4,782 research outputs found
Arbitrary Lagrangian–Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces
An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors
On Approximations of the Curve Shortening Flow and of the Mean Curvature Flow based on the DeTurck trick
In this paper we discuss novel numerical schemes for the computation of the
curve shortening and mean curvature flows that are based on special
reparametrizations. The main idea is to use special solutions to the harmonic
map heat flow in order to reparametrize the equations of motion. This idea is
widely known from the Ricci flow as the DeTurck trick. By introducing a
variable time scale for the harmonic map heat flow, we obtain families of
numerical schemes for the reparametrized flows. For the curve shortening flow
this family unveils a surprising geometric connection between the numerical
schemes in [5] and [9]. For the mean curvature flow we obtain families of
schemes with good mesh properties similar to those in [3]. We prove error
estimates for the semi-discrete scheme of the curve shortening flow. The
behaviour of the fully-discrete schemes with respect to the redistribution of
mesh points is studied in numerical experiments. We also discuss possible
generalizations of our ideas to other extrinsic flows
Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations
We introduce a new method, the Local Monge Parametrizations (LMP) method, to
approximate tensor fields on general surfaces given by a collection of local
parametrizations, e.g.~as in finite element or NURBS surface representations.
Our goal is to use this method to solve numerically tensor-valued partial
differential equations (PDE) on surfaces. Previous methods use scalar
potentials to numerically describe vector fields on surfaces, at the expense of
requiring higher-order derivatives of the approximated fields and limited to
simply connected surfaces, or represent tangential tensor fields as tensor
fields in 3D subjected to constraints, thus increasing the essential number of
degrees of freedom. In contrast, the LMP method uses an optimal number of
degrees of freedom to represent a tensor, is general with regards to the
topology of the surface, and does not increase the order of the PDEs governing
the tensor fields. The main idea is to construct maps between the element
parametrizations and a local Monge parametrization around each node. We test
the LMP method by approximating in a least-squares sense different vector and
tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply
the LMP method to two physical models on surfaces, involving a tension-driven
flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP
method thus solves the long-standing problem of the interpolation of tensors on
general surfaces with an optimal number of degrees of freedom.Comment: 16 pages, 6 figure
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
Modeling Shallow Water Flows on General Terrains
A formulation of the shallow water equations adapted to general complex
terrains is proposed. Its derivation starts from the observation that the
typical approach of depth integrating the Navier-Stokes equations along the
direction of gravity forces is not exact in the general case of a tilted curved
bottom. We claim that an integration path that better adapts to the shallow
water hypotheses follows the "cross-flow" surface, i.e., a surface that is
normal to the velocity field at any point of the domain. Because of the
implicitness of this definition, we approximate this "cross-flow" path by
performing depth integration along a local direction normal to the bottom
surface, and propose a rigorous derivation of this approximation and its
numerical solution as an essential step for the future development of the full
"cross-flow" integration procedure. We start by defining a local coordinate
system, anchored on the bottom surface to derive a covariant form of the
Navier-Stokes equations. Depth integration along the local normals yields a
covariant version of the shallow water equations, which is characterized by
flux functions and source terms that vary in space because of the surface
metric coefficients and related derivatives. The proposed model is discretized
with a first order FORCE-type Godunov Finite Volume scheme that allows
implementation of spatially variable fluxes. We investigate the validity of our
SW model and the effects of the bottom geometry by means of three synthetic
test cases that exhibit non negligible slopes and surface curvatures. The
results show the importance of taking into consideration bottom geometry even
for relatively mild and slowly varying curvatures
An adaptive octree finite element method for PDEs posed on surfaces
The paper develops a finite element method for partial differential equations
posed on hypersurfaces in , . The method uses traces of
bulk finite element functions on a surface embedded in a volumetric domain. The
bulk finite element space is defined on an octree grid which is locally refined
or coarsened depending on error indicators and estimated values of the surface
curvatures. The cartesian structure of the bulk mesh leads to easy and
efficient adaptation process, while the trace finite element method makes
fitting the mesh to the surface unnecessary. The number of degrees of freedom
involved in computations is consistent with the two-dimension nature of surface
PDEs. No parametrization of the surface is required; it can be given implicitly
by a level set function. In practice, a variant of the marching cubes method is
used to recover the surface with the second order accuracy. We prove the
optimal order of accuracy for the trace finite element method in and
surface norms for a problem with smooth solution and quasi-uniform mesh
refinement. Experiments with less regular problems demonstrate optimal
convergence with respect to the number of degrees of freedom, if grid
adaptation is based on an appropriate error indicator. The paper shows results
of numerical experiments for a variety of geometries and problems, including
advection-diffusion equations on surfaces. Analysis and numerical results of
the paper suggest that combination of cartesian adaptive meshes and the
unfitted (trace) finite elements provide simple, efficient, and reliable tool
for numerical treatment of PDEs posed on surfaces
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