Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Meshfree formulations of kinematic wave for channel flow routing

Abstract: This study concerns the development of various Meshfree formulations, namely Point Interpolation Method (PIM), Radial Point Interpolation Method (RPIM) and Element Free Galerkin (EFG) in solving numerically, St Venant’s kinematic wave equations for the hydrologic modeling of surface runoff and channel flow. It involves problem formulations derivation of governing equations, provision of the corresponding solutions by generating Matlab source codes, verification of results against established data, parametric study and assessment of performance of the newly derived Meshfree formulations against established numerical methods, namely Finite Element Method (FEM) and Finite Difference Method (FDM). The originality and the main contribution of the study are solving the Meshfree formulations of the kinematic wave equations numerically. The formulations are verified when it is found that the results produced by the source codes are in general in close agreement with the benchmark data. Although slight discrepancies have been observed in some cases, these are later validated as due to several factors, namely shape parameters values which are yet to be optimized, different number of nodes used for comparison and manual discretization of input data. In obtaining the best performance of the methods, optimum values of the shape parameters have been determined through a parametric study which once obtained are used in the performance assessment. RPIM and PIM are found to be less sensitive to the optimum values as compared to EFG. Two types of performance are assessed; the convergence rate and the computer resource consumption in terms of CPU time. Based on this study, it can be concluded that, in general, Meshfree methods perform comparably with the established methods in terms of convergence rate despite the fact it does not need the construction of mesh which can save modelling time. This shows the potential of Meshfree as numerical methods for its future development.

Adaptive modelling of finite strain shear band localization using the element-free Galerkin method

Abstract: Efficient computational modelling of problems with material and geometric nonlinearities is very challenging. These problems are often solved with the adaptive finite element method (FEM), which involves error estimation coupled with refinement strategies to automatically find regions for fine and coarse discretization. For these problems meshless methods offer a good alternative involving no remeshing, and only simple insertion or deletion of the nodes. An adaptive element-free Galerkin method (EFGM) is developed here for these types of problems, in which an existing error estimation procedure for linear elasto-static analysis is extended here to nonlinear problems. Maximum entropy (max-ent) shape functions are used instead of the conventional moving least squares (MLS) approximation in the EFGM, which allows implementation of the essential boundary conditions directly. A numerical example is given to demonstrate the implementation and performance of the current approach.

Smooth FEs on polyhedral domains

Abstract: Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, creating an exciting interplay between the subjects This is the first and only book to prove in a systematic and unifying way, stability, convergence and computing results for the different numerical methods for nonlinear elliptic problems The proofs use linearization, compact perturbation of the coercive principal parts, or monotone operator techniques, and approximation theory Examples are given for linear to fully nonlinear problems (highest derivatives occur nonlinearly) and for the most important space discretization methods: conforming and nonconforming finite element, discontinuous Galerkin, finite difference, wavelet (and, in a volume to follow, spectral and meshfree) methods A number of specific long open problems are solved here: numerical methods for fully nonlinear elliptic problems, wavelet and meshfree methods for nonlinear problems, and more general nonlinear boundary conditions We apply it to all these problems and methods, in particular to eigenvalues, monotone operators, quadrature approximations, and Newton methods Adaptivity is discussed for finite element and wavelet methods The book has been written for graduate students and scientists who want to study and to numerically analyze nonlinear elliptic differential equations in Mathematics, Science and Engineering It can be used as material for graduate courses or advanced seminars

Penetration modeling of ultra‐high performance concrete using multiscale meshfree methods

Abstract: Terminal ballistics of concrete is of extreme importance to the military and civil communities. Over the past few decades, ultra‐high performance concrete (UHPC) has been developed for various applications in the design of protective structures because UHPC has an enhanced ballistic resistance over conventional strength concrete. Developing predictive numerical models of UHPC subjected to penetration is critical in understanding the material's enhanced performance. This study employs the advanced fundamental concrete (AFC) model, and it runs inside the reproducing kernel particle method (RKPM)‐based code known as the nonlinear meshfree analysis program (NMAP). NMAP is advantageous for modeling impact and penetration problems that exhibit extreme deformation and material fragmentation. A comprehensive experimental study was conducted to characterize the UHPC. The investigation consisted of fracture toughness testing, the utilization of nondestructive microcomputed tomography analysis, and projectile penetration shots on the UHPC targets. To improve the accuracy of the model, a new scaled damage evolution law (SDEL) is employed within the microcrack informed damage model. During the homogenized macroscopic calculation, the corresponding microscopic cell needs to be dimensionally equivalent to the mesh dimension when the partial differential equation becomes ill posed and strain softening ensues. Results of numerical investigations will be compared with results of penetration experiments.

A radial point interpolation method for 1D contaminant transport modelling through landfill liners

Abstract: In the framework of meshfree methods, a new methodology is developed based on radial point interpolation method (RPIM). This methodology is applied to a one-dimensional contaminant transport modelling in the saturated porous media. The one-dimensional form of advection-dispersion equation involving reactive contaminant is considered in the analysis. The Galerkin weak form of the governing equation is formulated using 1D meshfree shape functions constructed using thin plate spline radial basis functions. MATLAB code is developed to obtain the numerical solution. Numerical examples representing various phenomena, which occur during migration of contaminants, are presented to illustrate the applicability of the proposed method and the results are compared with those obtained from the analytical and finite element solutions. The proposed RPIM has generated results with no oscillations and they are insensitive to Peclet constraints. In order to test the practical applicability and performance of the RPIM, three case studies of contaminant transport through the landfill liners are presented. A good agreement is obtained between the results of the RPIM and the field investigation data.