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Dams as one of the most important structures are always exposed to various hazards such as earthquake. As dam failure may lead to financial damages and fatalities, it should be designed with most economical and accurate methods. An earthquake causes hydrodynamic pressure waves exerting on the dam. This is one of the important factors in design of dams that are always considered by consulting engineers. Helmholtz equation is the governing relation on the propagation of hydrodynamic pressure waves in dam reservoirs during an earthquake. In order to solve the Helmholtz equation to calculate hydrodynamic pressures on dams, the reservoir’s boundary conditions (BCs) should be taken exactly into account. The BCs include (a) the interface boundary of dam and reservoir (as initial zone of reservoir excitation), (b) bottom boundary (with partial absorption of wave energy by accumulated sediments), (c) upstream boundary (with radiation of another part of the wave energy from the reservoir), and (d) formation of surface waves in the upper boundary of the reservoir. The purpose of present study is to model the mentioned physical phenomena in the frequency domain, using a new semi-analytical method, called Decoupled Equations Method (DEM). In the DEM, only the domain boundaries are discretized by specific high-order non-isoparametric elements. The main features used for modeling of geometry and physics of the problem consists of: (1) high-order Chebyshev polynomials as mapping functions, (2) special shape functions of 2n_η+1 degree polynomials for (n_η+1)-node elements , (3) Clenshaw-Curtis quadrature, and (4) integral forms produced by weighted residual method. By using these features and their properties, coefficient matrices of the system of governing equations become diagonal. This means that the governing partial differential equation for each degree of freedom (DOF) becomes independent from other DOFs of the domain to be analyzed. Therefore, this reduction in space dimensions of the main problem may significantly reduce computational costs in comparison with other available numerical methods. In this study, for the first time in order to provide a solution by low costs to calculate the hydrodynamic pressure distribution on the gravity dams, the relations of reservoir’s BCs are derived in local coordinates by using of the DEM and, the process of applying derived equations is then expressed into the solution of Helmholtz equation. To verify this method, an example of this field is solved by using the DEM, where dam and its rigid foundation are excited by horizontal harmonic vibration. The obtained responses from the solution of this example indicates that the present method for modeling of the potential problems with natural boundary conditions under earthquake excitations, by considering propagation of hydrodynamic waves in the reservoir, show acceptable accuracy and feasibility in comparison with the available analytical solution. The results of the DEM should be developed for more general condition of dam-reservoir interaction, which include flexible concrete gravity dams with inclined dam-reservoir interaction boundary conditions along with partial absorption of wave energy by accumulated sediments. These features are being followed by the authors, and will be disseminated in new papers soon.

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Due to the importance of acoustic response control of submerged vibrating structures, in this study,the optimization of acoustic power radiation from a square stiffened plate under harmonic loading was investigated.Since one face of the plate is in contact with water, a fully coupled analysis was used. The effect of fluid in the analysis was considered via added mass matrix. The added mass matrix was obtained based on both Rayleigh integral and the boundary element approaches.The obtained added mass matrix was then added to the mass matrix of the structure calculated from the finite element discretization of plate. Several variables such as acoustic pressure at specific points and also radiated power were calculated. Results show good agreement between obtained results from the Rayleigh integral and the boundary element. To reduce the radiation power, dynamic absorbers in the form of lumped mass and mass-springs in specific locations on the plate surface were considered. Because optimization procedure requires several evaluation of cost function in the design variable space, model reduction can save a great amount of computation efforts. Therefore, the truncated modal matrix was employed and its effectiveness and precision on the obtained results was studied. Finally, Genetic Algorithm (GA) was used for minimizing the appropriate goal function in three case studies: concentrated mass on cross-points, dynamic absorbers on cross-points and combination of two former cases.All the studied cases resulted on significant reduction in the goal function index.

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The existence of crack and notch is a significant and critical subject in the analysis and design of solids and structures. As most of damage problems do not have closed-form solutions, numerical methods are current approaches dealing with fracture mechanics problems. This study presents a novel application of the decoupled equations method (DEM) to model crack issues. Based on linear elastic fracture mechanics (LEFM), the J-integral is computed using the DEM. In this method, only the boundaries of problems are discretized using specific higher-order sub-parametric 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 (LCO) is located at the crack tip, the geometry of crack problems are directly implemented 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 this method is fully demonstrated through two benchmark problems. The numerical results agree very well with the results from existing experimental results and numerical methods available in literature.

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The elastostatic problems are a significant subject in the analysis and design of solids and structures. As most of the complicated elastostatic problems do not have closed-form solutions, numerical methods such as finite element method (FEM), boundary element method (BEM), discrete element method (DEM), meshless methods, scaled boundary finite element method (SBFEM), and hybrid methods are the current approaches dealing with these types of engineering problems. This study presents a novel application of the decoupled equations method (DEM) to assessment elastostatic issues. In the present method, the so-called local coordinate's origin (LCO) is selected at a point, from which the entire domain boundary may be observed. For the bounded domains, the LCO may be chosen on the boundary or inside the domain. Furthermore, only the boundaries which are visible from the LCO need to be discretized, while other remaining boundaries passing through the LCO are not required to be discretized. In this method, only the boundaries of problems are discretized using specific higher-order sub-parametric elements and higher-order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw-Curtis quadrature result in diagonal Euler’s differential equations. So, the coefficient matrices are diagonal, which provide a system of single Euler’s differential equations for the ith degree of freedom (DOF). If n indicates the number of DOFs of the problem assumed to be analyzed by the proposed method, only n Euler’s differential equations (with only one unknown differential equation for each DOF) should be solved. In the proposed method, the LCO is the same for all nodes, for which the LCO has the same displacement components. Therefore, the physical concept of this fact may be considered as some semi-parallel springs adjoining to each other at the LCO. Therefore, the proposed procedure is called “redistribution” of the stresses in the present method. At the final step, using the calculated displacement field along ξ, the displacement at any point of the problem’s domain is interpolated by using the proposed special shape functions. Although the governing equation of each DOF is decoupled from those of other DOFs, however the “redistribution” of the stresses at the LCO and resolving the problem for each DOF, represents the connection between all DOFs of the domain. In the solution procedure, the order of displacement function u(ξ) depends on nodal force function F^b (ξ). To analysis of elastostatic problems in the classical Decoupled Equations Method, F^b (ξ) varies in the undertaken domain like a body force. Therefore, F^b (ξ) is defined as a linear function. In this study by proposing new forms of force function, the response of elastostatic problems is assessed. In the following Sensitivity of this method via proposed nodal force functions is fully demonstrated through two benchmark problems. The results show that stress and displacement fields totally depend on the form of force function. Also, the results show to get optimum results, proposing an appropriate nodal force function corresponding to physical concept is necessary. For example in the cantilever beam which is subjected to a shear force at its free end, by considering the linear form for nodal force function results in minimum error. In the other hands, in the Kirsch’s problem with a central small circular hole, considering the nonlinear form for nodal force function leads to minimum error.