We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm is quite robust and converges in much fewer iterations and is more efficient than many existing methods.

Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a periodic potential. Using multiscale perturbation methods, the envelope equations of solitary waves near Bloch bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of the second-order dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch modes, the envelope equations lead to a host of solitary wave structures, including reduced-symmetry solitons, dipole-array solitons, vortex-cell solitons, and so on—many of which have not been reported before to our knowledge. It is also shown analytically that the centers of envelope solutions can be positioned at only four possible locations at or between potential peaks. Numerically, families of these solitary waves are directly computed both near and far away from the band edges. Near the band edges, the numerical solutions spread over many lattice sites, and they fully agree with the analytical solutions obtained from the envelope equations. Far away from the band edges, solitary waves are strongly localized, with intensity and phase profiles characteristic of individual families.

Let be the scattering relation on a compact Riemannian manifold M with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let . be the length of that geodesic ray. We study the question of whether the metric g is uniquely determined, up to an isometry, by knowledge of 冃 and . restricted on some subset D. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from D covers T M; and that those geodesics have no conjugate points. Under an additional topological assumption, we prove that 冃 and . restricted to D uniquely recover an isometric copy of g locally near generic metrics, and in particular, near real analytic ones.

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In this article, we present a second-order in time implicit-explicit (IMEX) local discontinuous Galerkin (LDG) method for computing the Cahn-Hilliard equation, which describes the phase separation phenomenon. It is well-known that the Cahn-Hilliard equation has a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. The discretized Cahn-Hilliard system modeled by the IMEX LDG method can inherit the nonlinear stability of the continuous model. We apply a stabilization technique and prove unconditional energy stability of our scheme. Numerical experiments are performed to validate the analysis. Computational efficiency can be significantly enhanced by using this IMEX LDG method with a large time step.