Analytical and numerical solutions for a class of optimization problems in elasticity

The subject of topology optimization methods in structural de- sign has increased rapidly since the publication of [?], where some ideas from homogenization theory were put into practice. Since then, several engineering applications have been developed successfully. However, in the literature, there is a lack of analytical solutions, even for simple cases, which might help in the validation of the numerical results. In this work, we develop analytical solu- tions for simple minimum compliance problems, in the framework of elasticity theory. We compare these analytical solutions with numerical results obtained via an algorithm proposed in [?]

A very high-order finite volume method for the time-dependent convection-diffusion problem with Butcher tableau extension

The time discretization of a very high-order finite volume method may give rise to new numerical difficulties resulting into accuracy degradations. Indeed, for the simple one-dimensional unstationary convection-diffusion equation for instance, a conflicting situation between the source term time discretization and the boundary conditions may arise when using the standard Runge-Kutta method. We propose an alternative procedure by extending the Butcher Tableau to overcome this specific difficulty and achieve fourth-, sixth- or eighth-order of accuracy schemes in space and time. To this end, a new finite volume method is designed based on specific polynomial reconstructions for the space discretization, while we use the Extended Butcher Tableau to perform the time discretization. A large set of numerical tests has been carried out to validate the proposed method.Fundação para a Ciência e a Tecnologia (FCT

Performance Evaluation of Portfolio Stocks Selected with the EU-EV Risk Model

In this paper, the performance of portfolios consisting of stocks selected with the recently proposed expected utility, entropy and variance (EU-EV) risk model is analysed. The portfolios were constructed using data of the PSI 20 index, from January 2019 to December 2020, by reducing the number of stock components to the half with the EU-EV risk model. The effciency of these portfolios in terms of the mean-variance model was shown to be approximately equal to the effciency of portfolios obtained from the whole set of stocks. The aim is to evaluate the performance of the constructed portfolios, by comparing their in-sample and out-of-sample results with those of the benchmark. For that purpose, cumulative returns in the in-sample period from January 2019 to December 2020 and in the out-of-sample period from January 2021 to December 2022, considering both an one-year and a two-year time horizon, as well as different performance metrics, such as Sharpe ratio, Sortino ratio, Beta and Alpha, are analysed. The results reveal that the portfolios constructed with the EU-EV risk model outperform the benchmark portfolio in the given periods, where a better performance was obtained in the one-year out-of-sample period. These results suggest that the strategy of constructing portfolios using the best ranked stocks according to the EU-EV risk model can be useful for short-term investment objectives.FCT -Fundação para a Ciência e a Tecnologia(UIDB/00013/2020

Automatic exercise generation for exploring connections between mathematics and music

Mathematics and Music are closely connected and their multifaceted relationship has been explored since ancient times. Pythagoras was one of the first who discovered and formalized one of those connections by studying the arithmetic of musical intervals, expressing them by numerical ratios and relating ratios to consonance/dissonance notions, being the Pythagorean tuning and scale based on arithmetic principles. Another interesting link between Mathematics and Music is the geometric approach to musical composition. Geometric patterns are present in different musical style compositions and some composing techniques, such as the 12-tone technique, use geometric transformations, involving also applications of modular arithmetic and set theory to Music. The aim of this work is to present existing relations between Mathematics and Music and to propose automatic exercise generation for exploring and studying those relations. The exercises are generated using the system MVGEN and the LaTeX package MusiXTeX. The generation process and the automatically generated exercises can be used for creating learning and assessment materials for education in Mathematics and Music, linking Science and Art.(undefined

Álgebra linear e geometria analítica

Apontamentos da disciplina de Álgebra Linear e Geometria Analítica das Licenciaturas em Electrónica Industrial e Computadores e Gestão Industrial (primeiro semestre do ano lectivo 2005/06)

Compact schemes in time with applications to partial differential equations

We propose a new class of fourth-and sixth-order schemes in time for parabolic and hyperbolic equations. The method follows the compact scheme methodology by elaborating implicit relations between the approximations of the function and its derivatives. We produce a series of A-stable methods with low dispersion and high accuracy. Several benchmarks for linear and non-linear Ordinary Differential Equations demonstrate the effectiveness of the method. Then a second set of numerical benchmarks for Partial Differential Equations such as convection-diffusion, Schrodinger equation, wave equation, Burgers, and Euler system give the numerical evidences of the superior advantage of the method with respect to the traditional Runge-Kutta or multistep methods.S. Clain and G.J. Machado acknowledge the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020. M.T. Malheiro acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM. S. Clain, G.J. Machado, and M.T. Malheiro acknowledge the fi-nancial support by FEDER - Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 - Programa Operacional Fatores de Competitividade, POCI-01-0145-FEDER-028118 and PTDC/MAT-APL/28118/2017

An a posteriori strategy for adaptive schemes in time for one-dimensional advection-diffusion transport equations

Stability condition is a more restrictive constraint that leads to unnecessary small-time steps with respect to the accuracy and results in computational time wastage. We propose a node by node adaptive time scheme to relax the stability constraint enabling a larger global time step for all the nodes. A nonlinear procedure for optimising both the schemes in time and space is proposed in view of increasing the numerical efficiency and reducing the computational time. The method is based on a four-parameter family of schemes we shall tune in function of the physical data (velocity, diffusion), the characteristic size in time and space, and the local regularly of the function leading to a nonlinear procedure. The a posteriori strategy we adopt consists in, given the solution at time t(n), computing a candidate solution with the highest accurate schemes in time and space for all the nodes. Then, for the nodes that present some instabilities, both the schemes in time and space are modified and adapted in order to preserve the stability with a large time step. The updated solution is computed with node-dependent schemes both in time and space. For the sake of simplicity, only convection-diffusion problems are addressed as a prototype with a two-parameters five-points finite difference method for the spatial discretisation together with an explicit time two-parameters four-stages Runge-Kutta method. We prove that we manage to obtain an optimal time-step algorithm that produces accurate numerical approximations exempt of non-physical oscillations.G.J. Machado and S. Clain acknowledge the financial support by FEDER -Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 -Programa Operacional Fatores de Competitividade, and the National Funds through FCT -Fundacao para a Ciencia e a Tecnologia, project no. UID/FIS/04650/2019.M.T. Malheiro acknowledge the financial support by Portuguese Funds through FCT (Fundacao para a Ciencia e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM.M.T. Malheiro, G.J. Machado, and S. Clain acknowledge the financial support by FEDER - Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 -Programa Operacional Fatores de Competitividade, and the National Funds through FCT -Fundacao para a Ciencia e a Tecnologia, project no. POCI-01-0145-FEDER-028118

6th-order finite volume approximation for the steady-state burger and euler equations: the mood approach

We propose an innovative method based on the MOOD technology (Multi-dimensional Optimal Order Detection) to provide a 6th-order finite volume approximation for the one-dimensional steady-state Burger and Euler equations. The main ingredient consists in using an 'a posteriori' limiting strategy to eliminate non physical oscillations deriving from the Gibbs phenomenon while keeping a high accuracy for the smooth part. A short overview of the MOOD method will be presented and numerical tests with regular or discontinuous solutions will assess the method capacity to produce excellent approximations. In the latter situation, the numerical results enable to detect the zone where it is necessary to reduce the degree of the polynomial reconstructions to preserve the scheme robustness.Fundação para a Ciência e a Tecnologia (FCT

New cell-vertex reconstruction for finite volume scheme : application to the convection-diffusion-reaction equation

The design of efficient, simple, and easy to code, second-order finite volume methods is an important challenge to solve practical problems in physics and in engineering where complex and very accurate techniques are not required. We propose an extension of the original Frink's approach based on a cell-to-vertex interpolation to compute vertex values with neighbor cell values. We also design a specific scheme which enables to use whatever collocation point we want in the cells to overcome the mass centre point restrictive choice. The method is proposed for two- and three-dimension geometries and a second-order extension time-discretization is given for time-dependent equation. A large number of numerical simulations are carried out to highlight the performance of the new method.Fundação para a Ciência e a Tecnologia (FCT

6th-order finite volume approximations for the stokes equations with a curved boundary

A new solver for the Stokes equations based on the finite volume method is proposed using very accurate polynomial reconstruction to provide a 6th-order scheme. We face two main difficulties: the gradient-divergence duality where the divergence free condition will impose the pressure gradient, and on the other hand, we assume that the domain has a regular curved boundary. The last point implies that a simple approximation of the boundary using piecewise segment lines dramatically reduces the scheme accuracy to at most a second-order one. We propose a new and simple technology which enables to restore the full scheme accuracy based on a specific polynomial reconstruction only using the Gauss points of the curved boundary and does not require any geometrical transformation.Fundação para a Ciência e a Tecnologia (FCT)This research was financed by FEDER Funds through Programa Operational Fatores de Competitividade — COMPETE and by Portuguese Funds FCT — Fundação para a Ciência e a Tecnologia, within the Projects PEst-C/MAT/UI0013/2014, PTDC/MAT/121185/2010 and FCT-ANR/MAT-NAN/0122/201