scholarly journals Quasi-Interpolation in a Space of C2 Sextic Splines over Powell–Sabin Triangulations

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2276
Author(s):  
Salah Eddargani ◽  
María José Ibáñez ◽  
Abdellah Lamnii ◽  
Mohamed Lamnii ◽  
Domingo Barrera
Keyword(s):  
Fourth Order ◽  
Interior Point ◽  
Small Area ◽  
Spline Functions ◽  
Interpolation Operators ◽  
Marsden’S Identity

In this work, we study quasi-interpolation in a space of sextic splines defined over Powell–Sabin triangulations. These spline functions are of class C2 on the whole domain but fourth-order regularity is required at vertices and C3 regularity is imposed across the edges of the refined triangulation and also at the interior point chosen to define the refinement. An algorithm is proposed to define the Powell–Sabin triangles with a small area and diameter needed to construct a normalized basis. Quasi-interpolation operators which reproduce sextic polynomials are constructed after deriving Marsden’s identity from a more explicit version of the control polynomials introduced some years ago in the literature. Finally, some tests show the good performance of these operators.

The Stresses Induced in a Half-Space by an Arbitrary Axisymmetric Pressure Distribution

1987 ◽  
Vol 109 (4) ◽  
pp. 630-633
Author(s):  
D. A. Hills ◽  
A. Sackfield
Keyword(s):  
Contact Problem ◽  
Pressure Distribution ◽  
Half Space ◽  
Fourth Order ◽  
Small Area ◽  
Hertz Contact ◽  
Hertz Contact Problem ◽  
General Method

A general method is described for deducing the stresses induced in a half-space by an axisymmetric pressure distribution applied over a small area of the surface. The technique is illustrated by re-evaluating the solution to Hertz’ contact problem, and deducing the stresses induced in a fourth-order contacting pair.

scholarly journals Bernstein Polynomials

2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Anandaram Mandyam N
Keyword(s):  
Fourth Order ◽  
Bernstein Polynomials ◽  
Basis Functions ◽  
Spline Functions ◽  
Bezier Curves ◽  
Bézier Curves ◽  
B Splines

Bernstein polynomials (aka, B-polys) have excellent properties allowing them to be used as basis functions in many applications of physics. In this paper, a brief tutorial description of their properties is given and then their use in obtaining B-polys, B-splines or Basis spline functions, Bezier curves and ODE solution curves, is computationally demonstrated.  An example is also described showing their application to solving a fourth-order BVP relating to the bending at the free end of a cantilever.

scholarly journals Some estimates of virtual element methods for fourth order problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qingguang Guan
Keyword(s):  
Error Estimates ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Optimal Error Estimates ◽  
Virtual Element Method ◽  
Regularity Estimates ◽  
Interpolation Operators ◽  
Virtual Element ◽  
The Right ◽  
Element Method

<p style='text-indent:20px;'>In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis bases on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates were obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions were proved.</p>

Rocking Beam Microarea Diffraction Applied to Metallic thin Films

1976 ◽  
Vol 34 ◽  
pp. 396-397
Author(s):  
R. H. Geiss
Keyword(s):  
Thin Films ◽  
Electron Diffraction ◽  
Small Area ◽  
Objective Lens ◽  
Electron Optics ◽  
Metallic Thin Films ◽  
Transmission Electron Diffraction ◽  
Transmission Electron ◽  
Sample Height ◽  
Scanning Transmission

The theory and practical limitations of micro area scanning transmission electron diffraction (MASTED) will be presented. It has been demonstrated that MASTED patterns of metallic thin films from areas as small as 30 Åin diameter may be obtained with the standard STEM unit available for the Philips 301 TEM. The key to the successful application of MASTED to very small area diffraction is the proper use of the electron optics of the STEM unit. First the objective lens current must be adjusted such that the image of the C2 aperture is quasi-stationary under the action of the rocking beam (obtained with 40-80-160 SEM settings of the P301). Second, the sample must be elevated to coincide with the C2 aperture image and its image also be quasi-stationary. This sample height adjustment must be entirely mechanical after the objective lens current has been fixed in the first step.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

Sign in / Sign up

Export Citation Format

Share Document