Journal articles: 'Meshfree methods (Numerical analysis)'

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Liu, Wing Kam, Sergio R. Idelsohn, and Eugenio O�ate. "Announcement ?Meshfree Methods?" International Journal for Numerical Methods in Engineering 49, no. 5 (2000): 721–23. http://dx.doi.org/10.1002/1097-0207(20001020)49:5<721::aid-nme92>3.0.co;2-4.

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Garg, Sahil, and Mohit Pant. "Meshfree Methods: A Comprehensive Review of Applications." International Journal of Computational Methods 15, no. 04 (2018): 1830001. http://dx.doi.org/10.1142/s0219876218300015.

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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.

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Dumont, Serge, Olivier Goubet, Tuong Ha-Duong, and Pierre Villon. "Meshfree methods and boundary conditions." International Journal for Numerical Methods in Engineering 67, no. 7 (2006): 989–1011. http://dx.doi.org/10.1002/nme.1659.

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Gu, Yuan Tong. "An Adaptive Local Meshfree Updated Lagrangian Approach for Large Deformation Analysis of Metal Forming." Advanced Materials Research 97-101 (March 2010): 2664–67. http://dx.doi.org/10.4028/www.scientific.net/amr.97-101.2664.

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The large deformation analysis is one of major challenges in numerical modelling and simulation of metal forming. Because no mesh is used, the meshfree methods show good potential for the large deformation analysis. In this paper, a local meshfree formulation, based on the local weak-forms and the updated Lagrangian (UL) approach, is developed for the large deformation analysis. To fully employ the advantages of meshfree methods, a simple and effective adaptive technique is proposed, and this procedure is much easier than the re-meshing in FEM. Numerical examples of large deformation analysis are presented to demonstrate the effectiveness of the newly developed nonlinear meshfree approach. It has been found that the developed meshfree technique provides a superior performance to the conventional FEM in dealing with large deformation problems for metal forming.

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Niemiec, Dominik, Roman Bulko, and Juraj Mužík. "The Meshfree Localized Petrov-Galerkin Approach in Slope Stability Analysis." Civil and Environmental Engineering 15, no. 1 (2019): 79–84. http://dx.doi.org/10.2478/cee-2019-0011.

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Abstract The article focuses on the use of the meshfree numerical method in the field of slope stability computations. There are many meshfree implementations of numerical methods. The article shows the results obtained using the meshfree localized Petrov-Galerkin method (MLPG) – localized weak-form of the equilibrium equations with an often used elastoplastic material model based on Mohr-Coulomb (MC) yield criterion. The most important aspect of MLPG is that the discretization process uses a set of nodes instead of elements. Node position within the computational domain is not restricted by any prescribed relationship. The shape functions are constructed using just the set of nodes present in the simple shaped domain of influence. The benchmark slope stability numerical model was performed using the developed meshfree computer code and compared with conventional finite element (FEM) and limit equilibrium (LEM) codes. The results showed the ability of the implemented theoretical preliminaries to solve the geotechnical stability problems.

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Belytschko, T., Y. Krongauz, J. Dolbow, and C. Gerlach. "On the completeness of meshfree particle methods." International Journal for Numerical Methods in Engineering 43, no. 5 (1998): 785–819. http://dx.doi.org/10.1002/(sici)1097-0207(19981115)43:5<785::aid-nme420>3.0.co;2-9.

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WANG, DONGDONG, and ZHENTING LIN. "A COMPARATIVE STUDY ON THE DISPERSION PROPERTIES OF HRK AND RK MESHFREE APPROXIMATIONS FOR KIRCHHOFF PLATE PROBLEM." International Journal of Computational Methods 09, no. 01 (2012): 1240015. http://dx.doi.org/10.1142/s0219876212400154.

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Dispersion analysis provides a rational way to examine the dynamic properties of numerical methods through comparing the numerical and continuum frequencies. In this paper a detailed comparative investigation is presented on the dispersion features of the Hermite reproducing kernel (HRK) and the conventional reproducing kernel (RK) meshfree methods for Kirchhoff plate problem with particular reference to the spatial discretizations. In the analysis the nodal variables of the semi-discretized meshfree Kirchhoff plate equations are assumed as harmonic wave functions to extract the numerical frequency. For the RK approximation, only the deflectional nodal variables are expressed by the harmonic wave functions, while unlike RK approximation, both deflectional and rotational nodal variables should be expressed by the harmonic wave functions for the HRK approximation. The dispersion analysis results uniformly evince that the HRK meshfree discretization has much smaller dispersion errors and performs superiorly compared to the conventional RK meshfree discretization for Kirchhoff plate problem.

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Racz, Donat, and Tinh Quoc Bui. "Novel adaptive meshfree integration techniques in meshless methods." International Journal for Numerical Methods in Engineering 90, no. 11 (2012): 1414–34. http://dx.doi.org/10.1002/nme.4268.

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Zhang, Hongjun, Guangsong Chen, Linfang Qian, and Jia Ma. "FE-Meshfree QUAD4 Element with Modified Radial Point Interpolation Function for Structural Dynamic Analysis." Shock and Vibration 2019 (January 8, 2019): 1–23. http://dx.doi.org/10.1155/2019/3269276.

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The partition-of-unity method based on FE-Meshfree QUAD4 element synthesizes the respective advantages of meshfree and finite element methods by exploiting composite shape functions to obtain high-order global approximations. This method yields high accuracy and convergence rate without necessitating extra nodes or DOFs. In this study, the FE-Meshfree method is extended to the free and forced vibration analysis of two-dimensional solids. A modified radial point interpolation function without any supporting tuning parameters is applied to construct the composite shape functions. The governing equations of elastodynamic problem are transformed into a standard weak formulation and then discretized into time-dependent equations which are solved via Bathe time integration scheme to conduct the forced vibration analysis. Several numerical test problems are solved and compared against previously published numerical solutions. Results show that the proposed FE-Meshfree QUAD4 element owns greater tolerance for mesh distortion and provides more accurate solutions.

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Alves, Carlos J. S., and Svilen S. Valtchev. "Numerical comparison of two meshfree methods for acoustic wave scattering." Engineering Analysis with Boundary Elements 29, no. 4 (2005): 371–82. http://dx.doi.org/10.1016/j.enganabound.2004.09.008.

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Mendonça, Flávio dos Ramos de Sousa, Wilber Humberto Vélez Gómez, and Artur Antônio de Almeida Portela. "A local meshless analysis of dynamics problems / Uma análise local desordenada dos problemas dinâmicos." Brazilian Journal of Development 7, no. 10 (2021): 96793–812. http://dx.doi.org/10.34117/bjdv7n10-134.

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This paper is concerned with new formulations of local meshfree numerical method, for the solution of dynamic problems in linear elasticity, Integrated Local Mesh Free (ILMF) method. The key attribute of local numerical methods is the use of a modeling paradigm based on a node-by-node calculation, to generate the rows of the global system of equations of the body discretization. In the local domain, assigned to each node of a discretization, the work theorem is kinematically formulated, leading thus to an equation of mechanical equilibrium of the local node, that is used by local meshfree method as the starting point of the formulation. The main feature of this paper is the use of a linearly integrated local form of the work theorem. The linear reduced integration plays a key role in the behavior of local numerical methods, since it implies a reduction of the nodal stiffness which, in turn, leads to an increase of the solution accuracy. As a consequence, the derived meshfree and finite element numerical methods become fast and accurate, which is a feature of paramount importance, as far as computational efficiency of numerical methods is concerned. The cantilever beam was analyzed with this technique, in order to assess the accuracy and efficiency of the new local numerical method for dynamic problems with regular and irregular nodal configuration. The results obtained in this work are in perfect agreement with Mesh-Free Local Petrov-Galerkin (MLPG) and the Finite Element Method (FEM) solutions.

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Wu, Junchao, and Dongdong Wang. "An accuracy analysis of Galerkin meshfree methods accounting for numerical integration." Computer Methods in Applied Mechanics and Engineering 375 (March 2021): 113631. http://dx.doi.org/10.1016/j.cma.2020.113631.

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LIU, S. J., H. WANG, and H. ZHANG. "SMOOTHED FINITE ELEMENTS LARGE DEFORMATION ANALYSIS." International Journal of Computational Methods 07, no. 03 (2010): 513–24. http://dx.doi.org/10.1142/s0219876210002246.

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The smoothed finite element method (SFEM) was developed in order to eliminate certain shortcomings of the finite element method (FEM). SFEM enjoys some of the flexibilities of meshfree methods. One advantage of SFEM is its applicability to modeling large deformations. Due to the absence of volume integration and parametric mapping, issues such as negative volumes and singular Jacobi matrix do not occur. However, despite these advantages, SFEM has never been applied to problems with extreme large deformation. For the first time, we apply SFEM to extreme large deformations. For two numerical problems, we demonstrate the advantages of SFEM over FEM. We also show that SFEM can compete with the flexibility of meshfree methods.

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Yang, Yongtao, Xuhai Tang, Hong Zheng, and Quansheng Liu. "Four-Node Quadrilateral Element with Continuous Nodal Stress for Geometrical Nonlinear Analysis." International Journal of Computational Methods 15, no. 02 (2017): 1850005. http://dx.doi.org/10.1142/s0219876218500056.

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In this paper, the performance of a hybrid ‘FE-Meshfree’ quadrilateral element with continuous nodal stress (Quad4-CNS) is investigated for geometrical nonlinear solid mechanic problems. By combining finite element method (FEM) and meshfree method, this Quad4-CNS synergizes the individual strengths of these two methods, which leads to higher accuracy, better convergence rate, as well as high tolerance to mesh distortion. Therefore, Quad4-CNS is attractive for geometrical nonlinear solid mechanic problems where excessive distorted meshes occur. For geometrical nonlinear analysis, numerical results show that the results of Quad4-CNS element are much better than those of four-node isoparametric quadrilateral element (Quad4), and are comparable to quadratic quadrilateral element (Quad8) and other hybrid ‘FE- Meshfree’ elements.

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Ishihara, Ken, Takehiro Noda, and Hiroyuki Sakurai. "Investigating Applicability of the Meshfree Method to the Structural Analysis of Tires." Tire Science and Technology 40, no. 2 (2012): 60–82. http://dx.doi.org/10.2346/1945-5852-40.2.60.

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ABSTRACT In contrast to the finite element method (FEM), which is widely used in the tire industry nowadays, some alternative methods have been proposed by academic communities over the past decade or so. The meshfree method is one of those new methodologies. Originally intended to remove the burden of creating the mesh that is inherent in FEM, the meshfree method relies on the point data rather than the mesh, which makes it much easier to discretize the geometry. In addition to those modeling issues, it has been found that the meshfree method has several advantages over FEM in handling geometrical nonlinearities, continuities, and so forth. In accordance with those emerging possibilities, the authors have been conducting research on the matter. This article describes the results of the authors' preliminary research on the applicability of the meshfree method to tire analyses, which include the theoretical outline, the strategy of tire modeling, numerical results, comparisons with results of FEM, and conclusions.

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Rabczuk, T., and T. Belytschko. "Adaptivity for structured meshfree particle methods in 2D and 3D." International Journal for Numerical Methods in Engineering 63, no. 11 (2005): 1559–82. http://dx.doi.org/10.1002/nme.1326.

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Duan, Qinglin, Xikui Li, Hongwu Zhang, and Ted Belytschko. "Second-order accurate derivatives and integration schemes for meshfree methods." International Journal for Numerical Methods in Engineering 92, no. 4 (2012): 399–424. http://dx.doi.org/10.1002/nme.4359.

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Chen, Jiun-Shyan, Michael Hillman, and Marcus Rüter. "An arbitrary order variationally consistent integration for Galerkin meshfree methods." International Journal for Numerical Methods in Engineering 95, no. 5 (2013): 387–418. http://dx.doi.org/10.1002/nme.4512.

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Talebi, Hossein, Cristóbal Samaniego, Esteban Samaniego, and Timon Rabczuk. "On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods." International Journal for Numerical Methods in Engineering 89, no. 8 (2011): 1009–27. http://dx.doi.org/10.1002/nme.3275.

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Saucedo-Zendejo, F. R., and A. C. Cortes-Vargas. "A meshfree method for analysis of thermo-elastic problems with moving heat sources in welding." Journal of Physics: Conference Series 2275, no. 1 (2022): 012009. http://dx.doi.org/10.1088/1742-6596/2275/1/012009.

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Abstract This manuscript presents the development and application of a meshfree procedure based on the finite pointset methods for thermo-elastic problems with moving heat sources, which are present in welding processes. The meshfree nature of this formulation gives the advantage of dealing with geometrical distortions and even fragmentations without the need of using computationally expensive remeshing approaches with a very simple implementation. A description of the implementation of this method and the solutions of some numerical examples are presented in order to show the potential of this formulation for dealing with thermoelasticity problems with moving heat sources and to introduce promising future fields of application.

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Song, Nan, Dong Qian, Jian Cao, Wing Kam Liu, and Shaofan Li. "Effective Models for Prediction of Springback In Flanging." Journal of Engineering Materials and Technology 123, no. 4 (2000): 456–61. http://dx.doi.org/10.1115/1.1395019.

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A study on the prediction of springback angle is presented, with focus on the straight flanging operation. The objective of this work is to evaluate the reliability of different methods of prediction. An experiment of straight flanging operation is conducted. Major prediction approaches such as analytical model, numerical simulation using the Finite Element Method (FEM) and the Meshfree Method using the Reproducing Kernel Particle Methods (RKPM) are discussed. A set of sample problems is computed and comparisons are made with the experiment. The numerical analysis shows that the prediction from the 3D meshfree contact code matches well with the data from the FEM 2D solid model. A material property described by the kinematic hardening law provides a better prediction of springback than the isotropic hardening law.

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Quaglino, A., and R. Krause. "kFEM: Adaptive meshfree finite-element methods using local kernels on arbitrary subdomains." International Journal for Numerical Methods in Engineering 114, no. 6 (2018): 581–97. http://dx.doi.org/10.1002/nme.5755.

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Yang, Judy P., and Jian-Yu Chen. "Strong-Form Formulated Generalized Displacement Control Method for Large Deformation Analysis." International Journal of Applied Mechanics 09, no. 07 (2017): 1750101. http://dx.doi.org/10.1142/s1758825117501010.

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The traditional analysis of geometric nonlinearity is mostly based on the weak-formulated Galerkin method such as the finite element method. The element nature has limited its application as a result of numerical integration in the governing equation and quality control of deformed mesh. In the middle of 1990s, the meshfree methods have been developed and become one leading research topic in computational mechanics. Especially, the strong form collocation methods require no additional efforts to process numerical integration and impose Dirichlet boundary condition, thereby making the collocation methods computationally efficient. In the incremental–iterative process, how to accurately reflect the change in the slope of the load–deflection curve of the structure and remain numerically stable are of major concerns. Thus, we propose a strong-form formulated generalized displacement control method to analyze geometric nonlinear problems, where the radial basis collocation method is adopted. The numerical examples demonstrate the ability of the proposed method for large deformation analysis.

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Arroyo, M., and M. Ortiz. "Localmaximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods." International Journal for Numerical Methods in Engineering 65, no. 13 (2006): 2167–202. http://dx.doi.org/10.1002/nme.1534.

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Silva‐Valenzuela, R., A. Ortiz‐Bernardin, N. Sukumar, E. Artioli, and N. Hitschfeld‐Kahler. "A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition." International Journal for Numerical Methods in Engineering 121, no. 10 (2020): 2174–205. http://dx.doi.org/10.1002/nme.6304.

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Tootoonchi, Arash, Arman Khoshghalb, and Nasser Khalili. "Meshfree Method Analysis of Biot's Consolidation Using Cell-Based Smoothed Point Interpolation Method." Applied Mechanics and Materials 846 (July 2016): 409–14. http://dx.doi.org/10.4028/www.scientific.net/amm.846.409.

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A set of cell-based smoothed point interpolation methods are proposed for the numerical analysis of Biot’s formulation. In the proposed methods, the problem domain is discretized using a triangular background mesh. Shape functions are constructed using either polynomial or radial point interpolation method (PIM), leading to the delta function property of shape functions and consequently, easy implementation of essential boundary conditions. The Biot’s equations are discretised in space and time. A variety of support domain selection schemes (T-schemes) are investigated. The accuracy and convergence rate of the proposed methods are examined by comparing the numerical results with the analytical solution for the benchmark problem of one dimensional consolidation.

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Yoon, Sangpil, Cheng-Tang Wu, Hui-Ping Wang, and Jiun-Shyan Chen. "Efficient Meshfree Formulation for Metal Forming Simulations." Journal of Engineering Materials and Technology 123, no. 4 (2000): 462–67. http://dx.doi.org/10.1115/1.1396349.

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A stabilized conforming (SC) nodal integration method is developed for elastoplastic contact analysis of metal forming processes. In this approach, strain smoothing stabilization is introduced to eliminate spatial instability in collocation meshfree methods. The gradient matrix associated with strain smoothing satisfies the integration constraint (IC) of linear exactness in the Galerkin approximation. Strain smoothing formulation and numerical procedures for history-dependent problems are introduced. Applications to metal forming analysis are presented, with the results demonstrating a significant improvement in computational efficiency without loss of accuracy.

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HASEGAWA, KYOKO, SUSUMU NAKATA, and SATOSHI TANAKA. "MESHFREE ELASTODYNAMIC ANALYSIS OF THREE-DIMENSIONAL SOLIDS USING RADIAL POINT INTERPOLATION METHOD." International Journal of Modeling, Simulation, and Scientific Computing 02, no. 01 (2011): 83–95. http://dx.doi.org/10.1142/s1793962311000372.

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Meshfree methods are effective tools for solving partial differential equations. The radial point interpolation method, a partial differential equation solver based on a meshfree approach, enables accurate imposition of displacement boundary conditions and has been successfully applied to elastostatic analysis of various kinds of three-dimensional solids. In this method, stiffness matrix construction accounts for the majority of CPU time required for the entire process, resulting in high computational costs, especially when higher-order numerical integration is applied for accurate matrix construction. An alternative method, modified radial point interpolation, was proposed to overcome this shortcoming and has accomplished fast computation of elastostatic solid analysis. The purpose of this study is to develop an algorithm for time-dependent simulation of three-dimensional elastic solids. We show that the modified radial point interpolation method also accelerates the construction of the mass matrix required for time-dependent analysis in addition to that of the stiffness matrix. In our approach, the problem domain is assumed to have an implicit function representation that can be constructed from a set of surface points measured using a three-dimensional scanning system. Several numerical tests for elastodynamic analysis of complex shape models are presented.

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Năstăsescu, Vasile. "FEM or SPH ?" Journal of Engineering Sciences and Innovation 1, no. 1 (2016): 34–48. http://dx.doi.org/10.56958/jesi.2016.1.1.34.

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This paper brings, in front of the reader, some aspects regarding the using of those numerical methods, perhaps most used, for analysis of the fluids and structures. Next to the FEM (Finite Element Method), new numerical methods appeared, among these, the methods named meshfree methods are nowadays most used. The SPH (Smoothed Particle Hydrodynamics) method belongs to this category of meshfree method, being the most used in different fields like astrophysical phenomena, fluid dynamics, structure dynamics and others. The paper put face to face some results obtained by FEM and SPH, so the reader can alone to appreciate which method is better in a given problem or other. For to facilitate analysis and to understand the results, the fundamentals of SPH method are presented. In contrast to FEM, the SPH method is less known and less used in Romania. This finding underlies the emergence of this article. The answer to the title question depends on every one and it is influenced by many factors. Finally, the author suggests an answer by a correction of the title question: FEM and SPH or FEM with SPH.

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SOARES, DELFIM. "DYNAMIC ELASTOPLASTIC ANALYSES BY SMOOTHED POINT INTERPOLATION METHODS." International Journal of Computational Methods 10, no. 05 (2013): 1350030. http://dx.doi.org/10.1142/s0219876213500308.

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In this work, meshfree techniques based on weakened weak formulations are presented for the solution of dynamic problems considering elastoplastic materials. Nonlinear internal forces are computed taking into account edge-based, cell-based, and node-based smoothed domains. T-schemes are applied for the construction of the support domains of the approximating shape functions, which are here formulated based on the radial point interpolation method. The mass matrix is also computed considering smoothed domains and their quadrature points. For the time-domain solution of the nonlinear system of equations, the Newmark/Newton–Raphson method is adopted. Numerical results illustrate the accuracy and efficiency of the discussed methodologies.

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Hesar, Goudarz Ghanizadeh, Yeliz Pekbey, Hasan Yildiz, and Farshid Khosravi Maleiki. "A mesh-free simulation of mode I delamination of composite structures." Science and Engineering of Composite Materials 21, no. 1 (2014): 137–49. http://dx.doi.org/10.1515/secm-2013-0019.

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AbstractA numerical and experimental investigation for the analysis of delamination problem under mode I loading in composite material is presented. Firstly, the simulation of the delamination under mode I loading and failure of composite materials based on the cohesive segments model is investigated by using the meshfree method. With the partition of unity of moving least-squares shape functions, the discontinuities at the cohesive segments are approximated with additional degrees of freedom at the nodes. An iterative solution scheme between the continuous and discontinuous fields is presented to solve mode I delamination growth. Secondly, to verify the meshfree method’s results, an experimental investigation and the finite element method were used for the simulation of delamination. The experimental study used a double-cantilever beam made of carbon/epoxy laminate (AS4/3501-6) which consists of 10 plies in [0]10 and [0/90/0/90/0]s layup with delamination inserted in the middle of the laminate. The critical fracture force, which can be experimentally measured, was used to calculate the mode I delamination fracture toughness of the carbon/epoxy laminate. Results obtained from the meshfree method showed very good agreement with experimental data for single-mode delamination under mode I loading. The meshfree method could also be used effectively to produce delamination growth in composite laminates and is especially suitable for the simulation of complex delamination patterns that are difficult to model using traditional numerical methods.

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Rohit, Gaurang R., Jagdish M. Prajapati, and Vikram B. Patel. "Coupling of Finite Element and Meshfree Method for Structure Mechanics Application: A Review." International Journal of Computational Methods 17, no. 04 (2019): 1850151. http://dx.doi.org/10.1142/s0219876218501517.

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In many engineering problems, the meshfree methods (MMs) have been dynamically projected and increasingly advanced in order to overwhelm some hitches in the predictable numerical methods. Over the past three decades in many different application area, MMs have found their way ranging from solid mechanics analysis, fluid problems, vibration analysis, heat transfer and optimization to numerical solutions of all kinds of (partial) differential equations. As every technique has shortcomings, the meshfree method also has drawbacks like higher computational cost and imposition of boundary condition which can be overruled by coupling it with the finite element method (FEM). In the past two decades, coupled MMs and FEM have appeared into a new session of computational methods with significant achievement. In addition, a noteworthy amount of growth has been made in addressing the major deficiencies that were present in the conventional methods and MMs at the premature phases. The objective of the present work is to provide a comprehensive review of various coupling techniques used for interface elements of MMs and FEM and general discussion on shape function formulation of FE and element free Galerkin method (EFGM). Key contribution of coupling techniques for coupled EFGM and FEM to structure mechanics application as primary concern.

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Li, Dong Feng, and Jian Tong Zhang. "A Review of Numerical Study of Micro-Scale Modeling for Asphalt Mixture." Applied Mechanics and Materials 716-717 (December 2014): 332–37. http://dx.doi.org/10.4028/www.scientific.net/amm.716-717.332.

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Combining with digital image processing and numerical simulation technology, X-ray computerized tomography (CT) was used to study the microstructure of asphalt mixture for analyzing internal structure of asphalt mixture. The microstructure modeling methods of asphalt mixture can be classified as continuum-based numerical method and discontinuum-based numerical method. This paper described a review of the work done by many researchers on the modeling of asphalt mixture. The simulation methods are included finite element network model (FENM), a micromechanical finite element model (FEM), a clustered discrete element model (DEM), disturbed state concept (DSC), DDA (Discontinuous Displacement Analysis), numerical manifold method (NMM) and meshfree manifold method (MMM) that were used in micromechanical modeling of asphalt mixture.

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Hillman, Michael, and Jiun-Shyan Chen. "An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics." International Journal for Numerical Methods in Engineering 107, no. 7 (2015): 603–30. http://dx.doi.org/10.1002/nme.5183.

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Rosolen, Adrian, Daniel Millán, and Marino Arroyo. "On the optimum support size in meshfree methods: A variational adaptivity approach with maximum-entropy approximants." International Journal for Numerical Methods in Engineering 82, no. 7 (2009): 868–95. http://dx.doi.org/10.1002/nme.2793.

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Ju, J. W., K. Y. Yuan, A. W. Kuo, and J. S. Chen. "Novel Strain Energy Based Coupled Elastoplastic Damage and Healing Models for Geomaterials – Part II: Computational Aspects." International Journal of Damage Mechanics 21, no. 4 (2011): 551–76. http://dx.doi.org/10.1177/1056789511407360.

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In Part I of this sequel (Ju, J.W., Yuan, K.Y. and Kuo, A.W. (2010). Novel Strain Energy Based Coupled Elastoplastic Damage and Healing Models for Geomaterials – Part I: Formulations, International Journal of Damage Mechanics, DOI: 10.1177/1056789511407359), we have developed innovative strain energy based coupled elastoplastic hybrid isotropic and anisotropic damage-healing formulations for geomaterials under complex 2D earth-moving processes. Emanating from a micromechanics-based brittle (tensile) damage characterization (P+) and a ductile (mixed tension–compression) damage-healing characterization ([Formula: see text]), the proposed hybrid isotropic and anisotropic damage-healing models for soils are implemented. Entirely new computational algorithms are systematically developed based on the two-step operator splitting methodology. The elastic damage-healing predictor and the plastic corrector are consistently implemented within the existing Nonlinear Meshfree Analysis Program at University of California, Los Angeles ( Chen, J.S., Wu, C.T., Yoon, S. and You, Y. (2001) . A Stabilized Conforming Nodal Integration for Galerkin Meshfree Methods, International Journal for Numerical Methods in Engineering, 50: 435–466). Several numerical simulations featuring sophisticated earth excavation, transport, compaction, and a numerical notched soil bar under cyclic tension–compression loading are presented to illustrate the salient elastoplastic damage and healing features of soils, such as shear band and partial recovery of soil stiffness due to compression (compaction) by the proposed innovative damage-healing models and step-by-step computational algorithms.

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Bui, Tinh Quoc. "Buckling analysis of simply supported composite laminates subjected to an in-plane compression load by a novel mesh-free method." Vietnam Journal of Mechanics 33, no. 2 (2011): 65–78. http://dx.doi.org/10.15625/0866-7136/33/2/39.

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Buckling analysis of composite laminates under an in-plane compression load based on the mesh-free Galerkin Kriging method is presented. The moving Kriging interpolation (MK) technique possessing the delta property is employed to construct the shape functions, and thus no special techniques for imposing the essential boundary conditions are required. The present formulation is based on the Kirchhoff plate theory. The applicability, the accuracy and the effectiveness of the method are illustrated through a number of numerical examples. The results calculated by the proposed method are compared with those of existing reference solutions available in the literature and very good agreements are observed. It can be said that the proposed method can be considered as an alternative numerical technique in terms of meshfree methods.

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Sarboland, Maryam, and Azim Aminataei. "On the Numerical Solution of One-Dimensional Nonlinear Nonhomogeneous Burgers’ Equation." Journal of Applied Mathematics 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/598432.

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The nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers’ equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. These methods are based on the multiquadric (MQ) quasi-interpolation operatorℒ𝒲2and direct and indirect radial basis function networks (RBFNs) schemes. In the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to show the efficiency of the present methods, several experiments are considered. Our numerical solutions are compared with the analytical solutions as well as the results of other numerical schemes. Furthermore, the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are efficient, robust, and reliable for solving Burgers’ equation.

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Wang, Lihua, and Zheng Zhong. "Radial Basis Collocation Method for the Dynamics of Rotating Flexible Tube Conveying Fluid." International Journal of Applied Mechanics 07, no. 03 (2015): 1550045. http://dx.doi.org/10.1142/s1758825115500453.

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A Meshfree Radial Basis Collocation Method (RBCM) associated with explicit and implicit time integration scheme is formulated to study the coupling dynamics of a rotating flexible tube conveying fluid, which involves a partial differential equation (PDE) with variable coefficients. Dispersion studies are performed and they indicate that the proposed RBCM has a very small dispersion error compared with conventional FEM and Galerkin-based meshfree methods. Numerical examples are conducted for the influence of initial flow rate of the fluid, discretization and shape parameter on the dispersion error. The critical time step is obtained from a Von Neumann stability analysis. For the eigenproblem, Hermite-type RBCM is proposed in order to construct square matrices and eigenvalue analysis gives the frequencies of the system. Subsequently, the influence of angular velocity, flow rate of the fluid and the time variation on the fundamental frequencies is studied. Though proposed for studying the dynamics of a rotating flexible tube conveying fluid, this solution scheme is applicable to other dynamical problems which have similar PDEs with variable coefficients.

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Behradnia, Sogol, Amir Khosravifard, Mohammad-Rahim Hematiyan, and Yui-Chuin Shiah. "Identification of Time Variations of Moving Loads Applied to Plates Resting on Viscoelastic Foundation Using a Meshfree Method." Aerospace 9, no. 7 (2022): 357. http://dx.doi.org/10.3390/aerospace9070357.

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Dynamic identification of the intensity of the moving loads applied to structures is an important task in aerospace, marine, and transportation industries. In the present work, a general technique is presented for identification of the time variations in moving loads applied to plate structures resting on viscoelastic foundation. The identification problem is formulated as an inverse problem, which utilizes dynamic responses. The direct analyses required for the identification problem are performed by a meshfree method based on the moving node technique. In this technique, a node, which travels with the applied force, is utilized in the meshfree method. Since there is no connectivity between the nodes of meshfree methods, this technique can be implemented easily, while reducing the computational labor. Another benefit of this technique is that any simple or complicated trajectory of the moving load can be handled without any additional concerns. Two numerical example problems are solved and the effects of several parameters, including the measurement error, and number of sensors on the accuracy of the results are investigated. Through the examples, it is shown that the presented technique can identify the time variations in moving loads efficiently and accurately.

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Zakrzewski, Nadia, Majidreza Nazem, Scott William Sloan, and Mark Cassidy. "On Application of the Maximum Entropy Meshless Method for Large Deformation Analysis of Geotechnical Problems." Applied Mechanics and Materials 846 (July 2016): 331–35. http://dx.doi.org/10.4028/www.scientific.net/amm.846.331.

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Traditional grid-based numerical techniques such as the Finite Element Method (FEM) are known to suffer when large deformations of the continuum are encountered. As such, there has been limited success using this class of methods to solve many of the complex problems encountered in computational geomechanics. The potential of Meshfree techniques for addressing this perceived deficiency has been recognised. This study presents a robust Maximum Entropy Meshless (MEM) method for the analysis of problems involving geometrical nonlinearity in computational geomechanics. The method is validated via simulation of an undrained layer of soil under a rigid and rough strip footing undergoing large deformations and its merit is demonstrated through a comparison of the results with those obtained via the FEM.

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Mirzaee, Farshid, Shadi Rezaei, and Nasrin Samadyar. "Numerical solution of two-dimensional stochastic time-fractional Sine–Gordon equation on non-rectangular domains using finite difference and meshfree methods." Engineering Analysis with Boundary Elements 127 (June 2021): 53–63. http://dx.doi.org/10.1016/j.enganabound.2021.03.009.

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Wu, C. T., W. Hu, and M. Koishi. "A Smoothed Particle Galerkin Formulation for Extreme Material Flow Analysis in Bulk Forming Applications." International Journal of Computational Methods 13, no. 03 (2016): 1650019. http://dx.doi.org/10.1142/s0219876216500195.

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This paper presents a new particle formulation for extreme material flow analyses in the bulk forming applications. The new formulation is first established by an introduction of a smoothed displacement field to the standard Galerkin formulation to eliminate zero-energy modes in conventional particle methods. The discretized system of linear equations is consistently derived and integrated using a direct nodal integration scheme. The linear formulation is next extended to the large deformation quasi-static analysis of inelastic materials. As quasi-static Lagrangian simulation proceeds in the severe deformation range, the analysis method is switched to explicit dynamics formulation and an adaptive Lagrangian kernel approach is preformed to reset the reference configuration and maintain the injective deformation mapping at the particles. Both nonconvex and convex meshfree approximations are investigated in this study. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method.

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PEKEDIS, MAHMUT, and HASAN YILDIZ. "NUMERICAL ANALYSIS OF A PROJECTILE PENETRATION INTO THE HUMAN HEAD VIA MESHLESS METHOD." Journal of Mechanics in Medicine and Biology 14, no. 04 (2014): 1450059. http://dx.doi.org/10.1142/s0219519414500596.

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In recent years, physicists, engineers and medical scientists have tried to demonstrate the biomechanics of gunshot wounds with numerical methods and experimental observations. Currently, the finite element method (FEM) is the most widely used numerical method among the studies related to ballistic wound injuries. However, when the FEM is used for the penetration analysis, the path of the projectile in the skull is subjected to extremely large deformations which will introduce errors due to distortion of elements. To overcome this error, the meshfree technique was established to simulate the gunshot wound as a preliminary study in which the skull was modeled by smoothed particle hydrodynamics (SPH) and the projectile was modeled by nondeformable rigid elements. In order to simulate a realistic penetration phenomenon, orthotropic material properties were defined for different regions (forehead, zygomatic and mandible) with material principal axis along the surface of the bones. Human response to the ballistics impacts were determined in terms of force occurring along the pathway of the bullet in the skull, residual velocity of the projectile and penetration depth. The obtained results were compared with the data reported in literature. As a result, mechanical behavior of the head under ballistic impacts simulated by the SPH, compared well with the results determined by the data given in literature, which indicates the applicability of the SPH method as a powerful technique in simulating different gunshot wound mechanisms.

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Zamolo, Riccardo, Davide Miotti, and Enrico Nobile. "Numerical analysis of thermo-fluid problems in 3D domains by means of the RBF-FD meshless method." Journal of Physics: Conference Series 2177, no. 1 (2022): 012007. http://dx.doi.org/10.1088/1742-6596/2177/1/012007.

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Abstract The use of CAE (Computer Aided Engineering) software, commonly applied to the design and verification of a great variety of manufactured products, is totally reliant on accurate numerical simulations. Classic mesh-based methods, e.g., Finite Element (FEM) and Finite Volume (FVM), are usually employed for such simulations, where the role of the mesh is crucial for both accuracy and time consumption issues. This is especially true for complex 3D domains which are typically encountered in most practical problems. Meshless, or meshfree, methods have been recently introduced in order to replace the usual mesh with much simpler node distributions, thus purifying the data structures of any additional geometric information. Radial Basis Function-Finite Difference (RBF-FD) meshless methods have been shown to be able to easily solve problems of engineering relevance over complex-shaped domains with great accuracy, with particular reference to fluid flow and heat transfer problems. In this paper the RBF-FD method is employed to solve heat transfer problems with incompressible, steady-state laminar flow over 3D complex-shaped domains. The required node distributions are automatically generated by using a meshless node generation algorithm, which has been specifically developed to produce high quality node arrangements over arbitrary 3D geometries. The presented strategy represents therefore a fully-meshless approach for the accurate and automatic simulation of thermo-fluid problems over 3D domains of practical interest.

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Rabczuk, Timon. "Computational Methods for Fracture in Brittle and Quasi-Brittle Solids: State-of-the-Art Review and Future Perspectives." ISRN Applied Mathematics 2013 (March 20, 2013): 1–38. http://dx.doi.org/10.1155/2013/849231.

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An overview of computational methods to model fracture in brittle and quasi-brittle materials is given. The overview focuses on continuum models for fracture. First, numerical difficulties related to modelling fracture for quasi-brittle materials will be discussed. Different techniques to eliminate or circumvent those difficulties will be described subsequently. In that context, regularization techniques such as nonlocal models, gradient enhanced models, viscous models, cohesive zone models, and smeared crack models will be discussed. The main focus of this paper will be on computational methods for discrete fracture (discrete cracks). Element erosion technques, inter-element separation methods, the embedded finite element method (EFEM), the extended finite element method (XFEM), meshfree methods (MMs), boundary elements (BEMs), isogeometric analysis, and the variational approach to fracture will be reviewed elucidating advantages and drawbacks of each approach. As tracking the crack path is of major concern in computational methods that preserve crack path continuity, one section will discuss different crack tracking techniques. Finally, cracking criteria will be reviewed before the paper ends with future research perspectives.

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WU, S. C., H. O. ZHANG, C. ZHENG, and J. H. ZHANG. "A HIGH PERFORMANCE LARGE SPARSE SYMMETRIC SOLVER FOR THE MESHFREE GALERKIN METHOD." International Journal of Computational Methods 05, no. 04 (2008): 533–50. http://dx.doi.org/10.1142/s0219876208001613.

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One main disadvantage of meshfree methods is that their memory requirement and computational cost are much higher than those of the usual finite element method (FEM). This paper presents an efficient and reliable solver for the large sparse symmetric positive definite (SPD) system resulting from the element-free Galerkin (EFG) approach. A compact mathematical model of heat transfer problems is first established using the EFG procedure. Based on the widely used Successive Over-Relaxation–Preconditioned Conjugate Gradient (SSOR–PCG) scheme, a novel solver named FastPCG is then proposed for solving the SPD linear system. To decrease the computational time in each iteration step, a new algorithm for realizing multiplication of the global stiffness matrix by a vector is presented for this solver. The global matrix and load vector are changed in accordance with a special rule and, in this way, a large account of calculation is avoided on the premise of not decreasing the solution's accuracy. In addition, a double data structure is designed to tackle frequent and unexpected operations of adding or removing nodes in problems of dynamic adaptive or moving high-gradient field analysis. An information matrix is also built to avoid drastic transformation of the coefficient matrix caused by the initial-boundary values. Numerical results show that the memory requirement of the FastPCG solver is only one-third of that of the well-developed AGGJE solver, and the computational cost is comparable with the traditional method with the increas of solution scale and order.

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Ling, Leevan, and Qi Ye. "On meshfree numerical differentiation." Analysis and Applications 16, no. 05 (2018): 717–39. http://dx.doi.org/10.1142/s021953051850001x.

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We combine techniques in meshfree methods and Gaussian process regressions to construct kernel-based estimators for numerical derivatives from noisy data. Specially, we construct meshfree estimators from normal random variables, which are defined by kernel-based probability measures induced from symmetric positive definite kernels, to reconstruct the unknown partial derivatives from scattered noisy data. Our developed theories give rise to Tikhonov regularization methods with a priori parameter, but the shape parameters of the kernels remain tunable. For that, we propose an error measure that is computable without the exact values of the derivative. This allows users to obtain a quasi-optimal kernel-based estimator by comparing the approximation quality of kernel-based estimators. Numerical examples in two dimensions and three dimensions are included to demonstrate the convergence behavior and effectiveness of the proposed numerical differentiation scheme.

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Bompadre, A., B. Schmidt, and M. Ortiz. "Convergence Analysis of Meshfree Approximation Schemes." SIAM Journal on Numerical Analysis 50, no. 3 (2012): 1344–66. http://dx.doi.org/10.1137/110828745.

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Suzuki, Yoshiro, and Kosuke Soga. "Seamless-domain method: a meshfree multiscale numerical analysis." International Journal for Numerical Methods in Engineering 106, no. 4 (2015): 243–77. http://dx.doi.org/10.1002/nme.5115.

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Journal articles on the topic 'Meshfree methods (Numerical analysis)'

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