40 research outputs found
-smooth isogeometric spline functions of general degree over planar mixed meshes: The case of two quadratic mesh elements
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 -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 -smooth isogeometric spline space over the domain described
by two elements. Several examples of interest are presented in detail
Machine Learning and Its Application to Reacting Flows
This open access book introduces and explains machine learning (ML) algorithms and techniques developed for statistical inferences on a complex process or system and their applications to simulations of chemically reacting turbulent flows. These two fields, ML and turbulent combustion, have large body of work and knowledge on their own, and this book brings them together and explain the complexities and challenges involved in applying ML techniques to simulate and study reacting flows. This is important as to the world’s total primary energy supply (TPES), since more than 90% of this supply is through combustion technologies and the non-negligible effects of combustion on environment. Although alternative technologies based on renewable energies are coming up, their shares for the TPES is are less than 5% currently and one needs a complete paradigm shift to replace combustion sources. Whether this is practical or not is entirely a different question, and an answer to this question depends on the respondent. However, a pragmatic analysis suggests that the combustion share to TPES is likely to be more than 70% even by 2070. Hence, it will be prudent to take advantage of ML techniques to improve combustion sciences and technologies so that efficient and “greener” combustion systems that are friendlier to the environment can be designed. The book covers the current state of the art in these two topics and outlines the challenges involved, merits and drawbacks of using ML for turbulent combustion simulations including avenues which can be explored to overcome the challenges. The required mathematical equations and backgrounds are discussed with ample references for readers to find further detail if they wish. This book is unique since there is not any book with similar coverage of topics, ranging from big data analysis and machine learning algorithm to their applications for combustion science and system design for energy generation
Blending techniques in Curve and Surface constructions
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Adaptive isogeometric methods with (truncated) hierarchical splines on planar multi-patch domains
Isogeometric analysis is a powerful paradigm which exploits the high
smoothness of splines for the numerical solution of high order partial
differential equations. However, the tensor-product structure of standard
multivariate B-spline models is not well suited for the representation of
complex geometries, and to maintain high continuity on general domains special
constructions on multi-patch geometries must be used. In this paper we focus on
adaptive isogeometric methods with hierarchical splines, and extend the
construction of isogeometric spline spaces on multi-patch planar domains
to the hierarchical setting. We introduce a new abstract framework for the
definition of hierarchical splines, which replaces the hypothesis of local
linear independence for the basis of each level by a weaker assumption. We also
develop a refinement algorithm that guarantees that the assumption is fulfilled
by splines on certain suitably graded hierarchical multi-patch mesh
configurations, and prove that it has linear complexity. The performance of the
adaptive method is tested by solving the Poisson and the biharmonic problems
G1-smooth Biquintic Approximation of Catmull-Clark Subdivision Surfaces
International audienceIn this paper a construction of a globally G1 family of BĂ©zier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface is presented. The resulting surface is a collection of BĂ©zier patches, which are bicubic C2 around regular vertices and biquintic G1 around extraordinary vertices (and C1 on their one-rings vertices). Each BĂ©zier point is computed using a locally defined mask around the neighboring mesh vertices. To define G1 conditions, we assign quadratic gluing data around extraordinary vertices that depend solely on their valence and we use degree five patches to satisfy these G1 constraints. We explore the space of possible solutions, considering several projections on the solution space leading to different explicit formulas for the masks. Certain control points are computed by means of degree elevation of the C0 scheme of Loop and Schaefer [22], while for others, explicit masks are deduced by providing closed-form solutions of the G1 conditions, expressed in terms of the masks. We come up with four different schemes and conduct curvature analysis on an extensive benchmark in order to assert the quality of the resulting surfaces and identify the ones that lead to the best result, both visually and numerically. We demonstrate that the resulting surfaces converge quadratically to the CC limit when the mesh is subdivided
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
Object Recognition
Vision-based object recognition tasks are very familiar in our everyday activities, such as driving our car in the correct lane. We do these tasks effortlessly in real-time. In the last decades, with the advancement of computer technology, researchers and application developers are trying to mimic the human's capability of visually recognising. Such capability will allow machine to free human from boring or dangerous jobs
Sensor Signal and Information Processing II
In the current age of information explosion, newly invented technological sensors and software are now tightly integrated with our everyday lives. Many sensor processing algorithms have incorporated some forms of computational intelligence as part of their core framework in problem solving. These algorithms have the capacity to generalize and discover knowledge for themselves and learn new information whenever unseen data are captured. The primary aim of sensor processing is to develop techniques to interpret, understand, and act on information contained in the data. The interest of this book is in developing intelligent signal processing in order to pave the way for smart sensors. This involves mathematical advancement of nonlinear signal processing theory and its applications that extend far beyond traditional techniques. It bridges the boundary between theory and application, developing novel theoretically inspired methodologies targeting both longstanding and emergent signal processing applications. The topic ranges from phishing detection to integration of terrestrial laser scanning, and from fault diagnosis to bio-inspiring filtering. The book will appeal to established practitioners, along with researchers and students in the emerging field of smart sensors processing
Mathematical foundations of adaptive isogeometric analysis
This paper reviews the state of the art and discusses recent developments in
the field of adaptive isogeometric analysis, with special focus on the
mathematical theory. This includes an overview of available spline technologies
for the local resolution of possible singularities as well as the
state-of-the-art formulation of convergence and quasi-optimality of adaptive
algorithms for both the finite element method (FEM) and the boundary element
method (BEM) in the frame of isogeometric analysis (IGA)