MODELING TANGENTIAL VECTOR FIELDS ON REGULAR SURFACES BY MEANS OF MIE POTENTIALS

2007 ◽  
Vol 05 (03) ◽  
pp. 417-449 ◽  
Author(s):  
WILLI FREEDEN ◽  
CARSTEN MAYER
Keyword(s):  
Vector Fields ◽  
Numerical Test ◽  
Scaling Functions ◽  
Refinement Equation ◽  
Regular Surface ◽  
Tangential Vector ◽  
Classical Potential ◽  
Classical Potential Theory ◽  
Complex Valued ◽  
Integration Rules

By means of the limit and jump relations of classical potential theory with respect to the vectorial Helmholtz equation, a wavelet approach is established on a regular surface. The multiscale procedure is constructed in such a way that the emerging scalar, vectorial and tensorial potential kernels act as scaling functions. Corresponding wavelets are defined via a canonical refinement equation. A tree algorithm for fast decomposition of a tangential complex-valued vector field given on a regular surface is developed based on numerical integration rules. Some numerical test examples conclude the paper.

scholarly journals Multiscale deformation analysis by Cauchy-Navier wavelets

2003 ◽  
Vol 2003 (12) ◽  
pp. 605-645 ◽  
Author(s):  
M. K. Abeyratne ◽  
W. Freeden ◽  
C. Mayer
Keyword(s):  
Deformation Analysis ◽  
Scaling Functions ◽  
Earth Surface ◽  
Regular Surface ◽  
Fast Wavelet ◽  
Linear Elastostatics ◽  
Spherical Boundary ◽  
Pyramid Scheme ◽  
Navier Equation ◽  
Integration Rules

A geoscientifically relevant wavelet approach is established for the classical (inner) displacement problem corresponding to a regular surface (such as sphere, ellipsoid, and actual earth surface). Basic tools are the limit and jump relations of (linear) elastostatics. Scaling functions and wavelets are formulated within the framework of the vectorial Cauchy-Navier equation. Based on appropriate numerical integration rules, a pyramid scheme is developed providing fast wavelet transform (FWT). Finally, multiscale deformation analysis is investigated numerically for the case of a spherical boundary.

scholarly journals Modeling Tangential Vector Fields on a Sphere

2018 ◽  
Vol 113 (524) ◽  
pp. 1625-1636 ◽  
Author(s):  
Minjie Fan ◽  
Debashis Paul ◽  
Thomas C. M. Lee ◽  
Tomoko Matsuo
Keyword(s):  
Vector Fields ◽  
Tangential Vector

scholarly journals Harmonic morphisms applied to classical potential theory

2011 ◽  
Vol 202 ◽  
pp. 107-126
Author(s):  
Bent Fuglede
Keyword(s):  
Potential Theory ◽  
Unit Sphere ◽  
Harmonic Morphisms ◽  
Harmonic Morphism ◽  
Open Set ◽  
Superharmonic Functions ◽  
Radial Projection ◽  
Classical Potential ◽  
Classical Potential Theory

AbstractIt is shown that ifϕdenotes a harmonic morphism of type Bl between suitable Brelot harmonic spacesXandY, then a functionf, defined on an open setV ⊂ Y, is superharmonic if and only iff ∘ ϕis superharmonic onϕ–1(V) ⊂ X. The “only if” part is due to Constantinescu and Cornea, withϕdenoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case whereϕis the projection from ℝNto ℝn(N > n ≥1) or whereϕis the radial projection from ℝN\ {0} to the unit sphere in ℝN(N≥ 2).

On the Dynamic Stability of Fluid-Conveying Pipes

1973 ◽  
Vol 40 (1) ◽  
pp. 48-52 ◽  
Author(s):  
D. S. Weaver ◽  
T. E. Unny
Keyword(s):  
Potential Theory ◽  
Dynamic Stability ◽  
Thin Shell ◽  
Critical Flow ◽  
Thin Shells ◽  
Classical Potential ◽  
Classical Potential Theory ◽  
Limiting Case ◽  
Beam Mode ◽  
Fluid Conveying Pipes

This paper presents a general analysis of the dynamic stability of a finite-length, fluid-conveying pipe. The Flu¨gge-Kempner equation is used in conjunction with classical potential theory so that circumferential modes as well as the usual beam modes may be considered. The cylinders are found to become unstable statically at first but flutter is predicted for higher velocities. The critical flow velocities for short, thin shells are associated with a number of circumferential waves. This number reduces for thicker and longer shells until the instability is in a beam mode. When the limiting case of a long thin shell is taken, it is found to agree with previous results obtained using a simpler beam approach.

scholarly journals INTEGER OPERATORS IN FINITE VON NEUMANN ALGEBRAS

2011 ◽  
Vol 03 (04) ◽  
pp. 433-450
Author(s):  
ANDREAS THOM
Keyword(s):  
Potential Theory ◽  
Spectral Properties ◽  
Von Neumann Algebras ◽  
Logarithmic Capacity ◽  
Integral Group ◽  
Von Neumann ◽  
Classical Potential ◽  
Classical Potential Theory ◽  
Adjoint Operators ◽  
Finite Von Neumann Algebras

Motivated by the study of spectral properties of self-adjoint operators in the integral group ring of a sofic group, we define and study integer operators. We establish a relation with classical potential theory and in particular the circle of results obtained by Fekete and Szegö, see [3, 4, 13]. More concretely, we use results by Rumely, see [12], on equidistribution of algebraic integers to obtain a description of those integer operator which have spectrum of logarithmic capacity less than or equal to one. Finally, we relate the study of integer operators to a recent construction by Petracovici and Zaharescu, see [10].

scholarly journals Analysis of finite element methods for vector Laplacians on surfaces

2019 ◽  
Vol 40 (3) ◽  
pp. 1652-1701 ◽  
Author(s):  
Peter Hansbo ◽  
Mats G Larson ◽  
Karl Larsson
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Vector Fields ◽  
Optimal Order ◽  
Order Error ◽  
Optimal Order Error Estimates ◽  
Elastic Membranes ◽  
Tangential Vector ◽  
Higher Order Finite Elements ◽  
Lagrange Elements

Abstract We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in ${\mathbb{R}}^3$. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a ${\mathbb{R}}^3$ vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and $L^2$ norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.

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