91,650 research outputs found
A computational method for the coupled solution of reaction–diffusion equations on evolving domains and manifolds: application to a model of cell migration and chemotaxis
In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk–surface reaction–diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk–surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane
A moving grid finite element method applied to a model biological pattern generator
Many problems in biology involve growth. In numerical simulations it can therefore be very convenient to employ a moving computational grid on a continuously deforming domain. In this paper we present a novel application of the moving grid finite element method to compute solutions of reaction–diffusion systems in two-dimensional continuously deforming Euclidean domains. A numerical software package has been developed as a result of this research that is capable of solving generalised Turing models for morphogenesis
Viscous Flow in Domains with Corners: Numerical Artifacts, their Origin and Removal
We show that an attempt to compute numerically a viscous flow in a domain
with a piece-wise smooth boundary by straightforwardly applying well-tested
numerical algorithms (and numerical codes based on their use, such as COMSOL
Multiphysics) can lead to spurious multivaluedness and nonintegrable
singularities in the distribution of the fluid's pressure. The origin of this
difficulty is that, near a corner formed by smooth parts of the piece-wise
smooth boundary, in addition to the solution of the inhomogeneous problem,
there is also an eigensolution. For obtuse corner angles this eigensolution (a)
becomes dominant and (b) has a singular radial derivative of velocity at the
corner. A method is developed that uses the knowledge about the eigensolution
to remove multivaluedness and nonintegrability of the pressure. The method is
first explained in the simple case of a Stokes flow in a corner region and then
generalised for the full-scale unsteady Navier-Stokes flow in a domain with a
free surface.Comment: Under consideration for publication in the Journal of Fluid
Mechanics. Figure bouding box problems resolve
Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space
In a previous paper, the authors presented an elemental enriched space to be used in a finite-element framework (EFEM) capable of reproducing kinks and jumps in an unknown function using a fixed mesh in which the jumps and kinks do not coincide with the interelement boundaries. In this previous publication, only scalar transport problems were solved (thermal problems). In the present work, these ideas are generalized to vectorial unknowns, in particular, the incompressible Navier-Stokes equations for multifluid flows presenting internal moving interfaces. The advantage of the EFEM compared with global enrichment is the significant reduction in computing time when the internal interface is moving. In the EFEM, the matrix to be solved at each time step has not only the same amount of degrees of freedom (DOFs) but also the same connectivity between the DOFs. This frozen matrix graph enormously improves the efficiency of the solver. Another characteristic of the elemental enriched space presented here is that it allows a linear variation of the jump, thus improving the convergence rate, compared with other enriched spaces that have a constant variation of the jump. Furthermore, the implementation in any existing finite-element code is extremely easy with the version presented here because the new shape functions are based on the usual finite-element method shape functions for triangles or tetrahedrals, and once the internal DOFs are statically condensed, the resulting elements have exactly the same number of unknowns as the nonenriched finite elements.Peer ReviewedPreprin
Combining Boundary-Conforming Finite Element Meshes on Moving Domains Using a Sliding Mesh Approach
For most finite element simulations, boundary-conforming meshes have
significant advantages in terms of accuracy or efficiency. This is particularly
true for complex domains. However, with increased complexity of the domain,
generating a boundary-conforming mesh becomes more difficult and time
consuming. One might therefore decide to resort to an approach where individual
boundary-conforming meshes are pieced together in a modular fashion to form a
larger domain. This paper presents a stabilized finite element formulation for
fluid and temperature equations on sliding meshes. It couples the solution
fields of multiple subdomains whose boundaries slide along each other on common
interfaces. Thus, the method allows to use highly tuned boundary-conforming
meshes for each subdomain that are only coupled at the overlapping boundary
interfaces. In contrast to standard overlapping or fictitious domain methods
the coupling is broken down to few interfaces with reduced geometric dimension.
The formulation consists of the following key ingredients: the coupling of the
solution fields on the overlapping surfaces is imposed weakly using a
stabilized version of Nitsche's method. It ensures mass and energy conservation
at the common interfaces. Additionally, we allow to impose weak Dirichlet
boundary conditions at the non-overlapping parts of the interfaces. We present
a detailed numerical study for the resulting stabilized formulation. It shows
optimal convergence behavior for both Newtonian and generalized Newtonian
material models. Simulations of flow of plastic melt inside single-screw as
well as twin-screw extruders demonstrate the applicability of the method to
complex and relevant industrial applications
Investigation of mixed element hybrid grid-based CFD methods for rotorcraft flow analysis
Accurate first-principles flow prediction is essential to the design and development of rotorcraft, and while current numerical analysis tools can, in theory, model the complete flow field, in practice the accuracy of these tools is limited by various inherent numerical deficiencies. An approach that combines the first-principles physical modeling capability of CFD schemes with the vortex preservation capabilities of Lagrangian vortex methods has been developed recently that controls the numerical diffusion of the rotor wake in a grid-based solver by employing a vorticity-velocity, rather than primitive variable, formulation. Coupling strategies, including variable exchange protocols are evaluated using several unstructured, structured, and Cartesian-grid Reynolds Averaged Navier-Stokes (RANS)/Euler CFD solvers. Results obtained with the hybrid grid-based solvers illustrate the capability of this hybrid method to resolve vortex-dominated flow fields with lower cell counts than pure RANS/Euler methods
Cauchy-characteristic Evolution of Einstein-Klein-Gordon Systems: The Black Hole Regime
The Cauchy+characteristic matching (CCM) problem for the scalar wave equation
is investigated in the background geometry of a Schwarzschild black hole.
Previously reported work developed the CCM framework for the coupled
Einstein-Klein-Gordon system of equations, assuming a regular center of
symmetry. Here, the time evolution after the formation of a black hole is
pursued, using a CCM formulation of the governing equations perturbed around
the Schwarzschild background. An extension of the matching scheme allows for
arbitrary matching boundary motion across the coordinate grid. As a proof of
concept, the late time behavior of the dynamics of the scalar field is
explored. The power-law tails in both the time-like and null infinity limits
are verified.Comment: To appear in Phys. Rev. D, 9 pages, revtex, 5 figures available at
http://www.astro.psu.edu/users/nr/preprints.htm
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