Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

Hirshikesh,; Natarajan, Sundararajan; Annabattula, Ratna Kumar; Bordas, Stéphane; Atroshchenko, Elena

doi:10.1186/s40540-017-0020-3

Cited by 6 publications

(6 citation statements)

References 35 publications

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“…Continuity or boundary conditions are incorporated into the interior domain by various ways, in which one of the ways is by hybrid method known as hybrid Trefftz finite element method (HT-FEM). This method uses the conforming functions of the element boundary (also known as frame) to link the interior of the elements together [140].…”

Section: Trefftz/hybrid Trefftz Polygonal Finite Element (T-fem or Htmentioning

confidence: 99%

“…HT-FEM has been successfully applied in linear elasticity problems [140] and found to be able to produce more accurate results and higher convergence rates compared to the conforming PFEM with Laplace/Wachspress shape functions. One of the advantages of the HT-FEM is that elements with embedded cracks or voids can be constructed.…”

Section: Trefftz/hybrid Trefftz Polygonal Finite Element (T-fem or Htmentioning

confidence: 99%

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A Brief Review on Polygonal/Polyhedral Finite Element Methods

Perumal

2018

Mathematical Problems in Engineering

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This paper provides brief review on polygonal/polyhedral finite elements. Various techniques, together with their advantages and disadvantages, are listed. A comparison of various techniques with the recently proposed Virtual Node Polyhedral Element (VPHE) is also provided. This review would help the readers to understand the various techniques used in formation of polygonal/polyhedral finite elements.

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Section: Trefftz/hybrid Trefftz Polygonal Finite Element (T-fem or Htmentioning

confidence: 99%

Section: Trefftz/hybrid Trefftz Polygonal Finite Element (T-fem or Htmentioning

confidence: 99%

A Brief Review on Polygonal/Polyhedral Finite Element Methods

Perumal

2018

Mathematical Problems in Engineering

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“…In order to establish the link between the neighboring elements, auxiliary interelement displacement fields, which are similar in form but not identical to the internal fields, are considered at the boundaries of the element assuming that each such boundary is a fictitious interelement and relating these auxiliary fields with the unknown constants of the internal fields, the continuity between the elements is enforced. Since its inception, the HTFE method has been successfully applied to various solid mechanics problems . Since the exact solutions of the governing equations of the element domain are derived a priori to form the Trefftz functions, the HTFE is highly convergent and locking free .…”

Section: Introductionmentioning

confidence: 99%

“…Although the HTFE is characterized with some great advantages as mentioned earlier, application of the HTFE method for the analysis of smart laminated composite structures is still not available in the open literature. In addition, a trend has been noticed from the open literature on HTFE that the simultaneous governing partial differential equations are manipulated to derive a single governing equation for which the homogeneous and particular solutions can be derived. However, deriving the single governing equation from a set of simultaneous partial differential equations is highly tedious and limited to simplified problems such as the structure made of isotropic material while the displacement field of the structures is modeled by simple displacement theory such as the Mindlin's plate theory.…”

Section: Introductionmentioning

confidence: 99%

A novel smart hybrid‐Trefftz finite element for smart laminated composite plates

Ray

2019

Numerical Meth Engineering

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A novel smart hybrid-Trefftz finite element (HTFE) has been developed for the analysis of smart laminated composite plates. The substrates of the smart plates are symmetric and antisymmetric cross-ply plates. The derivation of this HTFE is devoid of the complicated task of finding the particular solutions of simultaneous governing partial differential equations. The Trefftz functions are constructed from the finite number of free-field exact solutions of the homogeneous simultaneous governing partial differential equations of the element domain in a straightforward manner without transforming them into a single governing equation. The HTFE is validated with the exact solutions of the smart composite plates. It is observed that this HTFE is an efficient finite element and can be utilized for the analysis of active control of smart composite structures. KEYWORDS composites, finite element methods, smart materialsInt J Numer Methods Eng. 2019;120:707-726.wileyonlinelibrary.com/journal/nme

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“…The only exception to them is a triangular element. Other approaches that focus on the development of polygonal finite elements include the scaled boundary FEM, the virtual element method, strain smoothing technique, BEM‐based FEM, and h p −clouds . Sukumar et al showed a connection between the hourglass control and the virtual element method (VEM), and Natarajan et al showed a connection between the cell‐based smoothed FEM and the VEM.…”

Section: Introductionmentioning

confidence: 99%

Quadratic serendipity finite elements over convex polyhedra

Sinu

Natarajan

Shankar

2017

Numerical Meth Engineering

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SummaryThe conventional approach to construct quadratic elements for an n-sided polygon will yield n(n + 1)∕2 shape functions, which increases the computational effort. It is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion. Floater and Lai proposed a systematic way to construct higher-order approximations over arbitrary polygons using the generalized barycentric and triangular coordinates. This approach ensures 2n shape functions with nodes only on the boundary of the polygon. In this paper, we extend the polygonal splines approach to 3 dimensions and construct serendipity shape functions over hexahedra and convex polyhedra. This is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates. The quadratic shape functions possess Kronecker delta property and satisfy constant, linear, and quadratic precision. The accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests. The numerical results show that the quadratic serendipity elements pass the patch test, yield optimal convergence rate, and can tolerate extreme mesh distortion.

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Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

Cited by 6 publications

References 35 publications

A Brief Review on Polygonal/Polyhedral Finite Element Methods

A Brief Review on Polygonal/Polyhedral Finite Element Methods

A novel smart hybrid‐Trefftz finite element for smart laminated composite plates

Quadratic serendipity finite elements over convex polyhedra

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