Showing papers on "Meshfree methods published in 2022"

Parallel domain discretization algorithm for RBF-FD and other meshless numerical methods for solving PDEs

Abstract: In this paper, we present a novel parallel dimension-independent node positioning algorithm that is capable of generating nodes with variable density, suitable for meshless numerical analysis. A very efficient sequential algorithm based on Poisson disc sampling is parallelized for use on shared-memory computers, such as the modern workstations with multi-core processors. The parallel algorithm uses a global spatial indexing method with its data divided into two levels, which allows for an efficient multi-threaded implementation. The addition of bootstrapping enables the algorithm to use any number of parallel threads while remaining as general as its sequential variant. We demonstrate the algorithm performance on six complex 2- and 3-dimensional domains, which are either of non rectangular shape or have varying nodal spacing or both. We perform a run-time analysis of the algorithm, to demonstrate its ability to reach high speedups regardless of the domain and to show how well it scales on the experimental hardware with 16 processor cores. We also analyse the algorithm in terms of the effects of domain shape, quality of point placement, and various parallelization overheads.

A numerical integration strategy of meshless numerical manifold method based on physical cover and applications to linear elastic fractures

Abstract: The meshless numerical manifold method (MNMM) inherits two covers of the numerical manifold method. A mathematical cover is composed of nodes' influence domains and a physical cover consists of physical patches, which are produced through cutting mathematical cover by physical boundaries. Because two covers are adopted, MNMM can naturally and uniquely solve both the continuous and discontinuous problems under Galerkin's variational framework. However, Galerkin's meshless method needs background integration grids to realize solving, which often does not match the nodes' influence domains, so the accuracy of numerical integration is reduced. Consider that MNMM allows even distribution of nodes and the physical cover contains the characteristics of the boundary and nodes' influence domains, the study presents a new numerical integration strategy to ensure that the background integration grids match the nodes' influence domains. The method can be applied to continuous and discontinuous problems, and is proved to be equivalent to the influence domain integration. At the same time, a reasonable arrangement of mathematical nodes is made to assure the background integration grid that it is accordant and straightforward. In this way, the number of physical patches of each integral point is the same, which improves the accuracy of interpolation calculation. The effectiveness of the proposed method is verified by numerical examples of both continuous and discontinuous problems.

A numerical integration strategy of meshless numerical manifold method based on physical cover and applications to linear elastic fractures

Abstract: The meshless numerical manifold method (MNMM) inherits two covers of the numerical manifold method. A mathematical cover is composed of nodes' influence domains and a physical cover consists of physical patches, which are produced through cutting mathematical cover by physical boundaries. Because two covers are adopted, MNMM can naturally and uniquely solve both the continuous and discontinuous problems under Galerkin's variational framework. However, Galerkin's meshless method needs background integration grids to realize solving, which often does not match the nodes' influence domains, so the accuracy of numerical integration is reduced. Consider that MNMM allows even distribution of nodes and the physical cover contains the characteristics of the boundary and nodes' influence domains, the study presents a new numerical integration strategy to ensure that the background integration grids match the nodes' influence domains. The method can be applied to continuous and discontinuous problems, and is proved to be equivalent to the influence domain integration. At the same time, a reasonable arrangement of mathematical nodes is made to assure the background integration grid that it is accordant and straightforward. In this way, the number of physical patches of each integral point is the same, which improves the accuracy of interpolation calculation. The effectiveness of the proposed method is verified by numerical examples of both continuous and discontinuous problems.

Modeling fracture in viscoelastic materials using a modified incremental meshfree RPIM and DIC technique

Abstract: In the present work, fracture mechanics of linear viscoelastic materials is investigated numerically and experimentally. Herein, an effective and accurate meshfree method is developed to analyze fracture mechanics in viscoelastic problems. To this end, an incremental approach with higher-order finite difference (FD) schemes for analysis of viscoelastic materials in the time domain, along with an improved meshfree method based on the global weak formulation, i.e. the radial point interpolation method (RPIM), is employed. An accurate and efficient integration approach with minimum computational cost, i.e. the background decomposition method (BDM), is also used to compute the domain integrals. To evaluate the accuracy and efficiency of the proposed numerical method, first, two example problems are provided and the robustness of the presented method is assessed. Then, a non-contact full-field optical method, i.e. the digital image correlation technique, is utilized to investigate the viscoelastic fields of a polymer, polymethyl-methacrylate (PMMA), during crack propagation in mode I. Also, in order to properly model the fracture behavior of this material by the meshfree RPIM, the mechanical properties of the PMMA, including the Prony viscoelastic parameters, are obtained by an ASTM test method. Moreover, the obtained results from the DIC test and also the load-displacement curve of the cracked specimen are compared with the results of the proposed meshfree RPIM and a very close agreement is observed.

Modeling fracture in viscoelastic materials using a modified incremental meshfree RPIM and DIC technique

Abstract: In the present work, fracture mechanics of linear viscoelastic materials is investigated numerically and experimentally. Herein, an effective and accurate meshfree method is developed to analyze fracture mechanics in viscoelastic problems. To this end, an incremental approach with higher-order finite difference (FD) schemes for analysis of viscoelastic materials in the time domain, along with an improved meshfree method based on the global weak formulation, i.e. the radial point interpolation method (RPIM), is employed. An accurate and efficient integration approach with minimum computational cost, i.e. the background decomposition method (BDM), is also used to compute the domain integrals. To evaluate the accuracy and efficiency of the proposed numerical method, first, two example problems are provided and the robustness of the presented method is assessed. Then, a non-contact full-field optical method, i.e. the digital image correlation technique, is utilized to investigate the viscoelastic fields of a polymer, polymethyl-methacrylate (PMMA), during crack propagation in mode I. Also, in order to properly model the fracture behavior of this material by the meshfree RPIM, the mechanical properties of the PMMA, including the Prony viscoelastic parameters, are obtained by an ASTM test method. Moreover, the obtained results from the DIC test and also the load-displacement curve of the cracked specimen are compared with the results of the proposed meshfree RPIM and a very close agreement is observed.

Generalized Multiscale Finite Element Method for piezoelectric problem in heterogeneous media

Abstract: In this paper, we study multiscale methods for piezocomposites. We consider a model of static piezoelectric problem that consists of deformation with respect to components of displacements and a function of electric potential. This problem includes the equilibrium equations, the quasi-electrostatic equation for dielectrics, and a system of coupled constitutive relations for mechanical and electric fields. We consider a model problem that consists of coupled differential equations. The first equation describes the deformations and is given by the elasticity equation and includes the effect of the electric field. The second equation is for the electric field and has a contribution from the elasticity equation. In previous findings, numerical homogenization methods are proposed and used for piezocomposites. We consider the Generalized Multiscale Finite Element Method (GMsFEM), which is more general and is known to handle complex heterogeneities. The main idea of the GMsFEM is to develop additional degrees of freedom and can go beyond numerical homogenization. We consider both coupled and split basis functions. In the former, the multiscale basis functions are constructed by solving coupled local problems. In particular, coupled local problems are solved to generate snapshots. Furthermore, in the snapshot space, a local spectral decomposition is performed to identify multiscale basis functions. Our approaches share some common concepts with meshless methods as they solve the underlying problem on a coarse mesh, which does not conform heterogeneities and contrast. We discuss this issue in the paper. We show that with a few basis functions per coarse element, one can achieve a good approximation of the solution. Numerical results are presented.

Efficient image denoising technique using the meshless method: Investigation of operator splitting RBF collocation method for two anisotropic diffusion-based PDEs

Abstract: Images taken and stored digitally are often degraded by noise, so that the perceived image quality is significantly decreased in the presence of noise and human gaze behavior is affected by this noise as well. In recent years, image denoising has been modeled as partial differential equations (PDEs), which can be said to eliminate noise as well as preserve image detail is one of their main advantages. Numerical simulation of these PDEs, which are often nonlinear, high-order and high-dimensional, is one of the most challenging topics in this field. In this paper, one of the most powerful meshless numerical approaches, the operator splitting radial basis function (RBF) collocation method, is introduced to solve two modified versions of the Perona-Malik (PM) equation. The method splits each problem into several equations each of which is solved in one direction so as to make the major problem solvable in a smaller size. Furthermore, being in one direction makes it possible to solve the equations without any need for domain decomposition and its complexities. An unconditionally stable method (Crank-Nicolson) is used for discretizing time in this paper. Using meshless approaches in the image processing problems makes sense since the structure of digital images consist of pixels that corresponds to the distributed nodes of the meshless method so the complex time consuming domain construction of the finite element methods is avoided to get higher speeds in obtaining the desired denoised results. Some images with different noise levels are denoised using the presented method in the section of numerical results to show the desirable performance of the method. Also, the quality of the denoised images are concluded in tables which can be used to compare the obtained results.

Meshless Computational Strategy for Higher Order Strain Gradient Plate Models

Abstract: The present research focuses on the use of a meshless method for the solution of nanoplates by considering strain gradient thin plate theory. Unlike the most common finite element method, meshless methods do not rely on a domain decomposition. In the present approach approximating functions at collocation nodes are obtained by using radial basis functions which depend on shape parameters. The selection of such parameters can strongly influences the accuracy of the numerical technique. Therefore the authors are presenting some numerical benchmarks which involve the solution of nanoplates by employing an optimization approach for the evaluation of the undetermined shape parameters. Stability is discussed as well as numerical reliability against solutions taken for the existing literature.

A hybrid algorithm using the FEM-MESHLESS method to solve nonlinear structural problems

Abstract: In this paper, we present a hybrid numerical development to solve two-dimensional nonlinear structural problems. The proposed approach was developed by combining weak and strong formulations and using a High-Order Development, Continuation technique and Hybrid approximation (HODC-HYB). The hybrid approximation is based on meshless strong form method and Finite Element Method (FEM). This algorithm allows us to overcome several drawbacks such as the difficulties of implementing meshless strong form methods near the boundary of the structural domain, meshless methods can be unstable and less precise for problems with Neumann boundary conditions, but these methods can overcome the connectivity technique and numerical integration in a big part of the domain. Numerical tests are carried out to demonstrate the reliability and the performance of the proposed algorithm by setting up a comparative study with the solutions obtained by HODC-FEM and HODC-MESHLESS algorithms, which are based on the weak and strong forms, respectively.

A Novel Spatio-Temporal Fully Meshless Method for Parabolic PDEs

Abstract: We introduce a meshless method derived by considering the time variable as a spatial variable without the need to extend further conditions to the solution of linear and non-linear parabolic PDEs. The method is based on a moving least squares method, more precisely, the generalized finite difference method (GFDM), which allows us to select well-conditioned stars. Several 2D and 3D examples, including the time variable, are shown for both regular and irregular node distributions. The results are compared with explicit GFDM both in terms of errors and execution time.

Numerical analysis of thermo-fluid problems in 3D domains by means of the RBF-FD meshless method

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.

Groundwater flow simulation in a confined aquifer using Local Radial Point Interpolation Meshless method (LRPIM)

Abstract: Groundwater flow problems are generally solved using analytical or numerical methods. Though analytical solutions are exact and preferable, they are not available for complex field problems. Hence numerical methods such as Finite Element and Finite Difference methods are used to solve complex groundwater problems. These conventional mesh/ grid-based numerical methods need construction of a detailed mesh/ grid. On the other hand, the meshless approach creates a system of algebraic equations on a collection of distributed nodes in the problem area and the boundary. As a result, it is easy to incorporate any modifications to the model at a later time by simply adding nodes to the domain. In this study a weak form meshless method known as local radial point interpolation method (LRPIM) which uses radial basis functions for approximation or interpolation is developed to solve the groundwater flow problems in a confined aquifer. The results obtained from the LRPIM model has been compared with other numerical methods for benchmark and real field problems, and are found to be satisfactory. Implementation of the essential boundary conditions was relatively easier in LRPIM and gave good accuracy for the problems considered. LRPIM can potentially be used as an alternative to the other conventional methods, especially where the domain boundary is irregular or varying with time.

An improved RBF based differential quadrature method

Abstract: Differential quadrature (DQ) based on radial basis function (RBF) was provided by Shu [1] as a meshless method which discretizes any derivative by a weighted linear summation of functional values at randomly distributed particles. However, in the RBF-DQ technique by Shu, the particles are treated as fixed interpolation points and these points cannot move with fluid. Therefore, the conventional RBF-DQ technique is an Euler mesh-free method and lack of the Lagrangian characteristics. In this paper, RBF-DQ is improved to achieve the movement of particles with fluid. Inspired by smoothed particle hydrodynamics (SPH) technique, the particle shifting technology is applied into RBF-DQ to prevent scarce discretization error in the stretching direction and particle clustering within the compressing direction. Boundary and virtual particles are created to remedy particle deficiency near the boundary. An appropriate interpolation method is developed to solve the information of virtual particles and boundary in the Lagrangian frame. The findings indicated that the improved RBF-DQ is able to achieve Lagrangian characteristics, and has good numerical characteristics in terms of accuracy and convergence.

Fracture Toughness Determination on an SCB Specimen by Meshless Methods

Abstract: This work investigates fracture characteristics of a marble semi-circular bend (SCB) specimen with a pre-defined crack under a compressive loading condition. It aims at evaluating how the fracture toughness can be affected by the crack and span length variation. Numerically, the model is solved using meshless methods, extended to the linear elastic fracture mechanics (LEFM), resorting to radial point interpolation method (RPIM) and its natural neighbor versions (NNRPIMv1 and NNRPIMv2). Alternatively, to validate the meshless method results, the problem is resolved following the finite element method (FEM) model based on the standard 2D constant strain triangle elements. As a result, fracture toughness and the critical strain energy release rate are characterized following the testing method on the cracked straight through semi-circular bend specimen (CSTSCB). A comparison is drawn amongst the theoretical, meshless methods and FEM results to evaluate the capability of advanced numerical methods. Encouraging results have been accomplished leading to validate the supporting numerical methodologies.

Meshless Generalized Finite Difference Method for the Propagation of Nonlinear Water Waves under Complex Wave Conditions

Abstract: The propagation of nonlinear water waves under complex wave conditions is the key issue of hydrodynamics both in coastal and ocean engineering, which is significant in the prediction of strongly nonlinear phenomena regarding wave–structure interactions. In the present study, the meshless generalized finite difference method (GFDM) together with the second-order Runge–Kutta method (RKM2) is employed to construct a fully three-dimensional (3D) meshless numerical wave flume (NWF). Three numerical examples, i.e., the propagation of freak waves, irregular waves and focused waves, are implemented to verify the accuracy and stability of the developed 3D GFDM model. The results show that the present numerical model possesses good performance in the simulation of nonlinear water waves and suggest that the 3D “RKM2-GFDM” meshless scheme can be adopted to further simulate more complex nonlinear problems regarding wave–structure interactions in ocean engineering.

An upwind generalized finite difference method (GFDM) for meshless analysis of heat and mass transfer in porous media

Abstract: In this paper, an upwind GFDM is developed for coupled heat and mass transfer problems in porous media. GFDM is a meshless method that can obtain the difference schemes of spatial derivatives by using Taylor expansion in local node influence domains and the weighted least squares method. The first-order single-point upstream scheme in the FDM/FVM-based reservoir simulator is introduced to GFDM to form the upwind GFDM, based on which a sequential coupled discrete scheme of the pressure diffusion equation and the heat convection-conduction equation is solved to obtain pressure and temperature profiles. This paper demonstrates that this method can be used to obtain the meshless solution of the convection–diffusion equation with a stable upwind effect. For porous flow problems, the upwind GFDM is more practical and stable than the method of manually adjusting the influence domain based on the prior information of the flow field to achieve the upwind effect. Two types of calculation errors are analyzed, and three numerical examples are implemented to illustrate the good calculation accuracy and convergence of the upwind GFDM for heat and mass transfer problems in porous media and indicate the increase in the radius of the node influence domain will increase the calculation error of temperature profiles. Overall, the upwind GFDM discretizes the computational domain using only a point cloud that is generated with much less topological constraints than the generated mesh, but achieves good computational performance as the mesh-based approaches, and therefore has great potential to be developed as a general-purpose numerical simulator for various porous flow problems in domains with complex geometry.

The numerical meshless approach for solving the 2-D time nonlinear multi-term fractional cable equation in complex geometries

Abstract: The cable equation plays a prominent role in biological neuron models, for instance, in spiking neuron models and electrophysiology. Thus, the current investigation scrutinizes the 2D time nonlinear multi-term fractional cable equation. By adopting a valid meshfree technique, the nonlinear multi-term time-fractional cable equations (NM-TTFCEs) that consist of government equations and their boundary conditions are transformed into the boundary value problems. For this purpose, the finite difference method is derived for temporal discretization such that the considered NM-TTFCEs can be transformed into a sequence of boundary value problems in inhomogeneous Helmholtz-type equations. The dual reciprocity method (DRM) is implemented to obtain a particular solution and the improved singular boundary method (ISBM) is employed to evaluate the homogeneous solution. Moreover, we apply the meshless method for solving two-dimensional NM-TTFCEs on regular and irregular distribution points with several computational domains. The numerical results vouch for the accuracy and high efficiency of the proposed method. Finally, we will conclude that the ISBM/DRM method can be considered a potential alternative to existing meshless strong form approaches in solving multi-term fractional equation problems with complex geometries.

Groundwater flow simulation in a confined aquifer using Local Radial Point Interpolation Meshless method (LRPIM)

Abstract: Groundwater flow problems are generally solved using analytical or numerical methods. Though analytical solutions are exact and preferable, they are not available for complex field problems. Hence numerical methods such as Finite Element and Finite Difference methods are used to solve complex groundwater problems. These conventional mesh/ grid-based numerical methods need construction of a detailed mesh/ grid. On the other hand, the meshless approach creates a system of algebraic equations on a collection of distributed nodes in the problem area and the boundary. As a result, it is easy to incorporate any modifications to the model at a later time by simply adding nodes to the domain. In this study a weak form meshless method known as local radial point interpolation method (LRPIM) which uses radial basis functions for approximation or interpolation is developed to solve the groundwater flow problems in a confined aquifer. The results obtained from the LRPIM model has been compared with other numerical methods for benchmark and real field problems, and are found to be satisfactory. Implementation of the essential boundary conditions was relatively easier in LRPIM and gave good accuracy for the problems considered. LRPIM can potentially be used as an alternative to the other conventional methods, especially where the domain boundary is irregular or varying with time.

Meshless numerical solution for nonlocal integral differentiation equation with application in peridynamics

Abstract: The mid-point quadrature rule is often used to solve the nonlocal integral differentiation equations, in which a central material point is used to calculate the integral of the subdomain in the meshless geometry. However, it can only achieve the linear approximation and the first order convergence rate, and it is only accurate enough when the entire cell overlaps with the neighborhood of a material point. In this paper, to achieve the higher precision and the higher rate of convergence speed, a novel coupling method of reproducing kernel particle method and Gaussian-Legendre quadrature scheme is proposed to solve the nonlocal integral differentiation equations in the meshless geometry, in which the mid-point quadrature scheme is replaced with Gauss quadrature scheme to calculate the integral of the subdomain in the meshless geometry, the reproducing kernel particle method is employed to construct the shape functions for material points in the nonlocal model, and to approximate the value of the Gaussian quadrature node. Moreover, the meshless solution for the peridynamic equation is derived detailedly based on the proposed method. Compared with the traditional meshless solution for the nonlocal equations, the proposed solution has higher precision and higher rate of convergence speed in different problems. The numerical results demonstrate improved results in both mathematics and mechanical problems using the proposed method.

A meshfree-based topology optimization approach without calculation of sensitivity

Abstract: This paper presents a novel topology optimization approach without calculation of sensitivity for the minimum compliance problems, based on the meshfree Radial Point Interpolation Method (RPIM). Relying on the algorithm of Proportional Topology Optimization (PTO), material is distributed using only information of the objective function (which is the elastic strain energy). Material properties are interpolated by the well-known Solid Isotropic Material with Penalization (SIMP) technique; however the pseudo density (design variables) are not defined on the element center as usually encountered in finite element-based approaches, but on integration points. Since no element exists in meshfree analysis, this would be a natural choice. More importantly, the number of integration points is in general larger than that of elements or that of nodes, resulting in higher resolution of the density field. The feasibility and efficiency of the proposed approach are demonstrated and discussed via several numerical examples.