Immersed boundary-conformal isogeometric method for linear elliptic problems

Abstract

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.

Robust Numerical Integration on Curved Polyhedra Based on Folded Decompositions

“…The compromise we make in this work is to create J + cells for visualization wherever it is needed. For example, the boundary of a volume is decomposed according to quadrangulation [41] to ensure that all the surface cells have positive Jacobians, where quantities of interest are computed and visualized. This is done independently from creating volumetric cells, and it is only meant for visualization purposes.…”

“…The B-rep is embedded into a 8 × 8 Cartesian grid. We consider two ways of decomposition for trimmed elements: the J + decomposition by quadrangulation [41], and the folded decomposition through triangulation in Section 3.2. In either case, the degree of all the Bézier patches (i.e., T K,j in ( 17)) is set to be 2, the same as that of the B-spline curve.…”

“…This is meant to create highly distorted cells and it guarantees the presence of folded cells for this particular geometry. J + decompositions are generated using the quadrangulation method proposed in [41].…”

A fully non-invasive hybrid IGA/FEM scheme for the analysis of localized non-linear phenomena

“…As a result, numerous sophisticated methods have been developed over the years to make possible local simulations with IGA. For the representation of geometrical details, one solution, to avoid tedious (or even impossible) spline reparametrizations leading to the splitting of the geometry into several C 0 patches [6,7], may be to resort to immersed IGA where the geometry is given in terms of trimming entities while the numerical approximation space is built on an embedding spline cuboid [8,9,10]. Then, regarding for instance fracture and/or delamination, one may refer to the IG version of XFEM, namely XIGA [11,12,13], or to IG cohesive elements [14,15], or even to phase-field approaches [16,17,18], to name a few.…”

“…The remaining difficulty when considering domain coupling within IGA is the formulation and implementation of a possibly non-conforming coupling. Inspired from immersed methods [8,9,10], the usual coupling of the non-invasive strategy by means of Lagrange multipliers was replaced by a Nitsche-based coupling to answer this issue in the field of full global/local IGA [33,34]. However, such an approach appears inconsistent with the use of standard industrial FE codes.…”