Local maximum‐entropy approximation schemes: a seamless bridge between finite elements and meshfree methods

By Marino Arroyo, Magdalena Ortiz. 2006

410 citations recorded by the source corpus.

Subfields: Geophysics, Management, Monitoring, Policy and Law, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/b4d678051473d4e99837d11ff813072f034c5f5b

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In Geophysics

In Management, Monitoring, Policy and Law

In Mechanics of Materials

Citations in the corpus

Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment (2013, background): “Arroyo and Ortiz [30] reported local maximumentropy approximation schemes which can be considered as a convenient basis for the numerical solution of partial differential equations in the style of mesh-free Galerkin methods.”
Some basic formulations of the virtual element method (VEM) for finite deformations (2017, background): “, concave ones), harmonic shape functions [15, 16, 17] and max-entropy shape functions [18, 19, 20] need to be computed numerically at the quadrature points element by element.”
Phase-field modeling of fracture in linear thin shells (2014, positive): “As it was discussed in [39], the locality of the LME basis functions smoothness depends on a nondimensional parameter γ.”
Isogeometric analysis: An overview and computer implementation aspects (2012, neutral): “approximations [125].”
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, methodology): “A related approach is the local maximum-entropy (max-ent) meshfree method [48] that yield compactly-supported basis functions of exponential form that constitute a partition-ofunity [49], possess linear completeness, and provide a smooth transition from Delaunay basis functions [50] to global maximu”
Topology optimization using polytopes (2013, background): “, φn are a set of barycentric coordinates, for instance [4,7,12,15,16], which satisfy all the desired properties of a conforming interpolants such as partition of unity, Kronecker-delta, linear completeness and piecewise linear (C0 function) along the edges of e.”
New perspectives on polygonal and polyhedral finite element methods (2014, methodology): “The dual problem (unconstrained convex minimization) is efficiently solved using Newton’s method [40, 76, 21].”
Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy (2015, methodology): “We resort here to LME approximants [28], a meshfree method with non-negative and smooth basis functions.”
A generalized approximation for the meshfree analysis of solids (2011, methodology): “However, it is difficult to extend this method to higher order approximations since the first k moments of arbitrary fixed points do not belong to the moment space given by the convex hull of the nodal points [27, 31].”
Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons (2013, methodology): “For a Gaussian nodal prior weight function, the implicit function theorem is invoked to prove the smoothness of non-negative [37] and signed shape functions [1].”
Optimal transportation meshfree approximation schemes for fluid and plastic flows (2010, methodology): “[38, 39] and references therein) and the shape functions can then be reconstructed from the rearranged nodal sets [27].”
Smooth, second order, non‐negative meshfree approximants selected by maximum entropy (2009, methodology): “In [1], some properties of the so-called convex approximants (non-negative and reproducing exactly affine functions) were identified.”
Meshless methods: A review and computer implementation aspects (2008, methodology): “The recent inception of the smoothed finite element method (SFEM) [108,110–112,125,126], regained interest in polygonal interpolation [127] and, in particular, the recent discovery of maximum entropy approximations [123,124] may help bringing some more flexibility to the XFEM.”
Meshfree Methods: Progress Made after 20 Years (2017, positive): “Convex approximations for meshfree computation based on the principle of maximum entropy (MaxEnt) (Jaynes 1957) to achieve unbiased statistical in ﬂ uence of nodal data have been proposed for the Galerkin solution of PDEs (Arroyo and Ortiz 2006; Sukumar 2004).”