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Abstract
A gradient smoothing method (GSM) based on strong form of governing equations for solid mechanics problems is proposed in this paper, in which gradient smoothing technique is used successively over the relevant gradient smoothing domains to develop the first- and second-order derivative approximations by calculating weights for a set of field nodes surrounding a node of interest. The GSM is found very stable and can be easily applied to arbitrarily irregular triangular meshes for complex geometry. Unlike other strong form methods, the present method has excellent stability that is crucial for adaptive analysis. An effective and robust residual based error indicator and simple refinement procedure using Delaunay diagram are then implemented in our GSM for adaptive analyses. The reliability and performance of the proposed GSM for adaptive procedure are demonstrated in several solid mechanics problems including problems with singularities and concentrated loading, compared with the well-known finite element method (FEM).
... The studies have also been conducted to improve efficiency by combining the meshfree methods with traditional mesh-based methods (Liu, 2000;Hao and Belytschko, 2004;Bourantas et al., 2018;Chen, 2003). Among them, the meshfree collocation method is considered a true meshfree method because it does not require even the background cells needed for integration in the weak form method (Zhang et al., 2008). In addition, the advantage of the meshfree collocation method is that Dirichlet boundary condition, which requires special treatment due to the non-interpolatory property of meshfree shape function in the Galerkin meshfree method, can be directly applied. ...
... In the conventional finite difference or generalized finite difference methods, the node placement must be regular for the derivative approximation of the field function, but GSM can be easily applied to arbitrary irregular meshes for complex geometry. In GSM, since there is no overlap between the smoothing domains, the sum of smoothing domain areas corresponding to all nodes is equal to the area of the problem domain (Zhang et al., 2008;Liu et al., 2008). The smoothing domain is created by connecting the centroids of elements around the node of interest and the midpoints of the lines connecting the node of interest and the surrounding nodes. ...
... As in GSM, the weighted Shepard function was used as the smoothing function in LGSM (Zhang et al., 2008). ...
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.
... Liu et al. [11] also presented error estimation for Moving Least-square reproducing kernel (MLSRK) methods in which dilation parameter of window function (instead of mesh size) explicitly represented interpolation error with two other independent constant parameters. Liu and coworkers developed adaptive methods for the least-squares radial point collocation method [12][13], the point interpolation method [14][15], the gradient smoothing method [16] and the smoothed finite element method [17][18]. Several different error estimators were used, including the residual based error estimator, the strain energy error estimator and the error estimator based on interpolation error. ...
... The MLS shape function in Eq. (8) is employed as trial function in the weak form of Eqs (15)(16)(17), and the final discrete equations can be obtained. A detailed description of the procedure can be found in [4]. ...
... The relative error is defined as [16] ...
Meshless methods are suitable for adaptive analysis, as the nodes are unstructured, and can be added or deleted freely. However, the smooth shape functions may produce spurious oscillation away from the region containing error, which may result in addition of unnecessary nodes. In order to avoid the influence of spurious oscillation on adaptive analysis, a node-based error estimator is presented. The recovered nodal stress value is obtained from a reference solution using a double refinement technique. Numerical tests are presented illustrating the effectiveness of the proposed approach in the terms of the number and distribution of nodes compared with traditional approaches.
... Recently, the detailed investigations have been conducted on various types of meshfree method and their properties [56,57]. Compared to the weak-form method, the meshless collocation method is very simple and is considered a true meshless method because it doesn't even need a background cell for numerical integration [58]. On the other hand, the Galerkin meshfree method requires special treatment to apply Dirichlet boundary condition due to the non-interpolatory property of the meshfree shape function, but in the meshless collocation method it can be applied directly without any other treatment. ...
... In GSM, a welldesigned smoothing domain is generated around the node of interest, and the derivative of the field function at that node is approximated by the integration of the field function along the smoothing domain boundary. At this time, the sum of the areas of the smoothing domains for all nodes is equal to the area of the entire problem domain [58,64]. However, a fixed smoothing mesh domain which is time-consuming to create in the problem domain is required in GSM. ...
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.
... Local error estimation in the solution and refinement of nodes are two steps in every adaptation process. Several posteriori error estimators for mesh free methods can be found in the literature (Liszka et al., 1996;Zhang et al. 2008). One of the error estimations is based on obtaining a residual error which depends on the governing equations of the problem. ...
In this paper, a novel spatially adaptive Mesh free approach using local nodes and radial basis functions,
based on weak-form formulation is presented to solve the even-parity neutron transport equation. A
residual based error indicator is proposed and used in an adaptive scheme for error estimation. The local
balance of neutrons is used as an indicator to determine the appropriate regions of problem for node
refinement. In this regard our previous computer program, MFreent, is upgraded for solving neutron
transport equation using K+ variational principle in an adaptive scheme for two-dimensional X � Y geometry. The multi-quadrics radial basis function is used to construct the radial point interpolation shape
functions for spatial approximation of even-parity angular flux. Also, the angular part of flux is implemented via spherical harmonics expansion. Some numerical test cases are investigated to evaluate the
performance of the locally adaptive node-refinement approach for both fixed source and eigenvalue
problems solution. The obtained results confirm the solution accuracy and efficiency of the proposed
method.
... The resulting linear system of equations in terms of displacement are solved 10 at discrete source points, data which can then be used to approximate the solution at any point in the problem domain. Compared to weak form-based meshless methods, collocation methods remove the need for integration, and an associated mesh of integration cells, making implementation simple and straightforward [7]. However, greater complexity arises in strong form methods in the approximation of higher order derivatives than in the weak form case, which can lead to time-consuming calculation of 15 basis functions and their associated derivatives. ...
Point collocation methods are strong form approaches that can be applied to continuum mechanics problems and possess attractive features over weak form-based methods due to the absence of a mesh. While various adaptive strategies have been proposed to improve the accuracy of weak form-based methods, such techniques have received little attention for strong form-based methods. In this paper, combined rh-adaptivity, in which r-and h-adaptivities are adopted iteratively, is applied to the local maximum entropy point collocation method for the first time to solve linear elasticity problems. Material force residuals act as driving forces in r-adaptivity to relocate collocation points, reducing the error associated with a given point distribution. Physical equilibrium residuals are used as the error estimator in h-adaptivity to determine the insertion locations for new points, diminishing the error caused by inadequate degrees of freedom. Issues arising in mesh-based methods, such as mesh distortion and hanging nodes, are entirely absent from the proposed method. The paper introduces the approach for the first time and the study is therefore confined to 2D domains. Numerical examples are presented to demonstrate the performance of the proposed adaptive strategies, comparing convergence rates and computational costs using uniform refinement, pure r-, hand combined rh-adaptivities.
... Recently, Liu et al. proposed a generalized gradient smoothing technique (GST) [26][27][28] and the so-called weakened weak (W2) form [29][30][31] , which forms the theoretical foundations of the smoothed finite element method (SFEM) [32][33][34][35][36][37][38][39] and the smoothed radial point interpolation method (SRPIM) [40][41][42] . Among the families of SFEM, the edgebased smoothed finite element method (ES-FEM) is a "star " element due to its superior properties, such as (1) higher computational efficiency; ...
This paper presents a hybrid plane wave expansion/edge-based smoothed finite element method (PWE/ES-FEM) for band structures simulation of semi-infinite beam-like phononic crystals (PCs). The field variables are first approximated using linear shape functions in conjunction with the spatial Fourier series. Within the further formed edge-based smoothing domains, the gradient smoothing technique (GST) is employed to perform the strain smoothing operation. Based on the smoothed Galerkin weakform, the discretized system equations associated with the eigenvalue problem are finally obtained. Numerical examples demonstrate that the present method possesses higher computational accuracy in band structures simulation of semi-infinite beam-like PCs.
... GSM gradient operator is firstly often employed in stabilizing the nodal integrations for methods based on Galerkin weak forms [36], [37]. It is then used individually, named Gradient smoothing method (GSM), to handle the fluid dynamics problems [35], [38]- [41] and solid mechanics problems [42]- [44] governed by strong-form (differential form) N-S equation. Compare to the numerical methods based on weak-form equations, the GSM is much simple, straightforward and easy to implement. ...
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.
... The GSM gradient operator is an accurate and adaptive gradient approximation technique which works very well in the cases with a highly irregular unstructured grid, and it has been shown equivalent to the central difference scheme in FDM when particles distribute uniformly [27]. GSM has been used in handling the fluid dynamics problems [28], [29], solid mechanics problems [30], [31], and fluid-structure interaction problems [32]. ...
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.
... These numerical methods can be classified into two fundamental frames: the grid-based methods, e.g., the finite difference method (FDM), finite element method (FEM) [1], finite volume method (FVM), gradient smoothing method (GSM) [2]- [5], and recently the meshfree methods [6], e.g., the smoothed particle hydrodynamics (SPH) method [7], the particle in cell (PIC) method [11]- [13], and the material point method (MPM) ( [8]- [10]) which is a successor of the particle in cell method (PIC) and has been intensely developed most recently (e.g., [14]- [17]). Although, compared to the grids methods, some of the meshfree methods (SPH for example) may not perform well in accuracy and stability aspects and more tedious in handling boundary conditions, they will not suffer from issues related to meshing, which is critical for complex geometry. ...
A 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.
... Physically speaking, the parameter √ β is the speed of artificial pressure wave [52], but it can be tuned to speed up the convergence rate of the overall iterative solution procedures. Based on numerical experiments, the value of β is defined as [55] with β 2 min = 1 and C β = 2.5 for non-dimensionalized equations. ...
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.
... Enrichment technique was carried out by adding a node in the center of the Delaunay cell if the refinement criterion was met [18]. Gradient smoothing method (GSM) was also used for adaptive analysis of the elliptic partial differential equations [19]. Majority of the refinement techniques presented in the literature are based on node enrichment techniques. ...
In this paper, an adaptive refinement strategy based on a node-moving technique is proposed and used for the efficient solution of the steady-state incompressible Navier–Stokes equations. The value of a least squares functional of the residual of the governing differential equation and its boundary conditions at nodal points is regarded as a measure of error and used to predict the areas of poor solutions. A node-moving technique is then used to move the nodal points to the zones of higher numerical errors. The problem is then resolved on the refined distribution of nodes for higher accuracy. A spring analogy is used for the node-moving methodology in which nodal points are connected to their neighbors by virtual springs. The stiffness of each spring is assumed to be proportional to the errors of its two end points and its initial length. The new positions of the nodal points are found such that the spring system attains its equilibrium state. Some numerical examples are used to illustrate the ability of the proposed scheme for the adaptive solution of the steady-state incompressible Navier–Stokes equations. The results demonstrate a considerable improvement of the results with a reasonable computational effort by using the proposed adaptive strategy. Copyright © 2011 John Wiley & Sons, Ltd.
... The S-FEM is known to work well for solid mechanics problems. Though via a strong formulation, the S-FEM gets simplified with the implementation of the gradient smoothing technique (GST) in strong form, for details see [91,92]. Mathematically, the GST is similar to FVM, but uses gradient smoothing operations exclusively in nested fashions (e.g. ...
This paper presents a review on the key research areas in the design and analysis of ships and floating structures. The major areas of computer application are identified in several stages of ship/floating structure design and analysis with the principal emphasis on the methodologies, the modeling, and the integration of the design and analysis process. The discussion addresses some of the key challenges in computer applications for ship and floating structure design and analysis, and reports on the emerging trends in the research, design and industrial application.
... A number of meshless methods have been proposed [10,11]. Smoothed particle hydrodynamics method (SPH) [12], elementfree Galerkin method (EFGM) [13], reproducing kernel particle method (RKPM) [14], finite point method (FPM) [15], hybrid boundary node method (HBNM) [16], boundary knot method (BKM) [17], meshless local Petrov-Galerkin method (MLPGM) [18], general finite difference method (GFDM) [19] and gradient smoothing method (GSM) [20][21][22] are the well known meshless methods considered for fluid flow problems. In the past two decades, a new group of meshless method based on the radial basis function (RBF) has drawn much of the attention of many researchers in science and engineering. ...
An improved localized radial basis function collocation method is developed for computational aeroacoustics, which is based on an improved localized RBF expansion using Hardy multiquadrics for the desired unknowns. The method approximates the spatial derivatives by RBF interpolation using a small set of nodes in the neighborhood of any data center. This approach yields the generation of a small interpolation matrix for each data center and hence advancing solutions in time will be of comparatively lower cost. An upwind implementation is further introduced to contain the hyperbolic property of the governing equations by using flux vector splitting method. The 4–6 low dispersion and low dissipation Runge–Kutta optimized scheme is used for temporal integration. Corresponding boundary conditions are enforced exactly at a discrete set of boundary nodes. The performances of the present method are demonstrated through their application to a variety of benchmark problems and are compared with the exact solutions.
... You et al. [28] have utilized the reproducing kernel as a low-pass filter, and the corresponding high-pass filter is used to identify the locations of high gradient and serves as an operator for error indication. The residual-based error estimations have recently been applied to the PIMs by Kee et al. [8] and Zhang et al. [30] and the gradient smoothing method by Zhang et al. [32]. ...
This paper develops a smoothing domain-based energy (SDE) error indicator and an efficient adaptive procedure using edge-based smoothed point interpolation methods (ES-PIM), in which the strain field is constructed via the generalized smoothing operation over smoothing domains associated with edges of three-node triangular background cells. Because the ES-PIM can produce a close-to-exact stiffness and achieve ‘super-convergence’ and ‘ultra-accurate’ solutions, it is an ideal candidate for adaptive analysis. A SDE error indicator is first devised to make use of the features of the ES-PIM. A local refinement technique based on the Delaunay algorithm is then implemented to achieve high efficiency. The refinement of nodal neighbourhood is accomplished simply by adjusting a scaling factor assigned to control local nodal density. Intensive numerical studies, including the problems with stress concentration and solution singularity, demonstrate that the proposed adaptive procedure is effective and efficient in producing solutions with desired accuracy.
This paper proposes an element decomposition method (EDM) for elastic-static, free vibration and forced vibration analyses of three-dimensional solid mechanics. The problem domain is first discretized using eight-node hexahedral elements. Then, each hexahedron is further subdivided into a set of sub-tetrahedral cells, and the local strains in each sub-tetrahedron are obtained using linear interpolation functions. For each hexahedron, the strain of the whole element is the weighted average value of the local strains, which means only one integration point is adopted to establish the stiffness matrix. To cure the numerical instability of one-point quadrature and improve the accuracy, a variation gradient item is complemented by variance of the local strains. Numerical examples, including both benchmark and practical engineering cases, demonstrate that the present method possesses the following interesting properties compared with the traditional finite element method using the same mesh discretization (1) super accuracy and faster convergence rate; (2) higher computational efficiency; (3) more immune to mesh distortion.
In this paper, an improved node moving technique was developed for adaptive solution of some flow problems. Collocated Discrete Least Squares Meshless (CDLSM) method was applied for simulation. This method is a truly meshless method and also enjoys the symmetric and positive-definite property. Then, an improved node moving technique was developed based on the spring analogy. In this mechanism, each node was assumed to be connected to its neighboring nodes via some virtual springs. Then, higher values of stiffness were allocated to the springs located at higher error areas. In the previously published works, the corresponding system of equations was assumed to be linear. In this work, first, it was shown that this assumption may lead to inaccurate results in some cases and then, an improved method was proposed considering the nonlinear nature of the system of equations. Some numerical examples were used to illustrate the ability of the proposed node moving technique for some simple examples and benchmark problems in the Computational Fluid Dynamics (CFD) context. The Results of the study showed a considerable improvement in comparison with those of previously published works.
In this paper, the Cell-based smoothed finite element method (CS-FEM) and the asymptotic homogenization method (AHM) are used to accurately simulate the responses of 1–3 type magneto-electro-elastic (MEE) structure under dynamic load. The dynamic characteristics of micromechanics are discussed. Firstly, the properties of MEE materials at different composition ratios are calculated by AHM. In the next step, the CS-FEM equations for the multi-physics problems are deduced based on the MEE constitutive equations and the effective parameters. The transient reactions of the problem are solved by introducing the modified Wilson-θ approach. To illustrate the precision, reliability and convergence of CS-FEM, four numerical examples of various thin-walled structures are designed, results of CS-FEM are compared with those of the traditional finite element method (FEM) with denser elements. It shows that CS-FEM has higher accuracy, faster convergence speed and higher computational efficiency. This paper has some application value for the design and development of thin-walled intelligent sensors and energy harvesters, etc.
Thanks to its meshfree feature, the smoothed particle hydrodynamics (SPH) has apparent advantages over the traditional finite element method for addressing large deformation problems in geotechnical engineering, e.g., landslides. This chapter provides the essentials of SPH theory and the key ingredients of SPH modeling. Especially, the stability-related issues are investigated and discussed by carrying out some benchmarking examples. Then, the effectiveness of SPH modeling is validated by applying it to a series of granular column collapse and earthquake-induced landslides. Finally, the most recent development of SPH is reviewed, and the Pros/Cons of SPH modeling relative to the conventional grid-based methods and other meshfree methods are explored in detail. Some latest advice on the use of SPH in geotechnical engineering is also involved in this chapter.
In order to solve partial differential equations (PDEs) numerically, one first needs to approximate the field functions (such as the displacement functions), and then obtain the derivatives of the field functions (such as the strains), by directly differentiating the field functions. Using such direct-derivatives in formulating a numerical method is common and is used in the standard finite element method (FEM), but such models are often found to be “stiff”. In the weakened weak (W2) formulations, it is found that the use of properly re-constructed derivatives can be beneficial in ways because the model can become “softer”. This paper presents a novel “pick-out” theory and technique for re-constructing the derivatives (such as the strains) of functions defined in a local domain, using smoothing operations. The local domain can be a smoothing domain used in the smoothed finite element methods (S-FEMs), smoothed point interpolation methods (S-PIMs), and smoothed particle hydrodynamics (SPH). It is discovered that through the use of a set of linearly independent smoothing functions that are continuous in the local domain, one can simply pick out various orders of smoothed derivatives (at the center of a domain) from any given function that may discontinuous (strictly) inside the local domain. As long as the smoothing function is continuous in the smoothing domain, the picked out “smoothed derivatives” are equivalent (in a local integral sense) to the compatible direct-derivatives, which ensures the convergence of the smoothed model (such as the S-FEM) when the smoothing domains shrinking to zero. The pick-out technique can be used in strong, weak, local weak, weak-strong, or weakened weak formulations to create stable and convergent numerical models. It may offer a new window of opportunity to develop new effective numerical models using smoothed derivatives (strains) that are “softer” and can produce accurate solutions also in the derivatives (strains and stresses) of the field functions (displacements).
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.
This review paper presents a methodological study on possible and existing meshfree methods for solving the partial differential equations (PDEs) governing solid mechanics problems, based mainly on the research work in the past two decades at the authors group. We start with a discussion on the general steps in a meshfree method based on nodes, with the displacements as the primary variables. We then examine the major techniques used in each of these steps: (1) techniques for displacement function approximations using nodes, (2) approximation of the gradient of the displacements or strains based on nodes and a background T-cells that can be automatically generated and refined, and (3) formulation techniques for producing algebraic equations. The function approximation techniques include node-based interpolation methods, cell-based interpolation methods, function smoothing techniques, and moving least squares approximation techniques. The gradient approximation includes direct differentiation, gradient smoothing, and special strain construction. Formulation techniques include strong-form, weakform, local weakform, weak-strong-form, and weakened weakform (W2). In theory, a meshfree method can be developed using a combination of function approximation, gradient approximation, and formulation techniques, which can lead to matrix of a large number of possible methods. This review attempts to provide an overall methodological review, rather than a usual review of comparing different methods. We hope to show readers the differences between the forests, and just between the trees.
A simple and effective r-adaptive technique for unstructured grids based on the segment spring analogy method is proposed. The finite element method and a corresponding error estimate method using second derivatives are used for computation. The traditional segment spring analogy method is modified, based on an idea of controlling the equilibrium length of the fictitious springs, and used for mesh adjustment. The principle of making numerical errors distributed uniformly over all elements is applied. Three numerical examples involving the one-dimensional (1D) convection-diffusion equation, the two-dimensional (2D) linear parabolic equation and the 2D Euler equations are presented and the effectiveness of the proposed r-type grid adaptive technique is examined.
A procedure to locally refine and un-refine an unstructured computational grid of four-node quadrilaterals (in 2D) or of eight-node hexahedra (in 3D) is presented. The chosen refinement strategy generates only elements of the same type as their parents, but also produces so-called hanging nodes along nonconforming element-to-element interfaces. Continuity of the solution across such interfaces is enforced strongly by Lagrange multipliers. The element split and un-split algorithm is entirely integer-based. It relies only upon element connectivity and makes no use of nodal coordinates or other real-number quantities. The chosen data structure and the continuous tracking of the nature of each node facilitate the treatment of natural and essential boundary conditions in adaptivity. A generalization of the concept of neighbor elements allows transport calculations in adaptive fluid calculations. The proposed procedure is tested in structure and fluid wave propagation problems in explicit transient dynamics.
An efficient edge-based smoothed finite element method (ES-FEM) has been recently developed for solving solid mechanics problems. The ES-FEM uses triangular elements that can be generated easily for complicated domains. In this paper, the complexity study of the ES-FEM based on triangular elements is conducted in detail, which confirms the ES-FEM produces higher computational efficiency compared to the FEM. Therefore, the ES-FEM offers an excellent platform for adaptive analysis, and this paper presents an efficient adaptive procedure based on the ES-FEM. A smoothing domain based energy (SDE) error estimate is first devised making use of the features of the ES-FEM. The present error estimate differs from the conventional approaches and evaluates error based on smoothing domains used in the ES-FEM. A local refinement technique based on the Delaunay algorithm is then implemented to achieve high efficiency in the mesh refinement. In this refinement technique, each node is assigned a scaling factor to control the local nodal density, and refinement of the neighborhood of a node is accomplished simply by adjusting its scaling factor. Intensive numerical studies, including an actual engineering problem of an automobile part, show that the proposed adaptive procedure is effective and efficient in producing solutions of desired accuracy.
An adaptive numerical method called, the adaptive random differential quadrature (ARDQ) method is presented in this paper.
In the ARDQ method, the random differential quadrature (RDQ) method is coupled with a posteriori error estimator based on
relative error norm in the displacement field. An error recovery technique, based on the least square averaging over the local
interpolation domain, is proposed which improves the solution accuracy as the spacing, h → 0. In the adaptive refinement, a novel convex hull approach with the vectors cross product is proposed to ensure that the
newly created nodes are always within the computational domain. The ARDQ method numerical accuracy is successfully evaluated
by solving several 1D, 2D and irregular domain problems having locally high gradients. It is concluded from the convergence
values that the ARDQ method coupled with error recovery technique can be effectively used to solve the locally high gradient
initial and boundary value problems.
A novel gradient smoothing method (GSM) is proposed in this paper, in which a gradient smoothing together with a directional
derivative technique is adopted to develop the first- and second-order derivative approximations for a node of interest by
systematically computing weights for a set of field nodes surrounding. A simple collocation procedure is then applied to the
governing strong-from of system equations at each node scattered in the problem domain using the approximated derivatives.
In contrast with the conventional finite difference and generalized finite difference methods with topological restrictions,
the GSM can be easily applied to arbitrarily irregular meshes for complex geometry. Several numerical examples are presented
to demonstrate the computational accuracy and stability of the GSM for solid mechanics problems with regular and irregular
nodes. The GSM is examined in detail by comparison with other established numerical approaches such as the finite element
method, producing convincing results.
Presented modification of the FDM enables local condensation of the mesh and easy discretisation of the boundary conditions in the case of an arbitrary shape of the domain. As a result an essential reduction of the required number of nodal points may usually be achieved. Thus the FDM can be competitive in some fields to the finite element method.Several troubles arising from the use of an irregulr mesh have been solved, the most important being elimination of singular or ill-conditioned stars, and a successful way of automatic discretization of boundary conditions was proposed. As a result of this complete automatization of the FDM has been reached. Many problems connected with the theory of the method proposed and its computer implementation have been discussed.FIDAM code—a system of computer programs designed for the solution of two-dimensional, linear and nonlinear, elliptic problems and three-dimensional parabolic problems is presented.Problems to be solved should be layed down in a local formulation as a set of partial derivative equations of the second order with boundary conditions not exceeding the same order.Various particular problems of applied mechanics and physics such as: torsion of bars, plane elasticity problems, deflections of plates and membranes, fluid now and temperature distribution have been solved using the FIDAM procedures. For the solution of nonlinear problems various iterative methods have been adopted—with special attention payed to the self-correcting method where the Newton-Raphson and incremental procedures are particular cases.The present version of the method allows fully automatic calculations to be carried out as in advanced programs of the finite element method and may be preferred in non-linear, optimization and time-dependent problems.
A methodology to build discrete models of boundary-value problems (BVP) is presented. The method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. A version of moving least-square interpolation and the collocation method are used to discretize BVP equations, which results in a truly meshless method (i.e. without a background mesh of integration points). h- and p-adaptive strategies are tested and very good convergence of the method was observed. The improvements in the methodology, in particular the introduction of spectral degrees of freedom, result in a fast and accurate method, significantly more efficient than the Finite Element Method or Element Free Galerkin Method. Several practical applications of the method to solve various engineering problems are presented.
The discrete Laplacian operator is constructed using the meshfree point collocation method which will be called the strong meshfree Laplacian operator. To define the strong meshfree Laplacian operator, we use the generalized moving least square approximation(4,5,6), which can calculate the approximated derivatives of shape functions. Some types of the locally layered node distribution are defined in this paper and two specific domains are constructed on which we can distribute the locally layered nodes. On such types of nodes, the discrete maximum principle can be shown to hold through the representation formula for the strong meshfree Laplacian operator. The discrete maximum principle, together with the reproducing property of the meshfree approximations, results in discrete a priori estimate for the strong meshfree Laplacian operator on the nodal solution space. Furthermore, the a priori estimate we have obtained enables to prove the existence and the uniqueness of the numerical solution and plays a central role in achieving the convergence results for the Poisson problem with Dirichlet boundary condition on nodal solution space. The order of convergence of the nodal solutions can be raised up to $O(h^2)$ on the proposed type of nodes in specific domains. For the generally shaped domains immersed in the previous domains, we can obtain the first order convergence result of $O(h)$. We know that finite difference methods and finite element methods(1,3) have some sort of discrete maximum principle for elliptic partial differential equation. For meshfree strong form collocation methods, no one has established any theoretical results until this paper. This is an important aspect, and this is the first paper that deals with the theoretical foundation of meshfree collocation method. . REFERENCES
We prove a posteriori error estimates for a finite element method for small strain elastoplasticity based on Hencky’s and Prandtl-Reuss’ flow rules, and formulate corresponding adaptive methods. We also present the results of some numerical experiments.
Accurate stress computation is an essential step of the efficient Galerkin meshfree formulation with the stabilized conforming nodal integration (SCNI). In this work particular emphasis is placed on the stress computations of SCNI-based Galerkin meshfree methods. The requirement for variational consistency of the SCNI formulation is discussed. It is shown that the discrete SCNI formulation is consistent in the variational sense if a constant stress field based on the smoothing strain is employed in the nodal representative or smoothing domain. Subsequently three stress computation approaches, i.e., direct nodal stress evaluation (DNS), consistent nodal stress evaluation (CNS), and consistent centroid stress evaluation (CCS), are presented. It turns out that both CNS and CCS satisfy the condition of variational consistence whereas DNS does not. A comprehensive numerical comparison of nodal strain energy error reveals that the variational consistence does not necessarily lead to more accurate results, while the proposed CCS approach is uniformly confirmed to yield the most favorable results.
Presents introductory skills needed for prediction of heat transfer and fluid flow, using the numerical method based on physical considerations. The author begins by discussing physical phenomena and moves to the concept and practice of the numerical solution. The book concludes with special topics and possible applications of the method.
In this paper, the adaptive nodal generation procedure based on the estimated local and global error in the element-free Galerkin (EFG) method is proposed. To investigate the possibility of h-type adaptivity of EFG method, a simple nodal refinement scheme is used. By adding new node along the background cell that is used in numerical integration, both of the local and global errors can be controlled adaptively. These errors are estimated by calculating the difference between the values of the projected stresses and original EFG stresses. The ultimate goal of this study is to develop the reliable nodal generator based on the local and global errors that is estimated posteriori. To evaluate the performance of proposed adaptive procedure, the convergence behavior is investigated for several examples.
The sixth editions of these seminal books deliver the most up to date and comprehensive reference yet on the finite element method for all engineers and mathematicians. Renowned for their scope, range and authority, the new editions have been significantly developed in terms of both contents and scope. Each book is now complete in its own right and provides self-contained reference; used together they provide a formidable resource covering the theory and the application of the universally used FEM. Written by the leading professors in their fields, the three books cover the basis of the method, its application to solid mechanics and to fluid dynamics.* This is THE classic finite element method set, by two the subject's leading authors * FEM is a constantly developing subject, and any professional or student of engineering involved in understanding the computational modelling of physical systems will inevitably use the techniques in these books * Fully up-to-date; ideal for teaching and reference
In this paper, a stabilized radial point collocation method (RPCM) that uses locally supported nodes is proposed using least-squares stabilization technique. The focus of this work is to stabilize the solution of the RPCM in order to perform adaptive analysis. Good stiffness matrix properties such as symmetric and positive definite (SPD) are gained via a least-squares procedure, and yet the formulation procedure of the stabilized LS-RPCM is still kept simple and straightforward. Adaptive analysis study has then been successfully carried out using the stabilized LS-RPCM. Error indicator based on interpolation error is adopted in the adaptive scheme for nodal refinement. Thanks for the meshfree features of the RPCM, refinement process can be easily done by inserting additional nodes based on the Voronoi diagram, without worrying about nodal connectivity in the formulation of system equations. Good numerical performance has been shown in the numerical examples presented in this paper.
In this paper we present an error indicator for the Element Free Galerkin (EFG) method, whose evaluation is computationally so simple that it can be readily implemented in existing EFG codes. The error indicator works very well in all numerical examples for 2-D potential and elasticity problems that are presented here, for regular and irregular grid of nodes. Moreover, it was demonstrated that this method is very simple in terms of economy and gives a good performance. The results show the error in EFG approximation may be estimated via the error indicator described in this paper. The indicator allows the global energy norm error to be estimated and also gives a good evaluation of local errors. It can thus be combined with a full adaptive process of refinement or, more simply, provide guidance for cloud of points redesign.
A comprehensive treatment of fracture mechanics suitable as a graduate
text and as a reference for engineers and researchers is presented. The
general topics addressed include: fundamental concepts of linear elastic
and elastic-plastic fracture mechanics; dynamic and time-dependent
fracture mechanics; micromechanisms of fracture in metals and alloys;
fracture mechanisms in polymers, ceramics, and composites; applications
to fracture toughness testing of metals and nonmetals, to structures,
fatigue crack propagation, and computational fracture mechanics.
Reference materials usually found in fracture mechanics handbooks is
provided.
In many fields using empirical areal data there arises a need for interpolating from irregularly-spaced data to produce a continuous surface. These irregularly-spaced locations, hence referred to as “data points,” may have diverse meanings: in meterology, weather observation stations; in geography, surveyed locations; in city and regional planning, centers of data-collection zones; in biology, observation locations. It is assumed that a unique number (such as rainfall in meteorology, or altitude in geography) is associated with each data point. In order to display these data in some type of contour map or perspective view, to compare them with data for the same region based on other data points, or to analyze them for extremes, gradients, or other purposes, it is extremely useful, if not essential, to define a continuous function fitting the given values exactly. Interpolated values over a fine grid may then be evaluated. In using such a function it is assumed that the original data are without error, or that compensation for error will be made after interpolation.
A pseudo-spectral point collocation meshfree method is proposed. We apply a scheme of approximating derivatives based on the moving least-square reproducing kernel approximations. Using approximated derivatives, we propose a new point collocation method. Unlike other meshfree methods that require direct calculation of derivatives for shape functions, with the proposed scheme, approximated derivatives are obtained in the process of calculating the shape function itself without further cost. Moreover, the scheme does not require the regularity of the window function, which ensures the regularity of shape functions. In this paper, we show the reproducing property and the convergence of interpolation for approximated derivatives of shape functions. As numerical examples of the proposed scheme, Poisson and Stokes problems are considered in various situations including the case of randomly generated node sets. In short, the proposed scheme is efficient and accurate even for complicated geometry such as the flow past a cylinder. Copyright © 2003 John Wiley & Sons, Ltd.
The objective of this article is to give an overview of finite element methods that currently are used extensively in academia and industry. The method is described in general terms, the basic formulation is presented, and some issues regarding effective finite element procedures are summarized. Various applications are given briefly to illustrate the current use of the method. Finally, the article concludes with key challenges for the additional development of the method.
Keywords:
finite elements;
reliability and effectiveness;
solids;
structures;
statics and dynamics;
CFD;
CAE;
computer programs
Domain integration by Gauss quadrature in the Galerkin mesh-free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh-free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh-free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh-free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh-free method as presented in several numerical examples.
A reproducing kernel particle method with built-in multiresolution features in a very attractive meshfree method for numerical solution of partial di!erential equations. The design and implementation of a Galerkin-based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and e$cient implementation of boundary conditions. In this paper we present a point collocation method based on reproducing kernel approximations. We show that, in a point collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach. The point collocation method based on reproducing kernel approximations, however, requires the calculation of higher-order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point collocation method are derived. The point collocation method is shown to be accurate for several one and two-dimensional problems and the convergence rate of the point collocation method is addressed.
A posteriori error estimates and an adaptive refinement scheme of first-order least-squares meshfree method (LSMFM) are presented. The error indicators are readily computed from the residual. For an elliptic problem, the error indicators are further improved by applying the Aubin–Nitsche method. It is demonstrated, through numerical examples, that the error indicators coherently reflect the actual error. In the proposed refinement scheme, Voronoi cells are used for inserting new nodes at appropriate positions. Numerical examples show that the adaptive first-order LSMFM, which combines the proposed error indicators and nodal refinement scheme, is effectively applied to the localized problems such as the shock formation in fluid dynamics. Copyright © 2003 John Wiley & Sons, Ltd.
A radial point interpolation based finite difference method (RFDM) is proposed in this paper. In this novel method, radial point interpolation using local irregular nodes is used together with the conventional finite difference procedure to achieve both the adaptivity to irregular domain and the stability in the solution that is often encountered in the collocation methods. A least-square technique is adopted, which leads to a system matrix with good properties such as symmetry and positive definiteness. Several numerical examples are presented to demonstrate the accuracy and stability of the RFDM for problems with complex shapes and regular and extremely irregular nodes. The results are examined in detail in comparison with other numerical approaches such as the radial point collocation method that uses local nodes, conventional finite difference and finite element methods. Copyright © 2006 John Wiley & Sons, Ltd.
This paper presents a stabilized meshfree method formulated based on the strong formulation and local approximation using
radial basis functions (RBFs). The purpose of this paper is two folds. First, a regularization procedure is developed for
stabilizing the solution of the radial point collocation method (RPCM). Second, an adaptive scheme using the stabilized RPCM
and residual based error indicator is established. It has been shown in this paper that the features of the meshfree strong-form
method can facilitated an easier implementation of adaptive analysis. A new error indicator based on the residual is devised
and used in this work. As shown in the numerical examples, the new error indicator can reflect the quality of the local approximation
and the global accuracy of the solution. A number of examples have been presented to demonstrate the effectiveness of the
present method for adaptive analysis.
In this paper, an adaptive refinement procedure using the element free Galerkin method (EFGM) for the solution of 2D linear elastostatic problems is suggested. Based on the numerical experiments done in Part I of the current study, in the proposed adaptive refinement scheme, the Zienkiewicz and Zhu (Z–Z) error estimator using the T-Belytschko (TB) stress recovery scheme is employed for the a posteriori error estimation of EFGM solution. By considering the a priori convergence rate of the EFGM solution and the estimated error norm, an adaptive refinement strategy for the determination of optimal node spacing is proposed. A simple point mesh generation scheme using pre-defined templates to generate new nodes inside the integration cells for adaptive refinement is also developed. The performance of the suggested refinement procedure is tested by using it to solve several benchmark problems. Numerical results obtained indicate that the suggested procedure can lead to the generation of nearly optimal meshes and the effects of singular points inside the problem domain are largely eliminated. The optimal convergence rate of the EFGM analysis is restored and the effectivity indices of the Z–Z error estimator are converging towards the ideal value of unity as the meshes are refined.
Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enable p-adaptivity to be achieved. Methods for constructing discontinuous approximations and approximations with discontinuous derivatives are also described. Next, several issues in implementation are reviewed: discretization (collocation and Galerkin), quadrature in Galerkin and fast ways of constructing consistent moving least-square approximations. The paper concludes with some sample calculations.
The basis of the finite point method (FPM) for the fully meshless solution of elasticity problems in structural mechanics is described. A stabilization technique based on a finite calculus procedure is used to improve the quality of the numerical solution. The efficiency and accuracy of the stabilized FPM in the meshless analysis of simple linear elastic structural problems is shown in some examples of applications.
Several computational and mathematical features of the h-p cloud method are demonstrated in this paper. We show how h, p and h-p adaptivity can be implemented in the h-p cloud method without traditional grid concepts typical of finite element methods. The mathematical derivation of an a posteriori error estimate for the h-p cloud method is also presented. Several numerical examples illustrate the main ideas of the method.
Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or error indicators. After that, a new spatial discretization must be defined via h-, p- or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are clearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics. Peer Reviewed Postprint (author’s final draft)
The purpose of this book is to provide the fundamentals of MFree methods in as much detail as possible. Some typical MFree methods, such as EFG, MLPG, RPIM, and LRPIM, are discussed in great detail. The detailed numerical implementations and programming for these methods are also provided. In addition, the MFree collocation (strong-form) methods are also detailed. Many well-tested computer source codes for MFree methods are provided. The application and the performance of the codes provided can be checked using the examples attached. Input and output files are provided in table form for easy verification of the codes. All computer codes are developed by the authors based on existing numerical techniques for FEM and the standard numerical analysis. These codes consist of most of the basic MFree techniques, and can be easily extended to other variations of more complex procedures of MFree methods.
Releasing this set of source codes is to suit the needs of readers for an easy comprehension, understanding, quick implementation, practical applications of the existing MFree methods, and further improvement and development of their own MFree methods. All source codes provided in this book are developed and tested based on the MS Windows and MS Developer Studio 97 (Visual FORTRAN Professional Edition 5.0.A) on a personal computer. After slight revisions, these programs can also be executed in other platforms and systems, such as the UNIX system on workstations. In our research group these codes are frequently ported between the Windows and UNIX systems, and there has been no technical problem.
A stabilized conforming nodal integration for Galerkin mesh-free methods A two dimensional interpolation function for irregularly spaced points
- J S Chen
- C T Wu
- S Yoon
- Y You
J.S. Chen, C.T. Wu, S. Yoon, Y. You, A stabilized conforming nodal integration for Galerkin mesh-free methods, Int. J. Numer. Methods Eng. 50 (2001) 435–466. [22] D. Shepard, A two dimensional interpolation function for irregularly spaced points, in: Proceedings of ACM National Conference, 1968, pp. 517–524.
Special issue on adaptive meshing, Part II, Finite elements in analysis and design 25
- S H Lo
S.H. Lo (Ed.), Special issue on adaptive meshing, Part II, Finite elements in analysis and design 25.
Goal-oriented error estimation in solid mechanics
- H Steeb
- A Maute
- E Ramm
H. Steeb, A. Maute, E. Ramm, Goal-oriented error estimation in solid mechanics,
in: E. Stein (Ed.), Error-controlled Adaptive Finite Elements in Solid Mechanics,
Wiley, UK, 2002.
Numerical methods for conservation law on structured and unstructured meshes, VKI 2003 Lecture Series
- T Barth
T. Barth, Numerical methods for conservation law on structured and
unstructured meshes, VKI 2003 Lecture Series, 2003.
A two dimensional interpolation function for irregularly spaced points
- D Shepard
D. Shepard, A two dimensional interpolation function for irregularly spaced
points, in: Proceedings of ACM National Conference, 1968, pp. 517-524.
Meshless methods: an overview and recent developments
- Belytschko
Adaptive finite element strategies based on error assessment
- Huerta
Goal-oriented error estimation in solid mechanics
- Steeb