16 research outputs found
A characterization of linear independence of THB-splines in and application to B\'ezier projection
In this paper we propose a local projector for truncated hierarchical
B-splines (THB-splines). The local THB-spline projector is an adaptation of the
B\'ezier projector proposed by Thomas et al. (Comput Methods Appl Mech Eng 284,
2015) for B-splines and analysis-suitable T-splines (AS T-splines). For
THB-splines, there are elements on which the restrictions of THB-splines are
linearly dependent, contrary to B-splines and AS T-splines. Therefore, we
cluster certain local mesh elements together such that the THB-splines with
support over these clusters are linearly independent, and the B\'ezier
projector is adapted to use these clusters. We introduce general extensions for
which optimal convergence is shown theoretically and numerically. In addition,
a simple adaptive refinement scheme is introduced and compared to Giust et al.
(Comput. Aided Geom. Des. 80, 2020), where we find that our simple approach
shows promise.Comment: 28 pages, 11 figure
Almost- splines: Biquadratic splines on unstructured quadrilateral meshes and their application to fourth order problems
Isogeometric Analysis generalizes classical finite element analysis and
intends to integrate it with the field of Computer-Aided Design. A central
problem in achieving this objective is the reconstruction of analysis-suitable
models from Computer-Aided Design models, which is in general a non-trivial and
time-consuming task. In this article, we present a novel spline construction,
that enables model reconstruction as well as simulation of high-order PDEs on
the reconstructed models. The proposed almost- are biquadratic splines on
fully unstructured quadrilateral meshes (without restrictions on placements or
number of extraordinary vertices). They are smooth almost everywhere,
that is, at all vertices and across most edges, and in addition almost (i.e.
approximately) smooth across all other edges. Thus, the splines form
-nonconforming analysis-suitable discretization spaces. This is the
lowest-degree unstructured spline construction that can be used to solve
fourth-order problems. The associated spline basis is non-singular and has
several B-spline-like properties (e.g., partition of unity, non-negativity,
local support), the almost- splines are described in an explicit
B\'ezier-extraction-based framework that can be easily implemented. Numerical
tests suggest that the basis is well-conditioned and exhibits optimal
approximation behavior
An isogeometric finite element formulation for phase transitions on deforming surfaces
This paper presents a general theory and isogeometric finite element
implementation for studying mass conserving phase transitions on deforming
surfaces. The mathematical problem is governed by two coupled fourth-order
nonlinear partial differential equations (PDEs) that live on an evolving
two-dimensional manifold. For the phase transitions, the PDE is the
Cahn-Hilliard equation for curved surfaces, which can be derived from surface
mass balance in the framework of irreversible thermodynamics. For the surface
deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation.
Both PDEs can be efficiently discretized using -continuous interpolations
without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured
spline spaces with pointwise -continuity are utilized for these
interpolations. The resulting finite element formulation is discretized in time
by the generalized- scheme with adaptive time-stepping, and it is fully
linearized within a monolithic Newton-Raphson approach. A curvilinear surface
parameterization is used throughout the formulation to admit general surface
shapes and deformations. The behavior of the coupled system is illustrated by
several numerical examples exhibiting phase transitions on deforming spheres,
tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to
the text, added supplementary movie file
A Tchebycheffian extension of multi-degree B-splines: Algorithmic computation and properties
In this paper we present an efficient and robust approach to compute a
normalized B-spline-like basis for spline spaces with pieces drawn from
extended Tchebycheff spaces. The extended Tchebycheff spaces and their
dimensions are allowed to change from interval to interval. The approach works
by constructing a matrix that maps a generalized Bernstein-like basis to the
B-spline-like basis of interest. The B-spline-like basis shares many
characterizing properties with classical univariate B-splines and may easily be
incorporated in existing spline codes. This may contribute to the full
exploitation of Tchebycheffian splines in applications, freeing them from the
restricted role of an elegant theoretical extension of polynomial splines.
Numerical examples are provided that illustrate the procedure described
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Isogeometric Analysis : study of non-uniform degree and unstructured splines, and application to phase field modeling of corrosion
Isogeometric Analysis or IGA was introduced by Hughes et al. (2005) to facilitate efficient design-through-analysis cycles for engineered objects. The goal of this technology is the unification of geometric modeling and engineering analysis, and this is realized by exploiting smooth spline spaces used for the former as finite element spaces required for the latter. As intended, this allows the use of geometrically exact representations for the purpose of analysis. Several new spline constructions have been devised on grid-like meshes since IGA’s inception. The excellent approximation and robustness offered by them has rejuvenated the study of high order methods, and IGA has been successfully applied to myriad problems. However, an unintended consequence of adopting a splinebased design-through-analysis paradigm has been the inheritance of open problems that lie at the intersection of the fields of modeling and approximation using splines. The first two parts of this dissertation focus on two such problems: splines of non-uniform degree and splines on unstructured meshes. The last part of the dissertation is focused on phase field modeling of corrosion using splines. The development of non-uniform degree splines is driven by the observation that relaxing the requirement for a spline’s polynomial pieces to have the same degree would be very powerful in the context of both geometric modeling and IGA. This dissertation provides a complete solution in the univariate setting. A mathematically sound foundation for an efficient algorithmic evaluation of univariate non-uniform degree splines is derived. It is shown that the algorithm outputs a nonuniform degree B-spline basis and that, furthermore, it can be applied to create C¹ piecewise-NURBS of non-uniform degree with B-spline-like properties. In the bivariate setting, a theoretical study of the dimension of non-uniform degree splines on planar T-meshes and triangulations is carried out. Combinatorial lower and upper bounds on the spline space dimension are presented. For T-meshes, sufficient conditions for the bounds to coincide are provided, while for triangulations it is shown that the spline space dimension is stable in sufficiently high degree. Modeling complex geometries using only quadrilaterals leads, in general, to unstructured meshes. In locally structured regions of the mesh, smooth splines can be built following standard procedures. However, there is no canonical way of constructing smooth splines on an unstructured arrangement of quadrilateral elements. This dissertation proposes new spline constructions for the two types of unstructuredness that can be encountered – polar points (i.e., mesh vertices that are collapsed edges) and extraordinary points (i.e., mesh vertices shared by µ ≠4 quadrilaterals). On meshes containing polar points, smooth spline basis functions that form a convex partition of unity are built. Numerical tests presented to benchmark the construction indicate optimal approximation behavior. On meshes containing extraordinary points, two spline spaces are built, one for performing modeling and the other for approximation. The former is contained in the latter to ensure adherence to the philosophy of IGA. Excellent approximation behavior is observed during numerical benchmarking. Finally, a phase field model for corrosion is derived from first principles using Gurtin’s microforce theory and a Coleman–Noll type analysis. The derivation is general enough to include the effect of, for instance, mechanics on the process of corrosion, and an instance of such a coupled model is presentedComputational Science, Engineering, and Mathematic
A Geometric Approach Towards Momentum Conservation: Using design tools based on finite difference and integral methods
The equations governing fluid-flow are a set of partial differential equations, as is the case for a host of other continuous field problems. Analytical solutions to these problems are not always available and computers are unable to handle continuous representations of variables. This makes a finite-dimensional projection mandatory for all variables and this may result in a loss of information. At the same time, invoking the inherent association between physical field variables and geometric quantities, as seen in [1, 2, 3], it is known that stable discretisation schemes can be constructed. In this spirit, mimetic discretization strategies are based on minimizing the loss of information in going from a continuous to a discrete setting by making a clear distinction between exact/topological and approximate/constitutive relations in a physical law, and then focussing on an exact representation of the former and a suitable approximation of the latter. For further reading, please see [4, 5, 6, 7].Mechanical EngineeringAerospace Engineerin