We study the space of $C^{2}$-smooth isogeometric functions on bilinearly
parameterized multi-patch domains $\Omega \subset \mathbb{R}^{2}$, where the
graph of each isogeometric function is a ...multi-patch spline surface of bidegree
$(d,d)$, $d \in \{5,6 \}$. The space is fully characterized by the equivalence
of the $C^2$-smoothness of an isogeometric function and the $G^2$-smoothness of
its graph surface, cf. (Groisser and Peters,2015; Kapl et al.,2015). This is
the reason to call its functions $C^{2}$-smooth geometrically continuous
isogeometric functions. In particular, the dimension of this $C^{2}$-smooth
isogeometric space is investigated. The study is based on the decomposition of
the space into three subspaces and is an extension of the work (Kapl and
Vitrih, 2017) to the multi-patch case. In addition, we present an algorithm for
the construction of a basis, and use the resulting globally $C^{2}$-smooth
functions for numerical experiments, such as performing $L^{2}$ approximation
and solving triharmonic equation, on bilinear multi-patch domains. The
numerical results indicate optimal approximation order.
The design of globally \(C^s\)-smooth (\(s \geq 1\)) isogeometric spline spaces over multi-patch geometries is a current and challenging topic of research in the framework of isogeometric analysis. ...In this work, we extend the recent methods 25,28 and 31-33 for the construction of \(C^1\)-smooth and \(C^2\)-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of \(C^s\)-smooth isogeometric multi-patch spline spaces of an arbitrary selected smoothness \(s \geq 1\). More precisely, for any \(s \geq 1\), we study the space of \(C^s\)-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular \(C^s\)-smooth subspace of the entire \(C^s\)-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this \(C^s\)-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the \(C^s\)-smooth spline functions to perform \(L^2\) approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed \(C^s\)-smooth subspace.
In the paper, the geometric Lagrange interpolation by quadratic parametric patches is considered. The freedom of parameterization is used to raise the number of interpolated points from the usual 6 ...up to 10, i.e., the number of points commonly interpolated by a cubic patch. At least asymptotically, the existence of a quadratic geometric interpolant is confirmed for data taken on a parametric surface with locally nonzero Gaussian curvature and interpolation points based upon a three-pencil lattice. Also, the asymptotic approximation order 4 is established.
In this paper, a (
d
+ 1)-pencil lattice on a simplex in
is studied. The lattice points are explicitly given in barycentric coordinates. This enables the construction and the efficient evaluation of ...the Lagrange interpolating polynomial over a lattice on a simplex. Also, the barycentric representation, based on shape parameters, turns out to be appropriate for the lattice extension from a simplex to a simplicial partition.
In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple ...and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S3 on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.
In this paper we introduce a $C^1$ spline space over mixed meshes composed of
triangles and quadrilaterals, suitable for FEM-based or isogeometric analysis.
In this context, a mesh is considered to ...be a partition of a planar polygonal
domain into triangles and/or quadrilaterals. The proposed space combines the
Argyris triangle, cf. (Argyris, Fried, Scharpf; 1968), with the $C^1$
quadrilateral element introduced in (Brenner, Sung; 2005) and (Kapl, Sangalli,
Takacs; 2019) for polynomial degrees $p\geq 5$. The space is assumed to be
$C^2$ at all vertices and $C^1$ across edges, and the splines are uniquely
determined by $C^2$-data at the vertices, values and normal derivatives at
chosen points on the edges, and values at some additional points in the
interior of the elements.
The motivation for combining the Argyris triangle element with a recent $C^1$
quadrilateral construction, inspired by isogeometric analysis, is two-fold: on
one hand, the ability to connect triangle and quadrilateral finite elements in
a $C^1$ fashion is non-trivial and of theoretical interest. We provide not only
approximation error bounds but also numerical tests verifying the results. On
the other hand, the construction facilitates the meshing process by allowing
more flexibility while remaining $C^1$ everywhere. This is for instance
relevant when trimming of tensor-product B-splines is performed.
In the presented construction we assume to have (bi)linear element mappings
and piecewise polynomial function spaces of arbitrary degree $p\geq 5$. The
basis is simple to implement and the obtained results are optimal with respect
to the mesh size for $L^\infty$, $L^2$ as well as Sobolev norms $H^1$ and
$H^2$.
Analysis-suitable \(G^1\) (AS-\(G^1\)) multi-patch spline surfaces 4 are particular \(G^1\)-smooth multi-patch spline surfaces, which are needed to ensure the construction of \(C^1\)-smooth ...multi-patch spline spaces with optimal polynomial reproduction properties 16. We present a novel local approach for the design of AS-\(G^1\) multi-patch spline surfaces, which is based on the use of Lagrange multipliers. The presented method is simple and generates an AS-\(G^1\) multi-patch spline surface by approximating a given \(G^1\)-smooth but non-AS-\(G^1\) multi-patch surface. Several numerical examples demonstrate the potential of the proposed technique for the construction of AS-\(G^1\) multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, with optimal rates of convergence over them.
Splines over triangulations and splines over quadrangulations (tensor product splines) are two common ways to extend bivariate polynomials to splines. However, combination of both approaches leads to ...splines defined over mixed triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes are especially useful for representing complicated geometries obtained e.g. from trimming. As (bi-)linearly parameterized mesh elements are not flexible enough to cover smooth domains, we focus in this work on the case of planar mixed meshes parameterized by (bi-)quadratic geometry mappings. In particular we study in detail the space of \(C^1\)-smooth isogeometric spline functions of general polynomial degree over two such mixed mesh elements. We present the theoretical framework to analyze the smoothness conditions over the common interface for all possible configurations of mesh elements. This comprises the investigation of the dimension as well as the construction of a basis of the corresponding \(C^1\)-smooth isogeometric spline space over the domain described by two elements. Several examples of interest are presented in detail.
A particular class of planar two-patch geometries, called bilinear-like \(G^{2}\) two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and ...possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like \(G^2\) two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of \(C^{2}\)-smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the \(C^2\)-smoothness of an isogeometric function and the \(G^2\)-smoothness of its graph surface (cf. 12, 20). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like \(G^{2}\) two-patch parameterizations. Numerical results obtained by performing \(L^{2}\)-approximation indicate that already the subspace possesses optimal approximation properties.
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