International Journal of Computational Methods

The meshfree methods in computational mechanics have been actively proposed and increasingly developed in order to overcome some drawbacks in the conventional numerical methods. Over past three decades meshfree methods have found their way into many different application areas ranging from classical astronomical problems to solid mechanics analysis, fluid flow problems, vibration analysis, heat transfer and optimization to the numerical solution of all kind of (partial) differential equation problems. The present work is an effort to provide a comprehensive review of various Meshfree methods, their classification, underlying methodology, application area along with their advantages and limitations. Key contributions of mesh free techniques to the area of fracture mechanics have been discussed with applications of element free Galerkin method (EFGM) to fracture analysis as primary concern.

Reliability-based design optimization (RBDO) involves evaluation of probabilistic constraints which can be time-consuming in engineering structural design problems. In this paper, an efficient approach combined sequential optimization with approximate models is suggested for RBDO. The radial basis functions and Latin hypercube sampling are used to construct approximate models of the probabilistic constraints. Then, a sequential optimization with approximate models is carried out by the sequential optimization and reliability assessment method which includes a serial of cycles of deterministic optimization and reliability assessment. Three numerical examples are presented to demonstrate the efficiency of the proposed approach.

In the present study, for the first time, the flow and mass transfer in the rotary micropump-micromixers were investigated by the SPH method. In fact, the present work shows the ability of the SPH method to model the mixing process due to pumping action. The incompressible SPH method applied for modeling is improved by the kernel gradient corrective tensor, a particle shifting algorithm, and an improved periodic boundary condition. SPH is a proper method for modeling the mixing process because there is no modeling for the convective terms and so, the false diffusion is not observed in the SPH modeling. In the present study, first, a viscous micropump comprising a microchannel in which a circular cylinder rotates with special eccentricity is modeled and validated. Then, the geometry is manipulated in order to achieve a desirable micromixer.

In this paper, we propose fast iterative methods based on the Newton–Raphson–Kantorovich approximation in function space [Bellman and Kalaba, (1965)] to solve three kinds of the Lane–Emden type problems. First, a reformulation of the problem is performed using a quasilinearization technique which leads to an iterative scheme. Such scheme consists in an ordinary differential equation that uses the approximate solution from the previous iteration to yield the unknown solution of the current iteration. At every iteration, a further discretization of the problem is achieved which provides the numerical solution with low computational cost. Numerical simulation shows the accuracy as well as the efficiency of the method.

This paper describes a generic algorithm for concurrently solving multiple sub-domains that are selectively discretized in space and time. The mathematical background for this approach is largely based upon the fundamental principles of domain decomposition methods (DDM) and Lagrange multipliers. A proof of stability is provided using energy method and overall efficiency, accuracy and stability of multiple sub-domain coupling is evaluated using a series of numerical examples. Numerical stability is verified by ensuring energy balance at global as well as component sub-domain level. Discussed examples highlight the greatest advantage of MGMT method; which is high simulation speedups (at the cost of reasonably small errors).

This paper investigates the performance of the three-node triangular element with continuous nodal stress (Trig3-CNS) for geometry nonlinear solid mechanic problems. This Trig3-CNS element was recently proposed to improve accuracy of the finite element method (FEM). By synergizing the individual strengths of meshfree method and FEM, the Trig3-CNS element achieves higher accuracy and convergence rate. Furthermore, Trig3-CNS presents high tolerance to mesh distortion. Therefore, it is potentially useful for geometry nonlinear solid mechanics problems in which mesh distortion takes place. Compared with the traditional hybrid “FE-Meshfree” elements, Trig3-CNS naturally processes CNS without requiring any extra operation in post-processing. Numerical tests in the present work show that for geometry nonlinear analysis, the results of the Trig3-CNS element are better than the 3-node triangular element (Trig3) and 4-node isoparametric quadrilateral element (Quad4). In addition, the performance of Trig3-CNS is comparable to that of traditional hybrid “FE-Meshfree” elements.

Range Restricted GMRES (RRGMRES) can be considered as regularizing iterative method and its iterates as regularized solutions. In this paper, we use two preconditioned versions of this method for solving ill-posed inverse problems. Our proposed preconditioner matrix is appropriate to use directly for linear systems of the form $Ax=b$ with square ill-conditioned coefficient matrix. We show numerically that the Preconditioned RRGMRES method has more stable solution compared to RRGMRES.

The locking plate fixations have been developed to enhance bone healing by wide bridging of the fracture and allowing some level of interfragmentary movement (IFM) at the fracture site. However, the IFM induced by conventional locking plate constructs is not uniform at the fracture site and so result in asymmetric callus formation, and ultimately delayed healing. The far cortical locking technique has been recently innovated to address this issue by inducing a uniform IFM. However, the far cortical locking technique is still in its infancy and more research efforts are required before its practical clinical application. Using the theory of porous media and computational methods, this study investigated the effectiveness of far cortical locking technique in presence of different mechanical stiffness of locking plate. The research outcomes indicate that the application of far cortical looking technique enhances IFM at near cortex, and so reduce the difference of IFM between near and far cortex. Further, it shows that, under far cortical locking technique, the bending stiffness of a locking plate plays an important role in bone healing. The use of a stiffer locking plate together with far cortical locking screws encourages more uniform tissue development across the fracture gap. The current research underlines the importance of the optimal selection of plate stiffness for application of far cortical locking technique.

In this paper, the object of our study is two coupled partial differential equation which have a rich physical background. With a reasonable change to the initial value, we use central difference method and compact finite difference method to get the numerical results of the governing equations. In the analysis of the latter, we use the Hermitian and skew-Hermitian splitting (HSS) method to solve a large sparse non-Hermitian positive definite system of linear equations. In this way, the computational efficiency can be improved and the convergence is guaranteed which is proved by Bai [Bai et al. [2003] “Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,” SIAM J. Matrix Anal. Appl. 24, 603–626] and Chen [Chen et al. [2014] “Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition,” J. Comput. Appl. Math. 264, 115–130]. And we also pay attention to the effect of initial conditions, on which we add a small perturbation to observe its influence by comparing numerical results.

Aluminum has become a widely used material in the automotive industry due to the need for production of lighter automobiles. As the number and complexity of aluminum products increase, it becomes necessary to improve methods used to join these parts. Thermal interference fitting is one of these methods. It is not possible to calculate interference fit stresses accurately for all conditions using traditional calculation methods, and a method to calculate separation frequency of interference fitted aluminum parts is not available. This study investigates elastic–plastic interference fit stresses and separation frequencies of lightweight metal gear and steel shaft interference fitted joints (which have begun to be used widely in designs) using analytical and numerical calculation methods. It was found as a result of the study that plastic deformation occurred due to decreasing diameter, and increasing interference and numerical calculation methods produced accurate results. It was determined as a result of numerical analysis that the separation frequency of lightweight metal gears was lower in smaller interference fit diameters compared with steel gears, and that this increased in connection with increasing interference fit diameter and was found to be higher compared with steel gears.

The bending responses of the delaminated composite curved (single and doubly) shallow shell panel computed numerically and validated with subsequent experimental results. The laminated shell panel model is developed mathematically with the help of two higher-order mid-plane theories and finite element method. The generalized governing equations of the shell panel model is obtained via the variational method and solved numerically using a home-made computer code developed in MATLAB. The consistency of the developed models has been established through the convergence test and validated by comparing the present numerical results with available published data including experimental values. The experimental bending results are obtained via three-point bend test of the fabricated woven Glass/Epoxy composite plate including the delamination (different size and location) and utilized for the comparison purpose. The numerical and experimental comparison reveals the necessity of the current higher-order models for the delaminated structure. In order to report the comprehensive conclusions related to the flexural strength of laminated structure with internal damage, a detailed parametric study has been carried out numerically and discussed in detail including the effect delamination (size, position and location).

Hyperelastic models are of particular interest in modeling biomaterials. In order to implement them, one must derive the stress and elasticity tensors from the given potential energy function explicitly. However, it is often cumbersome to do so because researchers in biomechanics may not be well-exposed to systematic approaches to derive the stress and elasticity tensors as it is vaguely addressed in literature. To resolve this, we present a framework of a general approach to derive the stress and elasticity tensors for hyperelastic models. Throughout the derivation we carefully elaborate the differences between formulas used in the displacement-based formulation and the displacement/pressure mixed formulation. Three hyperelastic models, Mooney–Rivlin, Yeoh and Holzapfel–Gasser–Ogden models that span from first-order to higher order and from isotropic to anisotropic materials, are served as examples. These detailed derivations are validated with numerical experiments that demonstrate excellent agreements with analytical and other computational solutions. Following this framework, one could implement with ease any hyperelastic model as user-defined functions in software packages or develop as an original source code from scratch.