404 research outputs found
A direct procedure for interpolation on a structured curvilinear two-dimensional grid
A direct procedure is presented for locally bicubic interpolation on a structured, curvilinear, two-dimensional grid. The physical (Cartesian) space is transformed to a computational space in which the grid is uniform and rectangular by a generalized curvilinear coordinate transformation. Required partial derivative information is obtained by finite differences in the computational space. The partial derivatives in physical space are determined by repeated application of the chain rule for partial differentiation. A bilinear transformation is used to analytically transform the individual quadrilateral cells in physical space into unit squares. The interpolation is performed within each unit square using a piecewise bicubic spline
Representation and application of spline-based finite elements
Isogeometric analysis, as a generalization of the finite element method, employs spline methods to achieve the same representation for both geometric modeling and analysis purpose. Being one of possible tool in application to the isogeometric analysis, blending techniques provide strict locality and smoothness between elements. Motivated by these features, this thesis is devoted to the design and implementation of this alternative type of finite elements.
This thesis combines topics in geometry, computer science and engineering. The research is mainly focused on the algorithmic aspects of the usage of the spline-based finite elements in the context of developing generalized methods for solving different model problems.
The ability for conversion between different representations is significant for the modeling purpose. Methods for conversion between local and global representations are presented
Polynomial Meshes: Computation and Approximation
We present the software package WAM, written in Matlab, that generates Weakly
Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d
polynomial least squares and interpolation on compact sets with various geometries.
Possible applications range from data fitting to high-order methods for PDEs
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Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems
Grid generation for the solution of partial differential equations
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given
Cubature rules based on bivariate spline quasi-interpolation for weakly singular integrals
In this paper we present a new class of cubature rules with the aim of
accurately integrating weakly singular double integrals. In particular we focus
on those integrals coming from the discretization of Boundary Integral
Equations for 3D Laplace boundary value problems, using a collocation method
within the Isogeometric Analysis paradigm. In such setting the regular part of
the integrand can be defined as the product of a tensor product B-spline and a
general function. The rules are derived by using first the spline
quasi-interpolation approach to approximate such function and then the
extension of a well known algorithm for spline product to the bivariate
setting. In this way efficiency is ensured, since the locality of any spline
quasi-interpolation scheme is combined with the capability of an ad--hoc
treatment of the B-spline factor. The numerical integration is performed on the
whole support of the B-spline factor by exploiting inter-element continuity of
the integrand
Developments and trends in three-dimensional mesh generation
An intense research effort over the last few years has produced several competing and apparently diverse methods for generating meshes. Recent progress is reviewed and the central themes are emphasized which form a solid foundation for future developments in mesh generation
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