scholarly journals Magnetic micro-droplet in rotating field: numerical simulation and comparison with experiment

2017 ◽  
Vol 821 ◽  
pp. 266-295 ◽  
Author(s):  
J. Erdmanis ◽  
G. Kitenbergs ◽  
R. Perzynski ◽  
A. Cēbers
Keyword(s):  
Magnetic Field ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Dipolar Interaction ◽  
Boundary Integral ◽  
Rotating Magnetic Field ◽  
Three Dimensions ◽  
Triaxial Ellipsoid ◽  
Equilibrium Shapes ◽  
Magnetic Droplet

Magnetic droplets obtained by induced phase separation in a magnetic colloid show a large variety of shapes when exposed to an external field. However, the description of the shapes is often limited. Here, we formulate an algorithm based on three-dimensional boundary-integral equations for strongly magnetic droplets in a high-frequency rotating magnetic field, allowing us to find their figures of equilibrium in three dimensions. The algorithm is justified by a series of comparisons with known analytical results. We compare the calculated equilibrium shapes with experimental observations and find a good agreement. The main features of these observations are the oblate–prolate transition, the flattening of prolate shapes with the increase of magnetic field strength and the formation of starfish-like equilibrium shapes. We show both numerically and in experiments that the magnetic droplet behaviour may be described with a triaxial ellipsoid approximation. Directions for further research are mentioned, including the dipolar interaction contribution to the surface tension of the magnetic droplets, accounting for the large viscosity contrast between the magnetic droplet and the surrounding fluid.

Fast direct solvers for integral equations in complex three-dimensional domains

Acta Numerica ◽  
2009 ◽  
Vol 18 ◽  
pp. 243-275 ◽  
Author(s):  
Leslie Greengard ◽  
Denis Gueyffier ◽  
Per-Gunnar Martinsson ◽  
Vladimir Rokhlin
Keyword(s):  
Integral Equations ◽  
Degrees Of Freedom ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Three Dimensions ◽  
Computer Hardware ◽  
Matrix Vector Multiplication ◽  
Gradient Type ◽  
Mathematical Foundations

Methods for the solution of boundary integral equations have changed significantly during the last two decades. This is due, in part, to improvements in computer hardware, but more importantly, to the development of fast algorithms which scale linearly or nearly linearly with the number of degrees of freedom required. These methods are typically iterative, based on coupling fast matrix-vector multiplication routines with conjugate-gradient-type schemes. Here, we discuss methods that are currently under development for the fast, direct solution of boundary integral equations in three dimensions. After reviewing the mathematical foundations of such schemes, we illustrate their performance with some numerical examples, and discuss the potential impact of the overall approach in a variety of settings.

scholarly journals Solitary flexural–gravity waves in three dimensions

2018 ◽  
Vol 376 (2129) ◽  
pp. 20170345 ◽  
Author(s):  
Olga Trichtchenko ◽  
Emilian I. Părău ◽  
Jean-Marc Vanden-Broeck ◽  
Paul Milewski
Keyword(s):  
Gravity Waves ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Boundary Integral ◽  
Three Dimensions ◽  
Cosserat Theory ◽  
Theme Issue ◽  
Flexural Gravity Waves ◽  
Forced Waves ◽  
Ice Phenomena

The focus of this work is on three-dimensional nonlinear flexural–gravity waves, propagating at the interface between a fluid and an ice sheet. The ice sheet is modelled using the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis, presented in (Plotnikov & Toland. 2011 Phil. Trans. R. Soc. A 369 , 2942–2956 ( doi:10.1098/rsta.2011.0104 )). The fluid is assumed inviscid and incompressible, and the flow irrotational. A numerical method based on boundary integral equation techniques is used to compute solitary waves and forced waves to Euler's equations. This article is part of the theme issue ‘Modelling of sea-ice phenomena’.

Boundary Element Method Analysis of Three-Dimensional Thermoelastic Fracture Problems Using the Energy Domain Integral

2005 ◽  
Vol 73 (6) ◽  
pp. 959-969 ◽  
Author(s):  
R. Balderrama ◽  
A. P. Cisilino ◽  
M. Martinez
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Auxiliary Function ◽  
Boundary Integral ◽  
Crack Advance ◽  
Domain Integral ◽  
Energy Domain ◽  
Element Method

A boundary element method (BEM) implementation of the energy domain integral (EDI) methodology for the numerical analysis of three-dimensional fracture problems considering thermal effects is presented in this paper. The EDI is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains, temperatures, and derivatives of displacements and temperatures at internal points can be evaluated using the appropriate boundary integral equations. Special emphasis is put on the selection of the auxiliary function that represents the virtual crack advance in the domain integral. This is found to be a key feature to obtain reliable results at the intersection of the crack front with free surfaces. Several examples are analyzed to demonstrate the efficiency and accuracy of the implementation.

scholarly journals Development of boundary-element time-step scheme in solving 3D poroelastodynamics problems

2018 ◽  
Vol 183 ◽  
pp. 01042 ◽  
Author(s):  
Igor Vorobtsov ◽  
Aleksandr Belov ◽  
Andrey Petrov
Keyword(s):  
Boundary Element ◽  
Solid Phase ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Discrete Analogue ◽  
Boundary Integral ◽  
Step Method ◽  
Time Step ◽  
Element Scheme ◽  
Runge Kutta

The development of time-step boundary-element scheme for the three dimensional boundaryvalue problems of poroelastodynamics is presented. The poroelastic continuum is described using Biot’s mathematical model. Poroelastic material is assumed to consist of a solid phase constituting an elastic formdefining skeleton and carrying most of the loading, and two fluid phases filling the pores. Dynamic equations of the poroelastic medium are written for unknown functions of displacement of the elastic skeleton and pore pressures of the filling materials. Green’s matrices and, based on it, boundary integral equations are written in Laplace domain. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. Boundary element scheme is based on time-step method of numerical inversion of Laplace transform. A modification of the time-step scheme on the nodes of Runge-Kutta methods is considered. The Runge-Kutta scheme is exemplified with 2-and 3-stage Radau schemes. The results of comparing the two schemes in analyzing a numerical example are presented.

scholarly journals Minimum-resistance shapes in linear continuum mechanics

2013 ◽  
Vol 469 (2160) ◽  
pp. 20130206 ◽  
Author(s):  
Michael Zabarankin
Keyword(s):  
Optimality Condition ◽  
Boundary Integral Equations ◽  
Poisson Ratio ◽  
Integral Formula ◽  
Three Dimensional ◽  
Analytical Form ◽  
Boundary Integral ◽  
Minimum Drag ◽  
Isotropic Elastic Medium ◽  
Minimum Resistance

A necessary optimality condition for the problem of the minimum-resistance shape for a rigid three-dimensional inclusion displaced in an unbounded isotropic elastic medium subject to a constraint on the volume of the inclusion is obtained through Betti's reciprocal work theorem. It generalizes Pironneau's optimality condition for the minimum-drag shape for a rigid body immersed into a uniform Stokes flow and is specialized for axisymmetric inclusions in axisymmetric and transversal translations. In both cases of translation, the three-dimensional displacement field is represented in terms of generalized analytic functions, and the three-dimensional elastostatics problem is reduced to boundary-integral equations (BIEs) via the generalized Cauchy integral formula. Minimum-resistance shapes are found in the semi-analytical form of functional series from an iterative procedure coupling the optimality condition and the BIEs. They are compared with the minimum-resistance spheroids and with the minimum-resistance spindle-shaped and lens-shaped bodies. Remarkably, in the axisymmetric translation, the minimum-resistance shapes transition from spindle-like shapes to almost prolate spheroidal shapes as the Poisson ratio changes from 1/2 to 0, whereas in the transversal translation, they are close to oblate spheroidal shapes for any Poisson ratio.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

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