Keywords:Isogeometric analysis Injectivity of B-spline parameterization Constraint optimization method A posteriori error estimation r-Refinement a b s t r a c t Parameterization of computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis. In this paper, we investigate this problem in the 2D case, i.e., how to parametrize the computational domains by planar B-spline surface from the given CAD objects (four boundary planar B-spline curves). Firstly, two kinds of sufficient conditions for injective B-spline parameterization are derived with respect to the control points. Then we show how to find good parameterization of computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of planar B-spline parameterization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of planar B-spline parameterization. By using this method, the resulted parameterization has no self-intersections, and the isoparametric net has good uniformity and orthogonality. After introducing a posteriori error estimation for isogeometric analysis, we propose r-refinement method to optimize the parameterization by repositioning the inner control points such that the estimated error is minimized. Several examples are tested on isogeometric heat conduction problem to show the effectiveness of the proposed methods and the impact of the parameterization on the quality of the approximation solution. Comparison examples with known exact solutions are also presented.

In recent years, metasurfaces have emerged as revolutionary tools to manipulate the behavior of light at the nanoscale. These devices consist of nanostructures defined within a single layer of metal or dielectric materials, and they offer unprecedented control over the optical properties of light, leading to previously unattainable applications in flat lenses, holographic imaging, polarimetry, and emission control, amongst others. The operation principles of metaoptics include complex light-matter interactions, often involving insidious near-field coupling effects that are far from being described by classical ray optics calculations, making advanced numerical modeling a requirement in the design process. In this contribution, recent optimization techniques used in the inverse design of high performance metasurfaces are reviewed. These methods rely on the iterative optimization of a Figure of Merit to produce a final device, leading to freeform layouts featuring complex and non-intuitive properties. The concepts in numerical inverse designs discussed herein will push this exciting field toward realistic and practical applications, ranging from laser wavefront engineering to innovative facial recognition and motion detection devices, including augmented reality retro-reflectors and related complex light field engineering.

International audienceParameterization of computational domain is a key step in isogeometric analysis just as mesh generation is in finite element analysis. In this paper, we study the volume parameterization problem of multi-block computational domain in isogeometric version, i.e, how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Firstly, we show how to find good volume parameterization of single-block computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of B-spline volume parametrization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of B-spline volume parameterization. By using this method, the resulted volume parameterization has no self-intersections, and the isoparametric structure has good uniformity and orthogonality. Then we extend this method to the multi-block case, in which the continuity condition between the neighbor B-spline volume should be added to the constraint term. The effectiveness of the proposed method is illustrated by several examples based on three-dimensional heat conduction problem