11,140 research outputs found
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
Recent advances in parametric nonlinear model order reduction: treatment of shocks, contact and interfaces, structure-preserving hyper reduction, acceleration of multiscale formulations, and application to design optimization
International audienceParametric, projection-based, Model Order Reduction (MOR) is a mathematical tool for constructing a parametric Reduced-Order Model (ROM) by projecting a given parametric High Dimensional Model (HDM) onto a Reduced-Order Basis (ROB). It is rapidly becoming indispensable for a large number of applications including, among others, computational-based design and optimization, multiscale analysis, statistical analysis, uncertainty quantification, and model predictive control. It is also essential for scenarios where real-time simulation responses are desired. During the last two decades, linear, projection-based, parametric MOR has matured and made a major impact in many fields of engineering including electrical engineering, acoustics, and structural acoustics, to name only a few. By comparison, nonlinear, projection-based, parametric MOR remains somehow in its infancy. Nevertheless, giant strides have been recently achieved in many of its theoretical, algorithmic, and offline/online organizational aspects. The main purpose of this lecture is to highlight some of these advances, discuss their mathematical and computer science underpinnings, and report on their impact for an important class of problems in aerodynamics, fluid mechanics, nonlinear solid mechanics and structural dynamics, failure analysis, multiscale analysis, uncertainty quantification, and design optimization. To this effect, nonlinear, projection-based, parametric MOR will be first interpreted as a constrained semidiscretization on a subset of a compact Stiefel manifold, using a low-dimensional basis of global shape functions constructed a posteriori — that is, after some knowledge about the response of the system of interest has been developed. Usually, such a knowledge is gathered using the given parametric HDM and an offline training procedure where the model parameters are sampled with a greedy strategy based on a cost-effective ROM error indicator. Specifically, a set of problems related to the parametric problem of interest are solved at the sampled parameter points using the given HDM, and the computed solution snapshots are compressed to obtain the desired global ROB. Depending on the mathematical type of the governing equations underlying the given HDM, a dual ROB is also constructed and the sought-after nonlinear parametric ROM is constructed by Galerkin (or Petrov-Galerkin) projection of the HDM onto the global ROB (and its dual counterpart)
Dirichlet sigma models and mean curvature flow
The mean curvature flow describes the parabolic deformation of embedded
branes in Riemannian geometry driven by their extrinsic mean curvature vector,
which is typically associated to surface tension forces. It is the gradient
flow of the area functional, and, as such, it is naturally identified with the
boundary renormalization group equation of Dirichlet sigma models away from
conformality, to lowest order in perturbation theory. D-branes appear as fixed
points of this flow having conformally invariant boundary conditions. Simple
running solutions include the paper-clip and the hair-pin (or grim-reaper)
models on the plane, as well as scaling solutions associated to rational (p, q)
closed curves and the decay of two intersecting lines. Stability analysis is
performed in several cases while searching for transitions among different
brane configurations. The combination of Ricci with the mean curvature flow is
examined in detail together with several explicit examples of deforming curves
on curved backgrounds. Some general aspects of the mean curvature flow in
higher dimensional ambient spaces are also discussed and obtain consistent
truncations to lower dimensional systems. Selected physical applications are
mentioned in the text, including tachyon condensation in open string theory and
the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure
Solving the incompressible surface Navier-Stokes equation by surface finite elements
We consider a numerical approach for the incompressible surface Navier-Stokes
equation on surfaces with arbitrary genus . The approach is
based on a reformulation of the equation in Cartesian coordinates of the
embedding , penalization of the normal component, a Chorin
projection method and discretization in space by surface finite elements for
each component. The approach thus requires only standard ingredients which most
finite element implementations can offer. We compare computational results with
discrete exterior calculus (DEC) simulations on a torus and demonstrate the
interplay of the flow field with the topology by showing realizations of the
Poincar\'e-Hopf theorem on -tori
Hydrodynamic interactions in polar liquid crystals on evolving surfaces
We consider the derivation and numerical solution of the flow of passive and
active polar liquid crystals, whose molecular orientation is subjected to a
tangential anchoring on an evolving curved surface. The underlying passive
model is a simplified surface Ericksen-Leslie model, which is derived as a
thin-film limit of the corresponding three-dimensional equations with
appropriate boundary conditions. A finite element discretization is considered
and the effect of hydrodynamics on the interplay of topology, geometric
properties and defect dynamics is studied for this model on various stationary
and evolving surfaces. Additionally, we consider an active model. We propose a
surface formulation for an active polar viscous gel and exemplarily demonstrate
the effect of the underlying curvature on the location of topological defects
on a torus
Revisiting the problem of a crack impinging on an interface: A modeling framework for the interaction between the phase field approach for brittle fracture and the interface cohesive zone model
Artículo Open Access en el sitio web del editor. Pago por publicar en abierto.The problem of a crack impinging on an interface has been thoroughly investigated in the last three decades due to its important role in the mechanics and physics of solids. In the current investigation, this problem is revisited in view of the recent progresses on the phase field approach of brittle fracture. In this concern, a novel formulation combining the phase field approach for modeling brittle fracture in the bulk and a cohesive zone model for pre-existing adhesive interfaces is herein proposed to investigate the competition between crack penetration and deflection at an interface. The model, implemented within the finite element method framework using a monolithic fully implicit solution strategy, is applied to provide a further insight into the understanding of the role of model parameters on the above competition. In particular, in this study, the role of the fracture toughness ratio between the interface and the adjoining bulks and of the characteristic fracture-length scales of the dissipative models is analyzed. In the case of a brittle interface, the asymptotic predictions based on linear elastic fracture mechanics criteria for crack penetration, single deflection or double deflection are fully captured by the present method. Moreover, by increasing the size of the process zone along the interface, or by varying the internal length scale of the phase field model, new complex phenomena are emerging, such as simultaneous crack penetration and deflection and the transition from single crack penetration to deflection and penetration with subsequent branching into the bulk. The obtained computational trends are in very good agreement with previous experimental observations and the theoretical considerations on the competition and interplay between both fracture mechanics models open new research perspectives for the simulation and understanding of complex fracture patterns.Unión Europea FP/2007-2013/ERC 306622Ministerio de Economía y Competitividad DPI2012-37187, MAT2015-71036-P y MAT2015-71309-PJunta de Andalucía P11-TEP-7093 y P12-TEP- 105
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