Lecture Notes on the Bases for the Gradients of Powell Sabin Splines
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Powell-Sabin splines, designed for use in computer graphics, provide means of constructing continuously differentiable piecewise quadratic functions on a triangularization of a planar domain. Although this is not possible on arbitrary triangulations, it can be accomplished when elements are grouped into macroelements containing either 6 or 12 subtriangles in a specific way. The approach we shall adopt is based on the gradients of Powell-Sabin splines. This allows, in contrast to the work of Powell and Sabin, to construct an explicit basis in cases of interest to achieve the desired objective.
Izaskun Garrido was born in Bilbao, Spain, in 1967. She received the M.Sc. degree in applied mathematics from the University of the Basque Country (UPV/EHU), Bilbao, the M.Sc. degree in numerical analysis and programming and the Ph.D. degree in finite elements from the University of Dundee, Dundee, U.K., in 1999.
She has held several positions with the Potsdam Institute for Climate Impact Research, Potsdam, Germany, Zuse Institute Berlin, Berlin, Germany, and the University of Bergen, Bergen, Norway. She has been an Invited Researcher at Stanford University, Stanford, CA, USA, and the Lawrence Livermore National Laboratory, Livermore, CA, USA. She has been Vice Dean of Research and International Relations at UPV/EHU since 2009 and a Contracted Expert for the Research Executive Agency, and acts as an Expert Evaluator for the Spanish National Evaluation Agency and the EraNet projects. She is currently a Professor with the Department of Automatic Control and Systems Engineering, UPV/EHU. She has authored over 100 publications, and has supervised several Ph.D. theses. Her current research interests include numerical simulation and control applied to ocean power generation and fusion.
Aitor J. Garrido was born in Bilbao, Spain, in 1972. He received the M.Sc. degree in applied physics, the M.Sc. degree in electronic engineering, and the Ph.D. degree in control systems and automation from the University of the Basque Country, Bilbao, in 1999, 2001, and 2003, respectively.
He has held several research and teaching positions with the Department of Automatic Control and Systems Engineering, University of the Basque Country, where he is currently an Associate Professor (tenured) of Control Engineering and the Head of the Automatic Control Group. He has authored more than 100 papers in the main international conferences of the area, book chapters, and JCR (ISI)-indexed journals, and has supervised several Ph.D. theses. His current research interests include the applied control of dynamic systems, in particular, the control of ocean energy, nuclear fusion, and biological systems. He has served as a reviewer in a number of international indexed journals and conferences.
Chapter One Introduction
Chapter Two 6-Triangle Macro
Chapter Three 4-Triangle Macro
Chapter Four 12-Triangle Macro
Chapter Five Conclusions
Appendix A Matrix of Equations for One 6-Macro
Appendix B Change for the Midsides Tangential Velocities
Appendix C Change for the Middle Velocity
Appendix D Solving for the Midsides Normal Velocities of One 6-Macro
Appendix E Conditions for Joining Two 6-Macros
Appendix F Matrix of Equations for One 12-Macro
Appendix G Change for the Internal Mid-sides Nodal Values
Appendix H Change for the Central Nodal Value
Appendix I Treatement of External Tangential Mid-sides Nodal Values
Appendix J Geometric Substitution Verification of Constraints