It is widely recognized in engineering fracture mechanics that integral and differential forms of nonlocal damage models with a constant internal length scale suffer from an incorrect representation of failure mechanisms either by spurious damage growth or by incorrect damage initiation and propagation. In this regard, in this thesis, a displacement-based gradient-enhanced damage model (GEDM) with a transient internal length scale is formulated and used for failure analysis of quasi-brittle materials. Research is focused on mode-I and mode-II failure mechanisms. In a GEDM, a local field is enhanced into a nonlocal field and the nonlocal field is the output of the enhancement. Displacement-based GEDM enhances local displacement fields into nonlocal displacement fields instead of enhancing local equivalent strain fields into nonlocal equivalent strain fields, as in strain- and stressbased GEDMs. The key ingredient of the proposed extension is a transient internal length scale that tends to zero as the damage parameter tends to one. Various expressions for this transient internal length scale are proposed, formulated, and discussed. Also, the need for correction on the gradient activity operator in mode-II failure is demonstrated. To this end, an anisotropic formulation of the displacementbased GEDM is formulated and used to control the material failure mechanism in mode-II failure. Examples of the new model regularization capabilities are compared to the original/classical displacement-based GEDM with a constant internal length scale. Despite the existence of spurious damage growth in mode-I failure for two dimensional problems (4-point bending beam example) for both the transient isotropic and the transient anisotropic versions, spurious damage growth is eliminated for mode-I failure in one-dimensional problems. Also, the proposed transient isotropic model eliminates spurious damage growth for mode-II failure in two-dimensional problems. However, the damage migration issue is not solved. This issue is addressed by the implementation of the transient anisotropic model. The transient anisotropic model has no damage spreading and damage migration issues in mode-II failure and realistic damage initiation and propagation are guaranteed. These features enable the representation of failure patterns i.e., thin crack-like shear-band. In practical terms this leads to a non-broadening shear fracture process zone in the wake of the crack tip, addressing one of the main criticisms of existing gradient damage models. Applicability of the proposed models is demonstrated by representative one- and two-dimensional examples.@en

In this study, a three-dimensional (3D) non-ordinary state-based peridynamics (NOSB-PD) formulation for thermomechanical brittle and ductile fracture is presented. The Johnson-Cook (JC) constitutive and damage model is used to taken into account plastic hardening, thermal softening and fracture. The formulation is validated by considering two benchmark examples: 1) The Taylor-bar impact and 2) the Kalthoff-Winkler tests. The results show good agreements between the numerical simulations and the experimental results.

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Offshore wind turbine foundations are affected by cyclic loads due to oscillatory kinematic loads, such as those from wind, waves, and earthquakes. Monopiles are often used as a foundation concept for offshore windmill turbines. In this study, coupled dynamic equations with the u-P formulation for low-frequency load are considered for an offshore wind turbine monopile foundation, to present the response in terms of pore water pressure (PWP), stress and strain distribution in an elastic porous medium at regions around the monopile foundation. Different stress recovery techniques based on the Zienkeiwicz-Zhu (ZZ) error estimator namely, super-convergent patch recovery (SPR), weighted super-convergent patch recovery (WSPR), and L2-projection techniques are also investigated to recover the stresses at nodal points in the finite element method. To estimate errors in the time domain when performing transient simulations, three recovery processes are used with different meshes. The convergence of the dynamic problem is also studied. The results are verified with findings in the literature, revealing that the time period of effective stresses follows the applied load frequency. In conclusion, the history of the shear stress can have an important effect on the shear stress distribution, making it asymmetric in the time domain.@en

In finite element simulation of engineering applications, accuracy is of great importance considering that generally no analytical solution is available. Conventional error estimation methods aim to estimate the error in energy norms or the global L²-norm. These values can be used to estimate the accuracy of the model or to guide how to adapt the model to achieve more accuracy. However, in engineering applications specific quantities are required to be accurate. The novel error estimation approach which is called Dual-Weighted Residual (DWR) error estimation, approximates the error with respect to the quantity of interest which can be mean stress or displacement in a subspace or the solution ('s gradient) on a specific point, etc. DWR error estimation is a dual-based scheme which requires an adjoint (dual) problem. The dual problem is described by defining the quantity of interest in a functional form. Then by solving the primal and dual problems, errors in terms of the specified quantities are calculated. In this paper the DWR error estimation besides the conventional residual-based error estimation and a recovery-based error estimation are applied in a three-dimensional elasticity problem. Local estimated errors are exploited in order to accomplish the mesh adaptivity procedure. The goal-oriented mesh adaptivity control the local errors in terms of the prescribed quantities. Both refinement and coarsening processes are applied to raise the efficiency. The convergence rates are plotted to illustrate the superiorities of the goal-oriented adaptivity over the traditional ones.

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The node moving and multistage node enrichment adaptive refinement procedures are extended in mixed discrete least squares meshless (MDLSM) method for efficient analysis of elasticity problems. In the formulation of MDLSM method, mixed formulation is accepted to avoid second-order differentiation of shape functions and to obtain displacements and stresses simultaneously. In the refinement procedures, a robust error estimator based on the value of the least square residuals functional of the governing differential equations and its boundaries at nodal points is used which is inherently available from the MDLSM formulation and can efficiently identify the zones with higher numerical errors. The results are compared with the refinement procedures in the irreducible formulation of discrete least squares meshless (DLSM) method and show the accuracy and efficiency of the proposed procedures. Also, the comparison of the error norms and convergence rate show the fidelity of the proposed adaptive refinement procedures in the MDLSM method.@en

In this paper, the Mixed Discrete Least Squared Meshless (MDLSM) method is used for solving quadratic Partial Differential Equations (PDEs). In the MDLSM method, the domain is discretized only with nodes, and a minimization of a least squares functional is carried out. The least square functional is defined as the sum of the residuals of the governing differential equation and its boundary condition at the nodal points. In MDLSM, the main unknown parameter and its first derivatives are approximated independently with the same Moving Least Squares (MLS) shape functions. The solution of the quadratic PDE does not, therefore, require calculation of the complex second order derivatives of MLS shape functions. Furthermore, both Neumann and Dirichlet boundary conditions can be treated and imposed as a Dirichlet type boundary condition, which is applied using a penalty method. The accuracy and efficiency of the MDLSM method are tested against three numerical benchmark examples from one-dimensional and two-dimensional PDEs. The results are produced and compared with the irreducible DLSM method and exact analytical solutions, indicating the ability and efficiency of the MDLSM method in efficient and effective solution of quadratic PDEs.@en

A Mixed Discrete Least Square Meshless (MDLSM) method is proposed for the solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet-type which is easily incorporated via a penalty method. Because least squares based algorithm of MDLSM method, the proposed method does not need to be satisfied by the LBB condition. The performance of the proposed method is tested on a benchmark example from theory of elasticity namely the problem of infinite plate with a circular hole and the results are presented and compared with those of the analytical solution and the solutions obtained using the irreducible DLSM formulation. The results indicate that the proposed MDLSM method is more accurate than the DLSM method. The results show that the numerical solutions of the MDLSM method can be obtained with lower computational cost and with higher accuracy. Also its performance is marginally affected by the irregularity of the nodal distribution.@en

In this paper, the Artificial Neural Network (ANN) and the Adaptive Neuro-Fuzzy Inference System (ANFIS) are used to predict the shear strength of Reinforced Concrete (RC) beams, and the models are compared with American Concrete Institute (ACI) and Iranian Concrete Institute (ICI) empirical codes. The ANN model, with Multi-Layer Perceptron (MLP), using a Back-Propagation (BP) algorithm, is used to predict the shear strength of RC beams. Six important parameters are selected as input parameters including: concrete compressive strength, longitudinal reinforcement volume, shear span-to-depth ratio, transverse reinforcement, effective depth of the beam and beam width. The ANFIS model is also applied to a database and results are compared with the ANN model and empirical codes. The first-order Sugeno fuzzy is used because the consequent part of the Fuzzy Inference System (FIS) is linear and the parameters can be estimated by a simple least squares error method. Comparison between the models and the empirical formulas shows that the ANN model with the MLP/BP algorithm provides better prediction for shear strength. In adition, ANN and ANFIS models are more accurate than ICI and ACI empirical codes in prediction of RC beams shear strength.

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In this paper, an adaptive refinement procedure is proposed to be used with Discrete Least Squares Meshless (DLSM) method for accurate solution of planar elasticity problems. DLSM method is a newly introduced meshless method based on the least squares concept. The method is based on the minimization of a least squares functional defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries. A Moving Least Square (MLS) method is used to construct the shape function making the approach a fully least squares based approach. An error estimate and adaptive refinement strategy is proposed in this paper to increase the efficiency of the DLSM method. For this, a residual based error estimator is introduced and used to discover the region of higher errors. The proposed error estimator has the advantages of being available at the end of each analysis contributing to the efficiency of the proposed method. An enrichment method is then used by adding more nodes to the area of higher errors as indicated by the error estimator. A Voronoi diagram is used to locate the position of the nodes to be added to the current nodal configuration. Efficiency and effectiveness of the proposed procedure is examined by adaptively solving two benchmark problems. The results show the ability of the proposed strategy for accurate simulation of elasticity problems.@en

In this paper, an error indicator and adaptive refinement procedure in conjunction with the Discrete Least Squares Meshless (DLSM) method is presented for the effective and efficient analysis of planar elasticity problems. The DLSM method is a truly meshless method, which has been used for the solution of different problems ranging from solid to fluid mechanics problems. The method is based on the minimization of the least squares functional with respect to the nodal parameters. The least squares functional is formed as the weighted summation of the residual of the differential equation and its boundary condition. The DLSM method enjoys from providing a natural error indicator defined as the value of the functional at nodal points, which can be efficiently used to identify zones of larger numerical errors. The proposed error indicator has the additional advantage of easy computation since most of its components are already available from the main DLSM computation for analysis. Since both the approximation method and discretization method are meshless, repositioning of the nodes can be easily exploited for refining the numerical solution without any geometrical difficulties usually encountered in mesh based methods. Here, a node moving strategy based on spring analogy is proposed to displace the nodal points to the areas indicated by the higher values of the error indicator. A Voronoi diagram is used to identify neighboring nodes that should be connected to each other using springs. The efficiency and effectiveness of proposed adaptive refinement technique is tested on some benchmark examples with available analytical solutions and the results are presented. The results show that the proposed adaptive refinement technique is quiet effective for the accurate and efficient solution of elasticity problems.

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