Immersed boundary-conformal isogeometric method for linear elliptic problems

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 (Antolin et al in SIAM J Sci Comput 43(1):A330–A354, 2021). 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.

Results for Initial Trials Comparing Conformal with Non-Conformal Hypocenter Scanning Algorithms

This present paper sets out to perform a limited comparison between that method of P2P (point to point) scanning for Hypocenters, which uses the structure, imparted by a conformal mapping, to accelerate its ray tracing activity, and a method of scanning which incorporates the Earth spheroidal geometry directly within its ray tracing technique and which does not currently use a conformal mapping technique. The “conformal” method uses a mean-radius spherical Earth or can take a mean radius over the region covered by Epicenter and the sensor-field.In this comparison an attempt is made to compare and contrast a set of some possible functions (“indicator functions”) which serve to define the position of a Hypocenter within their response curve which is formed during the scan. Depending on the function, the expected indication may be either a minimum or a maximum, occurring within the range of the response.

York’s Solution to the Initial-Value Problem

In this chapter I briefly review York’s method (or the conformal method) for solving the initial value problem of (GR). This method, developed initially by Lichnerowicz and then generalized by Choquet-Bruhat and York, allows to find solutions of the constraints of (GR) (in particular the Hamiltonian, or refoliation constraint) by scanning the conformal equivalence class of spatial metrics for a solution of the Hamiltonian constraint, exploiting the fact that, in a particular foliation (CMC), the transverse nature of the momentum field is preserved under conformal transformations. This method allows to transform the initial value problem into an elliptic problem for the solution for which good existence and uniqueness theorems are available. Moreover this method allows to identify the reduced phase space of (GR) with the cotangent bundle to conformal superspace (the space of conformal 3-geometries), when the CMC foliation is valid. SD essentially amounts to taking this phase space as fundamental and renouncing the spacetime description when the CMC foliation is not available.