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…
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…
