Meshless Numerical Method Research Articles

Forecasting Disastrous Characteristics of Highway Landslides Using the Material Point Method: A Surcharge‐Induced Perspective

Precisely forecasting the disastrous characteristics of landslides can address the optimized preliminary design of the highway landslide stabilization. In this paper, taking the surcharge‐induced landslides as an example, we developed the models of the material point method (MPM), a novel meshless numerical method, to calculate the large deformation of the highway landslides, with the affordable computing power. After convincing verification of the MPM model to simulate a surcharge‐induced soil landslide, 32 typical postfailure scenarios were analyzed to obtain the highway landslide run‐out processes and disastrous characteristics, such as the sliding distance and speed of the leading edge and sliding body morphology. Moreover, linear regression equations of the maximal sliding distance and speed were deduced after verification, to forecast the reasonable avoiding distance. The maximal sliding distance and speed were found to be negatively linearly correlated with the internal frictional angle and cohesion of the soil, and positively linearly correlated with the surcharge and the slope angle, respectively. Optimized preliminary design, of the highways in the mountainous and hilly areas, can be performed based on those insights.

Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains

In this paper, we propose multi-scale deep neural networks (MscaleDNNs) using the idea of radial scaling in frequency domain and activation functions with compact support. The radial scaling converts the problem of approximation of high frequency contents of PDEs' solutions to a problem of learning about lower frequency functions, and the compact support activation functions facilitate the separation of frequency contents of the target function to be approximated by corresponding DNNs. As a result, the MscaleDNNs achieve fast uniform convergence over multiple scales. The proposed MscaleDNNs are shown to be superior to traditional fully connected DNNs and be an effective mesh-less numerical method for Poisson-Boltzmann equations with ample frequency contents over complex and singular domains.

Multi-Scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains

In this paper, we study a multi-scale deep neural network (MscaleDNN) as a meshless numerical method for computing oscillatory Stokes flows in complex domains. The MscaleDNN employs a multi-scale structure in the design of its DNN using radial scalings to convert the approximation of high frequency components of the highly oscillatory Stokes solution to one of lower frequencies. The MscaleDNN solution to the Stokes problem is obtained by minimizing a loss function in terms of L2 normof the residual of the Stokes equation. Three forms of loss functions are investigated based on vorticity-velocity-pressure, velocity-stress-pressure, and velocity-gradient of velocity-pressure formulations of the Stokes equation. We first conduct a systematic study of the MscaleDNN methods with various loss functions on the Kovasznay flow in comparison with normal fully connected DNNs. Then, Stokes flows with highly oscillatory solutions in a 2-D domain with six randomly placed holes are simulated by the MscaleDNN. The results show that MscaleDNN has faster convergence and consistent error decays in the simulation of Kovasznay flow for all four tested loss functions. More importantly, the MscaleDNN is capable of learning highly oscillatory solutions when the normal DNNs fail to converge.

Вклад силы присоединенных масс в формирование пропульсивной силы машущего профиля в вязкой жидкости

In the light of the recently proved general theorem on the added mass of bodies in a viscous incompressible fluid, the mechanism of the formation of the propulsive force of the flapping airfoil, which performs harmonic angular oscillations in the flow of a continuous medium, is investigated. The flow is described by the Navier-Stokes equations. The calculations were performed by a meshless numerical method of viscous vortex domains. The mechanism of the formation of a reversed vortex track in the wake behind the flapping profile in the positive traction mode is explained. The dominant contribution of the force of the added masses to the value of the propulsive force is revealed. The results obtained to some extent reveal the hydrodynamic mechanism of action of the caudal fin of underwater creatures.

Weak and strong from meshless methods for linear elastic problem under fretting contact conditions

Abstract We present numerical computation of stresses under fretting fatigue conditions derived from closed form expressions. The Navier-Cauchy equations, that govern the problem, are solved with strong and weak form meshless numerical methods. The results are compared to the solution obtained from well-established commercial package ABAQUS, which is based on finite element method (FEM). The results show that the weak form meshless solution exhibits similar behavior as the FEM solution, while, in this particular case, strong form meshless solution performs better in capturing the peak in the surface stress. This is of particular interest in fretting fatigue, since it directly influences crack initiation. The results are presented in terms of von Mises stress contour plots, surface stress profiles, and the convergence plots for all three methods involved in the study.

An accurate multi-regime SPH scheme for barotropic flows

This paper focuses on the use of a mesh-less numerical method to solve barotropic fluid flows: Smooth Particle Hydrodynamics (SPH). Such methods suffer from a severe lack of accuracy when evaluating state variables as the pressure field. SPH-ALE methods based on Riemann solvers significantly improve this evaluation but increase the scheme complexity and low-Mach issues are difficult to prevent. We propose an alternative scheme called γ−SPH−ALE relying on the combination of the SPH-ALE formalism and a finite volume stabilizing low-Mach scheme. Applied to Monophasic Barotropic Euler Equations, its characteristics are detailed and evaluated through a nonlinear stability analysis highlighting CFL-like conditions on the scheme parameters. As far as we know, it is the first time such nonlinear analysis is performed on a SPH-ALE scheme. Finally, its implementation on several academic test cases reveals that the proposed scheme actually increases both stability and accuracy, in reduced computation time, with respect to SPH ALE Riemann Solvers.

A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem

In this paper a numerical scheme based on the idea of radial basis function finite difference (RBF-FD) technique is considered to solve the backward heat conduction problems (BHCP). In the meshless numerical method of RBF-FD, according to the finite difference technique we approximate the required derivatives for every point xi ∈ Ω in the corresponding local-support domain Ωi. Then the partial differential equation problem is transformed into the problem of a linear system of algebraic equations. This method also belongs to localized radial basis function method or the closest point method. To compare RBF-FD method with another RBF technique, radial basis function collocation method (RBFCM) and the method of approximate particular solutions (MAPS) are also considered to solve such inverse problem, and in the computation the standard Tikhonov regularization technique with L-curve method for choose optional regularized parameter is used for solving the highly ill condition system of linear equations. Several numerical examples are presented to demonstrate the ability of the present approach for solving the backward heat conduction problem.

Development of a three dimensional meshless numerical method for the solution of the Boltzmann equation on complex geometries

We propose a three dimensional meshless numerical scheme based on the Boltzmann equation for the simulation of fluid flows in domains of complex geometries. The velocity space discretization in our method is the same as the standard lattice Boltzmann method, i.e. the same lattices are used to discretize microscopic velocities. The velocity-discrete Boltzmann equation with the BGK collision approximation is discretized in time using the Lax–Wendroff scheme, and subsequently in space using the meshless local Petrov–Galerkin method based on local radial basis functions augmented with polynomials. The method is first validated and tuned by solving two benchmark problems: The pressure-driven flow in an elliptical tube, and the three-dimensional lid-driven cavity flow. The error analysis for the first case shows the second order accuracy of the proposed method. The flow in a packed bed is then solved in order to illustrate the efficiency of the proposed method in the simulation of geometrically complex flows. The results show that both the computational time and the memory usage of our method are lower than those of the standard lattice Boltzmann method for the geometrically complex flows.

SOPHIA: Development of Lagrangian-based CFD code for nuclear thermal-hydraulics and safety applications

Lagrangian-based meshless CFD methods have recently been developed and are being gradually applied to various research areas. In nuclear engineering, the meshless methods are gaining attention in modeling natural disasters and severe accident phenomena, such as tsunami, molten corium behaviors, etc., because of its flexible computational domain and effective treatment of non-linear deformations. This paper introduces recent progress and on-going activities in Lagrangian-based CFD code development at Seoul National University (SNU). The code, named SOPHIA, is based on the smoothed particle hydrodynamics (SPH) method, one of the best-known meshless numerical methods, based on a Lagrangian framework that can easily handle various types of physics because of its simplicity in expressing and solving mathematical equations. The SOPHIA code currently incorporates basic conservation equations and various physical models, including fluid flow, heat transfer, turbulence, multi-phase, phase change, elastic solid, diffusion and so on. To handle multi-phase, multi-component, and multi-resolution flows simultaneously, the SOPHIA code formulates density and continuity equations based on the normalized-density form, which has been recently developed and is proposed in the current study. The SOPHIA code also adapts CUDA GPU architectures for parallelization and performance improvement. Based on the benchmark calculations, the parallelized GPU code shows much higher computational speed by two orders of magnitude compared to the single CPU code for 1.0 million particles. This paper summarizes the overall features of the SOPHIA code, including governing equations, algorithms, and parallelization methods, along with some benchmark simulations.

A meshless method for solving the nonlinear inverse Cauchy problem of elliptic type equation in a doubly-connected domain

In the paper a nonlinear inverse Cauchy problem of nonlinear elliptic type partial differential equation in an arbitrary doubly-connected plane domain is solved using a novel meshless numerical method. The unknown Dirichlet data on an inner boundary are recovered by over-specifying the Cauchy data on an outer boundary. A homogenization function is derived to annihilate the Cauchy data on the outer boundary, and then a homogenization technique generates a transformed equation in terms of a transformed variable, whose outer Cauchy boundary conditions are homogeneous. When the numerical solution is expanded by a sequence of boundary functions, which automatically satisfy the homogeneous Cauchy boundary conditions on the outer boundary, we can solve the transformed equation by the domain type meshless collocation method. For the nonlinear inverse Cauchy problems we require to iteratively solve the linear systems with the right-hand sides varying per iteration step. The accuracy and robustness of the homogenization boundary function method (HBFM) are examined through seven numerical examples, where we compare the exact Dirichlet data on the inner boundary to the ones recovered by the HBFM under a large noisy disturbance.

An effective semi-analytical method for solving telegraph equation with variable coefficients

In this paper, a new meshless numerical method is proposed for solving the second-order hyperbolic telegraph equations with coefficients variable in space and time. The temporal discretization is presented by the two-level time-stepping Crank-Nicolson scheme (CNS) with the meshless semi-analytical technique proposed by Reutskiy and Lin (Int. J. Numer. Methods Eng. 112 , 2004 (2017)) for spatial discretization. For the spatial discretization, the approximation is given to satisfy the boundary conditions in advance with any free parameters. Then the approximation is substituted back to the governing equations where the unknowns are determined by the collocation approach. The efficacy of the proposed approach has been confirmed by numerical experiments and comparisons with the results obtained by other techniques. Numerical results show that the proposed method is of high accuracy and efficacy for solving a wide class of telegraph equations. The method is very simple and does not require any specific technique in handling the singularity at the endpoints of the solution domain.

A GE/BC imbedded local polynomial collocation method for two dimensional multivariable problems

In this study, a meshless numerical method for 2D problems with 2 variables coupled together in 2 partial differential equations (PDEs) is proposed. This method employs the local polynomial approximation with the weighted-least-squares (WLS) approach. A very distinguishing feature of this method from other collocation methods is the governing equations are satisfied at boundary nodes as well as the boundary conditions. By doing so the strong-form collocation method would be more stable. The 2D elasto-statics analysis is chosen for demonstrating the performance. Three test cases are implemented and the results are compared with exact or analytic solutions. Very good agreements are found.

Prediction procedure for hail impact

The constant increase of composite materials’ performances makes them more and more used in recent aircrafts. Structures, as the wings or the fuselage, may suffer from hail impacts that can make critical damages or even perforate them. In order to guaranty the safety of passengers, aircrafts have to be certified and simulations have to demonstrate good agreements with real behaviour of the structures and the hail projectile. The aim of this work is to propose a procedure to analyse the home made manufacturing of the ice generally performed in laboratories, its mechanical characterization and a mechanical model that can predict the time-space profile of the impact force on a rigid structure. Because of the high strain level of the hail during the impact, the Smooth Particle Hydrodynamics (SPH) method will be used. Indeed, the finite elements method needs heavy remeshing that are time consuming to avoid mesh distortion. The SPH is a numerical meshless method that calculates interactions between particles at every time increment. Models available in the literature have been studied and the model of J.D. Tippmann (Tippmann, Kim, et Jennifer D. Rhymer 2013) is chosen. In this paper, the Tippman model is presented with its solving using the SPH. A parametric study is proposed in order to catch the relevant parts of this model. A simple experimental procedure is then proposed to feed the model and the results of impact simulations at different velocities are compared to experimental measurements realized in the laboratory.

An Accurate SPH Scheme for Dynamic Fragmentation modelling

We focus on the use of a meshless numerical method called Smooth Particle Hydrodynamics (SPH), to solve fragmentation issues as Hyper Velocity Impact (HVI) cases. Firstly applied to fluid flow simulations, this method can be extended to the solid dynamics framework. However it suffers from a lack of accuracy when evaluating state variables as the pressure field. And such inaccuracy generally generates non-physical processes (as numerical fragmentation). In the hydrodynamic context, SPH-ALE methods based on Riemann solvers significantly improve this evaluation, but increase the scheme complexity and low-Mach issues are difficult to prevent. We propose an alternative scheme called γ-SPH-ALE, firstly implemented to solve multi-regime barotropic flows, and secondly extended to solid dynamic cases. It relies on the combination of the SPH-ALE formalism and a finite volume stabilizing low-Mach scheme. Its characteristics are detailed and evaluated through a nonlinear stability analysis, highlighting CFL-like conditions on the scheme parameters. Finally, its implementation on several test cases reveals that the proposed scheme actually increases both stability and accuracy, in reduced computation time, with respect to classical solvers.

RBF-FD meshless optimization using direct search (GLODS) in the analysis of composite plates

An optimization technique, Global and Local Optimization using Direct Search (GLODS) is used to optimize stencil size and shape parameter of the multiquadric function used in the Radial Basis Function-Finite Difference (RBF-FD) meshless numerical method. The mechanical problem to be solved by the meshless method is a composite plate in bending, under sinusoidal load. Results show GLODS is capable of finding values for stencil size and shape parameter that produce excellent numerical solutions when compared with analytical solutions.

The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation

In this article, a meshless numerical method for solving the inverse source problem of the space-fractional diffusion equation is proposed. The numerical solution is approximated using the fundamental solution of the space-fractional diffusion equation as a basis function. Since the resulting matrix equation is extremely ill-conditioned, a regularized solution is obtained by adopting the Tikhonov regularization scheme, in which the choice of the regularization parameter is based on generalized cross-validation criterion. Two typical numerical examples are given to verify the efficiency and accuracy of the proposed method.

ISPH modelling of landslide generated waves for rigid and deformable slides in Newtonian and non-Newtonian reservoir fluids

A comprehensive modeling of landslide generated waves using an in-house parallel Incompressible Smoothed Particle Hydrodynamics (ISPH) code is presented in this paper. The study of landslide generated waves is challenging due to the involvement of several complex physical phenomena, such as slide-water interaction, turbulence and complex free surface profiles. A numerical tool that can efficiently calculate both slide motion, impact with the surface and the resulting wave is needed for ongoing study of these phenomena. Mesh-less numerical methods, such as Smoothed Particle Hydrodynamics (SPH), handle the slide motion and the complex free surface profile with ease. In this paper, an in-house parallel explicit ISPH code is used to simulate both subaerial and submarine landslides in 2D and in more realistic 3D applications. Both rigid and deformable slides are used to generate the impulsive waves. A landslide case is simulated where a slide falls into a non-Newtonian reservoir fluid (water-bentonite mixture). A new technique is also proposed to calculate the motion of a rigid slide on an inclined ramp implicitly, without using the prescribed motion in SPH. For all the test cases, results generated from the proposed ISPH method are compared with available experimental data and show good agreement.

Reproducing Kernel Method for Singularly Perturbed One-Dimensional Initial-Boundary Value Problems with Exponential Initial Layers

In this paper, a meshless numerical method is proposed for singularly perturbed one-dimensional initial-boundary value problems with exponential initial layers. The method is a combination of the domain decomposition method and the reproducing kernel method. A fitted reproducing kernel is constructed for initial layer domain problem. Some numerical results confirming the expected behavior of the method are shown.

A not-a-knot meshless method with radial basis functions for numerical solutions of Gilson–Pickering equation

In this paper, a numerical meshless method for solving the Gilson–Pickering equation is considered. The method is based on thin-plate radial basis function using collocation points. The scheme proposed works in a similar fashion as finite-difference method and a predictor–corrector scheme is proposed to avoid solving the nonlinear system. In addition, we use the super not-a-knot method for improving the accuracy at the boundaries. The results of numerical experiments are compared with the existing results in illustrative examples to confirm the accuracy and efficiency of the presented scheme and the norm of the error functions is obtained to show the convergence of the method.

A numerical scheme based on radial basis function finite difference (RBF-FD) technique for solving the high-dimensional nonlinear Schrödinger equations using an explicit time discretization: Runge–Kutta method

In this research, we investigate the numerical solution of nonlinear Schrödinger equations in two and three dimensions. The numerical meshless method which will be used here is RBF-FD technique. The main advantage of this method is the approximation of the required derivatives based on finite difference technique at each local-support domain as Ωi. At each Ωi, we require to solve a small linear system of algebraic equations with a conditionally positive definite matrix of order 1 (interpolation matrix). This scheme is efficient and its computational cost is same as the moving least squares (MLS) approximation. A challengeable issue is choosing suitable shape parameter for interpolation matrix in this way. In order to overcome this matter, an algorithm which was established by Sarra (2012), will be applied. This algorithm computes the condition number of the local interpolation matrix using the singular value decomposition (SVD) for obtaining the smallest and largest singular values of that matrix. Moreover, an explicit method based on Runge–Kutta formula of fourth-order accuracy will be applied for approximating the time variable. It also decreases the computational costs at each time step since we will not solve a nonlinear system. On the other hand, to compare RBF-FD method with another meshless technique, the moving kriging least squares (MKLS) approximation is considered for the studied model. Our results demonstrate the ability of the present approach for solving the applicable model which is investigated in the current research work.