Traditional Mesh-based Methods Research Articles

Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.

Analysis of geometric uncertainties in CFD problems solved by RBF-FD meshless method

In order to analyze incompressible and laminar fluid flows in presence of geometric uncertainties on the boundaries, the Non-Intrusive Polynomial Chaos method is employed, which allows the use of a deterministic fluid dynamic solver. The quantification of the fluid flow uncertainties is based on a set of deterministic response evaluations, which are obtained through a Radial Basis Function-generated Finite Differences meshless method. The use of such deterministic solver represents the key point of the analysis, thanks to the computational efficiency and similar accuracy over the traditional mesh-based numerical methods. The validation of the proposed approach is carried out through the solution of the flow past a 2D spinning cylinder near a moving wall and the flow over a backward-facing step, in presence of stochastic geometries. The applicability to practical problems is demonstrated through the investigation of geometric uncertainty effects on the forced convection of Al2O3-water nanofluid laminar flow in a grooved microchannel.

A generalized interpolation material point method for modelling coupled seepage-erosion-deformation process within unsaturated soils

In this paper, a computational framework based on the material point method (MPM) is developed to study the coupled seepage-erosion-deformation process within unsaturated soils. Based on the mixture theory, the unsaturated erodible soil is conceptualized as a three-phase multi-species porous medium, which is represented as a set of Lagrangian material points in a three-phase multi-species single point MPM framework. Governing equations as well as constitutive models for describing the coupled seepage-erosion-deformation behaviour within unsaturated erodible soils are presented: the solid skeleton is modelled as an elasto-plastic material; the pore water is treated as a weakly compressible fluid; while the fine particles in the solid matrix can be eroded and transferred into liquidised fine particles, and transported with the flowing liquid. This governing system is discretised with the generalized interpolation material point (GIMP) method and solved explicitly. The proposed coupled seepage-erosion-deformation GIMP is validated via three numerical examples and then employed for simulating the rainfall-induced slope failure process involving internal erosion. Thanks to the capability of the MPM in capturing large deformations, the evolution of the coupled seepage-erosion-deformation behaviour during the whole slope failure process can be obtained. This suggests that the proposed coupled GIMP framework is a promising approach for future studies of internal erosion problems in unsaturated soils involving complex multi-couplings of seepage flow, phase change and large deformations, which are difficult to be modelled by traditional mesh-based methods.

Numerical Study of 3-D Liquid Sloshing in an Elastic Tank by MPS-FEM Coupled Method

The sloshing phenomenon in a partially loaded oil tanker or liquid natural gas ship is a typical fluid-structure interaction problem involving multi-physics, violent free-surface flow, and nonlinear structural response. In the past decades, the complex phenomenon has been commonly investigated without consideration of the hydro-elastic behaviors of the bulkheads. In our previous work, the liquid sloshing phenomenon in a two-dimensional (2-D) elastic tank was numerically studied. However, the bulkheads of the tank will deform within the three-dimensional (3-D) space in reality. So, it is necessary to study the 3-D sloshing problem in an elastic tank. In this article, a hybrid approach is developed within the Lagrangian system. The moving particle semiimplicit (MPS) method is used to simulate the evolution of 3-D flow with a violent free surface, and the finite element method (FEM) is used for the numerical analysis of structural response due to the impact loads of the sloshing flow. To couple the MPS method and the FEM method, an interpolation scheme based on the kernel function of the particle method is proposed for the communication on the isomerous interface between the fluid and structure domains. The reliability of force and deformation interpolation modules is validated by two tests. Then, the sloshing phenomenon in a 3-D elastic tank is numerically investigated and compared against the previous published 2-D results. By varying the Young's modulus of the tank walls, characteristics regarding the evolutions of free surface, variation of impact pressures, and dynamic responses of the structures are presented. 1. Introduction To support the transportation demands of natural resources, more and more vessels, such as the very large crude carriers and the liquid natural gas carriers, are manufactured. For these huge structures, risks such as local deformation or even damage of cargo-containment systems resulting from sloshing phenomena subsequently increase. Therefore, it is necessary to take the elasticity of the tank walls into account in the research of sloshing phenomena (Dias & Ghidaglia 2018). However, the phenomena involving the vibrations of the tank walls are complex. In the process of sloshing wave interacting with elastic bulkheads, the fragmentation, splash, and fusion of water are observed. Meanwhile, the structures vibrate nonlinearly under the impact loads resulting from the sloshing wave. These phenomena are hard to simulate realistically by the traditional mesh-based methods.

An efficient and stabilised SPH method for large strain metal plastic deformations

Due to its simplicity and robustness, smooth particle hydrodynamics (SPH) has been widely used in the modelling of solid and fluid mechanics problems. Through the years, various formulations and stabilisation techniques have been adopted to enhance it. Recently, the authors developed JST–SPH, a mixed formulation based on the SPH method. Originally devised for modelling (nearly) incompressible hyperelasticity, the JST–SPH formulation is mixed in the sense that linear momentum and a number of strain definitions, instead of the displacements, act as main unknowns of the problem. The resulting governing system of conservation laws conveniently enables the application of the Jameson–Schmidt–Turkel (JST) artificial dissipation term, commonly employed in computational fluid dynamics, to solid mechanics. Coupled with meshless SPH discretisation, this novel scheme eliminates the shortcomings encountered when implementing fast dynamics explicit codes using traditional mesh-based methods. This paper focuses on the applicability of the JST–SPH mixed formulation to the simulation of high-rate, large metal elastic–plastic deformations. Three applications—including the simulation of an industry-relevant metal forming process—are examined under different loading conditions, in order to demonstrate the reliability of the method. Results compare favourably with both data from the previous literature, and simulations performed with a commercial finite elements package. Most noticeably, these results demonstrate that the total Lagrangian framework of JST–SPH, fundamental to reduce the computational effort associated with the scheme, retains its accuracy in the presence of large distortions. Moreover, an algorithmic flow chart is included at the end of this document, to facilitate the computer implementation of the scheme.

Method of fundamental solutions on simulating sloshing liquids in a 2D tank

A two dimensional numerical model for sloshing response of the partially filled container is developed by a meshless method in this study. Laplace equation is considered as the governing equation for potential flow, and fully non-linear free surface boundary conditions are implemented on the free surface. The numerical model is established in the moving coordinate system attached to the container. The present meshless method, method of fundamental solutions, only need to distribute nodes on the physical boundaries and the virtual boundaries outside the computational domain. Comparing with the traditional mesh-based numerical method, this method has the advantage of being easy to construct and flexible in dealing with the moving boundary problem. The program is written in Python. A simple method for tracking fully non-linear free surface is implemented, and the physical boundaries are updated with time. The numerical results agree very well with existing results in both resonance and off-resonance cases. The present numerical model is further applied to the sloshing response under coupled motions (horizontal and vertical). In order to study the stability of sloshing response (Benjamin and Ursell, 1954) under vertical motion, 25 cases are conducted in various sloshing parameters. In addition, the present numerical model has been successfully applied to investigating sloshing response under coupled motions.

Bioinspired superoleophobic/superhydrophilic functionalized cotton for efficient separation of immiscible oil-water mixtures and oil-water emulsions

Oil contamination is pressing environmental and natural resource issues. Oil-water separation technology can be used for water purification. Traditional methods are either time consuming or energy intensive. Bioinspired materials with combination of affinity and/or repellence towards oil and water in mesh-based methods have attracted widespread attention in the oil-water separation. Among existing materials, superhydrophobic/superoleophilic materials are prone to contamination by oil. Superhydrophilic/underwater superoleophobic materials can only operate underwater which restricts the practical use. In contrast, superoleophobic/superhydrophilic materials are desirable for oil-water separation. However, traditional mesh-based method cannot be used for separation of emulsions since the dispersed microdroplets can easily pass through the mesh pores. Superhydrophobic/superoleophilic cotton has been used for separation of water-in-oil emulsions. In this study, a facile method to fabricate bioinspired superoleophobic/superhydrophilic cotton was developed. The functionalized cotton exhibits an oil-repellent and water-wetting behavior and is capable of separating both immiscible oil-water mixtures and also oil-in-water emulsions without oil contamination. This developed cotton is of interest for various applications.

Smoothed Particle Hydrodynamics simulation of spudcan penetration based on two-phase mixture theory

A novel algorithm was developed based on the two-phase mixture theory within the framework of Smoothed Particle Hydrodynamics (SPH) aiming at simulating the complex soil-water-structure interaction when soil undergoes large deformation. In the algorithm, each constitute is assumed to occupy a certain part of the macroscopic mixture and satisfies its own conservation equations. The soil-water interaction force was composed of pore water pressure and viscous drag force. The spudcan was assumed to be rigid. Frictional sliding contact was employed to modify the momentum equation for particles located near or on the structure surface. With the proposed algorithm, it can investigate the pore water pressure, the effective stress and the porosity for soil undergoing extremely large deformation. Afterwards, its application to the simulation of continuous penetration of the spudcan was presented. The calculated pore water pressure and penetration resistance were compared to the previous research, indicating promising robustness and applicability of the present algorithm. Furthermore, the proposed algorithm could be a potentially efficient tool helping to solve other problems involving soil large deformation and complex soil-water-structure interaction such as deep penetration, underwater excavation, piping and so on, which is quite difficult dealing with traditional mesh-based methods.

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.

SPH Simulations of Solute Transport in Flows with Steep Velocity and Concentration Gradients

In this study, a meshless particle method, smoothed particle hydrodynamics (SPH), is adopted to solve the shallow water equations (SWEs) and the advection diffusion equations (ADEs) for simulating solute transport processes under 1D/2D conditions with steep gradients. A new SPH-SWEs-ADEs model is herein developed to focus on the numerical performance of solute transport in flows with steep velocity and concentration gradients, since the traditional mesh-based methods have numerical difficulties on solving such steep velocity/concentration gradient flows. The present model is validated by six benchmark study cases, including three steep concentration gradient cases and three coupled steep concentration/velocity gradient cases. The comparison between the simulated results and the exact solutions for the former three cases shows that complete mass concentration conservation in pure advection-dominated flows is preserved. The numerical oscillation in concentration and the negative concentration resulted from the discretization of the advection term of ADEs can be totally avoided. The other three cases confirm that this model can also well capture coupled steep gradients of velocities and concentrations. It is demonstrated that the presented solver is an effective and reliable tool to investigate solute transports in complex flows incorporating steep velocity gradients.

An augmented MFS approach for brain activity reconstruction

Weak electrical currents in the brain flow as a consequence of acquisition, processing and transmission of information by neurons, giving rise to electric and magnetic fields, which can be modeled by the quasi-stationary approximation of Maxwell’s equations. Electroencephalography (EEG) and magnetoencephalography (MEG) techniques allow for reconstructing the cerebral electrical currents and thus investigating the neuronal activity in the human brain in a non-invasive way. This is a typical electromagnetic inverse problem which can be addressed in two stages. In the first one a physical and geometrical representation of the head is used to find the relation between a given source model and the electromagnetic fields generated by the sources. Then the inverse problem is solved: the sources of measured electric scalar potentials or magnetic fields are estimated by using the forward solution. Thus, an accurate and efficient solution of the forward problem is an essential prerequisite for the solution of the inverse one. The authors have proposed the method of fundamental solutions (MFS) as an accurate, efficient, meshfree, boundary-type and easy-to-implement alternative to traditional mesh-based methods, such as the boundary element method and the finite element method, for computing the solution of the M/EEG forward problem. In this paper, further investigations about the accuracy of the MFS approximation are reported. In particular, the open question of how to efficiently design a good solution basis is approached with an algorithm inspired by the Leave-One-Out Cross Validation (LOOCV) strategy. Numerical results are presented with the aim of validating the augmented MFS with the state-of-the-art BEM approach. Promising results have been obtained.

Numerical studies to propose a ghost particle removed SPH (GR-SPH) method

This paper aims to propose a new modified SPH method with novel treatments at the boundaries. Although SPH methods decrease contradictions due to grid distortions compared to traditional mesh-based methods, the penetration of particles through boundaries and the consistency problem make the simulation of problems with definite boundaries a concern. The use of ghost boundary particles and the insertion of artificial forces at the boundaries are the most popular boundary consistency treatments proposed thus far. The use of artificial forces causes the mixing of molecular and finite theories, which can violate the conservation of momentum. This paper shows how the use of ghost boundary particles can violate the continuity equation in problems with non-zero velocity divergence. This study proposes a novel ghost particle removed SPH (GR-SPH) method that discards all ghost particles and artificial forces at the boundaries. Liner layers and liner particles have been defined inside the domain instead of ghost boundary particles in such a way that the so-called violations can partially be remedied. Based on the continuity equation and kernel function unity specification, a novel truncation correction factor has been defined for density renormalization to override the consistency problem at the boundaries. In addition, a new method is proposed to detect the particles near complex wall boundaries and evaluate the normal distance from boundaries. Finally, some benchmark problems have been solved to show the capabilities of the new modified SPH method for the prediction of both particle location and pressure distribution with acceptable accuracy. The GR-SPH method facilitates programming, with fewer particles contributing to the computations. Comparison of its outcomes with published results shows that the new treatments executed at the boundaries are effective.

Simulation of one dimension shock initiation of condensed explosive by SPH method

Purpose This paper aims to study the ability of SPH method in simulating shock initiation process. The initiation and subsequent explosion processes of condensed explosive involve high pressure propagation and material large deformation, which increase the simulation difficulty in using traditional mesh-based method. The study aims to take the SPH method as an alternative method to shock initiation simulation. Design/methodology/approach The SPH method combined with some correct aspects is applied to simulate the shock initiation process. The condensed explosive is ignited by the impact of high speed flyer. In order to avoid the non-physical penetration between particles of high velocity flyer and condensed explosive, a particle-to-particle contact algorithm is employed. After the ignition, the detonation process of condensed explosive is represented by the ignition and growth model. A modified SPH method based on Riemann-solver is applied to smooth the numerical oscillation at shock front. Two numerical examples are implemented to illustrate the capability of SPH method in shock initiation simulation. One is the interface velocity experiment of PBX-9501. Another is the plate push experiment of PBX-9502. Both of the examples include the shock initiation process of condensed explosive. Findings Numerical results show that the shock initiation process of condensed explosive can be well predicted by SPH method. The characteristics of detonation are captured in the simulation. The measured data in numerical examples are also in good agreement with the experimental data. Research limitations/implications Because of the research purpose is to study the ability of SPH for shock initiation simulation, only one-dimension numerical examples are discussed in the paper. Therefore, researchers are encouraged to extend and test the proposed method to two or three dimension shock initiation problems simulation. Originality/value This paper provides an alternative method for shock initiation simulation. The implemented method can overcome the weaknesses of traditional mesh based method in simulation of shock initiation problems.

Adaptive Radial Basis Function Methods for Pricing Options Under Jump-Diffusion Models

The aim of this paper is to show that option prices in jump-diffusion models can be computed using meshless methods based on radial basis function (RBF) interpolation instead of traditional mesh-based methods like finite differences or finite elements. The RBF technique is demonstrated by solving the partial integro-differential equation for American and European options on non-dividend-paying stocks in the Merton jump-diffusion model, using the inverse multiquadric radial basis function. The method can in principle be extended to Levy-models. Moreover, an adaptive method is proposed to tackle the accuracy problem caused by a singularity in the initial condition so that the accuracy in option pricing in particular for small time to maturity can be improved.

Three-dimensional modelling of coupled flow dynamics, heat transfer and residual stress generation in arc welding processes using the mesh-free SPH method

In this paper, a novel mesh-free approach is applied for modelling thermo-mechanical responses in a three-dimensional arc welding configuration using the Smoothed Particle Hydrodynamics (SPH) method. A fully coupled three-dimensional elastoplastic and heat transfer analysis is used to study the flow pattern of the filler material, and the resulting plastic strain development and temperature distributions for a simple arc welding configuration. The mesh-less and Lagrangian nature of SPH enables modelling of problems with large deformation and discontinuities, avoiding several disadvantages of the traditional mesh-based methods (e.g., FEM, FDM and FVM), and it is also able to implement coupled physics and complex constitutive behaviours due to the history tracking ability of the method.The plastic deformation and temperature distribution of the metal in the weld pool and the surrounding parent material are analysed during the cooling stage using SPH, and the resulting residual stresses are evaluated. This work establishes the capability of SPH as a three-dimensional modelling tool for gaining insights into the key physical processes of material deposition and its subsequent evolution during welding processes. Arc welding generates a non-uniform plastic strain distribution. The welding speed is found to be a crucial factor in controlling the plastic strain distribution and the quality of the welded joint, and there is a critical welding speed which produces the most uniform plastic strain distribution in the weld pool.The SPH method is able to predict the long term thermo-mechanical responses, namely heat transfer and residual stresses in the welded joint during the cooling stage. The SPH solutions demonstrate non-uniform cooling rates and temperature fields in a workpiece, as exhibited in practical welding processes. The solutions also show the existence of high temperature gradients around the boundary of the weld, which causes rapid microstructural and phase transformations. The SPH method also incorporated the evaluation of thermal residual stresses in the workpiece. In the early stages of welding, the residual stress varies rapidly with time and subsequently changes gradually. The residual stress variation over the welded joint depends on the local temperature gradients. The residual stress is very high at the interface between the filler and parent material and decreases rapidly further away from it. The typical thermo-mechanical behaviour found in the present study conforms to the observations of real life arc welding processes.

Modeling of complex dynamic systems using differential neural networks with the incorporation of a priori knowledge

In this paper, neural algorithms, including the multi-layered perceptron (MLP) differential approximator, generalized hybrid power series, discrete Hopfield neural network, and the hybrid numerical, are used for constructing models that incorporate a priori knowledge in the form of differential equations for dynamic engineering processes. The properties of these approaches are discussed and compared to each other in terms of efficiency and accuracy. The presented algorithms have a number of advantages over other traditional mesh-based methods such as reduction of the computational cost, speed up of the execution time, and data integration with the a priori knowledge. Furthermore, the presented techniques are applicable when the differential equations governing a system or dynamic engineering process are not fully understood. The proposed algorithms learn to compute the unknown or free parameters of the equation from observations of the process behavior, hence a more precise theoretical description of the process is obtained. Additionally, there will be no need to solve the differential equation each time the free parameters change. The parallel nature of the approaches outlined in this paper make them attractive for parallel implementation in dynamic engineering processes.

Numerical analysis of tsunami–structure interaction using a modified MPS method

Coastal structures are regarded as an effective defense for protecting the coastal area from tsunami disasters, considering that the tsunamis cannot be predicted. Therefore, much attention should be focused on tsunami–structure interaction (TSI). We must determine dynamic characteristics of the TSI such as wave height, free-surface elevation, dynamic wave pressure, and overtopping volume, all of which are essential to the design of coastal structures. The traditional mesh-based numerical method fails to accurately model TSI, because of large deformations that cause numerical diffusion. The moving particle simulation (MPS) is a pure Lagrangian mesh-less method, which can track free-surfaces with large deformations. The original MPS suffers from serious pressure fluctuations that affect the accuracy of the simulation. We modified three aspects of the original MPS: the kernel function, the source term, and the search of free-surface particles. We verified that these changes improved the pressure stability using two benchmark problems. Then, we applied the modified MPS to simulate the TSI, using a solitary wave to model the tsunami. We quantitatively and qualitatively compared our numerical results with the experimental data. The numerical results were consistent with the experimental data, which indicates that the modified MPS can capture the essential dynamic characteristics of the TSI and reproduce the entire interaction between the tsunami and structure.

Investigation on high-velocity impact of micron particles using material point method

Continuous high-velocity impact of micro space debris and micro-meteroids may cause significant accumulative damages to spacecrafts. The process of high-velocity impact of micron aluminum particles on the aluminum target is investigated with the material point method (MPM). As a meshfree particle method, MPM is very suitable for solving high-velocity impact problems owing to its prominent advantages of dealing with fracture, fragmentation and moving material interface over the traditional mesh-based methods. The target plate is modeled as semi-infinite media since its thickness is much larger than the characteristic length of the projectile particles. The micron particles are projected to the target individually and in group with different angles and different velocities. The predicted impact responses and dimensions of the craters agree well with the experimental results and the empirical equations. The influences of the flux density, the projectile angle and the impact velocity are thoroughly investigated, and the morphology modes of the crater group are concluded. Finally, an empirical formula is proposed for the crater depth under impact of particle group.

Numerical solution of the two-phase incompressible Navier–Stokes equations using a GPU-accelerated meshless method

Abstract This paper presents the development and implementation of a Meshless two-phase incompressible fluid flow solver and its acceleration using the graphics processing unit (GPU). The solver is formulated as a Localized Radial-Basis Function Collocation Meshless Method and the interface of the two-phase flow is captured using an implementation of the Level-Set method. The Compute Unified Device Architecture (CUDA) language for general-purpose computing on the GPU is used to accelerate the solver. Through the combined use of the LRC Meshless method and GPU acceleration this paper seeks to address the issue of robustness and speed in computational fluid dynamics. Traditional mesh-based methods require extensive and time-consuming user input for the generation and verification of a computational mesh. The LRC meshless method seeks to mitigate this issue through the use of a set of scattered points that need not meet stringent geometric requirements like those required by finite-volume and finite-element methods, such as connectivity and poligonalization. The method is shown to render very accurate and stable solutions and the implementation of the solver on the GPU is shown to accelerate the solution by several orders.

Image-Based Procedure for Biostructure Modeling

For high-resolution medical images, an image-based procedure is developed in strong form to perform microstructure analysis. Consider heterogeneous biomaterials such as bone tissue with porous composition—the associated microscopic cell problems and homogenized mechanical properties have been derived through the asymptotic homogenization to correlate the hierarchy in the macroscale and microscale. Nevertheless, for bioimages with highly irregular geometry, the process of model reconstruction by the traditional mesh-based methods unavoidably encounters issues such as mesh dependency and mesh distortion. Upon using the level set technique for model reconstruction, images of biological tissue showing complex topology can be identified and segmented into different phases effectively, such as the solid skeleton and pores in bone materials. In particular, the employment of the strong form collocation method takes advantage of point discretization and constitutes a seamlessly computational framework for solving level set equations and microscopic cell problems. The application to microstructure modeling of trabecular bone was demonstrated. The extension of the image-based biomaterial modeling includes prediction of bone fracture, bone remodeling process, and design of bone-implant systems.