Fabio Silva Botelho, PhD
Professor Adjunto, Departamento de Matemática
Universidade Federal de Santa Catarina - UFSC
About the first work to successfully apply convex analysis to the complementary energy concept for a model in non-linear mechanics, published in 1985.
My research on duality theory is, in some sense, an extension and generalization of the ideas of JJ Telega and WR Bielski contained in this article,
combined with some results on D.C. optimization.
Análise Funcional – Curso de Verão – 2020 – Resultados da P-2
Cálculo Avançado – 2019 -2 - Notas de Aula
Livros:
Functional Analysis and Applied Optimization in Banach Spaces, Springer 2014.
Real Analysis and Applications, Springer 2018.
This work is dedicated in memoriam to David Bohm and Jiddu Krishnamurti
Pre-prints – Physics
0.1- arXiv:2003.00325 [pdf, ps, other]
math.GM
A general variational formulation for relativistic mechanics based on fundamentals of differential geometry
Authors: Fabio Botelho
Abstract: This article develops a variational formulation for relativistic mechanics. The results are established through standard tools of variational analysis and differential geometry. The novelty here is that the main motion manifold has a n+1 dimensional range. Finally, it is worth emphasizing in a first approximation we have neglected the self-interaction energy part.
Submitted 29 February, 2020; originally announced March 2020.
Comments: 6 pages
MSC Class: 53Z05
2. A Variational Formulation for the Relativistic Klein-Gordon Equation
3- arXiv:1908.04611 [pdf, ps, other]
quant-ph
A variational formulation for relativistic mechanics, a new interpretation for the Bohr atomic model and some concerning applications
Authors: Fabio Botelho
Abstract: This article develops a variational formulation for the relativistic Klein-Gordon equation. The main results are obtained through an extension of the classical mechanics approach to a more general context, which in some sense, includes the quantum mechanics one. For the second part of the text, the definition of normal field and its relation with the wave function concept play a fundamental role in the main results establishment. Among the applications, we include a model with the presence of electromagnetic fields and also the modeling of a chemical reaction. Finally, in the last section, we present some results about the Spin operator in a relativistic context. △ Less
Submitted 16 October, 2019; v1 submitted 13 August, 2019; originally announced August 2019.
Comments: 38 pages, some minor mistakes and typos corrected
MSC Class: 81Q05
Pre-prints-Applied Mathematics
1-00 arXiv:1910.01118 [pdf, ps, other]
math.OC
On duality principles for one and three-dimensional non-linear models in elasticity
Authors: Fabio Botelho
Abstract: In this article, we develop duality principles applicable to primal variational formulations found in the non-linear elasticity theory. As a first application, we establish the concerning results in details for one and three-dimensional models. We emphasize such duality principles are applicable to a larger class of variational optimization problems, such as non-linear models of plates and shells and other models in elasticity. Finally, we formally prove there is no duality gap between the primal and dual formulations, in a local extremal context. △ Less
Submitted 3 October, 2019; originally announced October 2019.
Comments: 13 pages. arXiv admin note: text overlap with arXiv:1712.02701
MSC Class: 49N15
1- arXiv:1909.01790 [pdf, ps, other]
math.OC
A primal dual variational formulation suitable for a large class of non-convex problems in optimization
Authors: Fabio Botelho
Abstract: In this article we develop a new primal dual variational formulation suitable for a large class of non-convex problems in the calculus of variations. The results are obtained through basic tools of convex analysis, duality theory, the Legendre transform concept and the respective relations between the primal and dual variables. The novelty here is that the dual formulation is established also for the primal variables, however with a large domain region of concavity about a critical point. Finally, we formally prove there is no duality gap between the primal and dual formulations in a local extremal context. △ Less
Submitted 3 September, 2019; originally announced September 2019.
Comments: 8 pages
MSC Class: 49N15
Pre-print- On the solution of the Navier-Stokes system- announced, see it in the next line
arXiv:1908.09751 [pdf, ps, other]
math.GM
On the generalized method lines applied to the time-independent incompressible Navier-Stokes system
Authors: Fabio Botelho
Abstract: In the first part of this article, we obtain a linear system whose the solution solves the time-independent incompressible Navier-Stokes system for the special case in which the external forces vector is a gradient. In a second step we develop approximate solutions, also for the time independent incompressible Navier-Stokes system, through the generalized method of lines. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines and the concerning solution is written on these lines as functions of the boundary conditions and boundary shape. Finally, we emphasize these last main results are established through applications of the Banach fixed point theorem. △ Less
Submitted 11 August, 2019; originally announced August 2019.
Comments: 21 pages
MSC Class: 65N40; 65N06
0.1- arXiv:1907.02503 [pdf, ps, other]
math.OC
A numerical method for an inverse optimization problem through the generalized method of lines
Authors: Fabio Silva Botelho
Abstract: This article develops a solution for an inverse problem through the generalized method of lines. We consider a Laplace equation on a domain with internal and external boundaries with standard Dirichlet boundary conditions. Also, we specify a third non-homogeneous Newmann type boundary condition for the external boundary, and consider the problem of finding the optimal shape for the internal boundary such that all the prescribed boundary conditions are satisfied. The novelty here presented is the application of the generalized method of lines as a tool to compute a solution for such an inverse optimization problem. △ Less
Submitted 4 July, 2019; originally announced July 2019.
Comments: 7 pages
MSC Class: 49N45; 65N40
1.0-0arXiv:1907.00200 [pdf, ps, other]
math.OC
A duality principle and related numerical method for a class of shape optimization problems in elasticity
Authors: Fabio Botelho, Alexandre Molter
Abstract: In this article we develop a duality principle and concerning numerical method for a shape optimization problem in elasticity. We consider the problem of finding the optimal shape for an elastic solid which minimizes its structural inner energy resulting from the action of external loads to be specified. The main results are obtained through standard tools of convex analysis and duality theory. We emphasize our algorithm do not include a filter to process the results, so that the result obtained is indeed a critical point for the original optimization problem. Finally, in the last section, we present some numerical examples concerning applications of the theoretical results established. △ Less
Submitted 29 June, 2019; originally announced July 2019.
Comments: 7 pages
MSC Class: 49N15; 49Q10
1.0-arXiv:1906.07758 [pdf, ps, other]
math.OC
Duality suitable for a class of non-convex optimization problems
Authors: Fabio Botelho
Abstract: In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the critical points of the primal and dual formulations and formally prove there is no duality gap between such formulations, in a local extremal context.
Submitted 18 June, 2019; originally announced June 2019.
Comments: 7 pages
MSC Class: 49N15
1. arXiv:1904.12379 [pdf, ps, other]
math.NA
On the generalized method of lines and its proximal explicit and hyper-finite difference approaches
Authors: Fabio Botelho
Abstract: This article firstly develops a proximal explicit approach for the generalized method of lines. In such a method, the domain of the PDE in question is discretized in lines and the equation solution is written on these lines as functions of the boundary conditions and domain shape. The main objective of introducing a proximal formulation is to minimize the solution error as a typical parameter… ▽ More
Submitted 7 May, 2019; v1 submitted 28 April, 2019; originally announced April 2019.
Comments: 18 pages, some typos corrected, a new result added
MSC Class: 65N40; 65N06
1.1. arXiv:1904.02286 [pdf, ps, other]
math.OC
A primal dual variational formulation and a multi-duality principle for a non-linear model of plates
Authors: Fabio Botelho
Submitted 18 April, 2019; v1 submitted 3 April, 2019; originally announced April 2019.
Comments: 20 pages, some typos corrected, a new result added
MSC Class: 49N15; 74P99
Artigos completos publicados em periódicos
1. Fabio Botelho, A note on Riemannian geometry and the relativistic quantum mechanics context. CIÊNCIA E NATURA, v. 40, p. 58, 2018.
2. Fabio Botelho.,A Variational Formulation for the Relativistic Klein-Gordon Equation. CIÊNCIA E NATURA, v. 40, p. 57, 2018.
4. Lucas dos Santos Fernandez, Alexandre Molter, Fabio Silva Botelho, Simultaneous topology optimization and proportional actuators localization. SeMA Journal, v. 23, p. --, 2016.
5. Fabio Botelho, On the Lagrange multiplier theorem in Banach spaces. Matemática Aplicada e Computacional (Cessou em 1997. Cont. ISSN 1807-0302 Computational & Applied Mathematics)^{ }, v. 32, p. 135-144, 2013.
6. Fabio Botelho. Existence of solution for the Ginzburg Landau system, a related optimal control problem and its computation by the generalized method of lines. Applied Mathematics and Computation^{ }, v. 218, p. 11976-11989, 2012.
7. Fabio Botelho, On duality principles for scalar and vectorial multi-well variational problems. Nonlinear Analysis^{ }, v. 75, p. 1904-1918, 2012.
8. Fabio Botelho, Dual Variational Formulations for a Non-linear Model of Plates. JOURNAL OF CONVEX ANALYSIS^{ }, v. 17, p. 131-158, 2010.