“Simulation of PDEs on Geometries Obtained via Boolean Operations”

Abstract:

Geometric design uses splines representation of surfaces as the main building bloc, but it involves many other ingredients. Geometries are described as a combination of primitives and spline/NURBS boundary representations which are combined via boolean operations, such as intersections and unions. We aim to develop numerical methods that are robustly able to tackle the simulation of PDEs over such “unstructured” geometric representation. For example, dealing with trimming, which corresponds to an intersection operation in geometric modelling, falls into the category of immersed/unfitted discretizations (e.g., Finite Cell Method, cutFEM, immerse-geometric analysis, shifted boundary method, aggregated unfitted FEM), where computational meshes do not align with geometric boundaries/interfaces.

While geometric modelling is extremely flexible, various issues have to be addressed on the analysis side, such as stability, quadrature, imposition of boundary conditions, conditioning of the underlying linear system, etc.

In this talk, we discuss various aspects of this interesting challenge. Starting from defeaturing, which means a systematic removal of features from a geometric model, and its implication on accuracy, to stability issues in the presence of slim or small cut elements, to efficient quadrature, to the imposition of the boundary condition and to the extension of the framework to the assembly of several geometries via union.

Finally, we discuss also how to make these approaches viable in a shape optimization loop, by discussing their use within a reduced-order modelling framework.

I will conclude the talk by showing results and discussing challenges.

Citation:

 Professor Annalisa Buffa is among the most influential applied mathematicians of her generation; she has made pioneering contributions to modelling electromagnetism in non-smooth domains and the corresponding numerical methods. Moreover, she has contributed substantially to optimising the interplay of geometry and analysis in the simulation of solids and structures.  These results are of seminal importance as they impact engineering applications and industrial mathematics, as well as offer deep contributions to the development and analysis of numerical methods in their own right.

Biography:

Annalisa Buffa is a professor of Mathematics at Ecole Polytechnique Fédérale de Lausanne (EPFL) since 2016 and, prior to this, she has been the director and a research director of the Instituto di Matematica Applicata e Tecnologie informatiche of the Italian National Research Council (CNR). Corresponding member of the Accademia dei Lincei, foreign member of the Académie des Sciences, and member of Academia Europaea, Annalisa Buffa is a leading expert in the numerical analysis of partial differential equations. Her interests span from geometric design, computational mechanics, and computational electromagnetics to approximation theory, and functional analysis for PDEs. She received an ERC Starting grant in 2008, and an ERC Advanced grant in 2016, she is a recipient of the Collatz prize from the ICIAM (2015), and she will deliver the Sonia Kovalevsky Lecture at ICIAM 2024. She has been a plenary speaker at several venues, including the ECCOMAS conference in 2022, AIMS Conference on Dynamical Systems, Differential Equations and Applications in 2018, the International Congress of Mathematicians (section 15, 2014), the ICIAM in 2015, the GAMM conference and the FoCM conferences in 2014. Annalisa Buffa is a highly cited researcher, according to ISI (2019).