Reduced-Order Modeling for Complex Engineering Problems

Description

This event will focus on the numerical simulation of engineering problems in complex, possibly heterogeneous media, using computational approaches arising from homogenization theory, multiscale science, reduced order modeling, or a combination thereof. All aspects of mathematical modeling and numerical resolution and all aspects related to the practical implementation issues arising from the scientific questions addressed and numerical techniques employed, will be presented. The synergy between these aspects will be highlighted. A special emphasis will also be placed on the importance of this combination of ingredients for the successful solution of engineering problems in an industrial R&D context.

The overall purpose is to introduce, describe and study approaches that allow satisfactory approximation of solutions to these complex problems in a window of time acceptable in an industrial or, more generally, practical context. The focus is on methods that do not necessarily require elaborate and intrusive modifications of existing software that often capitalizes on a significant amount of year.person work (legacy problem). Modern approaches will be presented, that are not dedicated to endlessly obtaining more accurate results at any computational cost, but that achieve a suitable compromise between accuracy, computational cost, and implementation workload. The industrial context and the practical feasibility will be the driving motivation for the problems considered and the solutions proposed.

Winter School

The first part of this event (January 29-31, 2025) is a Winter School on these topics, aimed primarily at early career researchers and graduate and advanced undergraduate students. Students might come from various backgrounds (applied mathematics, computational mechanics, aerospace engineering, etc). The school will include three block courses:

• Sonia Fliss (CNRS-Inria-ENSTA, France) will speak about problems of diffraction by thin layers of periodic and random materials;
• Ulrich Hetmaniuk (founder of Shift-Invert Ltd, USA) will speak about concrete software issues for multiscale computational science;
• Anthony T. Patera (MIT, USA) will speak about perspectives on engineering estimation via the heat transfer dunking problem.

Applications for the Winter School closed on November 11, 2024. Higher priority will be given to applicants who are able to attend in-person. Decisions will be issued in early December.

Research Workshop

The second part of the event (February 3-7, 2025) is a Research Workshop devoted to this area. Participation in the winter school is not a prerequisite for attending the workshop (and vice versa), although the two parts of the event have been designed as a whole.

Review of funding requests for the workshop will begin the week of November 18, 2024.

The research workshop will include lightning talks and a poster session for early career researchers (including graduate students). If accepted, you will be asked to do both. In order to propose a lightning session talk and a poster, you must first register for the workshop, and then submit a proposal using the form that will become available on this page after you register. The registration form should not be used to propose a lightning session talk or poster.

The deadline for proposing is January 6, 2025. If your proposal is accepted, you should plan to attend the event in-person.

Winter School Lecture Descriptions

Sonia Fliss (CNRS-Inria-ENSTA, France): Problems of diffraction by thin layers of periodic and random materials

• The lectures are focused on problems of acoustic or electromagnetic diffraction by an obstacle covered by a thin layer of microstructured material with periodic or random physical properties. When the thickness of the layer and the typical size of the microstructure are of the same order and small compared to the wavelength, a numerical calculation in force can become prohibitive, especially when electromagnetic waves are involved. The course will explain how, with the help of multi-scale asymptotic developments, it is possible to build an effective model in which the thin layer is replaced by an equivalent boundary condition. The treatment of thin films is often referred to as surface homogenization, as opposed to the volume homogenization discussed in Course 2 below. In particular, the differences and links between the two types of problem and the associated numerical methods will be highlighted. It will be shown how the approximation error can also be quantified under certain assumptions about the microstructure (in particular, assumptions of stationarity and ergodicity are necessary in the case where the layer characteristics follow a probability law). Finally, the numerical resolution of this effective model is obviously simpler and much less costly, but requires additional calculations. Indeed, in these equivalent boundary conditions, a number of constants appear that are characteristic of the layer. These constants depend on “correctors”, solutions to static problems posed in unbounded domains. This raises some interesting numerical questions, which will also be addressed during the course.

Ulrich Hetmaniuk (founder of Shift-Invert Ltd, USA): Concrete software issues for multiscale computational science

• The lectures and the computer tutorials will illustrate state-of-the-art techniques to numerically solve problems where a small scale is present, such as computational homogenization with Representative Volume Element (RVE), the Multiscale Finite Element Method (MsFEM), the Heterogeneous Multiscale Method (HMM), and the Local Orthogonal Decomposition (LOD) method. These methods will be applied to the diffusion equation and to the heat equation. Extension to the Helmholtz equation will be discussed in coordination with Course 1. Practical aspects for transferring these techniques to industrial codes will be considered: non-intrusive implementation, implementation for heterogeneous architectures, and three-dimensional simulations.

Anthony T. Patera (MIT, USA): Perspectives on Engineering Estimation via the Heat Transfer Dunking Problem

• We consider the dunking problem: a solid body, initially at thermal equilibrium in a first environment at temperature $T_i$, is abruptly placed — “dunked” — at time $t = 0$ in a second environment, characterized by far field fluid/enclosure temperature $T_\infty$. The Quantities of Interest (QoI) describe the spatial distribution of the temperature as a function of time. The high-fidelity mathematical model for the dunking problem — a very detailed conjugate heat transfer formulation — is not practical for typical engineering studies. However, the QoI can be predicted, inexpensively and reasonably accurately, by well-established engineering estimation procedures: simplification of the high-fidelity mathematical model; subsequent approximate solution of the simplified mathematical model, often informed by archived experimental data. An important aspect of model simplification is the treatment of multiscale phenomena both in time — rapid variations in the fluid relative to slow variations in the solid — and in space — due to heterogeneous material composition. We shall first present the classical estimation procedures and then proceed to newer work on rigorous error estimation. The ultimate goal is Autonomous Heat Transfer Estimation (AHTE). We briefly describe and (in the hands-on part of the program) illustrate the AHTE framework with particular reference to underlying software components.Work in collaboration with Theron Guo (MIT), Kento Kaneko (MIT), and Claude Le Bris (ENPC). Research supported by the Office of Naval Research.

Registration

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