The Variational Theory of Complex Rays -  "something old, something new, something borrowed, something blue"

Abstract:

In computational mechanics, there are several inherently challenging problems that cannot be addressed solely through brute force approaches.
One such challenge is the dispersion phenomenon in Helmholtz equations, which are crucial for understanding noise and vibration. Many numerical techniques, such as the Finite Element Method (FEM), involve discretizing the analysis domain, leading to three distinct sources of error:

• Interpolation error: This arises from discretizing a continuous sinusoidal wave with polynomial discrete elements. It increases linearly with frequency and decreases linearly with element size. While this error can be minimized with sufficiently powerful computing resources, it remains a consideration.
• Machine precision error: This error, stemming from rounding errors inherent in the computational machinery, is typically negligible.
• Dispersion error: This error stems from discretizing a continuous medium. It increases cubically with frequency and decreases quadratically with element size. Consequently, it presents an intrinsic challenge in Helmholtz equations, as the number of required elements explodes with increasing frequency.

As a result, numerical analysis of mid-to-high frequency vibro-acoustic phenomena poses a significant challenge. Given that these frequencies directly impact comfort and fatigue resistance in systems, addressing this challenge is essential.

Trefftz methods offer a solution by representing the solution as a sum of richer, globally defined functions, thereby circumventing the dispersion error associated with domain discretization. Among these methods, the Variational Theory of Complex Rays (VTCR), first introduced by P. Ladevèze in 1996, stands out as particularly effective. The method can be summarized in four steps: domain decomposition into starred sub-domains, selection of shape functions for each sub-domain that a priori satisfy the equilibrium equation, weak formulation of the boundary conditions problem, and reconstruction of the solution field.

In this presentation, I will cover:

• Something old: A historical overview of the technique, justifying its relevance even after more than 20 years of development.
• Something borrowed: Recent advancements made by our research team towards a viable numerical tool for vibro-acoustic analysis.
• Something new: A vision for the mid-to-long-term future of VTCR.
• Something blue: An overview of pyVTCR, an open-source, object-oriented, High-Performance Computing (HPC) numerical platform designed for efficient vibro-acoustic computations.

Le mardi 10 septembre 2024 à 11h00 / Amphithéâtre François Canac, LMA

Voir en ligne : plus d’informations concernant l’orateur