Preface

Immersed Boundary Methods (IBM) have been attracting increased attention during the past two decades.Their central principle is to extend a domain of computation to a larger one, typically with a simple shape, which is easy to mesh.On this extended domain a finite element type computation is performed, distinguishing between regions interior and exterior to the original domain.Under the denotation 'fictitious domain' or 'embedded domain methods' the central principle has been followed already since the 1960ies.The recent new interest results from innovative and efficient algorithmic developments, from mathematical analysis showing optimal convergence despite the presence of cut elements, from the possibility to efficiently link these methods to various types of geometric models, and from many new engineering applications.Many variational versions of Immersed Boundary Methods have been developed, like CutFEM, the Finite Cell Method, Immersogeometric Analysis, Unfitted Finite Elements, the Shifted Boundary Method, Phi-FEM, just to name a few.This is Part I of a special issue focused on Immersed Boundary Methods of variational type, and the various aspects that make them successful in addressing complex problems, namely: mathematical analysis, a priori and a posteriori error estimation and adaptivity, advanced numerical integration procedures, data structures and parallel scaling of algorithms, integration with CAD models and non-standard geometric representations, and applications.The scope of this special issue is as broad as possible in terms of applications, such as, but not limited to: problems in solid mechanics, heat transfer, CFD, fluid/structure interaction, and any other types of domain coupling, as well as the connection between Immersed Boundary Methods, just to name a few.