Monica Torres

FINITE ELEMENT METHODS FOR NONLINEAR ACOUSTICS IN FLUIDS

In this paper, weak formulations and finite element discretizations of the governing partial differential equations of three-dimensional nonlinear acoustics in absorbing fluids are presented. The fluid equations are considered in an Eulerian framework, rather than a displacement framework, since in the latter case the corresponding finite element formulations suffer from spurious modes and numerical instabilities. When taken with the governing partial differential equations of a solid body and the continuity conditions, a coupled formulation is derived. The change in solid/fluid interface conditions when going from a linear acoustic fluid to a nonlinear acoustic fluid is demonstrated. Finite element discretizations of the coupled problem are then derived, and verification examples are presented that demonstrate the correctness of the implementations. We demonstrate that the time step size necessary to resolve the wave decreases as steepening occurs. Finally, simulation results are presented on a resonating acoustic cavity, and a coupled elastic/acoustic system consisting of a fluid-filled spherical tank.

PLANE-LIKE MINIMAL SURFACES IN PERIODIC MEDIA WITH EXCLUSIONS ∗

We consider minimal surfaces in a medium with exclusions (voids). This extends the results given in [Comm. Pure Appl. Math., 54 (2001), pp. 1403--1441] to the case of a degenerate metric such that the area of a surface of codimension 1 is measured by neglecting the parts inside the exclusions. We prove that, given any plane in the medium, there is at least one minimal surface that always stays at a bounded distance from the plane. We also explore the connections of this problem with the theory of homogenization of Hamilton--Jacobi equations.

On the distributional divergence of vector fields vanishing at infinity

The equation div v = F has a solution v in the space of continuous vector fields vanishing at infinity if and only if F acts linearly on BV m m−1 (Rm) (the space of functions in L m m−1 (Rm) whose distributional gradient is a vector valued measure) and satisfies the following continuity condition: F (uj) converges to zero for each sequence {uj} such that the measure norms of ∇uj are uniformly bounded and uj ⇀ 0 weakly in L m m−1 (Rm).

Level set methods to compute minimal surfaces in a medium with exclusions (voids)

In [59], periodic minimal surfaces in a medium with exclusions (voids) are constructed and in this paper we present two algorithms for computing these minimal surfaces. The two algorithms use evolution of level sets by mean curvature. The first algorithm solves the governing nonlinear PDE directly and enforces numerically an orthogonality condition that the surfaces satisfy when they meet the boundaries of the exclusions. The second algorithm involves h-adaptive finite element approximations of a linear convection-diffusion equation, which has been shown to linearize the governing nonlinear PDE for weighted mean curvature flow.

Morrey spaces and generalized Cheeger sets

Abstract We maximize the functional ∫ E h ⁢ ( x ) ⁢ 𝑑 x P ⁢ ( E ) , \frac{\int_{E}h(x)\,dx}{P(E)}, where E ⊂ Ω ¯ {E\subset\overline{\Omega}} is a set of finite perimeter, Ω is an open bounded set with Lipschitz boundary and h is nonnegative. Solutions to this problem are called generalized Cheeger sets in Ω. We show that the Morrey spaces L 1 , λ ⁢ ( Ω ) {L^{1,\lambda}(\Omega)} , λ ≥ n - 1 {\lambda\geq n-1} , are natural spaces to study this problem. We prove that if h ∈ L 1 , λ ⁢ ( Ω ) {h\in L^{1,\lambda}(\Omega)} , λ > n - 1 {\lambda>n-1} , then generalized Cheeger sets exist. We also study the embedding of Morrey spaces into L p {L^{p}} spaces. We show that, for any 0 < λ < n {0<\lambda<n} , the Morrey space L 1 , λ ⁢ ( Ω ) {L^{1,\lambda}(\Omega)} is not contained in any L q ⁢ ( Ω ) {L^{q}(\Omega)} , 1 < q < p = n n - λ {1<q<p=\frac{n}{n-\lambda}} . We also show that if h ∈ L 1 , λ ⁢ ( Ω ) {h\in L^{1,\lambda}(\Omega)} , λ > n - 1 {\lambda>n-1} , then the reduced boundary in Ω of a generalized Cheeger set is C 1 , α {C^{1,\alpha}} and the singular set has Hausdorff dimension at most n - 8 {n-8} (empty if n ≤ 7 {n\leq 7} ). For the critical case h ∈ L 1 , n - 1 ⁢ ( Ω ) {h\in L^{1,n-1}(\Omega)} , we demonstrate that this strong regularity fails. We prove that a bounded generalized Cheeger set E in ℝ n {\mathbb{R}^{n}} with h ∈ L 1 ⁢ ( ℝ n ) {h\in L^{1}(\mathbb{R}^{n})} is always pseudoconvex, and any pseudoconvex set is a generalized Cheeger set for some h.