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1.
We propose a $\theta$-$L$ approach for solving a sharp-interface model about simulating solid-state dewetting of thin films with isotropic/weakly anisotropic surface energies. The sharp-interface model is governed by surface diffusion and contact line migration. For solving the model, traditional numerical methods usually suffer from the severe stability constraint and/or the mesh distribution trouble. In the $\theta$-$L$ approach, we introduce a useful tangential velocity along the evolving interface and utilize a new set of variables (i.e., the tangential angle $\theta$ and the total length $L$ of the interface curve), so that it not only could reduce the stiffness resulted from the surface tension, but also could ensure the mesh equidistribution property during the evolution. Furthermore, it can achieve second-order accuracy when implemented by a semi-implicit linear finite element method. Numerical results are reported to demonstrate that the proposed $\theta$-$L$ approach is efficient and accurate. 相似文献
2.
《计算数学(英文版)》2023,41(4):771-796
We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a weak formulation for the problem, in which the contact angle condition is weakly enforced. By using piecewise linear elements in space and backward Euler method in time, we then discretize the formulation to obtain a parametric finite element approximation, where the interface and its contact line are evolved simultaneously. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form. Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting. 相似文献
3.
S. Morigi 《Journal of Computational and Applied Mathematics》2010,233(5):1277-1287
Surface processing tools based on Partial Differential Equations (PDEs) are useful in a variety of applications in computer graphics, digital animation, computer aided modelling, and computer vision. In this work, we deal with computational issues arising from the discretization of geometric PDE models for the evolution of surfaces, considering both normal and tangential velocities. The evolution of the surface is formulated in a Lagrangian framework. We propose several strategies for tangential velocities, yielding uniform redistribution of mesh points along the evolving family of surfaces, preventing computational instabilities and increasing the mesh regularity. Numerical schemes based on finite co-volume approximation in space will be considered. Finally, we describe how this framework may be employed in applications such as mesh regularization, morphing, and features preserving surface smoothing. 相似文献
4.
Robert Eymard Danielle Hilhorst Martin Vohralík 《Numerical Methods for Partial Differential Equations》2010,26(3):612-646
We propose and analyze in this paper a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and source terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell‐centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the conforming piecewise linear finite element method. In this way, the scheme is fully consistent and the discrete solution is naturally continuous across the interfaces between the subdomains with nonmatching grids, without introducing any supplementary equations and unknowns or using any interpolation at the interfaces. We allow for general inhomogeneous and anisotropic diffusion–dispersion tensors, propose two variants corresponding respectively to arithmetic and harmonic averaging, and use the local Péclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection‐dominated case. The scheme is robust, efficient since it leads to positive definite matrices and one unknown per element, locally conservative, and satisfies the discrete maximum principle under the conditions on the simplicial mesh and the diffusion tensor usual in the finite element method. We prove its convergence using a priori estimates and the Kolmogorov relative compactness theorem and illustrate its behavior on a numerical experiment. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 相似文献
5.
Anisotropic mesh refinement
in stabilized Galerkin methods 总被引:8,自引:0,他引:8
Summary.
The numerical solution of a convection-diffusion-reaction model problem is
considered in two and three dimensions. A stabilized finite element method
of Galerkin/Least-square type accomodates diffusion-dominated as well as
convection- and/or reaction-dominated situations. The resolution of
boundary layers occuring in the singularly perturbed case is achieved
using anisotropic mesh refinement in boundary layer regions.
In this paper, the
standard analysis of the stabilized Galerkin method on isotropic meshes
is extended to more general meshes with boundary layer refinement.
Simplicial Lagrangian elements of arbitrary order are used.
Received
March 6, 1995 / Revised version received August 18,
1995 相似文献
6.
The efficient generation of meshes is an important component in the numerical
solution of problems in physics and engineering. Of interest are situations where
global mesh quality and a tight coupling to the solution of the physical partial differential
equation (PDE) is important. We consider parabolic PDE mesh generation
and present a method for the construction of adaptive meshes in two spatial dimensions
using stochastic domain decomposition that is suitable for an implementation
in a multi- or many-core environment. Methods for mesh generation on periodic domains
are also provided. The mesh generator is coupled to a time dependent physical
PDE and the system is evolved using an alternating solution procedure. The method
uses the stochastic representation of the exact solution of a parabolic linear mesh generator
to find the location of an adaptive mesh along the (artificial) subdomain interfaces.
The deterministic evaluation of the mesh over each subdomain can then be
obtained completely independently using the probabilistically computed solutions as
boundary conditions. A small scaling study is provided to demonstrate the parallel
performance of this stochastic domain decomposition approach to mesh generation.
We demonstrate the approach numerically and compare the mesh obtained with the
corresponding single domain mesh using a representative mesh quality measure. 相似文献
7.
A Study on the Conditioning of Finite Element Equations with Arbitrary Anisotropic Meshes via a Density Function Approach 下载免费PDF全文
Lennard Kamenski & Weizhang Huang 《数学研究》2014,47(2):151-172
The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used to develop bounds on the smallest eigenvalue and the condition number that are sharper than existing estimates in one and two dimensions and comparable in three and higher dimensions. The new results reveal that the mesh concentration near the boundary has less influence on the condition number than the mesh concentration in the interior of the domain. This is especially true for the Jacobi preconditioned system where the former has little or almost no influence on the condition number. Numerical examples are presented. 相似文献
8.
Miyuki Koiso Bennett Palmer 《Calculus of Variations and Partial Differential Equations》2012,43(3-4):555-587
We study the stability of surfaces trapped between two parallel planes with free boundary on these planes. The energy functional consists of anisotropic surface energy, wetting energy, and line tension. Equilibrium surfaces are surfaces with constant anisotropic mean curvature. We study the case where the Wulff shape is of “product form”, that is, its horizontal sections are all homothetic and have a certain symmetry. Such an anisotropic surface energy is a natural generalization of the area of the surface. In particular, we study the stability of parts of anisotropic Delaunay surfaces which arise as equilibrium surfaces. They are surfaces of the same product form of the Wulff shape. We show that, for these surfaces, the stability analysis can be reduced to the case where the surface is axially symmetric and the functional is replaced by an appropriate axially symmetric one. Moreover, we obtain necessary and sufficient conditions for the stability of anisotropic sessile drops. 相似文献
9.
Victoria Hernández-Mederos Pedro L. del Ángel Jorge Estrada-Sarlabous 《Numerical Algorithms》2008,48(1-3):29-47
In this paper we propose an advancing front method for generating an isotropic triangular mesh on a regular parametric surface. Starting from a point on the surface, the method computes a set of points in the intersection curve between the surface and the sphere centered at that point with a prescribed radius. From this set we select the vertices of a cell composed by triangles approximately equilateral. The mesh grows repeating the described computation with boundary vertices of the cell as starting points. Compared to methods proposed by other authors, the current method may be considered as an improvement, since it is more efficient and flexible. Furthermore, the resulting mesh is closer to being isotropic. Additionally, we obtain a sufficient condition ensuring that a surface triangulation is of Delaunay type. 相似文献
10.
Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit
layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce
accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified
SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it
yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the
severe nonuniformity of the mesh. We give local error estimates that hold true uniformly in the perturbation parameter , provided only that , where mesh points are used. Numerical experiments support these theoretical results.
Received February 19, 1999 / Revised version received January 27, 2000 / Published online August 2, 2000 相似文献
11.
This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, the interface surface tension serves as the jump condition or the Dirichlet boundary condition of the pressure, and the pressure is solved with the interface-fitted finite element method developed in this work. In this study, a level-set function is used to capture the evolution of the interface and provide the interface location for the interface fitting. 相似文献
12.
We present a spline approximation method for a piece of a surface where jump discontinuities occur along curves. The data for the surface is assumed to be Fourier coefficients which are limited in order and possibly contaminated with noise. The support of the approximation is bounded by three sides of a rectangle with a fourth boundary possibly curved. Discontinuities of the surface may occur across the curved side and linear sides adjacent to it. The approximation uses a small number of lines through the support and parallel to the straight boundary lines that are adjacent to the curve. Along each line a one-dimensional spline approximation is done for a section of the surface over the line. This approximation uses two-dimensional Fourier coefficient data, localizing spline functions, and a technique which we developed earlier for one-dimensional analogues of the problem. We use a spline quasi-interpolation scheme to create a surface approximation from the section approximations. The result is accurate even when the surface is discontinuous across the curved boundary and adjacent side boundaries. 相似文献
13.
This paper presents a simple method for computing the intersection curve of a ruled surface and a free-form surface. The basic
idea is to reduce the problem of surface intersection to the one of projecting an appropriate curve such as a directrix of
the ruled surface, along its indicatrix curve (direction vector field of its generating lines), onto the free-form surface;
the projection curve is just the intersection curve. With techniques in classical differential geometry, we derive the differential
equations of the intersection curve in the cases of parametrically and implicitly defined free-form surfaces. The intersection
curve naturally inherits the parameter of the chosen directrix. Moreover, it is independent of the base surface geometry and
its parameterization, and is obtained by numerically solving the initial-value problem for a system of first-order ordinary
differential equations in the parametric domain associated to the surface representation for parametric case or in 3D space
for implicit case. Some experimental examples are also given to demonstrate that the presented method is effective and potentially
useful in computer aided design and computer graphics.
An erratum to this article can be found at 相似文献
14.
John W. Barrett Harald Garcke Robert Nürnberg 《Numerical Methods for Partial Differential Equations》2011,27(1):1-30
On the basis of our previous work, we introduce novel fully discrete, fully practical parametric finite element approximations for geometric evolution equations of curves in the plane. The fully implicit approximations are unconditionally stable and intrinsically equidistribute the vertices at each time level. We present iterative solution methods for the systems of nonlinear equations arising at each time level and present several numerical results. The ideas easily generalize to the evolution of curve networks and to anisotropic surface energies. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 相似文献
15.
Gerhard Dziuk 《manuscripta mathematica》1981,36(2):241-251
Consider a surface x, bounded by a smooth surface in Euclidean n-space and by a smooth curve having its end points on the boundary surface. Let x minimize Dirichlet's integral in the class of all such surfaces x. Then it is known that the trace of x on the boundary surface is a continuous curve. We treat the case where x is a stationary point of Dirichlet's integral and prove continuity in this case. 相似文献
16.
We describe a scheme for reconstructing a distribution of fluidvelocities from given values of the net flow across the facesof a rectangular mesh in two or three dimensions. Assuming thedata to be consistent, the reconstructed velocities will havethe properties that the continuity condition is everywhere satisfied,that the track of any fluid particle is a smooth curve and thatthere is no flow across any part of the boundary that has beenspecified as representing a solid wall. 相似文献
17.
18.
Norbert Heuer 《Applicable analysis》2013,92(3-4):411-437
We propose and analyze preconditioners for the p-version of the boundary element method in three dimensions. We consider indefinite hypersingular integral equations on surfaces and use quadrilateral elements for the boundary discretization. We use the GMRES method as iterative solver for the linear systems and prove for an overlapping additive Schwarz method that the number of iterations is bounded. This bound is independent of the polynomial degree of the ansatz functions and of the size of the underlying mesh. For a modified diagonal scaling, which uses special basis functions, we prove that the number of iterations grows only polylogarithmically in the polynomial degree. Here, a sufficiently fine mesh is required. Numerical results supporting the theory are presented. 相似文献
19.
The boundary element spline collocation method is studied for the time-fractional diffusion equation in a bounded two-dimensional
domain. We represent the solution as the single layer potential which leads to a Volterra integral equation of the first kind.
We discretize the boundary integral equation with the spline collocation method on uniform meshes both in spatial and time
variables. In the stability analysis we utilize the Fourier analysis technique developed for anisotropic pseudodifferential
equations. We prove that the collocation solution is quasi-optimal under some stability condition for the mesh parameters.
We have to assume that the mesh parameter in time satisfies
(ht=c h\frac2a)(h_t=c h^{\frac{2}{\alpha}}), where (h) is the spatial mesh parameter. 相似文献
20.
Edges and surface boundaries are often the most relevant features in images and multidimensional data. It is well known that multiscale methods including wavelets and their more sophisticated multidimensional siblings offer a powerful tool for the analysis and detection of such sets. Among such methods, the continuous shearlet transform has been especially successful. This method combines anisotropic scaling and directional sensitivity controlled by shear transformations in order to precisely identify not only the location of edges and boundary points but also edge orientation and corner points. In this paper, we show that this framework can be made even more flexible by controlling the scaling parameter of the anisotropic dilation matrix and considering non-parabolic scaling. We prove that, using ‘higher-than-parabolic’ scaling, the modified shearlet transform is also able to estimate the degree of local flatness of an edge or surface boundary, providing more detailed information about the geometry of edge and boundary points. 相似文献