The Helmholtz equation $-\nabla\cdot (a\nabla u) - \omega^2 u = f$ is considered in an unbounded wave guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$ , $S\subset \mathbb{R}^{d-1}$ a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction $x_1 \in \mathbb{R}$ or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies $\omega$ , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.

For a fixed k ∈ {1, …, d}, consider arbitrary random vectors X0, …, Xk ∈ ℝd such that the (k + 1)-tuples (UX0, …, UXk) have the same distribution for any rotation U. Let A be any nonsingular d × d matrix. We show that the k-dimensional volume of the convex hull of affinely transformed Xi satisfies \[|{\rm{conv}}(A{X_{\rm{0}}} \ldots ,A{X_k}){\rm{|}}\mathop {\rm{ = }}\limits^{\rm{D}} (|{P_\xi }\varepsilon |/{\kappa _k})|{\rm{conv}}\left( {{X_0}, \ldots ,{X_k}} \right)\], where ɛ:= {x ∈ ℝd : x┬ (A┬A)−1x ≤ 1} is an ellipsoid, Pξ denotes the orthogonal projection to a uniformly chosen random k-dimensional linear subspace ξ independent of X0, …, Xk, and κk is the volume of the unit k-dimensional ball. As an application, we derive the following integral geometry formula for ellipsoids: ck,d,p∫Ad,k |ɛ ∩ E|p+d+1μd,k(dE) = |ɛ|k+1∫Gd,k |PLɛ|pνd,k(dL), where $c_{k,d,p} = \big({\kappa_{d}^{k+1}}/{\kappa_k^{d+1}}\big) ({\kappa_{k(d+p)+k}}/{\kappa_{k(d+p)+d}})$. Here p > −1 and Ad,k and Gd,k are the affine and the linear Grassmannians equipped with their respective Haar measures. The p = 0 case reduces to an affine version of the integral formula of Furstenberg and Tzkoni (1971).

We study the error analysis of the weak Galerkin finite element method in [24, 38] (WG-FEM) for the Helmholtz problem with large wave number in two and three dimensions. Using a modified duality argument proposed by Zhu and Wu, we obtain the pre-asymptotic error estimates of the WG-FEM. In particular, the error estimates with explicit dependence on the wave number k are derived. This shows that the pollution error in the broken H1-norm is bounded by under mesh condition k7/2h2≤C0 or (kh)2+k(kh)p+1≤C0, which coincides with the phase error of the finite element method obtained by existent dispersion analyses. Here h is the mesh size, p is the order of the approximation space and C0 is a constant independent of k and h. Furthermore, numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the WG-FEM in reducing the pollution effect.

The coordinate transformation offers a remarkable way to design cloaks that can steer electromagnetic fields so as to prevent waves from penetrating into the cloaked region (denoted by Ω0, where the objects inside are invisible to observers outside). The ideal circular and elliptic cylindrical cloaked regions are blown up from a point and a line segment, respectively so the transformed material parameters and the corresponding coefficients of the resulted equations are highly singular at the cloaking boundary ∂Ω0. The electric field or magnetic field is not continuous across ∂Ω0. The imposition of appropriate cloaking boundary conditions (CBCs) to achieve perfect concealment is a crucial but challenging issue.

Based upon the principle that a well-behaved electromagnetic field in the original space must be well-behaved in the transformed space as well, we obtain CBCs that intrinsically relate to the essential “pole” conditions of a singular transformation. We also find that for the elliptic cylindrical cloak, the CBCs should be imposed differently for the cosine-elliptic and sine-elliptic components of the decomposed fields. With these at our disposal, we can rigorously show that the governing equation in Ω0 can be decoupled from the exterior region , and the total fields in the cloaked region vanish under mild conditions. We emphasize that our proposal of CBCs is different from any existing ones.

Using the exact circular (resp., elliptic) Dirichlet-to-Neumann (DtN) non-reflecting boundary conditions to reduce the unbounded domain to a bounded domain, we introduce an accurate and efficient Fourier-Legendre spectral-element method (FLSEM) (resp., Mathieu-Legendre spectral-element method (MLSEM)) to simulate the circular cylindrical cloak (resp., elliptic cylindrical cloak). We provide ample numerical results to demonstrate that the perfect concealment of waves can be achieved for the ideal circular/elliptic cylindrical cloaks under our proposed CBCs and accurate numerical solvers.

A coupled model, capable of simulating transonic flow, solid heat conduction, species transport, and gas radiation, is developed that provides better computational treatment of infrared radiation from hot exhaust nozzles. The modeling of gas radiation is based on a statistical narrow-band correlated-k analysis, whose parameters are deduced from the HITEMP line-by-line database. To improve computational efficiency, several methods are employed. A mixed analytical-numerical algorithm is described for the stiffness of the two-equation turbulence model and an alternating direction implicit pretreatment for the ill-conditioned matrix appearing in the coupled problem of flow and solid heat conduction. Moreover, an improved multigrid method and a symmetry plane treatment of the radiation transfer-energy equations are also introduced. Four numerical simulations are given to confirm the efficiency and accuracy of the numerical method. Finally, an account of the aerothermodynamics and infrared characteristics for two types of nozzles are presented. The infrared radiation intensity of the Chevron ejecting nozzle is clearly smaller than that of the common axisymmetric ejecting nozzle. All computations can be performed on a personal computer.

In a convex domain K in ℝd, a transmitter and a receiver are placed at random according to the uniform distribution. The statistics of the power received by the receiver is an important quantity for the design of wireless communication systems. Bounds for the moments of the received power are given, which depend only on the volume and the surface area of the convex domain.