728 research outputs found
New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow
This work proposes four novel hybrid quadrature schemes for the efficient and
accurate evaluation of weakly singular boundary integrals (1/r kernel) on
arbitrary smooth surfaces. Such integrals appear in boundary element analysis
for several partial differential equations including the Stokes equation for
viscous flow and the Helmholtz equation for acoustics. The proposed quadrature
schemes apply a Duffy transform-based quadrature rule to surface elements
containing the singularity and classical Gaussian quadrature to the remaining
elements. Two of the four schemes additionally consider a special treatment for
elements near to the singularity, where refined Gaussian quadrature and a new
moment-fitting quadrature rule are used.
The hybrid quadrature schemes are systematically studied on flat B-spline
patches and on NURBS spheres considering two different sphere discretizations:
An exact single-patch sphere with degenerate control points at the poles and an
approximate discretization that consist of six patches with regular elements.
The efficiency of the quadrature schemes is further demonstrated in boundary
element analysis for Stokes flow, where steady problems with rotating and
translating curved objects are investigated in convergence studies for both,
mesh and quadrature refinement. Much higher convergence rates are observed for
the proposed new schemes in comparison to classical schemes
The motion of a deforming capsule through a corner
A three-dimensional deformable capsule convected through a square duct with a
corner is studied via numerical simulations. We develop an accelerated boundary
integral implementation adapted to general geometries and boundary conditions.
A global spectral method is adopted to resolve the dynamics of the capsule
membrane developing elastic tension according to the neo-Hookean constitutive
law and bending moments in an inertialess flow. The simulations show that the
trajectory of the capsule closely follows the underlying streamlines
independently of the capillary number. The membrane deformability, on the other
hand, significantly influences the relative area variations, the advection
velocity and the principal tensions observed during the capsule motion. The
evolution of the capsule velocity displays a loss of the time-reversal symmetry
of Stokes flow due to the elasticity of the membrane. The velocity decreases
while the capsule is approaching the corner as the background flow does,
reaches a minimum at the corner and displays an overshoot past the corner due
to the streamwise elongation induced by the flow acceleration in the downstream
branch. This velocity overshoot increases with confinement while the maxima of
the major principal tension increase linearly with the inverse of the duct
width. Finally, the deformation and tension of the capsule are shown to
decrease in a curved corner
BINN: A deep learning approach for computational mechanics problems based on boundary integral equations
We proposed the boundary-integral type neural networks (BINN) for the
boundary value problems in computational mechanics. The boundary integral
equations are employed to transfer all the unknowns to the boundary, then the
unknowns are approximated using neural networks and solved through a training
process. The loss function is chosen as the residuals of the boundary integral
equations. Regularization techniques are adopted to efficiently evaluate the
weakly singular and Cauchy principle integrals in boundary integral equations.
Potential problems and elastostatic problems are mainly concerned in this
article as a demonstration. The proposed method has several outstanding
advantages: First, the dimensions of the original problem are reduced by one,
thus the freedoms are greatly reduced. Second, the proposed method does not
require any extra treatment to introduce the boundary conditions, since they
are naturally considered through the boundary integral equations. Therefore,
the method is suitable for complex geometries. Third, BINN is suitable for
problems on the infinite or semi-infinite domains. Moreover, BINN can easily
handle heterogeneous problems with a single neural network without domain
decomposition
Hydrodynamics of Monolayer Domains at the Air-Water Interface
Molecules at the air-water interface often form inhomogeneous layers in which
domains of different densities are separated by sharp interfaces. Complex
interfacial pattern formation may occur through the competition of short- and
long-range forces acting within the monolayer. The overdamped hydrodynamics of
such interfacial motion is treated here in a general manner that accounts for
dissipation both within the monolayer and in the subfluid. Previous results on
the linear stability of interfaces are recovered and extended, and a
formulation applicable to the nonlinear regime is developed. A simplified
dynamical law valid when dissipation in the monolayer itself is negligible is
also proposed. Throughout the analysis, special attention is paid to the
dependence of the dynamical behavior on a characteristic length scale set by
the ratio of the viscosities in the monolayer and in the subphase.Comment: 12 pages, RevTeX, 4 ps figures, accepted in Physics of Fluids
The boundary element method for elasticity problems with concentrated loads based on displacement singular elements
The boundary element method (BEM) has been implemented in elasticity problems very successfully because of its high accuracy. However, there are very few investigations about BEM with concentrated loads. The displacement at the concentrated load point is infinite and traditional elements will lead inaccurate results near the concentrated load point. This paper proposes two types of displacement singular elements to approximate the displacement near concentrated load point, and high accuracy can be obtained without refinement. The second type of displacement singular element is a general element type, which can work well for both problems with or without boundary concentrated loads. Numerical examples have been studied and compared with results obtained by traditional BEM and finite element method (FEM) to show the necessary of the proposed methods
Boundary integrated neural networks (BINNs) for 2D elastostatic and piezoelectric problems: Theory and MATLAB code
In this paper, we make the first attempt to apply the boundary integrated
neural networks (BINNs) for the numerical solution of two-dimensional (2D)
elastostatic and piezoelectric problems. BINNs combine artificial neural
networks with the well-established boundary integral equations (BIEs) to
effectively solve partial differential equations (PDEs). The BIEs are utilized
to map all the unknowns onto the boundary, after which these unknowns are
approximated using artificial neural networks and resolved via a training
process. In contrast to traditional neural network-based methods, the current
BINNs offer several distinct advantages. First, by embedding BIEs into the
learning procedure, BINNs only need to discretize the boundary of the solution
domain, which can lead to a faster and more stable learning process (only the
boundary conditions need to be fitted during the training). Second, the
differential operator with respect to the PDEs is substituted by an integral
operator, which effectively eliminates the need for additional differentiation
of the neural networks (high-order derivatives of neural networks may lead to
instability in learning). Third, the loss function of the BINNs only contains
the residuals of the BIEs, as all the boundary conditions have been inherently
incorporated within the formulation. Therefore, there is no necessity for
employing any weighing functions, which are commonly used in traditional
methods to balance the gradients among different objective functions. Moreover,
BINNs possess the ability to tackle PDEs in unbounded domains since the
integral representation remains valid for both bounded and unbounded domains.
Extensive numerical experiments show that BINNs are much easier to train and
usually give more accurate learning solutions as compared to traditional neural
network-based methods
The (2+1)-Dimensional Black Hole
I review the classical and quantum properties of the (2+1)-dimensional black
hole of Ba{\~n}ados, Teitelboim, and Zanelli. This solution of the Einstein
field equations in three spacetime dimensions shares many of the
characteristics of the Kerr black hole: it has an event horizon, an inner
horizon, and an ergosphere; it occurs as an endpoint of gravitational collapse;
it exhibits mass inflation; and it has a nonvanishing Hawking temperature and
interesting thermodynamic properties. At the same time, its structure is simple
enough to allow a number of exact computations, particularly in the quantum
realm, that are impractical in 3+1 dimensions.Comment: LaTeX, 34 pages, 4 figures in separate fil
Gravity and the Thermodynamics of Horizons
Spacetimes with horizons show a resemblance to thermodynamic systems and it
is possible to associate the notions of temperature and entropy with them.
Several aspects of this connection are reviewed in a manner appropriate for
broad readership. The approach uses two essential principles: (a) the physical
theories must be formulated for each observer entirely in terms of variables
any given observer can access and (b) consistent formulation of quantum field
theory requires analytic continuation to the complex plane. These two
principles, when used together in spacetimes with horizons, are powerful enough
to provide several results in a unified manner. Since spacetimes with horizons
have a generic behaviour under analytic continuation, standard results of
quantum field theory in curved spacetimes with horizons can be obtained
directly (Sections III to VII). The requirements (a) and (b) also put strong
constraints on the action principle describing the gravity and, in fact, one
can obtain the Einstein-Hilbert action from the thermodynamic considerations.
The latter part of the review (Sections VIII to X) investigates this deeper
connection between gravity, spacetime microstructure and thermodynamics of
horizons. This approach leads to several interesting results in the
semiclassical limit of quantum gravity, which are described.Comment: published version; references update
Prediction of interface failure probability in bi-material ceramic joints
The probabilistic framework that was developed constitutes an important step in the generalization of the Weakest-Link Approach to interface failure. This approach contributes to increase the reliability of industrial applications, related to using or manufacturing ceramic components with brittle interfaces, and facilitates a systematic planning of reliability experiments
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