Laplace transform

In mathematics, the 'Laplace transform, named after its discoverer Pierre-Simon Laplace (ləˈplɑ:s), is an integral transform that converts a function of a real variable (usually t, in the time domain) to a function of a complex variable s (in the complex frequency domain, also known as s-domain', or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions f, the Laplace transform is the integral \mathcal{L}{f}(s) = \int_0^\infty f(t)e^{-st} , dt. History The Laplace transform is named after mathematician and astronomer Pierre-Simon, marquis de Laplace, who used a similar transform in his work on probability theory. Laplace wrote extensively about the use of generating functions (1814), and the integr

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Isogeometric Analysis of PDEs and numerical implementation in the Finite Element Library LifeV

Nicolò Pagan

The aim of the project is double: to understand the flexibility of the Isogeometric Analysis tools through the solution of some PDEs problems; to test the improvement in the computational time given by a partial loops vectorization at compile-time of the LifeV IGA code. Three different appli- cations have been selected: the potential flow problem around an airfoil profile, the heat equation problem in a bent cylinder and the Laplace problem in a multi-patches geometry representing a blood vessels bifurcation. The geometries used are built through the NURBS package available with the software GeoPDEs. The numerical analysis of the first application is performed with both GeoPDEs and LifeV IGA code. The comparison between different implementations shows that the degrees of freedom loop vectorization at compile-time is able to reduce the matrix assembling time of around 20%. The automatic vectorization at compile-time of the loop on the elements requires too much computational effort without having a reasonable improvement in the running time performances. Unsteady problems and multi-patches geometry have not been tested with LifeV IGA code, but GeoPDEs results show the expected solutions.

2012

Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations

Espen Sande

We show that isogeometric Galerkin discretizations of eigenvalue problems related to the Laplace operator subject to any standard type of homogeneous boundary conditions have no outliers in certain optimal spline subspaces. Roughly speaking, these optimal subspaces are obtained from the full spline space defined on certain uniform knot sequences by imposing specific additional boundary conditions. The spline subspaces of interest have been introduced in the literature some years ago when proving their optimality with respect to Kolmogorov n-widths in L2-norm for some function classes. The eigenfunctions of the Laplacian - with any standard type of homogeneous boundary conditions - belong to such classes. Here we complete the analysis of the approximation properties of these optimal spline subspaces. In particular, we provide explicit L2 and H1 error estimates with full approximation order for Ritz projectors in the univariate and in the multivariate tensor-product setting. Besides their intrinsic interest, these estimates imply that, for a fixed number of degrees of freedom, all the eigenfunctions and the corresponding eigenvalues are well approximated, without loss of accuracy in the whole spectrum when compared to the full spline space. Moreover, there are no spurious values in the approximated spectrum. In other words, the considered subspaces provide accurate outlier-free discretizations in the univariate and in the multivariate tensor-product case. This main contribution is complemented by an explicit construction of B-spline-like bases for the considered spline subspaces. The role of such spaces as accurate discretization spaces for addressing general problems with non-homogeneous boundary behavior is discussed as well. c 2021 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE SA2022

Conception d'un complexe socio-culturel et sportif à Prizren (Kosovo)

Besir Gashi, Cédric Haldimann

Basé sur un travail théorique traitant de la multiethnicité sur un même territoire, le projet cherche à rassembler des peuples pour une nouvelle cohabitation, initiée par l'architecture, et s'appuyant sur la culture et le sport. Le projet s'implante sur le site de la gare désaffectée qui a laissé un vide spatial bien desservi et à proximité du centre-ville. Les traces laissées par l'ancienne affectation dessinent un parc public, qui devient un espace de rassemblement à l'échelle de la ville mais aussi du pays, grâce à la gare réactivée, génératrice de flux importants. Le parc devient la colonne vertébrale du projet, autour duquel viennent s'articuler des programmes publics. La conception d'un pôle culturel, composé principalement d'un grand théâtre, art dramatique ancré dans la culture locale et éradiqué avant la guerre, est pertinente. Le bâtiment profite de la hauteur de la cage de scène du théâtre pour devenir le nouveau signal d'une ville à constructions basses. L'orientation du bâtiment offre aux foyers une relation plus intime et privilégiée avec la rivière qui traverse la ville. Un nouveau stade de football pallie les infrastructures sportives désuètes. Il se rattache à la nouvelle gare, routière et ferroviaire, afin d'optimiser son accessibilité. Le stade est semi-enterré afin de suivre les gabarits bas de la ville, et couvert par une structure indépendante qui s'étend sur l'espace public et la gare.

2013

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transfo

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g) th

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ME-326: Control systems and discrete-time control

Ce cours inclut la modélisation et l'analyse de systèmes dynamiques, l'introduction des principes de base et l'analyse de systèmes en rétroaction, la synthèse de régulateurs dans le domain fréquentiel et dans l'espace d'état, et la commande de systèmes discrets avec une approche polynomiale.