SUMMARYThis paper proposes a novel Immersed Boundary Method where the embedded domain is exactly described by using its CAD boundary representation with NURBS or T-Splines. The common feature with other immersed methods is that the current approach substantially reduces the burden of mesh generation. In contrast, the exact boundary representation of the embedded domain allows to overcome the major drawback of existing immersed methods that is the inaccurate representation of the physical domain. A novel approach to perform the numerical integration in the region of the cut elements that is internal to the physical domain is presented and its accuracy and performance evaluated using numerical tests. The applicability, performance and optimal convergence of the proposed methodology is assessed by using numerical examples in three dimensions. It is also shown that the accuracy of the proposed methodology is independent on the CAD technology used to describe the geometry of the embedded domain.

This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order nature of the coupled PDE system is addressed by a sufficiently smooth hierarchical B-spline approximation on a background Cartesian mesh. The domain of interest is embedded into the background mesh and discretized in an unfitted fashion. The immersed boundary approach allows us to use B-splines on arbitrary domain shapes, regardless of their geometrical complexity, and could be directly extended, for instance, to shape and topology optimization. The domain boundary is represented by NURBS, and exactly integrated by means of the NEFEM mapping. Local adaptivity is achieved by hierarchical refinement of B-spline basis, which are efficiently evaluated and integrated thanks to their piecewise polynomial definition. Nitsche's formulation is derived to weakly enforce essential boundary conditions, accounting also for the non-local conditions on the non-smooth portions of the domain boundary (i.e. edges in 3D or corners in 2D) arising from Mindlin's strain gradient elasticity theory. Boundary conditions modeling sensing electrodes are formulated and enforced following the same approach. Optimal error convergence rates are reported using high-order B-spline approximations. The method is verified against available analytical solutions and well-known benchmarks from the literature.

-mail: manuel.tur@mcm.upv.es,jalbelda@mcm.upv.es,onmaral@upvnet.upv.es,jjrodena@mcm.upv.es Abstract This paper proposes a new formulation to impose Dirichlet boundary conditions on immersed boundary Cartesian Finite Element meshes. The method uses a recovered stress field calculated by Superconvergent Patch Recovery to stabilize the Lagrange multiplier formulation of the problem. The optimal convergence of the method and the convergence of the proposed iterative procedure are demonstrated. The proposed method is also suitable for problems with non-linear material behavior. Some numerical examples are included to confirm the theoretical results.

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