10,744 research outputs found
Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrability
Many engineering and physiological applications deal with situations when a
fluid is moving in flexible tubes with elastic walls. In the real-life
applications like blood flow, there is often an additional complexity of
vorticity being present in the fluid. We present a theory for the dynamics of
interaction of fluids and structures. The equations are derived using the
variational principle, with the incompressibility constraint of the fluid
giving rise to a pressure-like term. In order to connect this work with the
previous literature, we consider the case of inextensible and unshearable tube
with a straight centerline. In the absence of vorticity, our model reduces to
previous models considered in the literature, yielding the equations of
conservation of fluid momentum, wall momentum and the fluid volume. We show
that even when the vorticity is present, but is kept at a constant value, the
case of an inextensible, unshearable and straight tube with elastics walls
carrying a fluid allows an alternative formulation, reducing to a single
compact equation for the back-to-labels map instead of three conservation
equations. That single equation shows interesting instability in solutions when
the vorticity exceeds a certain threshold. Furthermore, the equation in stable
regime can be reduced to Boussinesq-type, KdV and Monge-Amp\`ere equations
equations in several appropriate limits, namely, the first two in the limit of
long time and length scales and the third one in the additional limit of the
small cross-sectional area. For the unstable regime, we numerical solutions
demonstrate the spontaneous appearance of large oscillations in the
cross-sectional area.Comment: 57 pages, 11 figures. arXiv admin note: text overlap with
arXiv:1805.1102
Variational Methods for Biomolecular Modeling
Structure, function and dynamics of many biomolecular systems can be
characterized by the energetic variational principle and the corresponding
systems of partial differential equations (PDEs). This principle allows us to
focus on the identification of essential energetic components, the optimal
parametrization of energies, and the efficient computational implementation of
energy variation or minimization. Given the fact that complex biomolecular
systems are structurally non-uniform and their interactions occur through
contact interfaces, their free energies are associated with various interfaces
as well, such as solute-solvent interface, molecular binding interface, lipid
domain interface, and membrane surfaces. This fact motivates the inclusion of
interface geometry, particular its curvatures, to the parametrization of free
energies. Applications of such interface geometry based energetic variational
principles are illustrated through three concrete topics: the multiscale
modeling of biomolecular electrostatics and solvation that includes the
curvature energy of the molecular surface, the formation of microdomains on
lipid membrane due to the geometric and molecular mechanics at the lipid
interface, and the mean curvature driven protein localization on membrane
surfaces. By further implicitly representing the interface using a phase field
function over the entire domain, one can simulate the dynamics of the interface
and the corresponding energy variation by evolving the phase field function,
achieving significant reduction of the number of degrees of freedom and
computational complexity. Strategies for improving the efficiency of
computational implementations and for extending applications to coarse-graining
or multiscale molecular simulations are outlined.Comment: 36 page
Empirical Determination of Bang-Bang Operations
Strong and fast "bang-bang" (BB) pulses have been recently proposed as a
means for reducing decoherence in a quantum system. So far theoretical analysis
of the BB technique relied on model Hamiltonians. Here we introduce a method
for empirically determining the set of required BB pulses, that relies on
quantum process tomography. In this manner an experimenter may tailor his or
her BB pulses to the quantum system at hand, without having to assume a model
Hamiltonian.Comment: 14 pages, 2 eps figures, ReVTeX4 two-colum
An efficient mixed variational reduced order model formulation for non-linear analyses of elastic shells
The Koiter-Newton method had recently demonstrated a superior performance for non-linear analyses of structures, compared to traditional path-following strategies. The method follows a predictor-corrector scheme to trace the entire equilibrium path. During a predictor step a reduced order model is constructed based on Koiter's asymptotic post-buckling theory which is followed by a Newton iteration in the corrector phase to regain the equilibrium of forces.
In this manuscript, we introduce a robust mixed solid-shell formulation to further enhance the efficiency of stability analyses in various aspects. We show that a Hellinger-Reissner variational formulation facilitates the reduced order model construction omitting an expensive evaluation of the inherent fourth order derivatives of the strain energy. We demonstrate that extremely large step sizes with a reasonable out-of-balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path. More importantly, the numerical effort of the corrector phase involving a Newton iteration of the full order model is drastically reduced thus revealing the true strength of the proposed formulation. We study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems
Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator
This paper introduces a novel method for the efficient and accurate
computation of the volume of a domain whose boundary is given by an orientable
hypersurface which is implicitly given as the iso-contour of a sufficiently
smooth level-set function. After spatial discretization, local approximation of
the hypersurface and application of the Gaussian divergence theorem, the volume
integrals are transformed to surface integrals. Application of the surface
divergence theorem allows for a further reduction to line integrals which are
advantageous for numerical quadrature. We discuss the theoretical foundations
and provide details of the numerical algorithm. Finally, we present numerical
results for convex and non-convex hypersurfaces embedded in cuboidal domains,
showing both high accuracy and thrid- to fourth-order convergence in space.Comment: 25 pages, 17 figures, 3 table
Fast Solvers for Cahn-Hilliard Inpainting
We consider the efficient solution of the modified Cahn-Hilliard equation for binary image inpainting using convexity splitting, which allows an unconditionally gradient stable time-discretization scheme. We look at a double-well as well as a double obstacle potential. For the latter we get a nonlinear system for which we apply a semi-smooth Newton method combined with a Moreau-Yosida regularization technique. At the heart of both methods lies the solution of large and sparse linear systems. We introduce and study block-triangular preconditioners using an efficient and easy to apply Schur complement approximation. Numerical results indicate that our preconditioners work very well for both problems and show that qualitatively better results can be obtained using the double obstacle potential
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