389 research outputs found
An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies
We present an immersed interface method for the vorticity-velocity form of
the 2D Navier Stokes equations that directly addresses challenges posed by
multiply connected domains, nonconvex obstacles, and the calculation of force
distributions on immersed surfaces. The immersed interface method is
re-interpreted as a polynomial extrapolation of flow quantities and boundary
conditions into the obstacle, reducing its computational and implementation
complexity. In the flow, the vorticity transport equation is discretized using
a conservative finite difference scheme and explicit Runge-Kutta time
integration. The velocity reconstruction problem is transformed to a scalar
Poisson equation that is discretized with conservative finite differences, and
solved using an FFT-accelerated iterative algorithm. The use of conservative
differencing throughout leads to exact enforcement of a discrete Kelvin's
theorem, which provides the key to simulations with multiply connected domains
and outflow boundaries. The method achieves second order spatial accuracy and
third order temporal accuracy, and is validated on a variety of 2D flows in
internal and free-space domains
Pseudo transient continuation and time marching methods for Monge-Ampere type equations
We present two numerical methods for the fully nonlinear elliptic
Monge-Ampere equation. The first is a pseudo transient continuation method and
the second is a pure pseudo time marching method. The methods are proven to
converge to a strictly convex solution of a natural discrete variational
formulation with conforming approximations. The assumption of existence
of a strictly convex solution to the discrete problem is proven for smooth
solutions of the continuous problem and supported by numerical evidence for non
smooth solutions
Singular function mortar finite element methods
This is the published version, also available here: http://dx.doi.org/10.2478/cmam-2003-0014.We consider the Poisson equation with Dirichlet boundary conditions on a polygonal domain with one reentrant corner. We introduce new nonconforming finite element discretizations based on mortar techniques and singular functions. The main idea introduced in this paper is the replacement of cut-off functions by mortar element techniques on the boundary of the domain. As advantages, the new discretizations do not require costly numerical integrations and have smaller a priori error estimates and condition numbers. Based on such an approach, we prove optimal accuracy error bounds for the discrete solution. Based on such techniques, we also derive new extraction formulas for the stress intensive factor. We establish optimal accuracy for the computed stress intensive factor. Numerical examples are presented to support our theory
Handling congestion in crowd motion modeling
We address here the issue of congestion in the modeling of crowd motion, in
the non-smooth framework: contacts between people are not anticipated and
avoided, they actually occur, and they are explicitly taken into account in the
model. We limit our approach to very basic principles in terms of behavior, to
focus on the particular problems raised by the non-smooth character of the
models. We consider that individuals tend to move according to a desired, or
spontanous, velocity. We account for congestion by assuming that the evolution
realizes at each time an instantaneous balance between individual tendencies
and global constraints (overlapping is forbidden): the actual velocity is
defined as the closest to the desired velocity among all admissible ones, in a
least square sense. We develop those principles in the microscopic and
macroscopic settings, and we present how the framework of Wasserstein distance
between measures allows to recover the sweeping process nature of the problem
on the macroscopic level, which makes it possible to obtain existence results
in spite of the non-smooth character of the evolution process. Micro and macro
approaches are compared, and we investigate the similarities together with deep
differences of those two levels of description
Second-order Shape Optimization for Geometric Inverse Problems in Vision
We develop a method for optimization in shape spaces, i.e., sets of surfaces
modulo re-parametrization. Unlike previously proposed gradient flows, we
achieve superlinear convergence rates through a subtle approximation of the
shape Hessian, which is generally hard to compute and suffers from a series of
degeneracies. Our analysis highlights the role of mean curvature motion in
comparison with first-order schemes: instead of surface area, our approach
penalizes deformation, either by its Dirichlet energy or total variation.
Latter regularizer sparks the development of an alternating direction method of
multipliers on triangular meshes. Therein, a conjugate-gradients solver enables
us to bypass formation of the Gaussian normal equations appearing in the course
of the overall optimization. We combine all of the aforementioned ideas in a
versatile geometric variation-regularized Levenberg-Marquardt-type method
applicable to a variety of shape functionals, depending on intrinsic properties
of the surface such as normal field and curvature as well as its embedding into
space. Promising experimental results are reported
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