Natural neighbour Galerkin methods

Sukumar, N.; Moran, Brian; Semenov, A. Yu.; Беликов, В. В.

doi:10.1002/1097-0207(20010110)50:1<1::aid-nme14>3.0.co;2-p

Cited by 247 publications

(47 citation statements)

References 19 publications

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“…The area associated with the target's Thiessen polygon that was taken from an existing polygon is called the 'borrowed area'. The natural neighbor interpolation algorithm uses a weighted average of the neighboring observations, where the weights are proportional to the 'borrowed area' (Sibson, 1981;Owen, 1992;Watson, 1994;Sukumar et al, 2001;Yanalak, 2003;Surfer 8 Software Online (Surfer 8 Manual, 2006)). …”

Section: Natural Neighbor Methodsmentioning

confidence: 99%

“…and the data collection method (classical surveying, global positioning system, photogrammetry, digitizing, scanning etc.). Transformation of the regular data to a scattered sampling pattern or from a scattered to a regular sampling pattern is possible by using a suitable interpolation algorithm (Petrie and Kennie, 1987;Watson, 1999;Sukumar et al, 2001;Yanalak, 2003).…”

Section: Introductionmentioning

confidence: 99%

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The effect of interpolation methods in surface definition: an experimental study

Yilmaz

2007

Earth Surf Processes Landf

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Mappings of the earth surface and their representation in 3D (three-dimensional) models are commonly used in most recent research. Modeling research, which starts with classical surveying methods, acquires new dimensions matching the modern technologies. 3D models of any object or earth surface can be used in much visual and scientific research. A digital model of the landscape is an important part within creation of geo-information systems used in the public administration and in the commercial sphere. It is an important tool in applications such as geomorphology, hydrology, geology, cartography, ecology, mining etc.Values of volume in terrains that do not have regular geometric structure can be obtained more accurately by using 3D models of surfaces with respect to developing technology. Basic data of 3D models must indicate 3D coordinates of the surveyed object in the reference frame. Distribution and intensity of points are important factors in modeling earth surfaces. A minimum number of points is desired in defining an object in 3D. Interpolation methods employing different mathematical models are used to obtain 3D models of terrain surfaces. In this study, the effect of interpolation methods in defining a terrain surface is investigated. For this purpose, a uniform surface, hill-shaped artificial object with a known volume is employed. The 3D surface and volume are calculated by using 12 different interpolation methods. Point distribution, point intensity and accuracy of point measurements are not considered. The same data set was used for all the interpolation methods. The interpolation methods are compared and evaluated based on the results.Minimum curvature interpolation was used by Briggs (1974) to interpolate gravity values on the corners of a regular grid using scattered gravity data and finally to produce a contour map. The theory of minimum curvature interpolation, which is a gridding method, is based on minimization of the total curvature at the grid points. The total square curvature of a surface is defined as C z z xThe effect of interpolation methods in surface definition: an experimental study 1351

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Section: Natural Neighbor Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The effect of interpolation methods in surface definition: an experimental study

Yilmaz

2007

Earth Surf Processes Landf

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“…So far, there have been over 20 types of meshless methods and several ways of classifying these methods . Represented by EFGM, the reproducing kernel particle method (RKPM) , the natural element method (NEM) , and the natural neighbor Galerkin method (NNGM) , however, the methods in the framework of the Galerkin variational formulation or its extensions are approaching maturity because of their solid mathematical foundations. In the following, brief reviews are made on those Galerkin‐based meshless methods.…”

Section: Introductionmentioning

confidence: 99%

Exact imposition of essential boundary condition and material interface continuity in Galerkin‐based meshless methods

Zheng

2016

Numerical Meth Engineering

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Summary Represented by the element free Galerkin method, the meshless methods based on the Galerkin variational procedure have made great progress in both research and application. Nevertheless, their shape functions free of the Kronecker delta property present great troubles in enforcing the essential boundary condition and the material continuity condition. The procedures based on the relaxed variational formulations, such as the Lagrange multiplier‐based methods and the penalty method, strongly depend on the problem in study, the interpolation scheme, or the artificial parameters. Some techniques for this issue developed for a particular method are hard to extend to other meshless methods. Under the framework of partition of unity and strict Galerkin variational formulation, this study, taking Poisson's boundary value problem for instance, proposes a unified way to treat exactly both the material interface and the nonhomogeneous essential boundary as in the finite element analysis, which is fit for any partition of unity‐based meshless methods. The solution of several typical examples suggests that compared with the Lagrange multiplier method and the penalty method, the proposed method can be always used safely to yield satisfactory results. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Generally, meshless methods can be divided in approximation meshless methods [34][35][36][37][38][39] and interpolation meshless methods. [40][41][42][43][44][45][46][47] The major advantage of using interpolator meshless methods is the possibility to impose directly the essential and natural boundary conditions, since the constructed test functions possess the delta Kronecker property, ' i ðx j Þ ¼ ij .…”

Section: Introductionmentioning

confidence: 99%

The Mandible Remodeling Induced by Dental Implants: A Meshless Approach

Belinha

Dinis

Jorge

2015

J. Mech. Med. Biol.

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This work aims to evaluate the prediction efficiency of a recently developed numerical approach in the mandible bone tissue remodeling process. The remodeling algorithm, seeking the minimization of the strain energy density, includes a phenomenological anisotropic material law capable of predicting the bone tissue mechanical properties based on the bone apparent density. The key factor of the proposed numerical approach is the inclusion of a flexible and efficient meshless method, which is used to obtain the strain energy density field. The inclusion of this advance discretization technique in the process is an asset since meshless methods produce smoother and more accurate strain energy density fields when compared with other numerical approaches. The bone tissue remodeling process of the molar region of the mandible, due to the inclusion of an implant system, is studied. The obtained results are in accordance with other numerical approach results available in the literature.

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Natural neighbour Galerkin methods

Cited by 247 publications

References 19 publications

The effect of interpolation methods in surface definition: an experimental study

The effect of interpolation methods in surface definition: an experimental study

Exact imposition of essential boundary condition and material interface continuity in Galerkin‐based meshless methods

The Mandible Remodeling Induced by Dental Implants: A Meshless Approach

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