Extensive research has been performed on damage modeling of composites and a number of models have been proposed to predict damage accumulation (Shokrieh and Lessard 2000; Dvorak and Zhang 2001; Van Paepegem and Degrieck 2003). Among those, progressive damage models that use one or more damage variables related to measurable manifestations of damage (interface debonding, transverse matrix cracks, delamination size, etc.) are considered the most promising because they quantitatively account for the accumulation of damage in the composite structure. Kumar et al. studied the effect of impactor and laminate parameters on the impact response and impact-induced damages in graphite/epoxy laminated cylindrical shells using three-dimensional (3D) **finite** **element** formulation (Kumar, Rao et al. 2007). Ghosh and Sinha (2004) developed a **finite** **element** analysis **procedure** to predict the initiation and propagation of damages as well as to analyze laminated composite plates damaged under forced vibration and impact loads. Zhao and Cho (2004) investigated the impact-induced damage initiation and propagation in the laminated composite shell under low-velocity impact. In addition, Abrate (1994) presented an overview of the work carried out by different researchers in the field of the optimum design of composite laminated plates and shells subjected to constraints on strength, stiffness, buckling loads and fundamental natural frequencies.

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Fig. 1 The representative volume **element** from microscale structure to a unit cell structure. In the case where RVE constituents are linear elastic and the material is assumed as isotropic, a single loading is sufficient to determine the effective shear and bulk modulus. In the **finite** **element** method, this can be easily conducted using the implicit **procedure**. Some studies [15,16] have applied simple load histories such as incremental uniaxial stretching or shear to the RVE in order to obtain homogenised stress-strain curves. Basically, the kinematic constraint is imposed to ensure that the motion of the RVE is driven by the history of the macroscopic deformation and enforced consistency between microscopic and macroscopic stress power [10].

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ABSTRACT: In this paper, we consider scattering by a guided wave incident obliquely on a surface breaking crack in a laminated composite plate, with a view to ultrasonic nondestructive assessment of cracks. The solution to this problem is the first step towards analyzing the general three-dimensional scattering problem. The method used for modeling is a **hybrid** method which combines **finite** **element** method with wave function expansion **procedure**. The reciprocity relations governing the reflection and transmission coefficients are established and are used to check the numerical accuracy. Also, the principle of energy conservation is used as another check for the accuracy of the numerical results. Numerical results for reflection and transmission coefficients are presented for an 8-layer cross-ply laminated plate.

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Abstract. A **hybrid** higher-order **finite** **element** boundary in- tegral (FE-BI) technique is discussed where the higher-order FE matrix elements are computed by a fully analytical pro- cedure and where the gobal matrix assembly is organized by a self-identifying **procedure** of the local to global trans- formation. This assembly **procedure** applys to both, the FE part as well as the BI part of the algorithm. The geometry is meshed into three-dimensional tetrahedra as **finite** elements and nearly orthogonal hierarchical basis functions are em- ployed. The boundary conditions are implemented in a strong sense such that the boundary values of the volume basis func- tions are directly utilized within the BI, either for the tangen- tial electric and magnetic fields or for the asssociated equiv- alent surface current densities by applying a cross product with the unit surface normals. The self-identified method for the global matrix assembly automatically discerns the global order of the basis functions for generating the matrix ele- ments. Higher order basis functions do need more unknowns for each single FE, however, fewer FEs are needed to achieve the same satisfiable accuracy. This improvement provides a lot more flexibility for meshing and allows the mesh size to raise up to λ/3. The performance of the implemented sys- tem is evaluated in terms of computation time, accuracy and memory occupation, where excellent results with respect to precision and computation times of large scale simulations are found.

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In addition, the finite element analysis model is in good agreement with the impact test results, therefore this finite element model and the analysis procedure will be [r]

In a wind turbine blade the chief structure which is responsible for the resistance of the force and bending moment is spar cap, its size has a significant impact on the blade mass and the stiffness of the blade. This paper presents the design and analysis for a spar cap for a wind turbine rotor blade. These studies are conducted parametrically, examining a range of Hexagonal basalt fibre material with glass and hence exhibiting a superior of structural performance of wind turbinespar cap and also reduce the usage of glass fibre material costs. The length of the beam,, the locations of the spar cap and the thicknesses of shear webs are taken as design variables. These parametric studies are used to determine the stress distribution and deformation of spar cap and resulting from the **hybrid** materials studied..The concepts' of parametric FEA was determined using ANSYS and Compared with the existing glass fibre

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Each span was modelled with a single **finite** **element** with 21 cross-sections and three integration modules. The connection was represented by one **element** with 10 sections distributed in two integration modules. The construction and loading history begins with the prestressing and self-weight of the pre-cast members, at 28 days. The top flange and the mild reinforcement were then added to the structure and new prestressing and self-weight loading cases applied, considering an age of 28 days for cast-in-place concrete. The live loads were applied and the prototype led to failure. Figure 12 presents the inner support moment x relative loading, results for prototype L-1 of Lopes et al. (1997) experimental work. Experimental and numerical results show a good agreement, even after the connection cracking. Figure 8 also shows a good estimate of the inner support moment at the experimental ultimate load level.

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The **hybrid** tube is positioned vertically. Then the moving plate is placed on the top of **hybrid** tube. The overall configurations of part located shown in Fig. 2. In all compression loading in this work, the bottom end of the **hybrid** tube was assumed to built-in and constrained in all degrees of freedom in rotations and translation. The moving plate at the upper end was constrained in X and Y direction. While the upper end was free in the Z direction and compressed by a rigid loading plate that moved at a downward initial velocity in quasi-static condition. In order to avoid penetration between upper end plate and composite tube, the interactions contact algorithms were defined. The “AUTOMATIC SURFACE TO SURFACE” algorithms contact was defined between upper end within **hybrid** tubes. On the one of contact algorithms was a self-contact algorithm to prevent interpenetration during the deformation.

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This paper presents an overview on applications of **hybrid** **finite** **element** method to thermal analysis of heterogeneous materials. Recent developments on the **hybrid** fundamental solution based **finite** **element** model (FEM) of heat transfer in nonlinear functionally graded materials (FGMs) and composite materials are described. Formulations for all cases are derived by means of modified variational functional and fundamental solutions. Generation of elemental stiffness equations from the modified variational principle is also discussed. Finally, a brief summary of the approach is provided.

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Analysis of the composite shell model of the afterbody was done with Ansys Composite PrepPost (ACP) and Ansys Mechanical Module, using **finite** **element** method. For structural tests, the surface of bulkhead no. 6 was chosen as the fixed support and the gravitational acceleration was defined. The weight of the empennage was applied at the surface under vertical stabilizer as remote load. The aerodynamic loads on vertical stabilizer was applied with the magnitude and position specified in Fig. 8 and Table 2. The aerodynamic loads on both side of the horizontal stabilizer was applied with the magnitude specified in Table 2. But apply only one direction (up or down) at a time. For example, the downward forces were applied on both side of the horizontal stabilizer as shown in Fig. 12.

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Reconstructed model is very accurate in terms of geometry, and according to the presented algorithm it reaches the theoretical limitations of MRI as there is no accuracy loss during the reconstruction process. However, in some cases, for example for testing solution or for any other proposes, where accuracy is not so important, less accurate and lighter model may be required. In those cases the Smoothing **Procedure** is developed. It is optional, and therefore, the accuracy of obtaining geometry was not tested. This **procedure** is based on temporary triangularization and NURBS smoothing algorithm, which could be performed using any existing reverse engineering software. The result after smoothing **procedure** being done is shown in Figure 10.

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The classical form s o f stiffening grooves were developed in the 1950s [4, 5], Newer studies have shown that the panel stiffness can be increased w ith the use o f autom ated optim ization o f the stiffening g ro o v e p attern s [6]. The program package ALTAIR/HyperWorks was used for the topology o p tim izatio n o f the car-floor panels rep o rted in th is paper. The n u m erical resu lts obtained by the **finite**-**element** solver MSC/Nastran [7] are the basis o f the car-floor panel optimization by the A L T A IR /H yperW orks. O nly the d isp la c e m e n ts, stre sse s and th e first n atu ral frequency can be defined as target values. And only the results o f one **finite**-**element** solver can be used fo r the o p tim iz a tio n . A lso, the o p tim iz a tio n algorithm is not publicly available.

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[106] W.-G. Jiang, R.-Z. Zhong, Q.H. Qin, Y.-G. Tong, Homogenized **Finite** **Element** Analysis on Effective Elastoplastic Mechanical Behaviors of Composite with Imperfect Interfaces, International Journal of Molecular Sciences, 15(12) (2014) 23389-23407. [107] H.-W. Wang, Q.H. Qin, H. Ji, Y. Sun, Comparison

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The key contributions of this work may be summarised as follows. We have introduced a new adaptive **procedure** for the fully-coupled EHL problem, based solely upon a local error estimate for the stress due to the elastic deformation, and demonstrated that this provides a robust mechanism for adapting the mesh in both the elasticity and the Reynolds discretizations. We have developed an adaptive strategy that combines h-refinement and r-refinement (node movement) in a manner that allows locally optimal mesh refinement. The combination of local adaptivity and our novel multigrid-based preconditioner, for the inner iterations of the Newton-Krylov solver, allow this fully-coupled EHL problem to be solved with linear time complexity for the first time: hence providing the first demonstration of the competitiveness of the fully-coupled approach with less general, but also optimal, half-space approximations such as [42]. Finally, and importantly, we have shown that the proposed technique is robust for heavily-loaded cases, which are by far the most computationally challenging.

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Thus, the engineering drawback with non-standard form and pure mathematics may be solved by **finite** **element** analysis wherever a closed type resolution isn't out there. The **finite** **element** analysis ways offer results ofdisplacements, stress distributionand reaction masses at supports etc. for the model.

The eight steps mentioned above have to be carried out before any meaningful information can be obtained regardless of the size and complexity of the problem to be solved. However, the specific commands and procedures that must be used for each of the steps will vary from one **finite** **element** package to another. The solution **procedure** for ANSYS is described in this tutor. Note that ANSYS (like any other FEM package) has numerous capabilities out of which only a few would be used in simple beam problems.

Information from a sufficient number of cases is gath- ered to conduct a feasibility and sensitivity study so as to select certain parameters from the geometry of the sys- tem. All the effective parameters used in the model analysis, such as end-plate width, bolt diameters, welds and all other significant variables, are incorporated in the analysis, and the ranges of the parameters have been re- stricted to practical ranges. A matrix of twenty five test cases was developed by varying the extended end-plate’s geometric variables within their practical ranges. **Finite** **element** analyses were carried out for the selected cases and the results in end-plate separation, end-plate stresses, and bolt forces were obtained.

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Fig. 6 - Minor principal stress propagation for smooth blasting in the top heading of a deep tunnel Thus, it can be seen from Figures 6 and 7 that the **hybrid** **finite**-discrete **element** method successfully modelled the last step of the top heading excavation by blast in a deep tunnel. A newly formed wall is produced and the EDZ around the tunnel wall is induced (Figure 7i).The EDZ can be divided into inner damage zone (Figure 7i-A between tunnel wall and the red dash line), outer damage zones (Figure 7i-B) where the rock properties are sharply changed, and disturbed zone (Figures 7i-C) where stress is redistributed. Therefore, the simulated result agrees well with the literatures [1, 22].

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does not depend on the number of mixture components. The linearization is performed with respect to the persistent variables – i.e. well defined independently of whether a given **element** is in a single phase or two phases. The derivatives in (34) are also well defined in both single phase and two phases. Therefore, our schemes here and in [14] perform well in both cases and no primary variables switching is needed for treating phase appearance/disappearance (cf. [3], [4], [15]). As the discretization of the transport equations is based on the approximation of the total component flux, the connection between the elements with different number of phases is treated in a natural way.

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The basic idea of the **finite** **element** method is to divide the region of interest into a large number of **finite** elements or sub-regions. These elements maybe one, two or three-dimensional. The idea of representing a given domain as a collection of discrete elements is not new, it is recorded that ancient mathematicians estimated the value of π by representing the circle as a polygon with a large number of sides. FEM has been used to solve complex engineering problems such as structural analysis in aircrafts, fluid flow, heat transfer and mass transport. Later on, this method found its way in solving electromagnetic field problems. Waveguide problems are described by using integral or differential equations. Then these equations are solved using numerical methods. FEM has established itself as one of the most powerful and accurate technique for solving problems associated with integrated optical waveguides and microwave devices. The versatility of the method allows elements of various shapes to be used to represent an arbitrary cross- section [49].

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