NumPI January 2020

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1 participants
2 discussions

Seminar on 29/01 (Simona Perotto)
by Leonardo Robol 29 Jan '20

29 Jan '20

Speaker: Simona Perotto Affiliation: Politecnico di Milano Wednesday, 29/01/2020, 11:00, Aula Tonelli, SNS. Title: Topology optimization: a new algorithm based on anisotropic mesh adaptation Topology optimization provides useful mathematical tools for the design of optimal (e.g., minimizing the compliance) structures for assigned loads and boundary conditions, and under suitable constraints (e.g., a given fraction of the initial volume). The SIMP (Solid Isotropic Material with Penalization) method represents one of the most investigated approaches for topology optimization. It belongs to the family of density-based methods, where an auxiliary (unknown) scalar field (the density), taking values between zero (void) and one (material), is adopted to describe the final layout of the optimized structure. SIMP exhibits many drawbacks, among which the non-uniqueness of the optimal structure and issues related to the adopted discretization for the optimization problem (e.g., checkerboard, greyscale, and staircase effects). In this talk, we present the new algorithm SIMPATY (SIMP+AdaptiviTY) which enriches the basic SIMP method with anisotropic mesh adaptivity to alleviate some of the above- mentioned issues. After introducing the mathematical background, we provide a detailed description of the SIMPATY algorithm and we verify its performance on benchmarks as well as on more challenging structure configurations. https://www.dm.unipi.it/webnew/it/seminari/topology-optimization-new-algori…

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Seminar on 20/01 (Fernando De Terán)
by Leonardo Robol 20 Jan '20

20 Jan '20

Speaker: Fernando De Terán Affiliation: Universidad Carlos III de Madrid Time: Monday, 20/01/2020, 14:00 Place: Aula Magna, Dipartimento di Matematica. Title: Flanders’ theorem for many matrices under commutativity assumptions Given two matrices $A \in C^{m \times n}$ and $B \in C^{n \times m}$ , it is known [1] that the Jordan canonical form of AB and BA can only differ in the sizes of the Jordan blocks associated with the eigenvalue zero, and the difference in the size of any two corresponding blocks is, at most, 1. Moreover, this change is exhaustive, in the sense that given two ordered lists of numbers such that the corresponding elements differ at most by 1, then there are two matrices A, B such that the fist list is the list of sizes of Jordan blocks associated with 0 in AB and the second one is the list of sizes of Jordan blocks associated with 0 in BA. The motivation of this work is the question: What happens with products of more than two matrices? First, we will see that, if we do not impose any restriction, the products ABC and CBA may have completely different eigenvalues. Therefore, some conditions must be imposed to the factors in order to be able to relate the Jordan canonical forms of any two products. By imposing some natural commutativity assumptions, we analyze the relationship between the Jordan canonical form of products, in different orders, of k square matrices $A_1, ..., A_k$. In particular, we study permuted products of $A_1, ..., A_k$ under the assumption that the graph of non-commutativity relations is a forest. Under this condition, we show that the Jordan structure of all nonzero eigenvalues is the same for all permuted products. For the eigenvalue zero, we obtain an upper bound on the difference between the sizes of Jordan blocks for any two permuted products, and we show that this bound is attainable. For k = 3 we show that, moreover, the bound is exhaustive. This is joint work with Ross A. Lippert, Yuji Nakatsukasa, and Vanni Noferini. https://www.dm.unipi.it/webnew/it/seminari/flanders%E2%80%99-theorem-many-m…

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NumPI January 2020