scholarly journals Global injectivity in second-gradient nonlinear elasticity and its approximation with penalty terms

2019 ◽  
Vol 24 (11) ◽  
pp. 3644-3673 ◽  
Author(s):  
Stefan Krömer ◽  
Jan Valdman
Keyword(s):  
Order Term ◽  
Two Dimensions ◽  
Penalty Term ◽  
Hyperelastic Materials ◽  
Matlab Code ◽  
Second Gradient ◽  
Global Injectivity ◽  
Global Invertibility ◽  
Theoretical Results ◽  
Higher Order Term

We present a new penalty term approximating the Ciarlet–Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher-order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet–Nečas condition. Moreover, the penalization can be chosen in such a way that for all low-energy deformations, self-interpenetration is avoided completely already at all sufficiently small finite values of the penalization parameter. We also present numerical experiments in two dimensions illustrating our theoretical results and provide own MATLAB code available for download and testing.

Derivation of Skyrme Lagrangian and higher-order term

1990 ◽  
Vol 41 (9) ◽  
pp. 2930-2932
Author(s):  
Akihiro Ito
Keyword(s):  
Order Term ◽  
Higher Order ◽  
Higher Order Term

scholarly journals Edge detection with trigonometric polynomial shearlets

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Kevin Schober ◽  
Jürgen Prestin ◽  
Serhii A. Stasyuk
Keyword(s):  
Edge Detection ◽  
Trigonometric Polynomial ◽  
Two Dimensions ◽  
Characteristic Functions ◽  
Numerical Examples ◽  
Special Cases ◽  
Periodic Characteristic ◽  
Boundary Curves ◽  
Theoretical Results ◽  
De La Vallée Poussin

AbstractIn this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

scholarly journals A rewriting calculus for cyclic higher-order term graphs

2007 ◽  
Vol 17 (3) ◽  
pp. 363-406 ◽  
Author(s):  
PAOLO BALDAN ◽  
CLARA BERTOLISSI ◽  
HORATIU CIRSTEA ◽  
CLAUDE KIRCHNER
Keyword(s):  
Order Term ◽  
Equational Theory ◽  
Term Rewriting ◽  
Higher Order ◽  
Graph Representation ◽  
First Order ◽  
Rewrite Rules ◽  
Evaluation Mechanism ◽  
Matching Constraints ◽  
Higher Order Term

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories.In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

Asymptotic expansions for laminar forced-convection heat and mass transfer Part 2. Boundary-layer flows

1966 ◽  
Vol 24 (2) ◽  
pp. 339-366 ◽  
Author(s):  
J. D. Goddard ◽  
Andreas Acrivos
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Mass Transfer ◽  
Prandtl Number ◽  
Order Term ◽  
Higher Order ◽  
Convection Heat ◽  
Boundary Layer Flows ◽  
Prandtl Numbers ◽  
Higher Order Term

This is the second of two articles by the authors dealing with asymptotic expansions for forced-convection heat or mass transfer to laminar flows. It is shown here how the method of the first paper (Acrivos & Goddard 1965), which was used to derive a higher-order term in the large Péclet number expansion for heat or mass transfer to small Reynolds number flows, can yield equally well higher-order terms in both the large and the small Prandtl number expansions for heat transfer to laminar boundary-layer flows. By means of this method an exact expression for the first-order correction to Lighthill's (1950) asymptotic formula for heat transfer at large Prandtl numbers, as well as an additional higher-order term for the small Prandtl number expansion of Morgan, Pipkin & Warner (1958), are derived. The results thus obtained are applicable to systems with non-isothermal surfaces and arbitrary planar or axisymmetric flow geometries. For the latter geometries a derivation is given of a higher-order term in the Péclet number expansion which arises from the curvature of the thermal layer for small Prandtl numbers. Finally, some applications of the results to ‘similarity’ flows are also presented.

scholarly journals Gibbs phenomena for Lq-best approximation in finite element spaces

2021 ◽  
Author(s):  
Sarah Roggendorf ◽  
Paul Houston ◽  
Kristoffer van der Zee
Keyword(s):  
Finite Element ◽  
Sufficient Conditions ◽  
Gibbs Phenomenon ◽  
Two Dimensions ◽  
Thin Layers ◽  
Jump Discontinuities ◽  
Recent Developments ◽  
Minimum Residual ◽  
Gibbs Phenomena ◽  
Theoretical Results

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretisation methods in non-standard function spaces, such as L q -type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of the numerical observations. In particular, we investigate the Gibbs phenomena for L q -best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon.

Accurate Description of Near-Crack-Tip Fields for the Estimation of Inelastic Zone Extent in Quasi-Brittle Materials

2012 ◽  
Vol 525-526 ◽  
pp. 529-532 ◽  
Author(s):  
Václav Veselý ◽  
Jakub Sobek ◽  
Lucie Šestáková ◽  
Stanislav Seitl
Keyword(s):  
Finite Element ◽  
Order Term ◽  
Higher Order ◽  
Fe Analysis ◽  
Displacement Fields ◽  
Deterministic Method ◽  
Stress And Displacement Fields ◽  
Higher Order Terms ◽  
Higher Order Term ◽  
Selection Of

A description of stress and displacement fields by means of the Williams power series using also higher-order terms is the focus of this paper. Coefficients of this series are determined via the over-deterministic method from the results of conventional finite element (FE) analysis. A study is conducted into the selection of the FE node set whose results are processed in this regression technique. Coefficients up to the twelfth term were determined with high precision. The effect of the position of the FE node set on the accuracy of the values of the higher-order term coefficients is reported.

The Deflection of Electron Reams Traversing A Crystal Face

1985 ◽  
Vol 43 ◽  
pp. 62-63
Author(s):  
J.M. Cowley ◽  
Z.L. Kang
Keyword(s):  
Electron Beam ◽  
Diffraction Pattern ◽  
Potential Field ◽  
Order Term ◽  
Image Force ◽  
Second Order ◽  
Crystal Face ◽  
The Face ◽  
Small X ◽  
Theoretical Results

A serious discrepancy exists between the experimental and theoretical results on the deflection of an electron beam passing parallel to, and just outside, the flat face of a crystal. In a previous paper we reported that for a beam of diameter about 15Å traversing the face of a small gold crystal within a distance of 20Å or less, the central spot of the diffraction pattern is seen to be displaced through angles of up to 10-2 radians. Similar observations had been made for MgO crystals. Rough agreement with the observations could be obtained by assuming the beam to be deflected by a potential field extending into the vacuum and having the formwhere ϕo is the inner potential of the crystal and A, B and C are positive constants. This model for the potential field is based on the assumption of an image force modified for small x by a second order term.

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