Quasi-Monte Carlo mesh-free integration for meshless weak formulations

Roșca, Victoria Elena; Leitão, Vitor M.A.

doi:10.1016/j.enganabound.2007.10.015

Cited by 15 publications

(8 citation statements)

References 17 publications

(23 reference statements)

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“…Recently, Rosca and Leitao efficiently used the Monte Carlo integration technique in meshless methods based on global and local weak forms [25]. They implemented this technique for the EFG method and meshless local Petrov-Galerkin method.…”

Section: Article In Pressmentioning

confidence: 99%

“…(25), vanishes on the lateral surface of the auxiliary domain. This is because the normal vector to the lateral surface of the auxiliary domain is perpendicular to the x-axis and n x ¼ 0 on this surface.…”

Section: D Domain Integralsmentioning

confidence: 99%

“…There are some approaches other than the use of a background mesh for the evaluation of the integrals in the EFG method and the RPIM. Nevertheless, this method of integration is still the standard integration technique in meshless methods based on the global weak form [25]. The issue of numerical integration in meshless methods has been of great importance to researchers and has been extensively studied [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

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A new method for meshless integration in 2D and 3D Galerkin meshfree methods

Khosravifard

Hematiyan

2010

Engineering Analysis with Boundary Elements

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Section: Article In Pressmentioning

confidence: 99%

Section: D Domain Integralsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new method for meshless integration in 2D and 3D Galerkin meshfree methods

Khosravifard

Hematiyan

2010

Engineering Analysis with Boundary Elements

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“…There are some approaches other than the use of cell structures or background meshes for the evaluation of the integrals in the EFG and RPIM methods. However, these approaches are still the standard integration technique based on the global weak form . The numerical integration in meshfree methods has been of great importance to scientists and has been extensively studied previously, for example, see .…”

Section: Introductionmentioning

confidence: 99%

Novel adaptive meshfree integration techniques in meshless methods

Racz

Bui

2012

Numerical Meth Engineering

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We propose two strategies of novel adaptive numerical integration based on mapping techniques for solving the complicated problems of domain integration encountered in meshfree methods. Several mapping methods are presented in detail that map a complex integration domain to much simpler ones, for example, squares, triangles or circles. The techniques described in the paper can be applied to both global and local weak forms, and the highly nonlinear meshfree integrands are evaluated with controlled accuracy. The necessity of the clumsy procedure of background mesh or cell structures used for integration purpose in existing meshfree methods is avoided, and many meshfree methods that require the domain integration can now become 'truly meshfree'. Various numerical examples in two dimensions are considered to demonstrate the applicability and the effectiveness of the proposed methods and it shows that the accuracy is improved significantly. Their obtained results are compared with analytical solutions and other approaches and very good agreements are found. Additionally, some three-dimensional cases applied by the present methods are also examined. ADAPTIVE MESHFREE INTEGRATION TECHNIQUES 1415 contrasted with results by FEM [14]. Hypervelocity impact of ceramic plates can be modeled not just experimentally but also computationally by SPH method for example. The solution for problems with large deformations is not always accurate when FEM is used [15]. Cracks in fracture mechanics problems can be represented by discontinuities and these problems can be modeled much easier by using meshfree methods. In the finite element analysis of a fracture mechanics problem the mesh needs to be redefined and updated at each stage of crack propagation, which is not necessary if meshfree methods are used [16,17]. The unknown field variables are approximated by a linear combination of shape functions that are built through the nodes scattered in the domain. Unlike the conventional FEM [18], all meshfree methods use a set of nodes scattered in the problem domain regardless of the connectivity of elements among nodes, see for example, [1-12, 19, 20], for details.Many meshfree methods, like the most common one, the EFG method uses the moving leastsquares (MLS) approximation to construct shape functions, and the global Galerkin weak form is adopted to discretize the governing partial differential equations. The MLPG method is formulated in a similar way as the EFG but a local weak form of the Petrov-Galerkin residual formulation is employed instead. Regardless of techniques used, all meshfree methods share the same problem related to numerical integration. Not all of the meshfree methods are truly meshless because some of them use background meshes or cell structures to evaluate the integrals in the global weak forms, for example, see EFG and RPIM. There are some approaches other than the use of cell structures or background meshes for the evaluation of the integrals in the EFG and RPIM methods. However, these approaches are still the sta...

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“…The Monte Carlo Integration (MCI) technique was suggested in [11,12] for electromagnetic scattering problems which takes care of the problem of singularity by avoiding a small region around the location of singularity. A similar approach is used in Boundary Element Method (BEM) [13] where the singular point is surrounded by a small circle of radius ε and then the solution is examined as ε → 0 However, the present paper proposes the Quasi Monte Carlo Integration (QMCI) technique [14][15][16][17] which takes care of the singularity problem inherently by judiciously choosing such methods for generation of quasi Monte Carlo points over the domain of integration, that the need for function evaluation at the singular point does not arise, thus proving to be a better option than MCI. Various methods have been used for extraction of capacitances of several capacitors [18,19].…”

Section: Introductionmentioning

confidence: 99%