Gradient Smoothing Method Research Articles

Reduced order model for simulation of air pollution model and application in 2D urban street canyons via the meshfree gradient smoothing method

Reduced order model for simulation of air pollution model and application in 2D urban street canyons via the meshfree gradient smoothing method

A gradient smoothing method and its multiscale variant for flows in heterogeneous porous media

In this paper, we study gradient smoothing methods (GSMs) with improved convergence behaviors for high-contrast problems such as the flow in heterogeneous porous media. We propose a GSM that is adaptive to the heterogeneity of the problem in the sense that gradient smoothing is applied inside each of the homogeneous regions separately. The smoothed gradient field of each element is constructed by utilizing the gradient fields of the adjacent elements within the corresponding homogeneous region. In addition, we propose a multiscale variant of the proposed GSM, which results in an accurate numerical solution even if a computational grid coarser than the scale of the heterogeneity is utilized. While existing GSMs have stiffness matrices with larger bandwidth than the standard finite element method (FEM) in general, the proposed multiscale GSM has the same bandwidth as the plain multiscale FEM. Hence, its online computational cost is essentially the same as that of the plain one. Improved performance of the proposed methods is demonstrated through various numerical examples.

A local gradient smoothing method for solving the free vibration model of functionally graded coupled structures

In this paper, a meshless method for analyzing the free vibration of four-parameter functionally graded composite shells is presented, which is based on moving least square (MLS) approximation and local gradient smoothing method (LGSM). Firstly, the displacement field model of the substructure is established by using the first order shear deformation theory (FSDT), and the governing equation of the substructure is obtained by using Hamilton principle. Starting from the MLS approximation the displacement field of substructure, the derivative of the shape function is approximated by the LGSM, and the vibration analysis models of each substructure are established respectively. Then, the vibration model of the whole coupling structure is established by coupling each substructure with the displacement coordination relation of the substructure, and the vibration model of the whole coupling structure is established. For the vibration problems considering different boundary conditions, the results obtained from the meshless vibration model are compared with the FEM results to verify the accuracy and applicability of the established analysis model. At the same time, in order to improve the accuracy of the solution, the convergence analysis is also carried out. Finally, in order to enrich the research content, the influence of geometric parameters and material parameters of the structure on the vibration characteristics of the whole structure is studied. The results show that this method can effectively analyze the free vibration of the coupled structure.

Vibration analysis of closed laminate conical, cylindrical shells and annular plates using meshfree method

A local gradient smoothing method(LGSM) in the framework of meshless theory is developed for implementing vibration analysis of closed laminate conical, cylindrical shells and annular plates. Based on the theoretical assumptions of the first-order laminated structure, the governing equations and boundary condition equations of the structure elements of revolution are derived and numerically discretized for the discrete vibration characteristic equations of the studied structures. Then, the meshless method is used to solve the vibration characteristic of the composite laminated shell, in which the derivatives of the displacement variables are approximated by employing LGSM. On this basis, the effects of the number of the involved discrete nodes and the dimensional coefficient related to the local smooth domain on the solution convergence are examined in detail. Meanwhile, for sake of checking the applicability of LGSM, several numerical results simulated by the presented method are compared with those from some published literature as well as FEM models. Moreover, some parametric studies with regard to the influence of structural boundary conditions, geometric dimensions, and fiber angle on the vibration behaviors of composite laminated shells are analyze. The results reveal that the developed LGSM meshless method has a good application potential in clarifying vibration phenomena of such laminated structure.

Angular particle impact on ductile materials using the Lagrangian gradient smoothing method

The impacts of single angular particles on ductile materials can result in large deformations and material chip separation, which cannot be effectively simulated using conventional mesh-based numerical methods. In this study, a novel particle-based mesh-free method called the Lagrangian gradient smoothing method (L-GSM) is first applied to simulate the dynamic process of single diamond-shaped particles impact on metallic surfaces. Based on the theory of L-GSM, a numerical model is established by incorporating the Johnson–Cook plasticity and failure models. Oxygen-free high-conductivity (OFHC) copper which is represented by a series of discrete particles, is selected as the material of the target block. Simulation results show that the L-GSM model can reproduce the typical erosion mechanisms such as cutting, micro-machining and ploughing. The large deformation of plastic craters and material removal are successfully simulated without using any additional techniques to handle the distorted domain. In addition, the computational accuracy and efficiency of the L-GSM are discussed by comparison with different mesh-free models. This study provides an effective numerical model with better accuracy and efficiency, helping to reveal the fundamental erosion mechanisms resulting from the impacts of angular particles.

A local gradient smoothing method for solving strong form governing equation

The gradient smoothing method(GSM) is used to approximate the derivatives of the meshfree shape function and it usually generates the smoothing domain by connecting the midpoints of sides and centroids of elements around the interested node. In order to simplify the generating process and geometric computation of smoothing domain, a local gradient smoothing method (LGSM) has been proposed in which the local smoothing domains corresponding to the nodes are independent of each other. The proposed method is very flexible to generate the smoothing domain, which can have a simple and regular shape. The derivatives of the meshfree shape function approximated by the proposed method was applied to the governing strong from of system equations at all nodes of problem domain. The convergence and accuracy of the proposed method according to the size of smoothing domain are examined. Numerical examples to some typical benchmark problems illustrated the efficiency of the proposed method.

Matrix-Free Time Domain Gradient Smoothing Method With Stretched-Coordinates Perfectly Matched Layer for Analysis of Photonic Devices

In this article, the gradient smoothing method is introduced for the first time to the time domain analysis of photonic devices. The proposed method combines the advantages of finite difference and finite element methods. It is capable of accurately modelling irregularly-shaped geometries by dividing the computational domain into elements like the finite element method. While it naturally produces a diagonal matrix for the field discretization similar to the finite difference method which makes it a perfect method for explicit time marching schemes. The beam propagation method with explicit time discretization is adopted for the propagation in time. Additionally, a novel stretched coordinate perfectly matched layer with two auxiliary equations is developed for the proposed method. The numerical aspects of the proposed method are demonstrated through the analysis of different photonic integrated structures.

Natural frequency calculation of open laminated conical and cylindrical shells by a meshless method

In this paper, a new and an efficient solution method based on local gradient smoothing method has been applied to free vibration problem of open composite laminated cylindrical and conical shells with elastic boundary conditions. The theoretical model is formulated by the first-order shear deformation theory, and the motion equation is obtained by the Hamilton’s principle. The motion equation is discretized by meshless shape function; in this process, the derivatives of the shape function are approximated by local gradient smoothing method. The accuracy, applicability and efficiency of this method are demonstrated for free vibrations of open composite laminated cylindrical and conical shells with different geometric, material parameters and boundary condition. The numerical results show good convergence characteristics and good agreement between the present method and the existing literature. And through several numerical examples, some useful results for free vibration results of open composite laminated cylindrical and conical shells are obtained, which may serve as a benchmark solutions for researchers to check their analytical and numerical methods.

A New TVD Scheme for Gradient Smoothing Method Using Unstructured Grids

An improved [Formula: see text]-factor algorithm for implementing total variation diminishing (TVD) scheme has been proposed for the gradient smoothing method (GSM) using unstructured meshes. Different from the methods using structured meshes, for the methods using unstructured meshes, generally the upwind point cannot be clearly defined. In the present algorithm, the value of upwind point has been successfully approximated for unstructured meshes by using the GSM with different gradient smoothing schemes, including node GSM (nGSM) midpoint GSM (mGSM) and centroid GSM (cGSM). The present method has been used to solve hyperbolic partial differential equation discontinuous problems, where three classical flux limiters (Superbee, Van leer and Minmod) were used. Numerical results indicate that the proposed algorithm based on mGSM and cGSM schemes can avoid the numerical oscillation and reduce the numerical diffusion effectively. Generally the scheme based on cGSM leads to the best performance among the three proposed schemes in terms of accuracy and monotonicity.

A 3D Lagrangian gradient smoothing method framework with an adaptable gradient smoothing domain‐constructing algorithm for simulating large deformation free surface flows

SummaryA novel smoothing particle hydrodynamics (SPH)‐like Lagrangian meshfree method, named as Lagrangian gradient smoothing method (L‐GSM), has been proposed to avoid the “tensile instability” issue in SPH simulation by replacing the SPH particle‐summation gradient approximation technique with a local grid‐based GSM gradient smoothing operator. The L‐GSM model has been proven effective and efficient when applied to a wide range of large deformation problems for fluids and flowing solids in two‐dimensional case. In this study, a three‐dimensional (3D) L‐GSM numerical framework is proposed for simulating large deformation problems with the existence of free surfaces through developing a widely adaptable 3D gradient smoothing domain (GSD) constructing algorithm. It includes three key novel ingredients: (i) the localized GSD based on an efficient distance‐oriented particle‐searching algorithm enabling both easy implementation and efficient computation; (ii) a novel algorithm for constructing 3D GSD to guarantee the effectiveness of the 3D GSM gradient operator adaptable to any extreme cases; (iii) a robust normalized 3D GSM gradient operator formulation that can restore the accuracy of gradient approximation even on boundary interface. The effectiveness of the proposed 3D GSD‐constructing algorithm is first verified under various distribution conditions of particles. The accuracy of the proposed adaptable 3D GSM gradient algorithm is then examined through conducting a series of numerical experiments with different spacing ratios. Finally, the 3D L‐GSM numerical framework is applied to solve a practical problem of free surface flows with large deformation: collapse of a soil column. The results reveal that the present adaptable 3D L‐GSM numerical framework can effectively handle the large deformation problems, like flowing solids, with a constantly changing arbitrary free surface profile.

A local Lagrangian gradient smoothing method for fluids and fluid-like solids: A novel particle-like method

The Lagrangian Gradient Smoothing Method (L-GSM) proposed in our earlier work overcomes effectively the ‘tensile instability’ problem inherently existed in the widely used smoothed particle hydrodynamics (SPH) method. However, the employment of background grid in L-GSM for the construction of gradient smoothing domains leads some drawbacks to the L-GSM framework: low computational efficiency, complex implementation procedure and technical challenges for parallel computing algorithm development. Accordingly, this study proposes a novel particle-like method, termed as local L-GSM (LL-GSM), through constructing gradient smoothing domains locally for LL-GSM particles. The present LL-GSM consists of three unique ingredients: (1) Only locally constructed gradient smoothing domains are used; (2) an efficient localized neighbor-searching algorithm is developed for the search of supporting particles; (3) a simple and effective free surface technique is adopted for accurate application of free surface effect. The accuracy, stability and efficiency of the newly proposed LL-GSM framework are then investigated comprehensively by conducting thorough theoretical and numerical analyses. At last, benchmark problems including two incompressible flows and two free surface flows are used to verify the capability of LL-GSM in handling large deformation of fluids and fluid-like solids. The LL-GSM results are evaluated carefully by comparing with experimental results, theoretical solutions, and numerical solutions. Results comparison demonstrates that the proposed LL-GSM method can give very accurate numerical solutions in all these problems with a much better computational efficiency and easier implementation. It is also evidenced that, same to L-GSM, the LL-GSM is free from the ‘tensile instability’ issue.

A conservative and consistent Lagrangian gradient smoothing method for simulating free surface flows in hydrodynamics

A novel particle method, Lagrangian gradient smoothing method (L-GSM), has been proposed in our earlier work to avoid the tensile instability problem inherently existed in SPH, through replacing the SPH gradient operator with a robust GSM gradient operator. However, the nominal area of each L-GSM particle determined by the relative location of particles is always inconsistent with the real representative area of it in simulation, especially in large-deformation problems. This is why the earlier L-GSM model has to be limited to the solid-like flow simulations where the deformation is not very serious. In this work, a conservative and consistent Lagrangian gradient smoothing method (CCL-GSM) is developed for handling large-deformation problems in hydrodynamics with an arbitrarily changing free surface profile. This is achieved by deriving a conservative and consistent form for the discretized Navier–Stokes governing equations in L-GSM, which even holds in the neighbor-updating or ‘remeshing’ process. Special techniques are also devised for free surface treatment, which is important to restore the conservation and consistency manner of CCL-GSM simulation on free surface boundary. The effectiveness of the proposed CCL-GSM framework is evaluated with a number of benchmarking examples, including dam break, wall impacts of breaking dam, water discharge and water splash. It shows that the CCL-GSM model can handle the incompressible flows with complicated free surfaces effectively and easily. The results comparison with experiments and SPH solutions demonstrates that the CCL-GSM can give a desirable result for all these examples.

A conservative and consistent Lagrangian gradient smoothing method for earthquake-induced landslide simulation

Numerical study of earthquake-induced landslides plays an important role in the disaster prevention and post-earthquake managements. Thanks to its mesh-free feature, the SPH (Smoothed Particle Hydrodynamics) method has significant advantages over the traditional grid-based methods in dealing with extremely large deformation problems such as high-speed fluid-like landslides. However, SPH is unstable when the material is in tension, which is widely known as the ‘tensile instability’ problem in SPH. Although the instability issue can be removed to certain stand by particular ad-hoc techniques, the root cause of the problem is not addressed. In addition, such ad-hoc techniques require additional user-defined parameters, which is difficult to make universally applicable. In this study, a conservative and consistent Lagrangian Gradient Smoothing Method (L-GSM) is proposed for fluid-like landslide simulations based on the recently developed meshfree L-GSM. In the present L-GSM, the strong-form of governing equations is discretized using the GSM gradient operator in a conservative and consistent manner. Moreover, a simple and accurate boundary treatment for free surface and solid boundary is newly proposed for L-GSM to handle the complicated free surface and solid boundary existed in the landslide problems. Finally, the proposed conservative and consistent L-GSM framework is validated by a soil column collapse benchmark and three real landslide problems. Results show that our L-GSM can give an accurate and reliable solution to the fluid-like landslide problem with a high efficiency. The treatments for free surface and solid boundary are also proven very effective in producing desired boundary conditions. Both cohesive soil column collapse example and Wangjiayan landslide example demonstrate that L-GSM is indeed free from the tensile instability problem.

Numerical simulation of flow past a bluff body of two different shapes using gradient smoothing method with unstructured grids

In this paper, flow across a bluff cylinder with circular and equilateral triangular sections shapes at low Reynolds number (10 ≤ Re ≤ 250) is numerically investigated using the gradient smoothed method (GSM). The method was originally developed based on unstructured grids, which could be generated easily for complicated domains. For solving the incompressible flow, artificial compressibility terms are introduced in Navier–Stokes equations. The spatial derivatives of convective and viscous fluxes are obtained using the gradient smoothing operation. And the time marching is implemented based on the dual time stepping technique. The situations of steady flows with Re = 10–40 and unsteady flows with Re = 50∼250 are simulated. Analysis of the drag coefficients, root mean square (rms) value of lift coefficients and Strouhal number of circular and triangular cylinders has been carried out. Compared with both experimental and numerical reference solutions in the literatures, the accuracy of GSM results has been demonstrated.

Numerical study of advection schemes for interface-capturing using gradient smoothing method

Based on the newly developed gradient smoothing method (GSM), three interface-capturing schemes have been implementing using unstructured mesh. The volume of fluid (VOF) model is solved without explicitly interface reconstructing in the framework of GSM. The variables on upwind points are successfully approximated using centroid GSM (cGSM) scheme, which generally cannot be clearly defined on unstructured mesh. Compressive interface-capturing scheme for arbitrary meshes (CICSAM), flux-blending interface-capturing scheme (FBICS), and cubic upwind interpolation-based blending scheme (CUIBS) have been employed through three benchmark cases. The results indicate that FBICS and CUIBS can produce more accurate predictions using unstructured meshes.

A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh

SummaryA novel Lagrangian gradient smoothing method (L‐GSM) is developed to solve “solid‐flow” (flow media with material strength) problems governed by Lagrangian form of Navier‐Stokes equations. It is a particle‐like method, similar to the smoothed particle hydrodynamics (SPH) method but without the so‐called tensile instability that exists in the SPH since its birth. The L‐GSM uses gradient smoothing technique to approximate the gradient of the field variables, based on the standard GSM that was found working well with Euler grids for general fluids. The Delaunay triangulation algorithm is adopted to update the connectivity of the particles, so that supporting neighboring particles can be determined for accurate gradient approximations. Special techniques are also devised for treatments of 3 types of boundaries: no‐slip solid boundary, free‐surface boundary, and periodical boundary. An advanced GSM operation for better consistency condition is then developed. Tensile stability condition of L‐GSM is investigated through the von Neumann stability analysis as well as numerical tests. The proposed L‐GSM is validated by using benchmarking examples of incompressible flows, including the Couette flow, Poiseuille flow, and 2D shear‐driven cavity. It is then applied to solve a practical problem of solid flows: the natural failure process of soil and the resultant soil flows. The numerical results are compared with theoretical solutions, experimental data, and other numerical results by SPH and FDM to evaluate further L‐GSM performance. It shows that the L‐GSM scheme can give a very accurate result for all these examples. Both the theoretical analysis and the numerical testing results demonstrate that the proposed L‐GSM approach restores first‐order accuracy unconditionally and does not suffer from the tensile instability. It is also shown that the L‐GSM is much more computational efficient compared with SPH, especially when a large number of particles are employed in simulation.

Development of Three-Dimensional GSM-CFD Solver for Compressible Flows

A three-dimensional (3D) Computational Fluid Dynamics (CFD) solver based on the gradient smoothing method (GSM) is developed for compressible flows based on previous research. The piecewise constant smoothing function with one-point integration scheme is implemented for gradient approximation of field variables and convective fluxes. The matrix-based method for gradient approximations is also developed to improve the numerical efficiency. Numerical examples of gradient approximations of several given functions have shown that the proposed GSM is more accurate and robust to mesh distortion. A transonic ONERA M6 wing is used to demonstrate the effectiveness of the proposed GSM-CFD solver.

A development of a GSM-CFD solver for non-Newtonian flows

Abstract An incompressible GSM-CFD solver based on the gradient smoothing method (GSM) is developed for solving non-Newtonian fluids in this paper. Node-based and edge-based smoothing domains are constructed to approximate the first- and second-order derivatives in the strong form Navier-Stokes equations. The non-Newtonian constitute law is implemented by using a generalized Newtonian fluid model with viscosity obeying the power-law. The widely used benchmark problem of flow past a circular cylinder is analyzed to validate the proposed solver by comparing it with numerical and experimental results available in the literature. The detailed comparison study on the results of wake length, separation angle and the drag coefficient demonstrates the accuracy and robustness of the present GSM-CFD solver. Our study also revealed the strong dependence of the flow behavior on the power index.

Coupling GSM/ALE with ES-FEM-T3 for fluid–deformable structure interactions

In light of the effectiveness of the edge-based smoothed finite element method (ES-FEM-T3) and arbitrary Lagrangian–Eulerian gradient smoothing method (GSM/ALE) in, respectively, solving the pure solid and fluid flow problems using three-node triangular elements, they are coupled together in the present study to solve the more challenging fluid–deformable structure interaction (FSI) problems based on the weak coupling algorithm. Specifically, the fluid flow is tracked over the moving mesh with the well developed GSM/ALE and the transient response of the solid part is solved by the newly developed explicit ES-FEM-T3. The solutions from these two parts are “linked” together by the carefully formulated FSI coupling conditions on the FSI interface. Detailed procedures are summarized to illustrate the implementations of the GSM/ALE with ES-FEM-T3 in an FSI analysis. Three benchmarks are employed to validate the proposed coupled smoothed method in solving both transient and steady-state FSI problems. The mesh sensitivity analysis is further carried out showing that the results of an FSI system appear more sensitive to the change in the solid mesh as compared to the fluid mesh, thus suggesting a more refined mesh for the solid part. Another significant finding is that the present method can still produce reliable results even on the extremely distorted mesh near the FSI interface. The successful coupling GSM/ALE with ES-FEM-T3 for solving FSI problems serves as a good start for further implementing the family of smoothed methods in solving more complex cross-area problems.

A matrix-form GSM–CFD solver for incompressible fluids and its application to hemodynamics

A GSM---CFD solver for incompressible flows is developed based on the gradient smoothing method (GSM). A matrix-form algorithm and corresponding data structure for GSM are devised to efficiently approximate the spatial gradients of field variables using the gradient smoothing operation. The calculated gradient values on various test fields show that the proposed GSM is capable of exactly reproducing linear field and of second order accuracy on all kinds of meshes. It is found that the GSM is much more robust to mesh deformation and therefore more suitable for problems with complicated geometries. Integrated with the artificial compressibility approach, the GSM is extended to solve the incompressible flows. As an example, the flow simulation of carotid bifurcation is carried out to show the effectiveness of the proposed GSM---CFD solver. The blood is modeled as incompressible Newtonian fluid and the vessel is treated as rigid wall in this paper.