Lightning Solvers for Potential Flows

We present a method for computing potential flows in planar domains. Our approach is based on a new class of techniques, known as “lightning solvers”, which exploit rational function approximation theory in order to achieve excellent convergence rates. The method is particularly suitable for flows in domains with corners where traditional numerical methods fail. We outline the mathematical basis for the method and establish the connection with potential flow theory. In particular, we apply the new solver to a range of classical problems including steady potential flows, vortex dynamics, and free-streamline flows. The solution method is extremely rapid and usually takes just a fraction of a second to converge to a high degree of accuracy. Numerical evaluations of the solutions are performed in a matter of microseconds and can be compressed further with novel algorithms.

Regional Boundary Exact Controllability of the Wave Equation by Strategic Actuators on a Polygonal Domain with Cracks

In this work we prove the exact controllability of the wave equation by acting on a strategic zone of the border of a non-convex polygonal domain with crack. Indeed, by combining two methods: that of Grisvard on the exact controllability on domains with corners and that of EL. Jai on the boundary strategic actutors, this exact controllability result has been proven.

New Laplace and Helmholtz solvers

Numerical algorithms based on rational functions are introduced that solve the Laplace and Helmholtz equations on 2D domains with corners quickly and accurately, despite the corner singularities.

Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians

In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine, in particular, the relationship between the positivity of the Jacobian det ∇u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps. We exhibit explicit twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer $u_{\sigma }: \Omega \to {\open R}^2$ in a model, two-dimensional case. We exploit the Jacobian constraint $\det \nabla u_{\sigma} \gt 0$ a.e. to obtain regularity results that apply ‘up to the boundary’ of domains with corners. It is shown that the unique shear map minimizer has the properties that (i) $\det \nabla u_{\sigma }$ is strictly positive on one part of the domain Ω, (ii) $\det \nabla u_{\sigma } = 0$ necessarily holds on the rest of Ω, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma }$ is not continuous on the whole domain.