NURBS-based parametric mesh-free methods

Shaw, Amit; Roy, Debasish

doi:10.1016/j.cma.2007.11.024

Cited by 26 publications

(10 citation statements)

References 37 publications

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“…On the other hand, the computational domain of such problem is non-convex. Therefore, special treatments such as the parametric approach [20] for approximating the non-convex domain is required for the meshfree analysis using convex approximation. Another simple approach which is used in this study to maintain the convexity in meshfree approximation and therefore avoids any special treatments is to properly adjust the nodal supports of the nodes near the reentrant corner.…”

Section: L-shaped Waveguidementioning

confidence: 99%

Convex Meshfree Solutions for Arbitrary Waveguide Analysis in Electromagnetic Problems

Wang¹

2013

PIER B

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Abstract-This paper presents a convex meshfree framework for solving the scalar Helmholtz equation in the waveguide analysis of electromagnetic problems. The generalized meshfree approximation (GMF) method using inverse tangent basis functions and cubic spline weight functions is employed to construct the first-order convex approximation which exhibits a weak Kronecker-delta property at the waveguide boundary and allows a direct enforcement of homogenous Dirichlet boundary conditions for the transverse magnetic (TM) mode analyses. Four arbitrary waveguide examples are analyzed to demonstrate the accuracy of the presented formulation, and comparison is made with the analytical, finite element and meshfree solutions.

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Section: L-shaped Waveguidementioning

confidence: 99%

Convex Meshfree Solutions for Arbitrary Waveguide Analysis in Electromagnetic Problems

Wang¹

2013

PIER B

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“…A brief account of RKPM shape functions [35,36,25,33] is provided in Appendix B. Application of the DNM must tackle the issue of invertibility of the linearized problem (e.g.…”

Section: Equations Of Equilibrium and Discretizationmentioning

confidence: 99%

A micropolar cohesive damage model for delamination of composites

Rahaman

Deepu

Roy

et al. 2015

Composite Structures

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a b s t r a c tA micropolar cohesive damage model for delamination of composites is proposed. The main idea is to embed micropolarity, which brings an additional layer of kinematics through the micro-rotation degrees of freedom within a continuum model to account for the micro-structural effects during delamination. The resulting cohesive model, describing the modified traction separation law, includes micro-rotational jumps in addition to displacement jumps across the interface. The incorporation of micro-rotation requires the model to be supplemented with physically relevant material length scale parameters, whose effects during delamination of modes I and II are brought forth using numerical simulations appropriately supported by experimental evidences.

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“…It must, nevertheless, be noted that, while implementing the ERIKM in the weak form, proper measures to enforce conformability in a seamless manner may become necessary. Indeed, the authors have recently developed a parametric version of the ERIKM, wherein the construction of shape functions and integration is carried out over a parametric space via NURBS such that the geometric map between the parametric space and the original geometry is always preserved [43]. Since the same 'NURBS cells' may be used to construct the basis functions and perform integration, it is possible to completely bypass the issue of non-conformability owing to a misalignment of supports and integration cells.…”

Section: Construction Of the Error Function Once Nurbs Basis Functionmentioning

confidence: 99%

A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

Shaw

Bendapudi

Roy

2007

Numerical Meth Engineering

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SUMMARYAn error-reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error-reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non-uniform rational B-splines (NURBS)-based mesh-free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40(1):127-148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non-interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non-linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent.

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NURBS-based parametric mesh-free methods

Cited by 26 publications

References 37 publications

Convex Meshfree Solutions for Arbitrary Waveguide Analysis in Electromagnetic Problems

Convex Meshfree Solutions for Arbitrary Waveguide Analysis in Electromagnetic Problems

A micropolar cohesive damage model for delamination of composites

A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

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