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1.
    
A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi‐sphere in the 3‐D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2‐D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

2.
    
Combining a modified functional with the moving least‐squares (MLS) approximation, the hybrid boundary node method (Hybrid BNM) is a truly meshless, boundary‐only method. The method may have advantages from the meshless local boundary integral equation (MLBIE) method and also the boundary node method (BNM). In fact, the Hybrid BNN requires only the discrete nodes located on the surface of the domain. The Hybrid BNM has been applied to solve 2D potential problems. In this paper, the Hybrid BNM is extended to solve potential problems in three dimensions. Formulations of the Hybrid BNM for 3D potential problems and the MLS approximation on a generic surface are developed. A general computer code of the Hybrid BNM is implemented in C++. The main drawback of the ‘boundary layer effect’ in the Hybrid BNM in the 2D case is circumvented by an adaptive face integration scheme. The parameters that influence the performance of this method are studied through three different geometries and known analytical fields. Numerical results for the solution of the 3D Laplace's equation show that high convergence rates with mesh refinement and high accuracy are achievable. Copyright © 2004 John Wiley & Sons, Ltd.  相似文献   

3.
    
This paper presents a fast formulation of the hybrid boundary node method (Hybrid BNM) for solving problems governed by Laplace's equation in 3D. The preconditioned GMRES is employed for solving the resulting system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is accelerated by the fast multipole method. Green's kernel function is expanded in terms of spherical harmonic series. An oct‐tree data structure is used to hierarchically subdivide the computational domain into well‐separated cells and to invoke the multipole expansion approximation. Formulations for the local and multipole expansions, and also conversion of multipole to local expansion are given. And a binary tree data structure is applied to accelerate the moving least square approximation on surfaces. All the formulations are implemented in a computer code written in C++. Numerical examples demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.  相似文献   

4.
    
Element‐free Galerkin (EFG) methods are based on a moving least‐squares (MLS) approximation, which has the property that shape functions do not satisfy the Kronecker delta function at nodal locations, and for this reason imposition of essential boundary conditions is difficult. In this paper, the relationship between corrected collocation and Lagrange multiplier method is revealed, and a new strategy that is accurate and very simple for enforcement of essential boundary conditions is presented. The accuracy and implementation of this new technique is illustrated for one‐dimensional elasticity and two‐dimensional potential field problems. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

5.
    
In this paper the meshless local boundary integral equation (LBIE) method for numerically solving the non‐linear two‐dimensional sine‐Gordon (SG) equation is developed. The method is based on the LBIE with moving least‐squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. The approximation functions are constructed entirely using a set of scattered nodes, and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes. A time‐stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non‐linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of method to deal with the unsteady non‐linear problems in large domains. Copyright © 2009 John Wiley & Sons, Ltd.  相似文献   

6.
    
The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least‐squares (MLS) approximations for generating the trial and test functions. In this approach, boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Besides, the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The GBNM is developed in this paper for solving three‐dimensional stationary incompressible Stokes flows in primitive variables. The numerical scheme is based on variational formulations for the first‐kind integral equations, which are valid for both interior and exterior problems simultaneously. A rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in Sobolev spaces. The capability of the method is also illustrated and assessed through some selected numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.  相似文献   

7.
    
In this paper, we present a direct meshless method of boundary integral equation (BIE), known as the boundary element‐free method (BEFM), for two‐dimensional (2D) elastodynamic problems that combines the BIE method for 2D elastodynamics in the Laplace‐transformed domain and the improved moving least‐squares (IMLS) approximation. The formulae for the BEFM for 2D elastodynamic problems are given, and the numerical procedures are also shown. The BEFM is a direct numerical method, in which the basic unknown quantities are the real solutions of the nodal variables, and the boundary conditions can be implemented directly and easily that leads to a greater computational precision. For the purpose of demonstration, some selected numerical examples are solved using the BEFM. Copyright © 2005 John Wiley & Sons, Ltd.  相似文献   

8.
    
In this paper, we present a procedure to estimate the error in elliptic equations using the element‐free Galerkin (EFG) method, whose evaluation is computationally simple and can be readily implemented in existing EFG codes. The estimation of the error works very well in all numerical examples for 2‐D potential problems that are presented here, for regular and irregular clouds of points. Moreover, it was demonstrated that this method is very simple in terms of economy and gives a good performance. The results show that the error in EFG approximation may be estimated via the error estimator described in this paper. The quality of the estimation of the error is demonstrated by numerical examples. The implemented procedure of error approximation 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 redesign of cloud of points. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

9.
    
A new efficient meshfree method is presented in which the first‐order least‐squares method is employed instead of the Galerkin's method. In the meshfree methods based on the Galerkin formulation, the source of many difficulties is in the numerical integration. The current method, in this respect, has different characteristics and is expected to remove some of the integration‐related problems. It is demonstrated through numerical examples that the present formulation is highly robust to integration errors. Therefore, numerical integration can be performed with great ease and effectiveness using very simple algorithms. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

10.
    
A new meshless method based on a regular local integral equation and the moving least‐squares approximation is developed. The present method is a truly meshless one as it does not need a ‘finite element or boundary element mesh’, either for purposes of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three‐dimensional problems) and their boundaries. No derivatives of the shape functions are needed in constructing the system stiffness matrix for the internal nodes, as well as for those boundary nodes with no essential‐boundary‐condition‐prescribed sections on their local boundaries. Numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable, and the computational results for the unknown variable and its derivatives are very accurate. No special post‐processing procedure is required to compute the derivatives of the unknown variable, as the original result, from the moving least‐squares approximation, is smooth enough. Copyright © 1999 John Wiley & Sons, Ltd.  相似文献   

11.
    
Recently, considerable effort has been devoted to the development of the so‐called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the ‘a posteriori error’ should be approximated. An ‘a posteriori error’ approximation based on the moving least‐squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least‐squares approximations. Different selection strategies of the moving least‐squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error. The performance of the developed approximation of the error is illustrated by analysing different examples for two‐dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors. Copyright © 2003 John Wiley & Sons, Ltd.  相似文献   

12.
    
A finite point method, least‐squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least‐squares sense. The moving least‐squares interpolant is used to construct the trial functions. The computational effort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution significantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational effort. Copyright © 2001 John Wiley & Sons, Ltd.  相似文献   

13.
    
The current work presents an improved immersed boundary method based on the ideas proposed by Vanella and Balaras (M. Vanella, E. Balaras, A moving‐least‐squares reconstruction for embedded‐boundary formulations, J. Comput. Phys. 228 (2009) 6617–6628). In the method, an improved moving‐least‐squares approximation is employed to build the transfer functions between the Lagrangian points and discrete Eulerian grid points. The main advantage of the improved method is that there is no need to obtain the inverse matrix, which effectively eliminates numerical instabilities caused by matrix inversion and reduces the computational cost significantly. Several different flow problems (Taylor‐Green decaying vortices, flows past a stationary circular cylinder and a sphere, and the sedimentation of a free‐falling sphere in viscous fluid) are simulated to validate the accuracy and efficiency of the method proposed in the present paper. The simulation results show good agreement with previous numerical and experimental results, indicating that the improved immersed boundary method is efficient and reliable in dealing with the fluid–solid interaction problems. Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

14.
    
Based on the moving least‐squares (MLS) approximation, we propose a new approximation method—the complex variable moving least‐squares (CVMLS) approximation. With the CVMLS approximation, the trial function of a two‐dimensional problem is formed with a one‐dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than in the trial function of the MLS approximation, and we can thus select fewer nodes in the meshless method that is formed from the CVMLS approximation than are required in the meshless method of the MLS approximation with no loss of precision. The meshless method that is derived from the CVMLS approximation also has a greater computational efficiency. From the CVMLS approximation, we propose a new meshless method for two‐dimensional elasticity problems—the complex variable meshless method (CVMM)—and the formulae of the CVMM for two‐dimensional elasticity problems are obtained. Compared with the conventional meshless method, the CVMM has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVMM. Copyright © 2006 John Wiley & Sons, Ltd.  相似文献   

15.
    
Previous work by the author has shown that the consistency of the SPH method can be improved to acceptable levels by substituting MLS interpolants for SPH interpolants, that the SPH inconsistency drives the tension instability and that imposition of consistency via MLS severely retards tension instability growth. The new method however was not conservative, and made no provision for boundary conditions. Conservation is an essential property in simulations where large localized mass, momentum or energy transfer occurs such as high‐velocity impact or explosion modeling. A new locally conservative MLS variant of SPH that naturally incorporates realistic boundary conditions is described. In order to provide for the boundary fluxes one must identify the boundary particles. A new, purely geometric boundary detection technique for assemblies of spherical particles is described. A comparison with SPH on a ball‐and‐plate impact simulation shows qualitative improvement. Copyright © 2000 John Wiley & Sons, Ltd.  相似文献   

16.
    
An accelerated boundary cloud method (BCM) for boundary‐only analysis of exterior electrostatic problems is presented in this paper. The BCM uses scattered points instead of the classical boundary elements to discretize the surface of the conductors. The dense linear system of equations generated by the BCM are solved by a GMRES iterative solver combined with a singular value decomposition based rapid matrix–vector multiplication technique. The accelerated technique takes advantage of the fact that the integral equation kernel (2D Green's function in our case) is locally smooth and, therefore, can be dramatically compressed by using a singular value decomposition technique. The acceleration technique greatly speeds up the solution phase of the linear system by accelerating the computation of the dense matrix–vector product and reducing the storage required by the BCM. Copyright © 2002 John Wiley & Sons, Ltd.  相似文献   

17.
    
Stress‐point integration provides significant reductions in the computational effort of mesh‐free Galerkin methods by using fewer integration points, and thus facilitates the use of mesh‐free methods in applications where full integration would be prohibitively expensive. The influence of stress‐point integration on the convergence and stability properties of mesh‐free methods is studied. It is shown by numerical examples that for regular nodal arrangements, good rates of convergence can be achieved. For non‐uniform nodal arrangements, stress‐point integration is associated with a mild instability which is manifested by small oscillations. Addition of stabilization improves the rates of convergence significantly. The stability properties are investigated by an eigenvalue study of the Laplace operator. It is found that the eigenvalues of the stress‐point quadrature models are between those of full integration and nodal integration. Stabilized stress‐point integration is proposed in order to improve convergence and stability properties. Copyright © 2007 John Wiley & Sons, Ltd.  相似文献   

18.
    
The Smooth‐Particle‐Hydrodynamics (SPH) method is derived in a novel manner by means of a Galerkin approximation applied to the Lagrangian equations of continuum mechanics as in the finite‐element method. This derivation is modified to replace the SPH interpolant with the Moving‐Least‐Squares (MLS) interpolant of Lancaster and Saulkaskas, and define a new particle volume which ensures thermodynamic compatibility. A variable‐rank modification of the MLS interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent flow reduces the number of neighbours of a particle to less than the minimum required. A surprise benefit of the Galerkin SPH derivation is a theoretical justification of a common ad hoc technique for variable‐h SPH. The new MLSPH method is conservative if an anti‐symmetric quadrature rule for the stiffness matrix elements can be supplied. In this paper, a simple one‐point collocation rule is used to retain similarity with SPH, leading to a non‐conservative method. Several examples document how MLSPH renders dramatic improvements due to the linear consistency of its gradients on three canonical difficulties of the SPH method: spurious boundary effects, erroneous rates of strain and rotation and tension instability. Two of these examples are non‐linear Lagrangian patch tests with analytic solutions with which MLSPH agrees almost exactly. The examples also show that MLSPH is not absolutely stable if the problems are run to very long times. A linear stability analysis explains both why it is more stable than SPH and not yet absolutely stable and an argument is made that for realistic dynamic problems MLSPH is stable enough. The notion of coherent particles, for which the numerical stability is identical to the physical stability, is introduced. The new method is easily retrofitted into a generic SPH code and some observations on performance are made. Copyright © 1999 John Wiley & Sons, Ltd.  相似文献   

19.
    
In this study, we first discuss the moving least‐square approximation (MLS) method. In some cases, the MLS may form an ill‐conditioned system of equations so that the solution cannot be correctly obtained. Hence, in this paper, we propose an improved moving least‐square approximation (IMLS) method. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill‐conditioned system of equations. Combining the boundary integral equation (BIE) method and the IMLS approximation method, a direct meshless BIE method, the boundary element‐free method (BEFM), for two‐dimensional elasticity is presented. Compared to other meshless BIE methods, BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied easily; hence, it has higher computational precision. For demonstration purpose, selected numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd.  相似文献   

20.
    
This paper examines the interaction between coplanar square cracks by combining the moving least‐squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element‐free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

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