Trefftz-Lekhnitskii Grains (TLGs) for efficient Direct Numerical Simulation (DNS) of the micro/meso mechanics of porous piezoelectric materials

Bishay, Peter L.; Atluri, Satya N.

doi:10.1016/j.commatsci.2013.10.038

Cited by 15 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finite element results are sensitive to mesh density; hence it could be a difficult task to find appropriate meshes for the RVE [Berger et al 2005]. The disadvantages of the numerical models can be avoided if we resort to the newly developed techniques such as those presented in this article and in [Bishay and Atluri 2014;Dong and Atluri 2012a;2012b;2012c;2012d] because these advanced methods can model a grain with its inclusion using only one element or region whose geometric shape is arbitrary, hence any statistical random distribution of fibers or particles can be accounted for with relatively very small computational cost and with high resolution in local fields' calculation.…”

Section: On Using Representative Volume Element (Rve) To Predict the mentioning

confidence: 99%

“…Here we summarize the expressions used to describe the primary and secondary fields in absence of body force and free-charge density ( b = 0, ρ f = 0) using the Lekhnitskii formalism. For more details, the reader is referred to [Bishay and Atluri 2014;Sheng et al 2006]. However, if the material is not piezoelectric and there is no coupling (g i j = 0), the expressions presented in the aforementioned references break down.…”

Section: General Solution Of Coupled/uncoupled Plane Electroelasticitmentioning

confidence: 99%

“…It can be seen from the figure that controlling the shape of the inclusions can result in varying the distribution and the maximum absolute values of the circumferential stress and electric displacement, as well as the circumferential and radial electric field, along the inclusion periphery. The effect of varying on the aforementioned variables along the inclusion periphery is presented in [Bishay et al 2014].…”

Section: Infinite Piezoelectric Domain With Elliptical Inclusionmentioning

confidence: 99%

“…This is the case for a domain with only 20 voids/inclusions. If it is required to analyze a domain with 200, 2000 or more grains with voids/inclusions in a direct numerical simulation (DNS), the regular finite element method would be highly expensive and impractical as compared to the MTCGs method and the newly developed methods presented in [Bishay and Atluri 2014;Dong and Atluri 2012b;2012c;2012d].…”

Section: Infinite Piezoelectric Domain With Elliptical Inclusionmentioning

confidence: 99%

See 3 more Smart Citations

Untitled

2014

J. Mech. Mater. Struct.

View full text Add to dashboard Cite

We continue our research on the performance of CAD-based global approximation to the analysis of 2D acoustic problems. In addition to previous "boundary-only" Coons and transfinite Gordon-Coons interpolations, we now investigate the quality of the solution when utilizing "tensor product B-splines" interpolation. For the latter, we propose a global collocation method that is successfully compared with the well known Galerkin-Ritz formulation. Particular attention is paid to the handling of Neumann boundary conditions as well as to the role of multiplicity of internal knots. The theory is supported by two numerical examples, one for a rectangular and the other for a circular acoustic cavity in which the approximate solution rapidly converges towards the exact solution.259 260 CHRISTOPHER G. PROVATIDIS literature therein. Our conclusion in that article is that rapid CAD-based global collocation methods have to be applied instead of time-consuming domain Galerkin-Ritz methods.For the first time, in 2005 the author expressed the above general idea of implementing a global collocation scheme in conjunction with CAD-based approximation [Provatidis 2006, p. 6704], while he continued with numerical applications in potential problems under Dirichlet [Provatidis 2008b] and Neumann [Provatidis 2009b] boundary conditions, as well as in plane stress elastostatics [Provatidis and Ioannou 2010]. Concerning eigenvalue and time response structural analysis, in 2008 he published a couple of papers [Provatidis 2008a; 2008c] (1D problems), and also he supervised a thesis concerning 2D acoustics and 2D elastodynamics in conjunction with Lagrange polynomials [Filippatos 2010]. It is worth mentioning that isogeometric collocation methods were recently presented by others [Auricchio et al. 2010].It is well known that B-splines collocation methods were initially developed by mathematicians, well before engineers understood that CAD-based Galerkin-Ritz methods require high computing effort, a fact that motivated them to investigate their replacement by isogeometric collocation schemes . For a detailed review of 273 papers covering the period 1934-1989 we refer to [Fairweather and Meade 1989], whereas recent works are cited in a survey [Bialecki et al. 2011]. A fair comparison between B-spline collocation and Galerkin methods on the basis of the same bandwidth (and not the same degree of shape functions) in one spatial dimension is [Kwok et al. 2001]. However, most works are limited to one-dimensional problems [de Boor 2001, Chapter XV], where, for cubic B-splines, the key point is to use two collocation points between any two breakpoints, thus resulting in as many equations as the number of unknowns; this is strictly related to using double knots (de Boor, personal communication, 2007). On the other hand, excellent results have been obtained in some particular examples using for collocation the images of either Greville or Demko abscissae [Auricchio et al. 2010].It is remarkable that de Boor's software, aiming at the solution of one-dimen...

show abstract

Section: On Using Representative Volume Element (Rve) To Predict the mentioning

confidence: 99%

Section: General Solution Of Coupled/uncoupled Plane Electroelasticitmentioning

confidence: 99%

Section: Infinite Piezoelectric Domain With Elliptical Inclusionmentioning

confidence: 99%

Section: Infinite Piezoelectric Domain With Elliptical Inclusionmentioning

confidence: 99%

See 2 more Smart Citations

Untitled

2014

J. Mech. Mater. Struct.

View full text Add to dashboard Cite

show abstract

“…For piezoelectric materials, Wang et al (2004) developed a hybrid finite element with a hole based on Lekhnitskii formalism, while Cao et al (2013) developed a hybrid finite element with defects based on the extended Stroh formalism. Computational cells or grains for direct numerical micromechanical simulation (DNMS) of micro/meso structures were developed by Dong and Atluri (2012b, 2012c, 2012d for heterogeneous and functionally graded isotropic elastic composite and porous materials, and by (Bishay and Atluri, 2014;Bishay et al, 2014b) for composite and porous piezoelectric materials. In these methods, each cell models an entire grain of the material, with elastic/rigid inclusions or voids.…”

Section: Introductionmentioning

confidence: 99%