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We investigate a numerical procedure based on extended isogeometric elements in combination with second-order cone programming (SOCP) for assessing collapse limit loads of cracked structures. We exploit alternative basis functions, namely B-splines or non-uniform rational B-splines (NURBS) in the context of limit analysis. The optimization formulat...
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... o as N A ðn; gÞ ¼ N i;p ðnÞM j;q ðgÞ ð 16Þ Fig. 1 illustrates the set of one-dimensional and two-dimensional B-spline basis ...
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... collapse limit factor of a grooved cracked plate with mesh of 20 Â 25 elements. Tables 5 and 6. The structure is discretized into a coarse mesh and control net corresponding to quadratic and cubic elements as shown in Fig. 10. Meshes of 6 Â 8; 12 Â 16; 18 Â 24 and 24 Â 32 NURBS elements are described in Fig. 11. Table 7 confirms the convergence of the present method. A comparison between the obtained results and those of other analytical and numerical approaches are reported in Table 8. As expected, the present solutions shows high reliability. The present ...
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... collapse limit factor of a grooved cracked plate with mesh of 20 Â 25 elements. Tables 5 and 6. The structure is discretized into a coarse mesh and control net corresponding to quadratic and cubic elements as shown in Fig. 10. Meshes of 6 Â 8; 12 Â 16; 18 Â 24 and 24 Â 32 NURBS elements are described in Fig. 11. Table 7 confirms the convergence of the present method. A comparison between the obtained results and those of other analytical and numerical approaches are reported in Table 8. As expected, the present solutions shows high reliability. The present method produces the solutions belonging to the reliable interval of the analytical ...
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... us consider the 2D plate with dimension b  H having an initial crack at center with length a as shown in Fig. 12a. Firstly, the convergence of the limit load factor k ¼ r lim it =r in case of cracked length ratio a/b = 0.5 which a mesh is plotted in Fig. 12b is tabulated in Table 9 (see Table 10). The present results calculated from XIGA with p = 2, 3, 4 are compared to that of XFEM and analytical solution from [11]. It can be seen that, combining ...
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... us consider the 2D plate with dimension b  H having an initial crack at center with length a as shown in Fig. 12a. Firstly, the convergence of the limit load factor k ¼ r lim it =r in case of cracked length ratio a/b = 0.5 which a mesh is plotted in Fig. 12b is tabulated in Table 9 (see Table 10). The present results calculated from XIGA with p = 2, 3, 4 are compared to that of XFEM and analytical solution from [11]. It can be seen that, combining with SOCP, numerical methods (XFEM and XIGA) give the upper bound convergence. Herein, XIGA gains the better result although it uses coarser ...
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... accuracy. However, higher order of NURBS increases significantly number of variables according to number of control points which is identified in Fig. 4. Therefore, in this study, we prefer to use quadratic and cubic NURBS functions. At the collapse state, the collapse mechanism and plastic dissipation distribution in the plate are plotted in Fig. 13. It is seen that XFEM cannot produce as smoothly plastic dissipation as XIGA although it uses extremely finer mesh. Because XIGA utilizes NURBS basis function with C pÀ1 continuity through element boundaries. Table 11 summarizes the limit load factor of a center cracked plate via cracked length ratio a/b using XIGA in comparison with ...
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... example is a plate of length H width b as shown in Fig. 14a and a single edge cracked length a subjected to tension is studied. The plate is discreted into 15 Â 31 elements as shown in Fig. 14b. This benchmark problem was solved by the analytical method [10] and the limit load factor is given ...
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... example is a plate of length H width b as shown in Fig. 14a and a single edge cracked length a subjected to tension is studied. The plate is discreted into 15 Â 31 elements as shown in Fig. 14b. This benchmark problem was solved by the analytical method [10] and the limit load factor is given ...
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... c ¼ 2= Fig. 15a provides results of analytical and numerical approaches. Both XFEM and XIGA solutions mach well with the analytical one. Moreover, it is also seen that the results obtained from the XIGA using a fewer number of DOFs are slightly closer to the exact one than the XFEM. The limit load factors for various ratios a=b and H=b are exhibited ...
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... Fig. 15a provides results of analytical and numerical approaches. Both XFEM and XIGA solutions mach well with the analytical one. Moreover, it is also seen that the results obtained from the XIGA using a fewer number of DOFs are slightly closer to the exact one than the XFEM. The limit load factors for various ratios a=b and H=b are exhibited in Fig. 15b. It seems that in case of short -crack, numerical results are slightly lower the analytical solution [10]. The limit loads also depend strongly on the ratio a=b. The results obtained agree very well with the exact solution. Fig. 16 illustrates the plastic dissipation distribution and collapse mechanism of the rectangular plate at ...
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... closer to the exact one than the XFEM. The limit load factors for various ratios a=b and H=b are exhibited in Fig. 15b. It seems that in case of short -crack, numerical results are slightly lower the analytical solution [10]. The limit loads also depend strongly on the ratio a=b. The results obtained agree very well with the exact solution. Fig. 16 illustrates the plastic dissipation distribution and collapse mechanism of the rectangular plate at ...
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... we consider a grooved plate with a single edge cracked length a subjected to in-plane tension load p as shown in Fig. 17. We believe that this investigation is useful to recognize a case study of limit load estimation of structures with defects [11]. The data are as same as subSection 5.1.2. The analytical solution was not available. The aim of this study is to estimate the load bearing capacity of structures involving holes and cracks. The full geometry ...
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... estimation of structures with defects [11]. The data are as same as subSection 5.1.2. The analytical solution was not available. The aim of this study is to estimate the load bearing capacity of structures involving holes and cracks. The full geometry of a grooved plate is modeled into a coarse mesh of 4 Â 5 quadratic NURBS elements as shown in Fig. 18a. A fine mesh of 20 Â 25 NURBS elements is then obtained from a coarse mesh as plotted in Fig. 18b. At the collapse state, the collapse mechanism in the plate is plotted in Fig. 18c. Table 11 shows that the presence of the crack affects very significantly the load bearing capacity of this structure. The same conclusion is obtained as p ...
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... solution was not available. The aim of this study is to estimate the load bearing capacity of structures involving holes and cracks. The full geometry of a grooved plate is modeled into a coarse mesh of 4 Â 5 quadratic NURBS elements as shown in Fig. 18a. A fine mesh of 20 Â 25 NURBS elements is then obtained from a coarse mesh as plotted in Fig. 18b. At the collapse state, the collapse mechanism in the plate is plotted in Fig. 18c. Table 11 shows that the presence of the crack affects very significantly the load bearing capacity of this structure. The same conclusion is obtained as p increases the limit load decreases. Finally, Fig. 19 shows the plastic dissipation in the plate ...
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... capacity of structures involving holes and cracks. The full geometry of a grooved plate is modeled into a coarse mesh of 4 Â 5 quadratic NURBS elements as shown in Fig. 18a. A fine mesh of 20 Â 25 NURBS elements is then obtained from a coarse mesh as plotted in Fig. 18b. At the collapse state, the collapse mechanism in the plate is plotted in Fig. 18c. Table 11 shows that the presence of the crack affects very significantly the load bearing capacity of this structure. The same conclusion is obtained as p increases the limit load decreases. Finally, Fig. 19 shows the plastic dissipation in the plate through the change of the crack ...
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... is then obtained from a coarse mesh as plotted in Fig. 18b. At the collapse state, the collapse mechanism in the plate is plotted in Fig. 18c. Table 11 shows that the presence of the crack affects very significantly the load bearing capacity of this structure. The same conclusion is obtained as p increases the limit load decreases. Finally, Fig. 19 shows the plastic dissipation in the plate through the change of the crack ...
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... Two main steps in numerical approaches of the limit analysis are a discretization method to approximate the problem fields and the solution of the optimization problem to determine the limit load factor. The first step, which is relied on both static and kinematic theorems, can be developed using a variety of numerical approaches, including finite element methods [Belytschko and Hodge, 1970;Nguyen-Xuan et al., 2012;Tin-Loi and Ngo, 2003;Ho et al., 2017;Madan and Bhowmick, 2021], meshfree methods [Chen et al., 2008;Liu and Zhao, 2013], boundary element method [Liu et al., 2004[Liu et al., , 2005 and isogeometric analysis (IGA) [Nguyen-Xuan et al., 2014;Do and Nguyen-Xuan, 2019;Nguyen et al., 2022]. The second step aims to find the efficient approaches for solving a optimization problem. ...
This study proposes a pseudo-lower bound method for direct limit analysis of two-dimensional structures and safety evaluation based on isogeometric analysis integrated through Bézier extraction. The key idea in this approach is that the stress field is separated into two parts: fictitious elastic and residual, and then the equilibrium conditions are recast by the weak form. Being different from the displacement approach which employs the kinematic formulation, the approximations based on the stress field satisfy automatically volumetric locking phenomena. Dealing with optimization problems, a second-order cone programming, providing significant advantages of the conic representation for yield criteria, is employed. The examination of various numerical benchmark problems shows an efficient and reliable method for the proposed approach.
... Initially, Hughes et al. [43] developed isogeometric analysis (IGA), which was later improved by Ghorashi et al. [44] by adding enrichment functions, and is now known as XIGA. In the last few years, this method was successfully applied to the vibration of cracked plates [45], bi-material problems [46], cracked shell structure [47], and collapse load of cracked plates [48]. ...
In this work, a higher-order XFEM approach is presented and implemented to analyze free flexural vibration in cracked composite laminated plate using higher-order shear deformation theory. The composite laminated plate experiences flexural vibration during its service life due to transverse shear loading. Out-of-plane modulus of the plate is much lesser than the in-plane modulus, so the free vibration study of the plate has become a crucial parameter for a safe service environment. The proposed methodology introduce higher-order tip enrichment functions (r2 terms) for crack simulation in laminated plate. These enrichment functions efficiently capture all conceivable displacements and rotations near the crack tip region. A comparative analysis between higher-order XFEM and conventional XFEM approach is presented in terms of computational accuracy. Several numerical examples are presented to demonstrate the accuracy of these higher-order enrichment functions, which contain variations in natural frequency with respect to crack sizes for different symmetric ply composite laminate plates under varied boundary conditions. The numerical results demonstrate that the proposed higher-order XFEM approach is significantly enhancing computational accuracy than the classical XFEM approach.
... Recently, kinematic limit analysis of cracked structures has gained some interest and it is combined with discretization schemes from fracture mechanics and computational geometry. For example in [32,33], the discretization of the displacement field is based on B-splines and additional enrichment functions at the tip of the crack. In this section, the structures of figure 12 were examined under plane stress conditions with von Mises criterion which reads ...
... Depending on the value of a/L very coarse meshes of 24-29 elements were applied (figure 13a). The results were obtained in less than 0.05 s, and they are compared with those of [32,33] and the analytical solution in table 4. It can be seen that even with these very coarse meshes the limit loads are almost identical to the exact analytical solution. It is also important that they are rigorous lower bounds. ...
... It is also important that they are rigorous lower bounds. On the other hand, the error in [32,33] is sometimes higher than 2% even though elegant isogeometric interpolation functions were employed. For a/L = 0.1, their result is below the exact value, which should not happen for a kinematic formulation. ...
There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.
... It has been proven that IGA preserves the exactness of the domains of conic sections and offers high-order continuity across elements [15][16][17]. Nguyen-Xuan et al. [18] integrated the XIGA with SOCP for the limit analysis of cracked structures. Nevertheless, the XIGA limit analysis suffers from its significant computational cost due to the excessive overhead of control points with increasing refinement. ...
... where ndofs is the total number of degree of freedoms for the entire domain. The Mosek optimization package is employed as an effective tool to solve Eq. (47) [18]. ...
... Only the region of a cracked structure containing sufficiently large dissipated work would be automatically captured and refined. Consequently, the computational cost incurred by the excessive overhead of IGA control points with increasing refinement can be significantly reduced for limit analysis problems [18]. ...
An adaptive isogeometric-meshfree (AIMF) coupling approach is proposed to predict the limit load factors of cracked structures. The concept of the present approach, which relies on forming an equivalence between the isogeometric analysis (IGA) and moving least-squares meshfree method, is developed based on reproducing conditions, resulting in a unified formulation of the basis functions. Thereby, the refinement of IGA can be implemented in a straightforward meshfree manner for limit analysis problems. The adaptivity of refinement is realized by adopting an indicator for plastic dissipation to automatically identify the material regions corresponding to dissipated work greater than a predefined threshold. Subsequently, the marked meshes are refined through the insertion of linear reproducing points. Enrichment functions are further introduced to the present approach for crack modelling. The resulting optimization formulation of limit analysis is re-expressed in the form of second-order cone programming which can be effectively tackled by the interior-point solvers. Through a series of numerical examples, the proposed approach has been proven to achieve both high convergence rates and accurate simulation results.
... Among researchers who used meshless method to formulate limit and shakedown problems are Liu et al. [32], Vaghefi et al. [33], Xia et al. [34], Le et al. [35], Ho et al. [36]. The shakedown problems also are formulated using iso-geometric analysis such as [37], [38]. ...
... Based on Andersen's study, Vu et al. [48], developed an dual algorithm to calculate shakedown loads of structures in which the upper bound shakedown load and lower bound shakedown load can be computed simultaneously. Algorithms based on second order cone programming have been used for direct methods in recent year such as [37], [49]- [52]. ...
Limit and shakedown analysis solve the plasticity problems by mathematical programming. If the characteristics of structures such as strength and loads are considered as random variables, shakedown analysis can be stated as a stochastic programming problem. The thesis contributes an approach to show that direct structural reliability design can be achieved on the basis of the required failure probabilities by chance constrained programming, which is an effective approach in stochastic programming. In the general case this is a hard problem because probabilities have to be calculated as high dimensional integrals during the optimization algorithm. The thesis developed successfully three algorithms to treat large-scale shakedown analysis problems with random strength and load variables. For random loads or the case of random strength and loads a kinematic algorithm (algorithm A1) has been developed. It allows to compute limit and shakedown loads for deterministic strength and loads; random strength with normal or lognormal distribution and normally distributed loads. Algorithm A2 has been developed to calculate lower bound and upper bound shakedown loads simultaneously in case of random strength random and deterministic loads acting on the structure. Algorithm A3 is a dual algorithm permitting the computation of limit and shakedown loads of a Kirchhoff-Love plate under uncertain conditions of strength. It calculates simultaneously upper and lower bound shakedown loads for normally or for lognormally distributed strength.
The First Order Reliability Method (FORM) is used for an independent check of the chance constrained programming solutions. If the deterministic problem has an analytical solution and strength and load are both either normally or lognormally distributed, then the so-called reliability index can also be computed analytically and the failure probability obtained. The latter is the starting point of the probabilistic limit state design proposed in this dissertation.
... The second step is to solve optimisation problems. Various approaches have previously been studied and implemented in existing optimisation packages such as basic reduction technique [29], interior-point method [33,73,76], linear matching method (LMM) [69,71,72,74], second order cone programming (SOCP) [32,45,54,65]. Relied on discreatisation methods and optimisation problems, lower bound or upper bound load M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 3 factor can be separately found. ...
... Besides, B-splines (or NURBS) give a flexible way to carry out both refinement and degree elevation [56]. IGA has been presented a lot in the literature [55][56][57][58][59][60][61][62][63][64][65][66][67][68] and so on. As known, Finite Element Method (FEM) uses Lagrange basis functions, which are confined to a local domain; on the contrary, the conventional IGA NURBS (or B-splines) basis functions are adopted for the entire domain of structures. ...
... analysis. An overview of these functions is presented in many of previous works using IGA [55][56][57][58][59][60][61][62][63][64][65]. In this section, we present the Bézier extraction operator for NURBS. ...
An effective numerical approach is proposed for calculating limit and shakedown load multipliers of structures. The key idea is to integrate isogeometric analysis (IGA) based on Bézier extraction into an associated primal-dual algorithm. Such an associated primal-dual algorithm based upon the von Mises yield criterion and a Newton-like iteration is used to compute simultaneously both quasi-lower and upper bounds of the collapse multiplier. The method requires a low number of degrees of freedom for the desired accuracy of solutions. Numerical results of this method demonstrate the efficiency, and show the better accuracy and convergence than several finite element solutions.
... Consequently, computational yield design procedures based on the finite element method and mathematical programming have been developed over past decades [1][2][3][4][5][6][7][8]24,34,38]. In the effort to further develop the direct method, various numerical discretization techniques have been considered, for instance mesh-free methods [9][10][11]29], smoothed finite elements [26][27][28], extended finite element method [12] and isogeometric elements [13]. Considering optimization algorithms, the primal-dual interior-point method has been found to be the most efficient and robust. ...
Error indicator plays an essential role in developing automatically adaptive analysis procedures that can maximize the rate of convergence of the approximation. For direct limit state analysis problems, the accuracy of numerical solutions is highly affected by local singularities arising from localized yield stresses. It turns out that these local singularity regions should be refined in computational process, making sure that errors in the stress fields can be reduced. In this paper, an error indicator based on the localized yield stresses for adaptive quasi-static yield design of structures will be described. The non-linear yield criterion is formulated as a quadratic conic constraint, allowing that the obtained optimization problem can be solved using highly efficient solvers. Various numerical examples are examined to illustrate the performance of the proposed adaptive procedure.
... The IGA method has been used with great success and successfully applied to many problems [25][26][27][28][29][30][31][32][33][34]. Among them which quoted based on some of the representative, for cracked problem, Phu-Nguyen et al. [25] developed isogeometric cohesive element for composite delamination by knot insertion algorithm directly from CAD. Verhoosel et al. [26] used IGA for the analysis of cohesive cracks by means of knot insertion. ...
In the present paper, extended isogeometric analysis (XIGA) is used to compute the stress intensity factor (SIF) in straight lugs of Aluminum 7075-T6. XIGA uses the non-uniform rational B-spline (NURBS) functions and enrichment functions through the partition of unity. The Heaviside function is enriched to capture jump at the crack faces, while the analytical asymptotic solutions are incorporate with NURBS to perform the crack tip singularity. The XIGA-based SIF of edge-cracked plate and straight attachment lug are compared with analytical solution and extended finite element method capability available in ABAQUS. Also, crack growth and fatigue life of single crack in attachment lug are estimated and then compared with the available experimental data for two different load ratios equal to 0.1 and 0.5. The SIF calculated from XIGA are in reasonable agreement with the available data.
... FG cracked plates under different loads and boundary conditions were numerically simulated using NURBS-based XIGA [43]. XIGA has been applied to stationary and propagating cracks in 2D [44], plastic collapse load analysis of cracked plane structures [45], and cracked plate/shell structures [46]. ...
... If they are updated at each step, the crack orientation is also updated. This is of particular interest when large deformation cannot be neglected [45]. ...
Based on the finite element software ABAQUS and graded element method, we developed a dummy node fracture element, wrote the user subroutines UMAT and UEL, and solved the energy release rate component of functionally graded material (FGM) plates with cracks. An interface element tailored for the virtual crack closure technique (VCCT) was applied. Fixed cracks and moving cracks under dynamic loads were simulated. The results were compared to other VCCT-based analyses. With the implementation of a crack speed function within the element, it can be easily expanded to the cases of varying crack velocities, without convergence difficulty for all cases. Neither singular element nor collapsed element was required. Therefore, due to its simplicity, the VCCT interface element is a potential tool for engineers to conduct dynamic fracture analysis in conjunction with commercial finite element analysis codes.
... DeLuycker et al. (2011) showed that XIGA possesses good accuracy and higher convergence rate for solving fracture mechanics problems. Nguyen-Xuan et al. (2014) combined the extended isogeometric analysis with second order cone programming for assessing the collapse limit loads of cracked structures. Tran et al. (2015) used the XIGA with higher order shear deformation theory to perform the free vibration analysis of cracked FGM plates. ...
In this work, the fatigue life of an interfacial cracked plate is evaluated in the presence of flaws by homogenized extended isogeometric analysis (XIGA). In XIGA, the crack faces are modeled by discontinuous Heaviside jump functions, whereas the singularity in the stress field at the crack tip is modeled by crack tip enrichment functions. Holes and inclusions are modeled by Heaviside function and distance function respectively. The discontinuities are modeled in a selected region near the major crack whereas the region away from the crack is modeled by an equivalent homogeneous material. The stress intensity factors (SIFs) for the interface cracks are numerically evaluated using the domain based interaction integral approach. Paris law of fatigue crack growth is employed for computing the fatigue life of an interfacial cracked plate. The results show that the defects/ discontinuities away from the main crack barely influence the fatigue life.
