64 research outputs found
Multiscale Problems in Solidification Processes
Our objective is to describe solidification phenomena in alloy systems. In the classical approach, balance equations in the phases are coupled to conditions on the phase boundaries which are modelled as moving hypersurfaces. The Gibbs-Thomson condition ensures that the evolution is consistent with thermodynamics. We present a derivation of that condition by defining the motion via a localized gradient flow of the entropy.
Another general framework for modelling solidification of alloys with multiple phases and components is based on the phase field approach. The phase boundary motion is then given by a system of Allen-Cahn type equations for order parameters. In the sharp interface limit, i.e., if the smallest length scale ± related to the thickness of the diffuse phase boundaries converges to zero, a model with moving boundaries is recovered. In the case of two phases
it can even be shown that the approximation of the sharp interface model by the phase field model is of second order in ±. Nowadays it is not possible to simulate the microstructure evolution in a whole workpiece. We present a two-scale model derived by homogenization methods including a mathematical justification by an estimate of the model error
A finite element method for a fourth order surface equation with application to the onset of cell blebbing
A variational problem for a fourth order parabolic surface partial differential equation is discussed. It contains nonlinear lower order terms, on which we only make abstract assumptions, and which need to be defined for specified problems.We derive a semi-discrete scheme based on the surface finite element method, show a-priori error estimates, and use the analytical results to prove well-posedness. Furthermore, we present a computational framework where specific problems can be conveniently implemented and, later on, altered with relative ease. It uses a domain specific language implemented in Python. The high level program control can also be done within the Python scripting environment. The computationally expensive step of evolving the solution over time is carried out by binding to an efficient C++ software back-end. The study is motivated by cell blebbing, which can be instrumental for cell migration. Starting with a force balance for the cell membrane, we derive a continuum model for some mechanical and geometrical aspects of the onset of blebbing in a form that fits into the abstract framework.
It is flexible in that it allows for amending force contributions related to membrane tension or the presence of linker molecules between membrane and cell cortex. Cell membrane geometries given in terms of a parametrisation or obtained from image data can be accounted for by the software. The use of a domain specific language to describe the model makes is straightforward to add additional effects such as reaction-diffusion equations modelling some biochemistry on the cell membrane.Some numerical simulations illustrate the approach
Analysis of the discontinuous Galerkin method for elliptic problems on surfaces
We extend the discontinuous Galerkin (DG) framework to a linear second-order
elliptic problem on a compact smooth connected and oriented surface. An
interior penalty (IP) method is introduced on a discrete surface and we derive
a-priori error estimates by relating the latter to the original surface via the
lift introduced in Dziuk (1988). The estimates suggest that the geometric error
terms arising from the surface discretisation do not affect the overall
convergence rate of the IP method when using linear ansatz functions. This is
then verified numerically for a number of test problems. An intricate issue is
the approximation of the surface conormal required in the IP formulation,
choices of which are investigated numerically. Furthermore, we present a
generic implementation of test problems on surfaces.Comment: 21 pages, 4 figures. IMA Journal of Numerical Analysis 2013, Link to
publication: http://imajna.oxfordjournals.org/cgi/content/abstract/drs033?
ijkey=45b23qZl5oJslZQ&keytype=re
An abstract framework for parabolic PDEs on evolving spaces
We present an abstract framework for treating the theory of well-posedness of
solutions to abstract parabolic partial differential equations on evolving
Hilbert spaces. This theory is applicable to variational formulations of PDEs
on evolving spatial domains including moving hypersurfaces. We formulate an
appropriate time derivative on evolving spaces called the material derivative
and define a weak material derivative in analogy with the usual time derivative
in fixed domain problems; our setting is abstract and not restricted to
evolving domains or surfaces. Then we show well-posedness to a certain class of
parabolic PDEs under some assumptions on the parabolic operator and the data.Comment: 38 pages. Minor typos correcte
Elastic flow interacting with a lateral diffusion process : the one-dimensional graph case
A finite element approach to the elastic flow of a curve coupled with a diffusion equation on the curve is analysed. Considering the graph case, the problem is weakly formulated and approximated with continuous linear finite elements, which is enabled thanks to second-order operator splitting. The error analysis builds up on previous results for the elastic flow. To obtain an error estimate for the quantity on the curve a better control of the velocity is required. For this purpose, a penalty approach is employed and then combined with a generalized Gronwall lemma. Numerical simulations support the theoretical convergence results. Further numerical experiments indicate stability beyond the parameter regime with respect to the penalty term that is covered by the theory
Curve shortening flow coupled to lateral diffusion
We present and analyze a semi-discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve. More precisely, curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field. The scheme is based on ideas of [Dziuk, G. Discrete anisotropic curve shortening flow, SIAM J. Numer. Anal. 36, 6 (1999), 1808–1830] for the curve shortening flow and [Dziuk, G., and Elliott, C. M. Finite elements on evolving surfaces, IMA J. Numer. Anal. 27, 2 (2007), 262–292] for the parabolic equation on the moving curve. Additional estimates are required in order to show convergence, most notably with respect to the length element: While in [Dziuk, G. Discrete anisotropic curve shortening flow, SIAM J. Numer. Anal. 36, 6 (1999), 1808–1830] an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation. Numerical simulation results support the theoretical findings
Analysis of the diffuse domain approach for a bulk-surface coupled PDE system
We analyse a diffuse interface type approximation, known as the diffuse domain approach, of a linear coupled bulk-surface elliptic \textcolor{blue}{PDE} system. The well-posedness of the diffuse domain approximation is shown using weighted Sobolev spaces and we prove that the solution to the diffuse domain approximation converges weakly to the solution of the coupled bulk-surface elliptic system as the approximation parameter tends to zero. Moreover, we can show strong convergence for the bulk quantity, while for the surface quantity, we can show norm convergence and strong convergence in a weighted Sobolev space. %Our analysis also covers a second order surface elliptic partial differential equation and a bulk elliptic partial differential equation with Dirichlet, Neumann and Robin boundary condition
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