 2 p (the conventional elements) to order 4 p (the new elements) where p is the order of the polynomial approximations. It was shown that the new approach yields the maximum order of the dispersion error for the stencil equations related to the high-order isogeometric elements. The analysis of the dispersion error of the high-order isogeometric elements with the Lumped Mass Matrix has also shown that independent of the procedures for the calculation of the Lumped Mass Matrix, the second order of the dispersion error cannot be improved with the conventional stiffness Matrix. However, the dispersion error for the Lumped Mass Matrix can be improved from the second order to order 2 p by the modification of the stiffness Matrix. The numerical examples confirm the computational efficiency of the new high-order isogeometric elements with reduced dispersion. We h ave also showed that numerical results obtained by the new and conventional high-order isogeometric elements may include spurious oscillations due to the dispersion error. These oscillations can be quantified and filtered by the two-stage time-integration technique recently developed in our papers. The approach developed in the paper can be directly applied to other space-discretization techniques with similar stencil equations.
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Optimal reduction of numerical dispersion for wave propagation problems. Part 2: Application to 2-D isogeometric elements
Computer Methods in Applied Mechanics and Engineering, 2017Co-Authors: A Idesman, Bikash DeyAbstract:Abstract Based on the optimal coefficients of the stencil equation, a numerical technique for the reduction of the numerical dispersion error has been suggested. New isogeometric elements with the reduced numerical dispersion error for wave propagation problems in the 2-D case have been developed with the suggested approach. By the minimization of the order of the dispersion error of the stencil equation, the order of the dispersion error is improved from order 2 p (the conventional isogeometric elements) to order 4 p (the isogeometric elements with reduced dispersion) where p is the order of the polynomial approximations. Because all coefficients of the stencil equation are obtained from the minimization procedure, the obtained accuracy is maximum possible. The corresponding elemental Mass and stiffness matrices of the isogeometric elements with reduced dispersion are calculated with help of the optimal coefficients of the stencil equation. The analysis of the dispersion error of the isogeometric elements with the Lumped Mass Matrix has also shown that independent of the procedures for the calculation of the Lumped Mass Matrix, the second order of the dispersion error cannot be improved with the conventional stiffness Matrix. However, the dispersion error with the Lumped Mass Matrix can be improved from the second order to order 2 p by the modification of the stiffness Matrix. The numerical examples confirm the computational efficiency of the isogeometric elements with reduced dispersion. The numerical results obtained by the new and conventional isogeometric elements may include spurious oscillations due to the dispersion error. These oscillations can be quantified and filtered by the two-stage time-integration technique developed recently. The approach developed in the paper can be directly applied to other space-discretization techniques with similar stencil equations.
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Accurate finite-element modeling of wave propagation in composite and functionally graded materials
Composite Structures, 2014Co-Authors: A IdesmanAbstract:Abstract For the first time, we have obtained accurate numerical solutions for wave propagation in inhomogeneous materials under impact loading. We have extended the earlier developed numerical approach for elastodynamics problems in homogeneous materials to inhomogeneous materials. The approach includes the two-stage time-integration technique with the quantification and the filtering of spurious oscillations, the special design of non-uniform meshes as well as includes the standard finite elements and the elements with reduced dispersion. Similar to wave propagation in homogeneous materials in the 1-D case, we have obtained very accurate results for composite and functionally graded materials using the linear elements with Lumped Mass Matrix and the explicit central difference method. We have also shown that specific non-uniform meshes yield much more accurate results compared to uniform meshes. We have also shown the efficiency of the finite elements with reduced dispersion compared with the standard finite elements.
Hong Zheng - One of the best experts on this subject based on the ideXlab platform.
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Numerical manifold method for vibration analysis of Kirchhoff's plates of arbitrary geometry
Applied Mathematical Modelling, 2019Co-Authors: Hong Zheng, Xiaoying ZhuangAbstract:Abstract 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.
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A Mass lumping scheme for the second-order numerical manifold method
Computers & Structures, 2019Co-Authors: Guohua Zhang, Yongtao Yang, Hong ZhengAbstract:Abstract The numerical manifold method (NMM) has been employed to solve many types of engineering problems. To improve accuracy of the traditional NMM, a second-order NMM with six-node triangular mesh was recently proposed. This second-order NMM is further applied for structural dynamic problems in this study. In order to reduce time consumption in solving large scale simultaneous algebraic equations, a Mass lumping scheme which is suitable for the second-order NMM is proposed. A series of numerical examples show that natural frequencies of structures assessed from the proposed Lumped Mass Matrix (LMM) are more accurate than those from the consistent Mass Matrix (CMM). In addition, even in the implicit time integration scheme to predict dynamic responses of structures, the CMM can be replaced by the proposed LMM without losing accuracy.
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A Mass lumping scheme for the 10-node tetrahedral element
Engineering Analysis with Boundary Elements, 2019Co-Authors: Guohua Zhang, Yongtao Yang, Guanhua Sun, Hong ZhengAbstract:Abstract In the dynamic analysis with explicit time integration scheme, a Lumped Mass Matrix (LMM) is always preferable than the consistent Mass Matrix (CMM), since the solving of large scale simultaneous algebraic equations can be avoided if a LMM is employed. In this study, a Mass lumping scheme which is suitable for the 10-node tetrahedral element is proposed. A series of numerical examples show that the natural frequencies assessed from the proposed LMM are very close to those from CMM. In addition, the dynamic responses of structure predicted by the proposed LMM are in very good agreement with those from CMM if an implicit time integration scheme is adopted. More importantly, the proposed LMM is less sensitive to mesh distortion than the CMM. The proposed LMM can supersede the CMM at any cases.
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Explicit Discontinuous Deformation Analysis Method with Lumped Mass Matrix for Highly Discrete Block System
International Journal of Geomechanics, 2018Co-Authors: Yongtao Yang, Hong ZhengAbstract:AbstractIn the traditional discontinuous deformation analysis (DDA) method, the implicit time integration scheme is used to integrate equations of motion for modeling the mechanical behavior of a h...
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A rigorous and unified Mass lumping scheme for higher-order elements
Computer Methods in Applied Mechanics and Engineering, 2017Co-Authors: Yongtao Yang, Hong Zheng, Mettupalayam V. SivaselvanAbstract:Abstract In dynamic analysis with explicit time integration schemes, a Lumped Mass Matrix (LMM) is preferable, because LMM can avoid solving the large scale simultaneous algebraic equations. Mathematically rigorous Mass lumping schemes, such as the Mass lumping by nodal quadrature and the row-sum technique, are applicable to only linear or bilinear elements. For higher-order elements, such as 8-node serendipity elements, the diagonal scaling procedure is the only lumping method that can be recommended to generate positive definite diagonal element Mass matrices. Unfortunately, there is no mathematical theory in support of this approach. This study proposes a general Mass lumping scheme applicable to higher order elements, where the virtual work of initial force is integrated over the problem domain that is viewed as the manifold covered by the finite element patches. By a series of numerical experiments, both free and forced vibration problems, it is suggested that even in the implicit time integration scheme the consistent Mass Matrix (CMM) can be superseded by the proposed LMM. Furthermore, the proposed LMM has much stronger adaptability to distorted meshes.
Guangjun Sun - One of the best experts on this subject based on the ideXlab platform.
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A time-domain spectral element method with C1 continuity for static and dynamic analysis of frame structures
Structures, 2020Co-Authors: Lu Han, Jingxiong Wang(, Guangjun SunAbstract:Abstract This paper presents a new C1-type beam spectral element method, which aims to ensure the continuity of the deflection and its first derivative at the junction of the adjacent spectral elements. The proposed method is based on the Euler-Bernoulli beam theory. The main novelty of the method is that two different sets of interpolation functions are adopted for the axial displacement and deflection in order to meet the C1 continuous requirement. By removing the rotational degrees of freedom of the internal nodes, a Lumped Mass Matrix can be formed in this spectral element method. Static and dynamic analysis of various beams and frame structures are performed to examine the validity of the proposed spectral beam element model. The numerical results show that the proposed method is accurate and effective. A single element for each component of the structure is able to provide good results.
Lu Han - One of the best experts on this subject based on the ideXlab platform.
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A time-domain spectral element method with C1 continuity for static and dynamic analysis of frame structures
Structures, 2020Co-Authors: Lu Han, Jingxiong Wang(, Guangjun SunAbstract:Abstract This paper presents a new C1-type beam spectral element method, which aims to ensure the continuity of the deflection and its first derivative at the junction of the adjacent spectral elements. The proposed method is based on the Euler-Bernoulli beam theory. The main novelty of the method is that two different sets of interpolation functions are adopted for the axial displacement and deflection in order to meet the C1 continuous requirement. By removing the rotational degrees of freedom of the internal nodes, a Lumped Mass Matrix can be formed in this spectral element method. Static and dynamic analysis of various beams and frame structures are performed to examine the validity of the proposed spectral beam element model. The numerical results show that the proposed method is accurate and effective. A single element for each component of the structure is able to provide good results.
Hauke Gravenkamp - One of the best experts on this subject based on the ideXlab platform.
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Critical assessment of different Mass lumping schemes for higher order serendipity finite elements
Computer Methods in Applied Mechanics and Engineering, 2019Co-Authors: Sascha Duczek, Hauke GravenkampAbstract:Abstract An efficient and robust finite element-based transient analysis of structures is important in many engineering applications. In this context, a diagonal or Lumped Mass Matrix is an essential prerequisite. In the last decades, several methods to construct such a Mass Matrix have been proposed and therefore a comprehensive and unbiased evaluation of these approaches is needed. In the present article, we consequently investigate established Mass lumping schemes, such as the row-sum method and the diagonal scaling method, as well as the recently proposed manifold-based method. Furthermore, the latter approach is rigorously extended to higher order serendipity finite elements in two and three dimensions. Note that the manifold-based method and the diagonal scaling method are very general approaches, in the sense that they are applicable to arbitrary nodal-based shape functions with arbitrary polynomial orders, while guaranteeing the positive-definiteness of the Lumped Mass Matrix. With respect to the row-sum method the positivity is not ensured and negative or zero diagonal components might be computed. Consequently, a detailed analysis of the influence and performance of these Mass lumping schemes on the numerical results is conducted. To this end, several dynamic benchmark models (including modal, harmonic, and transient analyses) are selected, showing that only suboptimal rates of convergence are attained if serendipity finite elements based on a diagonal Mass Matrix are employed.
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Mass lumping techniques in the spectral element method: On the equivalence of the row-sum, nodal quadrature, and diagonal scaling methods
Computer Methods in Applied Mechanics and Engineering, 2019Co-Authors: Sascha Duczek, Hauke GravenkampAbstract:Abstract In the context of wave propagation analysis, the spectral element method (SEM) in conjunction with a diagonal Mass Matrix is often the method of choice. Therefore, it is of high importance to investigate the influence of different Mass lumping schemes on the accuracy of the numerical results. To this end, we compare the performance of three established methods including the row-sum method, the nodal quadrature method, and the diagonal scaling method. The theoretical analysis of these methods reveals a close connection between them. Under certain conditions, that are discussed in detail in this article, we are able to show a direct equivalence between these three approaches. In this regard, the attainable accuracy of the numerical integration of the Mass Matrix plays an important role. By means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the Lumped Mass Matrix SEM in comparison to a formulation based on the consistent Mass Matrix.
