Meshless Approach Research Articles

A meshless computational approach for solving two-dimensional inverse time-fractional diffusion problem with non-local boundary condition

This paper is devoted to investigating a two-dimensional inverse anomalous diffusion problem. The missing solely time-dependent Dirichlet boundary condition is recovered by imposing an additional integral measurement over the domain. An efficient computational technique based on a combination of a time integration scheme and local meshless Petrov–Galerkin method is implemented to solve the governing inverse problem. Firstly, an implicit time integration scheme is used to discretize the model in the temporal direction. To fully discretize the model, the primary spatial domain is represented by a set of distributed nodes and data-dependent basis functions are constructed by using the radial point interpolation method. Then, the local meshless Petrov–Galerkin method is used to discretize the problem in the spatial direction. Numerical examples are presented to verify the accuracy and efficiency of the proposed technique. The stability of the method is examined when the input data are contaminated with noise.

Continuous density-based topology optimization of cracked structures using peridynamics

Peridynamics (PD) is a meshless approach that addresses some of the difficulties and limitations associated with mesh-based topology optimization (TO) methods. This study investigates topology optimization of structures with and without embedded cracks using peridynamics (PD-TO). To this end, PD is coupled with two different continuous density-based topology optimization methods, namely the optimality criteria and the proportional optimization. The optimization results are compared for a continuous definition of the design variables, which are the relative densities defined for PD particles. The checkerboard issue has been removed using filtering schemes. The accuracy of the proposed PD-TO approach is validated by solving benchmark problems and comparing the optimal topologies with those obtained using a FEM-based topology optimization. Various problems are solved with and without defects (cracks) under different loading and constraint boundary conditions. Topology optimization for an unstructured discretization problem has also been investigated applied to a complex geometry. The optimal topology of a cracked structure may change for different optimization methods. The numerical results demonstrate the accuracy, high efficiency, and robustness of the PD-TO approach.

A meshless generalized finite difference method for 2D elasticity problems

In this paper, a meshless generalized finite difference (FD) method is developed and presented for solving 2D elasticity problems. Different with other types of generalized FD method (GFDM) commonly constructed with moving least square (MLS) or radial basis function (RBF) shape functions, the present method is developed based upon B-spline based shape function. The method is a truly meshless approach. Key aspects attributed to the method are: B-spline basis functions augmented with polynomials are employed to construct its shape function. This allows B-splines with lower order to be chosen for the approximation and keeping the efficiency of computation related to tensor product operation of B-spline basis functions. In addition, as distribution of stencil nodes affects numerical performance of generalized FD method, neighboring nodes from triangle cells surrounding a center node are selected for building the supporting domains. While meeting compact stencil requirement, the selection eliminates necessity for determining appropriate number of supporting nodes or size of supporting domains. As a result, the proposed method shows good numerical approximation and accuracy for 2D elasticity problems. Numerical examples are presented to show the effectiveness of the proposed method for solving several 2D elasticity problems in various geometries.

RBF methods in a Stochastic Volatility framework for Greeks computation

In this paper we aim to apply a new, proposed meshless approach for Heston PDE resolution. In Mathematical Finance, most classical models may be reformulated as partial differential equations (PDE), which are commonly solved by finite difference and Monte Carlo methods. Alternatively, bio-inspired methods may be considered to select parameters and to solve said PDEs. In particular, we focus on Heston model PDE and we solve it via Radial Basis Functions (RBF) methods. Moreover some considerations on the computation of model parameters through learning approaches have been discussed. We aim to price a vanilla call option, focusing on the derivation of the formulae for its Greeks profiles. Eventually, we conclude RBFs are an accurate alternative method to price derivatives in a stochastic volatility framework, especially since they allow for a fast and precise computation of Greeks.

Hybrid Method Incorporated with Meshless Approach for Electromagnetic Wave Simulation

Hybrid Method Incorporated with Meshless Approach for Electromagnetic Wave Simulation

Hybrid meshless displacement discontinuity method (MDDM) in fracture mechanics: Static and dynamic

This paper investigated a hybrid Meshless Displacement Discontinuity Method (MDDM) for a cracked plate subjected to static and dynamic loadings. The purpose of MDDM is to model displacement discontinuity on a cracked surface by the displacement discontinuity method in an infinite plate. This was achieved by considering a meshless approach, the equilibrium equations, and the boundary conditions for a domain with an irregular nodes distribution. Also, by imposing the principle of superposition, accurate and convergent solutions can be obtained. In this paper, the static and dynamic stress intensity factors, and the crack growth for different initial crack length and crack slant angles are investigated. The Laplace transform method is applied to deal with dynamic problems and the time-dependent values are obtained by the Durbin inversion technique. Validations of the presented technique are demonstrated by four numerical examples of plates with a central embedded crack.

A review on the theories and numerical methods for injection molding simulations

Over the past two decades, numerical simulation of injection molding has made great progress, not only with its increasingly widespread use, but also with problems and challenges such as unreliable simulated performance, difficulty integrating commercial software with new methods, and inadequate consideration of microscopic mechanisms. This paper thoroughly explores the historical development of injection molding theoretical models and the key results of each point, and briefly introduces multiscale simulation methods. The characteristics and spectrum of application of the substance constitutive models are studied, and the effects of crystallization on the state and constitutive equations are demonstrated. Therefore, the characteristics of numerical methods for solving problems with injection molding are analyzed, and the variations between viscous and viscoelastic methods are studied. Analysis of the advantages and disadvantages of methods of flow front monitoring was performed. Also, simulations were implemented with the meshless approach for filling flow for certain special physical phenomena such as the fountain flow. The paper also forecasts the growth patterns and urgent problems of simulating injection molding based on examining and testing existing theories, algorithms, and the real demands of the plastics processing industry.

A stress recovery procedure for laminated composite plates based on strong-form equilibrium enforced via the RBF Kansa method

Often the employment of the full potentialities of laminated composite materials is restrained by the scarceness of cost-effective simulation strategies in the design stage. Albeit cheap modelling techniques exist, they typically prove ineffective in retrieving interlaminar stresses, the first responsible for the damage of laminates. In this paper, we propose a strategy to reconstruct the out-of-plane stress components in composite laminates, post-processing FEM data obtained with coarse models simplified according to the classical theory of laminates. The described approach relies on radial basis function interpolation and Kansa method to enforce equilibrium in strong form, within a meshless approach. The obtained results successfully compare with those from complete layup modeling for test cases where an analytical solution is available.

Implicit RBF Meshless Method for the Solution of Two-dimensional Variable Order Fractional Cable Equation

In the present work, the numerical solution of two-dimensional variable-order fractional cable (VOFC) equation using meshless collocation methods with thin plate spline radial basis functions is considered. In the proposed methods, we first use two schemes of order O(τ2) for the time derivatives and then meshless approach is applied to the space component. Numerical results obtained from solving considered model on regular and irregular domains, demonstrate the accuracy and efficiency of the proposed schemes.

A note on the numerical resolution of Heston PDEs

In this paper we aim to compare a popular numerical method with a new, recently proposed meshless approach for Heston PDE resolution. In finance, most famous models can be reformulated as PDEs, which are solved by finite difference and Monte Carlo methods. In particular, we focus on Heston model PDE and we solve it via radial basis functions (RBF) methods and alternating direction implicit. RBFs have become quite popular in engineering as meshless methods: they are less computationally heavy than finite differences and can be applied for high-order problems.

A local mesh-less collocation method for solving a class of time-dependent fractional integral equations: 2D fractional evolution equation

The local radial basis function (LRBF) method is an excellent tool for solving variable-order time-fractional evolution equations. This method reduces the expensive computational cost of the conventional global RBF collocation (GRBFC) approach. The present paper represents an efficient mesh-less LRBF collocation approach for solving the two-dimensional (2D) fractional evolution equation for the arbitrary fractional order in complex-shaped domains. Having been used an appropriate finite difference (FD) technique to discrete the time variable, the LRBF method is applied in order to solve the equation. Unlike the conventional GRBFC method, consideration of the local nodes in a subdomain surrounding the given local point is needed for the proposed LRBF method. This method can reduce the ill-conditioning of the global resultant RBF interpolation matrix while it is also well efficient for large scale problems. Some illustrative examples are given to demonstrate the efficiency and proficiency of the method.

Family Member Search Algorithms for Peridynamic Analysis

Peridynamic equation of motion is usually solved numerically by using meshless approaches. Family search process is one of the most time-consuming parts of a peridynamic analysis. Especially for problems which require continuous update of family members inside the horizon of a material point, the time spent to search for family members becomes crucial. Hence, efficient algorithms are required to reduce the computational time. In this study, various family member search algorithms suitable for peridynamic simulations are presented including brute-force search, region partitioning, and tree data structures. By considering problem cases for different number of material points, computational time between different algorithms is compared and the most efficient algorithm is determined.

A meshless method for the numerical solution of the seventh-order Korteweg-de Vries equation

This article describes a meshless method for the numerical solution of the seventh-order nonlinear one-dimensional non-stationary Korteweg-de Vries equation. The meshless scheme is based on the use of the collocation method and radial basis functions. In this approach, the solution is approximated by radial basis functions, and the collocation method is used to compute the unknown coefficients. The meshless method uses the following radial basis functions: Gaussian, inverse quadratic, multiquadric, inverse multiquadric and Wu’s compactly supported radial basis function. Time discretization of the nonlinear one-dimensional non-stationary Korteweg-de Vries equation is obtained using the θ-scheme. This meshless method has an advantage over traditional numerical methods, such as the finite difference method and the finite element method, because it doesn’t require constructing an interpolation grid inside the domain of the boundary-value problem. In this meshless scheme the domain of a boundary-value problem is a set of uniformly or arbitrarily distributed nodes to which the basic functions are “tied”. The paper presents the results of the numerical solutions of two benchmark problems which were obtained using this meshless approach. The graphs of the analytical and numerical solutions for benchmark problems were obtained. Accuracy of the method is assessed in terms of the average relative error, the average absolute error, and the maximum error. Numerical experiments demonstrate high accuracy and robustness of the method for solving the seventh-order nonlinear one-dimensional non-stationary Korteweg-de Vries equation.

Novel numerical method based on cubic B-splines for a class of nonlinear generalized telegraph equations in irregular domains

In this work, a cubic B-spline method based on finite difference and meshless approaches for solving 2D generalized telegraph equations in irregular single and multi-connected domains is presented. The three-layer Crank-Nicolson time-stepping scheme is applied for temporal derivatives and systems of second order elliptic partial differential equations (EPDEs) of general type with mixed derivatives and variable coefficients are obtained. The Dirichlet and Robin boundary conditions are considered. The approximate solution on each time layer is sought as series over basis functions which are taken in the form of the tensor products of cubic B-splines. The coefficients of the series are chosen to satisfy the governing EPDEs in the solution domain. The stability and accuracy of the proposed scheme is demonstrated by several examples.

The MAPS with polynomial basis functions for solving axisymmetric time-fractional equations

This study presents a parallel meshless approach for time-fractional equations in the axisymmetric geometry. In the present parallel meshless approach, the method of approximate particular solutions (MAPS), in conjunction with Laplace transform, extended precision arithmetic (EPA) and multiple scale technique (MST) is implemented. In this application, the Laplace transform is employed to eliminate the dependence on time. The MAPS is used to obtain the solution of time-independent equation in Laplace space. Furthermore, the solution of time dependent equation is restored by Stehfest’s algorithm. Here the multiple scale technique (MST) and extended precision arithmetic (EPA) are applied to alleviate the effect of an ill-conditioned matrix on a numerical inverse Laplace transform. Then multi-cores parallel computing can improve effectively computing speed of the proposed meshless approach. In the numerical implementation, both the 2D axisymmetric equations and the original 3D equations are considered and compared.

A meshless multiscale approach to modeling severe plastic deformation of metals: Application to ECAE of pure copper

Severe plastic deformation (SPD), occurring ubiquitously across metal forming processes, has been utilized to significantly improve bulk material properties such as the strength of metals. The latter is achieved by ultra-fine grain refinement at the polycrystalline mesoscale via the application of large plastic strains on the macroscale. We here present a multiscale framework that aims at efficiently modeling SPD processes while effectively capturing the underlying physics across all relevant scales. At the level of the macroscale boundary value problem, an enhanced maximum-entropy (max-ent) meshless method is employed. Compared to finite elements and other meshless techniques, this method offers a stabilized finite-strain updated-Lagrangian setting for improved robustness with respect to mesh distortion arising from large plastic strains. At each material point on the macroscale, we describe the polycrystalline material response via a Taylor model at the mesoscale , which captures discontinuous dynamic recrystallization through the nucleation and growth/shrinkage of grains. Each grain, in turn, is modeled by a finite-strain crystal plasticity model at the microscale . We focus on equal-channel angular extrusion (ECAE) of polycrystalline pure copper as an application, in which severe strains are generated by extruding the specimen around a 90 ° -corner. Our framework describes not only the evolution of strain and stress distributions during the process but also grain refinement and texture evolution, while offering a computationally feasible treatment of the macroscale mechanical boundary value problem. Though we here focus on ECAE of copper, the numerical setup is sufficiently general for other applications including SPD and thermo-mechanical processes (e.g., rolling, high-pressure torsion, etc.) as well as other materials systems.

Reduction of discretisation-induced anisotropy in the phase-field modelling of dendritic growth by meshless approach

A numerical procedure is developed to assess and reduce the discretisation-induced anisotropy in the solution of the phase-field model for dendritic growth. The meshless radial basis function-generated finite difference (RBF-FD) method and the forward Euler scheme are used for the spatial and temporal discretisation of the phase-field equations, respectively. A second-order accurate RBF-FD method is ensured by the use of augmentation with monomials up to the second order, while shape-parameter-free polyharmonic splines of the fifth-order are used as the radial basis functions. The performance of the RBF-FD method is assessed on regular and scattered node distributions by observing the mean phase field, the size of the primary dendrite arm, and the growth velocity. The observables at different orientation angles are compared to assess the orientation dependence of the solution. We show for the first time that the use of the RBF-FD method on a scattered node distribution provides a robust approach for the solution of the phase-field model for dendritic solidification with respect to an arbitrary preferential growth direction.

A localized meshless approach using radial basis functions for conjugate heat transfer problems in a heat exchanger

An understanding of conjugate heat transfer is important for enhancing the thermal performance of heat transfer devices. This study discusses the numerical solutions of the conjugate natural convective heat transfer problem using a local radial basis function meshless method. The temperature difference is assumed to be small enough to ensure the validity of the Boussinesq approximation. An artificial compressibility method is used to couple pressure to the continuity equation. The meshless method requires only one set of nodes, which are distributed in the spatial domain. The semi-algebraic equations are integrated using the second-order Runge–Kutta method. Three benchmark problems of conjugate natural convection with one vertical wall, conjugate natural convection with centrally vertical partition, and natural convection with a conducting solid in the middle of the square cavity are analyzed. The numerical results are compared with three benchmark solutions and experimental results; it was observed that the experimental and predicted results showed good agreement with each other. This numerical study is to contribute to understanding the knowledge of heat and fluid flow mechanisms for conjugate heat transfer in a heat exchanger acquired through experiments.

Numerical Simulations of MitraClip Placement: Clinical Implications

Mitral regurgitation (MR) is the most common type of valvular heart disease in patients over the age of 75 in the US. Despite the prevalence of mitral regurgitation in the elderly population, however, almost half of patients identified with moderate-severe MR are turned down for traditional open heart surgery due to frailty and other existing co-morbidities. MitraClip (MC) is a recent percutaneous approach to treat mitral regurgitation by placement of MC in the center of the mitral valve to reduce MR. There are currently no computational simulations to elucidate the role of MC on both the fluid and solid mechanics of the mitral valve. Here, we use the Smoothed Particle Hydrodynamics (SPH) approach to study various positional placements of the MC in the mitral valve and its impact on reducing MR. SPH is a particle based (meshless) approach that handles flow through narrow regions quite efficiently. Fluid and surrounding anatomical structure interactions is handled via contact and hence can be used for studying fluid-structure interaction problems such as blood flow with surrounding tissues/structure. This method is available as part of the Abaqus/Explicit solver. Regurgitation was initiated by removing targeted chordae tendineae that are attached to specified leaflets of the mitral valve and, subsequently, MC implants are placed in various locations, starting from the region near where the chordae tendineae were removed and moving away from the location towards the center of the valve. The MC implant location closest to where the chordae tendineae were removed showed the least amount of residual MR post-clip implantation amongst all other locations of MC implant considered. These findings have important implications for strategic placement of the MC depending on the etiology of MR to optimize clinical outcome.

Singular functions for heterogeneous composites with cracks and notches; the use of equilibrated singular basis functions

In this paper a novel method is presented to solve problems with weak singularities in two-dimensional heterogeneous media using equilibrated singular basis functions. This especially includes crack problems in composites of functionally graded material types. The method is mainly presented in a boundary formulation, although it may be found quite useful in other mesh-based or mesh-less approaches as the eXtended Finite Element Method (XFEM). The present paper considers harmonic and elasticity problems. The most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem. To this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach. After developing the formulation in a mapped polar co-ordination, some primary basis functions generated from Chebyshev polynomials and trigonometric functions, along with corresponding weight functions are employed. The numerical examples, either selected from the well-known literature or solved by well-established techniques, will demonstrate the capability of the method in problems related to composite materials.