arXiv subjects

Yannis Voet

Publications and source records attributed to Yannis Voet.

11 recordsLinked to original sources

arXiv2026

Deflation-based preconditioning for immersed finite element methods and immersogeometric analysis

Trimming is a ubiquitous operation in computer-aided-design whereby parts of a geometry are merged, intersected, or simply discarded. While it grants virtually unlimited flexibility in geometric design, it introduces a plethora of other difficulties when such geometries are used within immersed finite element methods. In particular, small cut elements lead to severely ill-conditioned system matrices requiring dedicated penalization, stabilization, or preconditioning techniques. In this work, we highlight the limitations of existing preconditioning strategies by first carefully examining the condition number of the diagonally scaled matrix and later providing realistic counter-examples for some well-established preconditioning strategies. Building on those insights, we propose a robust deflation-based preconditioning technique tailored to immersed finite element methods.

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arXiv2025

Identifying Kronecker product factorizations

The Kronecker product is an invaluable tool for data-sparse representations of large networks and matrices with countless applications in machine learning, graph theory and numerical linear algebra. In some instances, the sparsity pattern of large matrices may already hide a Kronecker product. Similarly, a large network, represented by its adjacency matrix, may sometimes be factorized as a Kronecker product of smaller adjacency matrices. In this article, we determine all possible Kronecker factorizations of a binary matrix and visualize them through its decomposition graph. Such sparsity-informed factorizations may later enable good (approximate) Kronecker factorizations of real matrices or reveal the latent structure of a network. The latter also suggests a natural visualization of Kronecker graphs.

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arXiv2025

On the prospects of interpolatory spline bases for accurate mass lumping strategies in isogeometric analysis

While interpolatory bases such as the Lagrange basis form the cornerstone of classical finite element methods, they have been replaced in the more general finite element setting of isogeometric analysis in favor of other desirable properties. Yet, interpolation is a key property for devising accurate mass lumping strategies that are ubiquitous in explicit dynamic analyses of structures. In this article, we explore the possibility of restoring interpolation for spline bases within isogeometric analysis for the purpose of mass lumping. Although reminiscent of the spectral element method, this technique comes with its lot of surprises and challenges, which are critically assessed.

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arXiv2025

Stabilization techniques for immersogeometric analysis of plate and shell problems in explicit dynamics

Finite element plate and shell formulations are ubiquitous in structural analysis for modeling all kinds of slender structures, both for static and dynamic analyses. The latter are particularly challenging as the high order nature of the underlying partial differential equations and the slenderness of the structures all impose a stringent constraint on the critical time step in explicit dynamics. Unfortunately, badly cut elements in immersed finite element discretizations further aggravate the issue. While lumping the mass matrix often increases the critical time step, it might also trigger spurious oscillations in the approximate solution thereby compromising the numerical solution. In this article, we extend our previous work in \cite{voet2025stabilization} to allow stable immersogeometric analysis of plate and shell problems with lumped mass matrices. This technique is based on polynomial extensions and restores a level of accuracy comparable to boundary-fitted discretizations.

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arXiv2025

Mass lumping and stabilization for immersogeometric analysis

Trimmed (multi-patch) geometries are the state-of-the-art technology in computer-aided design for industrial applications such as automobile crashworthiness. In this context, fast solution techniques extensively rely on explicit time integration schemes in conjunction with mass lumping techniques that substitute the consistent mass with a (usually diagonal) approximation. For smooth isogeometric discretizations, Leidinger [1] first showed that mass lumping removed the dependency of the critical time-step on the size of trimmed elements. This finding has attracted considerable attention but has unfortunately overshadowed another more subtle effect: mass lumping may disastrously impact the accuracy of low frequencies and modes, potentially inducing spurious oscillations in the solution. In this article, we provide compelling evidence for this phenomenon and later propose a stabilization technique based on polynomial extensions that restores a level of accuracy comparable to boundary-fitted discretizations.

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arXiv2024

A theoretical analysis of mass scaling techniques

Mass scaling is widely used in finite element models of structural dynamics for increasing the critical time step of explicit time integration methods. While the field has been flourishing over the years, it still lacks a strong theoretical basis and mostly relies on numerical experiments as the only means of assessment. This contribution thoroughly reviews existing methods and connects them to established linear algebra results to derive rigorous eigenvalue bounds and condition number estimates. Our results cover some of the most successful mass scaling techniques, unraveling for the first time well-known numerical observations.

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arXiv2024

A theoretical study on the effect of mass lumping on the discrete frequencies in immersogeometric analysis

In structural dynamics, mass lumping techniques are commonly employed for improving the efficiency of explicit time integration schemes and increasing their critical time step constrained by the largest discrete frequency of the system. For immersogeometric methods, Leidinger \cite{leidinger2020explicit} first showed in 2020 that for sufficiently smooth spline discretizations, the largest frequency was not affected by small trimmed elements if the mass matrix was lumped, a finding later supported by several independent numerical studies. This article provides a rigorous theoretical analysis aimed at unraveling this property. By combining linear algebra with functional analysis, we derive sharp analytical estimates capturing the behavior of the largest discrete frequency for lumped mass approximations and various trimming configurations. Additionally, we also provide estimates for the smallest discrete frequency, which has lately drawn closer scrutiny. Our estimates are then confirmed numerically for 1D and 2D problems.

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arXiv2024

Mass lumping and outlier removal strategies for complex geometries in isogeometric analysis

Mass lumping techniques are commonly employed in explicit time integration schemes for problems in structural dynamics and both avoid solving costly linear systems with the consistent mass matrix and increase the critical time step. In isogeometric analysis, the critical time step is constrained by so-called "outlier" frequencies, representing the inaccurate high frequency part of the spectrum. Removing or dampening these high frequencies is paramount for fast explicit solution techniques. In this work, we propose mass lumping and outlier removal techniques for nontrivial geometries, including multipatch and trimmed geometries. Our lumping strategies provably do not deteriorate (and often improve) the CFL condition of the original problem and are combined with deflation techniques to remove persistent outlier frequencies. Numerical experiments reveal the advantages of the method, especially for simulations covering large time spans where they may halve the number of iterations with little or no effect on the numerical solution.

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arXiv2023

Preconditioning techniques for generalized Sylvester matrix equations

Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they now arise in an increasingly large number of applications. In this work, we consider algebraic parameter-free preconditioning techniques for the iterative solution of generalized multiterm Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. While the former requires solving standard Sylvester equations in each iteration, the latter only requires matrix-matrix multiplications, which are highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby enhancing their performance with little or no damage to their effectiveness.

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arXiv2022

A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis

Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.

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arXiv2022

On the fast assemblage of finite element matrices with application to nonlinear heat transfer problems

The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is frequently pointed out as one of the bottlenecks in the computations. The second bottleneck being the large and numerous linear systems to be solved arising from the use of Newton's method to solve nonlinear systems of equations. In this paper, we will address the first issue. We will see how under mild assumptions the assemblage procedure may be rewritten using a completely loop-free algorithm. Our approach leads to a small matrix-matrix multiplication for which we may count on highly optimized algorithms.

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