  We present a novel isogeometric method, namely the Immersed
Boundary-Conformal Method (IBCM), that features a layer of discretization
conformal to the boundary while employing a simple background mesh for the
remaining domain. In this manner, we leverage the geometric flexibility of the
immersed boundary method with the advantages of a conformal discretization,
such as intuitive control of mesh resolution around the boundary, higher
accuracy per degree of freedom, automatic satisfaction of interface kinematic
conditions, and the ability to strongly impose Dirichlet boundary conditions.
In the proposed method, starting with a boundary representation of a geometric
model, we extrude it to obtain a corresponding conformal layer. Next, a given
background B-spline mesh is cut with the conformal layer, leading to two
disconnected regions: an exterior region and an interior region. Depending on
the problem of interest, one of the two regions is selected to be coupled with
the conformal layer through Nitsche's method. Such a construction involves
Boolean operations such as difference and union, which therefore require proper
stabilization to deal with arbitrarily cut elements. In this regard, we follow
our precedent work called the minimal stabilization method [1]. In the end, we
solve several 2D benchmark problems to demonstrate improved accuracy and
expected convergence with IBCM. Two applications that involve complex
geometries are also studied to show the potential of IBCM, including a spanner
model and a fiber-reinforced composite model. Moreover, we demonstrate the
effectiveness of IBCM in an application that exhibits boundary-layer phenomena.
