Meshless Method Research Articles

Evaluation of a Coupled CFD and Multi-Body Motion Model for Ice-Structure Interaction Simulation

The interaction of water flow, ice, and structures is common in fluvial ice processes, particularly around Ice Control Structures (ICSs) that are used to manage and prevent ice jam floods. To evaluate the effectiveness of ICSs, it is essential to understand the complex interaction between water flow, ice and the structure. Numerical modeling is a valuable tool that can facilitate such understanding. Until now, classical Eulerian mesh-based methods have not been evaluated for the simulation of ice interaction with ICS. In this paper we evaluate the capability, accuracy, and efficiency of a coupled Computational Fluid Dynamic (CFD) and multi-body motion numerical model, based on the mesh-based FLOW-3D V.2023 R1 software for simulation of ice-structure interactions in several benchmark cases. The model’s performance was compared with results from meshless-based models (performed by others) for the same laboratory test cases that were used as a reference for the comparison. To this end, simulation results from a range of dam break laboratory experiments were analyzed, encompassing varying numbers of floating objects with distinct characteristics, both in the presence and absence of ICS, and under different downstream water levels. The results show that the overall accuracy of the FLOW-3D model under various experimental conditions resulted in a RMSE of 0.0534 as opposed to an overall RMSE of 0.0599 for the meshless methods. Instabilities were observed in the FLOW-3D model for more complex phenomena that involve open boundaries and a larger number of blocks. Although the FLOW-3D model exhibited a similar computational time to the GPU-accelerated meshless-based models, constraints on the processors speed and the number of cores available for use by the processors could limit the computational time.

Peridynamic modelling of cryogenic deuterium pellet fragmentation for shattered pellet injection in tokamaks

Abstract Shattered pellet injection (SPI) is a promising method for controlling plasma disruptions in tokamaks. In this study, we present numerical modelling of the fragmentation of cryogenic deuterium pellets within the context of SPI, using the peridynamic (PD) theory. A dedicated in-house code has been developed, leveraging the meshfree method and GPU parallelization. The mechanical properties of cryogenic solid deuterium are obtained from available literature, and calibrated based on the shatter threshold along with the remaining solid mass fraction after shatter. The results from the bond-based peridynamics (BB-PD) successfully reproduce the main experimental results reported in the literature, both qualitatively and quantitatively.

Analysis of a meshless generalized finite difference method for the time-fractional diffusion-wave equation

In this paper, a generalized finite difference method (GFDM) is proposed and analyzed for meshless numerical solution of the time-fractional diffusion-wave equation. Two (3−α)-order accurate temporal discretization schemes are presented by using the L1 formula and the original H2N2 or fast H2N2 formulas to discretize the time-fractional derivative of order α∈(1,2). The stability of the temporal discretization schemes is analyzed. Then, the time-fractional diffusion-wave initial-boundary value problem is transformed into a series of time-independent integer-order boundary value problems, and discrete linear algebraic systems are built by the application of the GFDM. Accuracy analysis of the GFDM with both original H2N2 and fast H2N2 formulas is presented in theory, and numerical experimental results are provided to verify the theoretical results and the effectiveness of the proposed meshless method.

Deep Learning-Meshless Method for Inverse Potential Problems

This paper combines the deep learning method with the meshless method to propose a new numerical method, which is the deep learning-improved element-free Galerkin (DL-IEFG) method, for solving inverse potential problems. In this method, the unknown term in the governing equation of the inverse potential problem is represented by the feedforward neural network (FNN). By employing the improved element-free Galerkin (IEFG) method to solve the inverse potential problem, the solution equations are established to obtain numerical solutions. The training set is constructed on the valid values obtained from discretized observation spatial points. The predicted values at the training sample points are calculated by combining the numerical solutions with the approximation function built by the improved moving least-squares (MLS) approximation. Then, the FNN representing the unknown term is iterated using the loss function. The effectiveness of the DL-IEFG method for solving potential inverse problems is validated through numerical examples. In addition, the factors impacting the calculation accuracy and efficiency of the DL-IEFG method are investigated.

A novel meshless method in conjunction with a regularization technique for solving the transient heat source with additive noise

We consider the problem of determination to ill-conditioned heat sources in unsteady 2-D heat conduction problems. The method we consider is truly mesh-free and is particularly suited to the inverse problem and is called the backward substitution method (BSM). We derive that when this method is coupled with the Crank–Nicholson (CN) discretizations of time and Tikhonov regularization, one demonstrates the proposed method is easy to implement and pretty accurate at an appropriate parameter concerning noise in the data. Also, a trigonometric basis function (TBF) that enforces approximate interpolation on boundary conditions yields an ill-conditional matrix. Finally, we assess the feasibility of the proposed method by numerical simulations and experiments.

NUMERICAL SOLUTIONS OF BOUSSINESQ TYPE EQUATIONS BY MESHLESS METHODS

In this paper, two different meshfree method with radial basis functions (RBFs) is proposed to solve Boussinesq-type (Bq) equations. The basic conservative properties of the equation are investigated by computing the numerical values of the motion’s invariants. The accuracy of the method is tested using computational tests to simulate solitary waves in terms of L_∞ error norm. The outcomes are contrasted with analytical solution and a few other earlier studies in the literature. The results show that meshless methods are very effective and accurate.

The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system

In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time.

An improved natural stabilized nodal integration for locking‐related materials in meshfree methods

An improved natural stabilized nodal integration for locking‐related materials in meshfree methods

Adaptive topology optimization for enhancing resistance to brittle fracture using the phase field model

Computational cost is one of the challenges in the field of fracture resistance topology optimization. An efficient topology optimization approach is proposed for enhancing resistance to structural fracture based on the adaptive isogeometric–meshfree method. The mesh can be adaptively refined in the computational and design domains simultaneously to capture the fracture and structural boundary delicately, which significantly reduces the degree of freedom and improves the efficiency of the topology optimization problems related to crack propagation. The proposed approach naturally inherits the advantages of both the computer-aided design and continuity of the higher-order basis functions, where the geometry and the analysis models in the process of topology optimization are integrated. The problem is formulated to maximize the absorbed energy throughout the crack process using the solid isotropic material with penalization method under volume constraints. The phase field method is used for modeling crack propagation, thereby eliminating the need of an explicit representation of the crack surface and complex tracking procedures, while a delicate mesh is still required. With this approach, the external work required during the fracture process is maximized, and the crack growth is delayed remarkably. Several representative examples are demonstrated to verify the effectiveness of the present approach in fracture-resistance-enhanced topology optimization, and the computational cost shows remarkable improvement.

Meshless method for wave propagation in poroelastic transversely isotropic half‐space with the use of perfectly matched layer

AbstractNumerical investigation of wave propagation in transversely isotropic poroelastic half‐space with the use of a new stretched coordinate system through the Meshless Local Petrov–Galerkin (MLPG) formulation is presented in this paper. To this end, the u−p formulation of Biot is adopted as the framework of the porous media. One approach to numerically solve the infinite domain problems is the use of an absorber layer in which the whole half‐space is divided into two parts, that is (i) a finite part, in which the responses are interested, and (ii) the remaining semi‐infinite part, which is replaced by a Perfectly Matched Layer (PML). The stretched coordinates in the PML are introduced in such a way that the wave propagating in it does not generate spurious reflection to the finite part. Comparing the numerical results with some existing exact solutions and evaluating the norm of error demonstrate that the response functions in the finite part are achievable as precise as desired. Some new results are also presented which show the validity of the numerical approach in poroelastic transversely isotropic domain.

A stable meshfree method for simulations of munition penetration into earth

A stable meshfree method for simulations of munition penetration into earth

A meshless approach based on fractional interpolation theory and improved neural network bases for solving non-smooth solution of 2D fractional reaction–diffusion equation with distributed order

The primary objective of this research paper is to present a novel and effective meshless numerical approach for solving the 2D time fractional reaction diffusion system with distributed order on an arbitrary domain. Gauss–Legendre quadrature formula is applied to discretize distributed-order derivative integral. We establish the piecewise parabolic fractional interpolation theory and with its assistance, the proposed approach can proficiently solve the non-smooth solutions of the equations and more accurately approximate the Caputo fractional derivative. This meshless method based on improved neural network bases combines the high accuracy approximation advantage and strong express ability of neural network to construct the basis functions set on arbitrary domains, which significantly reduces a computational consumption. The bases constructed based on the neural network allow the selection of 12 bases numbers to achieve an approximation equivalent to that of 1000 ordinary bases. The theoretical analyses of error and convergence order for the meshless approach are carried out. Numerical examples are implemented to validate the high precision and capability of the meshless numerical approach.

Hybrid Haar wavelet and meshfree methods for hyperbolic double interface problems: Numerical implementations and comparative performance analysis

This paper introduces a variety of approaches for solving 2D and 3D hyperbolic double interface problems. The methods are based on the Haar wavelet method, multiquadric radial basis function method, and integrated multiquadric radial basis function method. Temporal derivatives are handled using the second central difference and the Houbolt method. Various numerical approaches based on these methods are developed, and their implementations are discussed in complete detail. The paper evaluates and compares the performances of these approaches using both linear and nonlinear 2D and 3D double interface hyperbolic problems. Error analysis, conducted using the L-infinity norm, and efficiency assessments measured through CPU times contribute to a comprehensive understanding of the applicability and comparative effectiveness of the proposed methods. This study provides valuable insights for researchers and practitioners dealing with the challenges posed by interface problems in general.

Solving nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov equation using two meshless methods

Solving nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov equation using two meshless methods

Smoothed point interpolation methods for phase-field modelling of pressurised fracture

The problem of hydraulic fracturing is of great relevance to various areas and is characterised by the occurrence of complex crack patterns with bifurcations and branches. For this reason, an interesting approach is the modelling of hydraulic fracture using a phase-field model. In addition to the discretisation using the Finite Element Method (FEM), some works have already explored the discretisation of the phase-field model with meshfree methods, including the Smoothed Point Interpolation Methods (SPIM) family. Seeking to take advantage of the good convergence results of SPIM for phase-field modelling of brittle fractures, this paper proposes the use of SPIM for phase-field modelling of pressurised fractures. In order to limit the computational cost, a prescribed SPIM-FEM coupling is employed, with the purpose of concentrating the meshless discretisation only in the regions of expected crack propagation. The model is characterised by a constant internal pressure load along the fracture that is applied indirectly from the formulation of the phase-field model. A series of numerical simulations is presented. The aim is to evaluate the proposed model, verify the results and point out characteristics of the phase-field model with internal pressure.

Three-dimensional elastoplastic post-buckling analysis of functionally graded plates using a novel meshfree Tchebychev-radial point interpolation approach

This paper presents a three-dimensional (3D) analysis of the post-buckling behavior of functionally graded (FG) plates for the first time using an elastoplasticity-based meshfree formulation. A novel 3D Tchebychev-radial point interpolation approach, which combines radial basis functions with Tchebychev polynomials, is developed to solve the nonlinear post-buckling problem. The incremental plastic deformation is modeled by the Prandtl–Reuss flow rule along the isotropic hardening von Mises criterion. The governing equations are derived using the principle of virtual work by considering the 3D full Green–Lagrange strain components. The Newton–Raphson technique along with the arc-length method is used to determine post-buckling equilibrium paths of FG plates. The effective elastoplastic properties of the functionally graded material are assessed by exploiting the homogenization method, named Tamura–Tomota–Ozawa (TTO) model. It has been demonstrated that the accuracy of the solution is improved by incorporating Tchebychev polynomials into the radial point interpolation method (RPIM) shape function, and the stability and robustness of the results are independent of variations in the shape parameter. Furthermore, it is confirmed that TRPIM, with a slightly higher CPU time compared to RPIM, exhibits a higher convergence rate. The excellent agreement of the results with those existing in the literature shows that the proposed meshfree method can be used as a robust and accurate numerical tool to predict the elastoplastic post-buckling behavior of FG plates. Further numerical assessments indicate that post-buckling paths are significantly affected by factors such as geometric parameters, material gradient, loading ratio, and boundary conditions (BCs).

Prediction of the Dangerous Range of Dumps Based on a Dynamic Process

A dump is a loose accumulation of solid waste from mining operations that easily leads to disasters such as landslides and debris flows. Taking a dump in the Panzhihua region of China as an example, based on the MPM and SPH meshless methods, the dynamic calculation of the instability process of the spoil slope was carried out according to the realistic three-dimensional calculation model of the generated terrain. Firstly, the strain and displacement of the dump under normal conditions and heavy rainfall conditions were simulated by the MPM material point method. The maximum spatial displacement of the dump under heavy rainfall conditions reached up to 386 m. Then, the motion and morphology of the accumulation of the dump under ordinary working conditions and heavy rainfall conditions were analyzed using the smooth particle hydrodynamics (SPH) method. Under heavy rainfall conditions, the maximum horizontal displacement of the dump was approximately 394 m. The research results are conducive to the risk assessment of the spoil slope and provide theoretical support for the calculation of the range of potential threats from the dump.

High-precision meshfree methods for solving 3D radiation diffusion problem in irregular spaces

Radiation diffusion is a significant phenomenon in inertial confinement fusion, geology and so on. Solving nonlinear radiation diffusion equation in irregular spaces is challenging. In this paper, the direct linearisation algorithm and the successive permutation iteration algorithm are constructed for solving 3D radiation diffusion equation, which are full-implicit discretization on time. The high-precision meshfree methods are provided by the Kansa's non-symmetric collocation method with the compactly supported radial basis function. Ultimately, the accuracy and efficiency of the presented algorithms are verified through the comparison of the fixed point, Newton and SOR methods in 3D numerical experiments. These novel meshfree methods not only avoid the complexity of grid generation, but also provide high-precision numerical solution for nonlinear radiation diffusion equation.

Simulation of a Pelton turbine using the moving particle simulation method: application to two challenging situations

ABSTRACT Simulation of the flow in a Pelton turbine is a challenging task for mesh based methods due to the air/water mixture and the rotation of the runner. Meshless based methods are better suited for such flows and take advantage of the graphical power unit to speed up the calculation. Simulations of a Pelton turbine are carried out using the software Particleworks based on moving particle simulation, which is a meshless method. The efficiency drop due to the ‘falaise’ effect and the erosion of the splitter are computed and compared with either experimental measurements or the literature. The results show that this method is able to capture the tendency of the efficiency drop in agreement with the available data and at a lower computational cost than the mesh based methods. These encouraging results should motivate the community to test and validate such an approach.

Free vibration and stationary stochastic response analyses of heated composite laminated plate with varying thickness and curvilinear fiber using a meshfree method

Free vibration and stationary stochastic response analyses of heated composite laminated plate with varying thickness and curvilinear fiber using a meshfree method