Modeling transient elastodynamic problems using a novel semi-analytical method yielding decoupled partial differential equations

Khodakarami, Mohammad Iman; Khaji, N.; Ahmadi, Mohammad Taghi

doi:10.1016/j.cma.2011.11.016

Cited by 18 publications

(14 citation statements)

References 42 publications

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“…Further development of the DSBFEM for the analysis of 3D elastodynamic problems is left as a future topic, whose 2D case has been recently published (Khodakarami et al, 2012). This topic is currently being followed by the authors and its results will appear soon.…”

Section: Discussionmentioning

confidence: 94%

A semi-analytical method with a system of decoupled ordinary differential equations for three-dimensional elastostatic problems

Khaji

Khodakarami

2012

International Journal of Solids and Structures

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Keywords:3D elastostatic problems Semi-analytical method Diagonal coefficient matrices Chebyshev polynomials Subparametric elements Clenshaw-Curtis quadrature Decoupled ordinary differential equations a b s t r a c tIn this paper, a new semi-analytical method is presented for modeling of three-dimensional (3D) elastostatic problems. For this purpose, the domain boundary of the problem is discretized by specific subparametric elements, in which higher-order Chebyshev mapping functions as well as special shape functions are used. For the shape functions, the property of Kronecker Delta is satisfied for displacement function and its derivatives, simultaneously. Furthermore, the first derivatives of shape functions are assigned to zero at any given node. Employing the weighted residual method and implementing Clenshaw-Curtis quadrature, coefficient matrices of equations' system are converted into diagonal ones, which results in a set of decoupled ordinary differential equations for solving the whole system. In other words, the governing differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain. To evaluate the efficiency and accuracy of the proposed method, which is called Decoupled Scaled Boundary Finite Element Method (DSBFEM), four benchmark problems of 3D elastostatics are examined using a few numbers of DOFs. The numerical results of the DSBFEM present very good agreement with the results of available analytical solutions.

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Section: Discussionmentioning

confidence: 94%

A semi-analytical method with a system of decoupled ordinary differential equations for three-dimensional elastostatic problems

Khaji

Khodakarami

2012

International Journal of Solids and Structures

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“…In other words, emphasis is basically devoted to those important aspects of the method which are subjected to remarkable modifications in comparison with the previous works of the authors [32][33][34][35][36][37][38]. For more details on the formulation of the method, the readers are referred to Ref.…”

Section: Fundamentals Of the Semi-analytical Methodsmentioning

confidence: 98%

“…In other words, the present method offers an efficient procedure for solving various problems as already reported in Refs. [32][33][34][35][36][37][38]. In this section, the required procedure for solving Euler's differential equations is illustrated, and the extension of the procedure to derive the SIFs is extensively discussed in the next section.…”

Section: Solution Proceduresmentioning

confidence: 99%

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Development of a new semi-analytical method in fracture mechanics problems based on the energy release rate

2016

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The existence of crack and notch is a considerable and serious subject in the failure study of solids and structures. As most of cracked domain problems do not have closed-form solutions, numerical methods are dominant approaches dealing with fracture mechanics problems. In fracture mechanics and failure analysis, cracked media energy and consequently stress intensity factors (SIFs) play a crucial and significant role. Based on linear elastic fracture mechanics, the SIFs may be estimated using the energy release rate, as an indirect approach. This study presents the development of a new semi-analytical method to model cracked media. In this method, only the boundaries of problems are discretized using specific higher-order subparametric elements and higher-order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw-Curtis quadrature yield diagonal Euler's differential equations. Therefore, when the local coordinate origin is located at the crack tip, the geometry of crack problems is implemented directly without further processing. In order to present crack boundary conditions, a modified form of nodal force function is presented. Validity and accuracy of the proposed method is fully illustrated by four benchmark problems, whose results agree very well with those of other numerical and/or analytical solutions existing in the literature.

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“…A new semi‐analytical method has been recently developed by Khaji and coworkers . In this method, only boundaries of problems are discretized using specific subparametric elements and higher‐order Chebyshev mapping functions.…”

Section: Introductionmentioning

confidence: 99%

Determination of stress intensity factors of 2D fracture mechanics problems through a new semi‐analytical method

Khaji

Yazdani

2015

Fatigue Fract Eng Mat Struct

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This study presents a novel development of a new semi‐analytical method with diagonal coefficient matrices to model crack issues. Accurate stress intensity factors based on linear elastic fracture mechanics are extracted directly from the semi‐analytical method. In this method, only the boundaries of problems are discretized using specific subparametric elements and higher‐order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw–Curtis numerical integration result in diagonal Euler's differential equations. Consequently, when the local coordinates origin is located at the crack tip, the stress intensity factors can be determined directly without further processing. In order to present infinite stress at the crack tip, a new form of nodal force function is proposed. Validity and accuracy of the proposed method is fully demonstrated through four benchmark problems, which are successfully modeled using a few numbers of degrees of freedom. The numerical results agree very well with the analytical solution, experimental outcomes and the results from existing numerical methods available in the literature.

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Modeling transient elastodynamic problems using a novel semi-analytical method yielding decoupled partial differential equations

Cited by 18 publications

References 42 publications

A semi-analytical method with a system of decoupled ordinary differential equations for three-dimensional elastostatic problems

A semi-analytical method with a system of decoupled ordinary differential equations for three-dimensional elastostatic problems

Development of a new semi-analytical method in fracture mechanics problems based on the energy release rate

Determination of stress intensity factors of 2D fracture mechanics problems through a new semi‐analytical method

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