Phase field method for mean curvature flow with boundary constraints

International audienceThis paper is concerned with the numerical approximation of mean curvature flow $t \to \Omega(t)$ satisfying an additional inclusion-exclusion constraint $\Omega_1 \subset \Omega(t) \subset \Omega_2$. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a $\Gamma$-convergence result and then show some numerical comparisons of these two different models

Some Anisotropic Viscoelastic Green Functions

In this paper, we compute the closed form expressions of elastody- namic Green functions for three different viscoelastic media with simple type of anisotropy. We follow Burridge et al. [Proc. Royal Soc. of London. 440(1910): (1993)] to express unknown Green function in terms of three scalar functions $\phi_i$, by using the spectral decomposition of the Christoffel tensor associated with the medium. The problem of computing Green function is, thus reduced to the resolution of three scalar wave equations satisfied by $\phi_i$, and subsequent equations with $\phi_i$ as source terms. To describe viscosity effects, we choose an empirical power law model which becomes well known Voigt model for quadratic frequency losses

Approximation of multiphase mean curvature flows with arbitrary nonnegative mobilities

This paper is devoted to the robust approximation with a variational phase field approach of multiphase mean curvature flows with possibly highly contrasted mobilities. The case of harmonically additive mobilities has been addressed recently using a suitable metric to define the gradient flow of the phase field approximate energy. We generalize this approach to arbitrary nonnegative mobilities using a decomposition as sums of harmonically additive mobilities. We establish the consistency of the resulting method by analyzing the sharp interface limit of the flow: a formal expansion of the phase field shows that the method is of second order. We propose a simple numerical scheme to approximate the solutions to our new model. Finally, we present some numerical experiments in dimensions 2 and 3 that illustrate the interest and effectiveness of our approach, in particular for approximating flows in which the mobility of some phases is zero.Comment: 22 pages, 8 figure

Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and it proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.Comment: 38 pages. A new subsection 6.4 is added where we consider the case of random Lam\'e coefficients. We thought this would corrupt the statistical stability of the imaging functional but our calculus shows that this is not the case as long as the random fluctuation is weak so that Born approximation is vali

Discrete maximum principle for a space-time least squares formulation of the transport equation with finite element.

Finite element methods are known to produce spurious oscillations when the transport equation is solved. In this paper, a variational formulation for the transport equation is proposed, and by introducing a positivity constraint combined with a penalization of the total variation of the solution, a discrete maximum principle is verified for lagrange first order finite element methods. Moreover, the oscillations are cancelled

A multiphase Cahn-Hilliard system with mobilities and the numerical simulation of dewetting

We propose in this paper a new multiphase Cahn-Hilliard model with doubly degenerate mobilities. We prove by a formal asymptotic analysis that it approximates with second order accuracy the multiphase surface diffusion flow with mobility coefficients and surface tensions. To illustrate that it lends itself well to numerical approximation, we propose a simple and effective numerical scheme together with a very compact Matlab implementation. We provide the results of various numerical experiments to show the influence of mobility and surface tension coefficients. Thanks to its second order accuracy and its good suitability for numerical implementation, our model is very handy for tackling notably difficult surface diffusion problems. In particular, we show that it can be used very effectively to simulate numerically the dewetting of thin liquid tubes on arbitrary solid supports without requiring nonlinear boundary conditions.Comment: 35 page