42 research outputs found
Projection-based stabilization of interface Lagrange multipliers in immersogeometric fluid–thin structure interaction analysis, with application to heart valve modeling
This paper discusses a method of stabilizing Lagrange multiplier fields used to couple thin immersed shell structures and surrounding fluids. The method retains essential conservation properties by stabilizing only the portion of the constraint orthogonal to a coarse multiplier space. This stabilization can easily be applied within iterative methods or semi-implicit time integrators that avoid directly solving a saddle point problem for the Lagrange multiplier field. Heart valve simulations demonstrate applicability of the proposed method to 3D unsteady simulations. An appendix sketches the relation between the proposed method and a high-order-accurate approach for simpler model problems
Incompressible image registration using divergence-conforming B-splines
Anatomically plausible image registration often requires volumetric
preservation. Previous approaches to incompressible image registration have
exploited relaxed constraints, ad hoc optimisation methods or practically
intractable computational schemes. Divergence-free velocity fields have been
used to achieve incompressibility in the continuous domain, although, after
discretisation, no guarantees have been provided. In this paper, we introduce
stationary velocity fields (SVFs) parameterised by divergence-conforming
B-splines in the context of image registration. We demonstrate that sparse
linear constraints on the parameters of such divergence-conforming B-Splines
SVFs lead to being exactly divergence-free at any point of the continuous
spatial domain. In contrast to previous approaches, our framework can easily
take advantage of modern solvers for constrained optimisation, symmetric
registration approaches, arbitrary image similarity and additional
regularisation terms. We study the numerical incompressibility error for the
transformation in the case of an Euler integration, which gives theoretical
insights on the improved accuracy error over previous methods. We evaluate the
proposed framework using synthetically deformed multimodal brain images, and
the STACOM11 myocardial tracking challenge. Accuracy measurements demonstrate
that our method compares favourably with state-of-the-art methods whilst
achieving volume preservation.Comment: Accepted at MICCAI 201
Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids
In this paper, we first construct a nonconforming finite element pair for the
incompressible Stokes problem on quadrilateral grids, and then construct a
discrete Stokes complex associated with that finite element pair. The finite
element spaces involved consist of piecewise polynomials only, and the
divergence-free condition is imposed in a primal formulation. Combined with
some existing results, these constructions can be generated onto grids that
consist of both triangular and quadrilateral cells
Correct energy evolution of stabilized formulations: The relation between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric analysis. II: The incompressible Navier-Stokes equations
This paper presents the construction of a correct-energy stabilized finite
element method for the incompressible Navier-Stokes equations. The framework of
the methodology and the correct-energy concept have been developed in the
convective--diffusive context in the preceding paper [M.F.P. ten Eikelder, I.
Akkerman, Correct energy evolution of stabilized formulations: The relation
between VMS, SUPG and GLS via dynamic orthogonal small-scales and isogeometric
analysis. I: The convective--diffusive context, Comput. Methods Appl. Mech.
Engrg. 331 (2018) 259--280]. The current work extends ideas of the preceding
paper to build a stabilized method within the variational multiscale (VMS)
setting which displays correct-energy behavior. Similar to the
convection--diffusion case, a key ingredient is the proper dynamic and
orthogonal behavior of the small-scales. This is demanded for correct energy
behavior and links the VMS framework to the streamline-upwind Petrov-Galerkin
(SUPG) and the Galerkin/least-squares method (GLS).
The presented method is a Galerkin/least-squares formulation with dynamic
divergence-free small-scales (GLSDD). It is locally mass-conservative for both
the large- and small-scales separately. In addition, it locally conserves
linear and angular momentum. The computations require and employ NURBS-based
isogeometric analysis for the spatial discretization. The resulting formulation
numerically shows improved energy behavior for turbulent flows comparing with
the original VMS method.Comment: Update to postprint versio
H1-conforming finite element cochain complexes and commuting quasi-interpolation operators on cartesian meshes
A finite element cochain complex on Cartesian meshes of any dimension based
on the H1-inner product is introduced. It yields H1-conforming finite element
spaces with exterior derivatives in H1. We use a tensor product construction to
obtain L2-stable projectors into these spaces which commute with the exterior
derivative. The finite element complex is generalized to a family of arbitrary
order