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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. [ABSTRACT FROM AUTHOR]

In this paper, a new semi-analytical method is developed with introducing a new global nonreflecting boundary condition at medium−structure interface, in which the coefficient matrices, as well as dynamic-stiffness matrix are diagonal. In this method, only the boundary of the problem's domain is discretized with higher-order sub-parametric elements, where special shape functions and higher-order Chebyshev mapping functions are employed. Implementing the weighted residual method and using Clenshaw–Curtis quadrature lead to diagonal Bessel's differential equations in the frequency domain. This method is then developed to calculate the dynamic-stiffness matrix throughout the unbounded medium. This method is a semi-analytical method which is based on substructure approach. Solving two first-order ordinary differential equations (i.e., interaction force–displacement relationship and governing differential equation in dynamic stiffness) allows the boundary condition of the medium−structure interface and radiation condition at infinity to be satisfied, respectively. These two differential equations are then diagonalized by implementing the proposed semi-analytical method. The interaction force–displacement relationship may be regarded as a nonreflecting boundary condition for the substructure of bounded domain. Afterwards, this method is extended to calculate the asymptotic expansion of dynamic-stiffness matrix for high frequency and the unit-impulse response coefficient of the unbounded media. Finally, six benchmark problems are solved to illustrate excellent agreements between the results of the present method and analytical solutions and/or other numerical methods available in the literature. [ABSTRACT FROM AUTHOR]

This paper proposes a semi-analytical method for modeling short-crested wave diffraction around a vertical cylinder of arbitrary cross-section, in an unbounded domain. In this method, only the boundaries of domain are discretized using special sub-parametric elements. The formulation of elements is constructed by employing higher-order Chebyshev mapping functions and special shape functions. The shape functions are introduced to satisfy Kronecker Delta property for the potential function and its derivative, corresponding to the governing Helmholtz equation of the problem. Furthermore, the first derivative of shape functions of any given control point are set to zero. By implementing weighted residual method and using Clenshaw–Curtis numerical integration, the coefficient matrices of equations system become diagonal, yielding a set of decoupled governing Bessel differential equations for the whole system. In other words, the governing equation for each degree of freedom (DOF) is independent of other DOFs of the domain. Accuracy and efficiency of present method are fully demonstrated through three short-crested wave diffraction problems which are successfully modeled using a few numbers of DOFs (or nodes), with excellent agreements between the results of the present method and those of other analytical/numerical solutions. [ABSTRACT FROM AUTHOR]

In this paper, a novel semi-analytical method, called Decoupled Equations Method (DEM), is presented for modeling of elastic wave propagation in the semi-infinite two-dimensional (2D) media which are involved surface topography. In the DEM, only the boundaries of the problem are discretized by specific subparametric elements, in which special shape functions as well as higher-order Chebyshev mapping functions are implemented. For the shape functions, Kronecker Delta property is satisfied for displacement function. Moreover, the first derivatives of displacement function with respect to the tangential coordinates on the boundaries are assigned to zero at any given node. Employing the weighted residual method and using Clenshaw–Curtis numerical integration, coefficient matrices of the system of equations are transformed into diagonal ones, which leads to a set of decoupled partial differential equations. To evaluate the accuracy of the DEM in the solution of scattering problem of plane waves, cylindrical topographical features of arbitrary shapes are solved. The obtained results present excellent agreement with the analytical solutions and the results from other numerical methods. [ABSTRACT FROM AUTHOR]

Abstract: In 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. [Copyright &y& Elsevier]

Abstract: In this paper, a novel semi-analytical method is proposed for modeling of two-dimensional (2D) elastodynamic problems. In this regard, the domain boundary of the problem is discretized by specific non-isoparametric elements that are proposed for the first time in this paper. In these elements, special shape functions as well as higher-order Chebyshev mapping functions are implemented. For the shape functions, Kronecker Delta property is satisfied for displacement function and its derivatives, simultaneously. Moreover, the first derivatives of shape functions are assigned to zero at any given node. Finally, using a weak form of weighted residual method and implementing Clenshaw–Curtis numerical integration, coefficient matrices of the system of equations are converted into diagonal ones, which leads to a set of decoupled partial differential equations for solving the whole system. This means that the governing partial differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain. Also, to assess the accuracy and efficiency of the proposed method, four benchmark problems of 2D elastodynamics are solved using a few numbers of DOFs. The numerical results present excellent agreement with the analytical solutions and the results from other numerical methods. [Copyright &y& Elsevier]