Gastvortrag von Yannis Voet: An outlook on mass lumping for isogeometric analysis

Abstract
Explicit time integration schemes coupled with Galerkin discretizations of time dependent PDEs in structural dynamics require solving a linear system with the mass matrix at each time step. The repeated solution of those linear systems has long been acknowledged as one of the most expensive steps in the solution process and is further exacerbated in isogeometric analysis (IGA). Moreover, the stringent constraint on the step size stemming from the outlier eigenvalues leads to an increasingly large number of linear systems. Instead of solving those linear systems “exactly”, practitioners resort to ad hoc approximations, with mass lumping being one of its best known examples.
Mass lumping techniques have a long history in classical FEM and consist in replacing the mass matrix in the time integration scheme by some diagonal approximation. Unfortunately, not all of them are applicable to more general bases encountered for instance in IGA and even if they are, applying them often causes a staggering loss of accuracy [1]. This is particularly true for trimmed isogeometric geometries [2, 3, 4, 5]. In this lecture, I examine some of the most significant recent developments [6, 7, 8, 9] before discussing critical challenges that still lie ahead.

References
[1] J. A. Cottrell, A. Reali, Y. Bazilevs, T. J. Hughes, Isogeometric analysis of structural vibrations, Computer methods in applied mechanics and engineering 195 (41-43) (2006) 5257–5296.
[2] L. Radtke, M. Torre, T. J. Hughes, A. Düster, G. Sangalli, A. Reali, An analysis of high order FEM and IGA for explicit dynamics: Mass lumping and immersed boundaries, International Journal for Numerical Methods in Engineering (2024) e7499.
[3] L. Coradello, Accurate isogeometric methods for trimmed shell structures., Ph.D. thesis, École polytechnique fédérale de Lausanne (2021).
[4] Y. Voet, E. Sande, A. Buffa, Mass lumping and stabilization for immersogeometric analysis, arXiv preprint arXiv:2502.00452 (2025).
[5] G. Guarino, Y. Voet, P. Antolin, A. Buffa, Stabilization techniques for immersogeometric analysis of plate and shell problems in explicit dynamics, arXiv preprint arXiv:2509.00522 (2025).
[6] Y. Voet, E. Sande, A. Buffa, Mass lumping and outlier removal strategies for complex geometries in isogeometric analysis, Mathematics of Computation 95 (357) (2026) 105–146.
[7] R. Hiemstra, T.-H. Nguyen, S. Eisenträger, W. Dornisch, D. Schillinger, Higher-order accurate mass lumping for explicit isogeometric methods based on approximate dual basis functions, Computational Mechanics (2025) 1–22.
[8] X. Li, D. Wang, On the significance of basis interpolation for accurate lumped mass isogeometric formulation, Computer Methods in Applied Mechanics and Engineering 400 (2022) 115533.
[9] Y. Voet, E. Sande, On the prospects of interpolatory spline bases for accurate mass lumping strategies in isogeometric analysis, arXiv preprint arXiv:2510.13510 (2025).

Zum Vortragenden
Yannis Voet ist als wissenschaftlicher Mitarbeiter am Chair of Numerical Modelling and Simulation der École polytechnique fédérale de Lausanne in der Schweiz unter der Leitung von Annalisa Buffa tätig.

Yannis Voet
[Bild: Yannis Voet]

Outlook on the lecture
[Bild: Yannis Voet]