A variational approach based on perturbed eigenvalue analysis for improving spectral properties of isogeometric multipatch discretizations

Nguyen, Thi Hoa; Hiemstra, René R.; Stoter, Stein K. F.; Schillinger, Dominik

doi:10.1016/j.cma.2022.114671

Cited by 9 publications

(7 citation statements)

References 26 publications

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“…There are various potential directions for future research. The proposed mass-scaling term may be extended to target the higher-order derivatives across element interfaces for higher-order polynomial basis functions, i.e., an application of the formulation from [21] for C 0 -continuous basis functions. This would likely mitigate the plateauing behavior observed in Section 3.1 and Fig.…”

Section: Discussionmentioning

confidence: 99%

“…to the largest ξT M Γ ξ [28,21]. Based on this empirical observation, we propose the following approximation:…”

Section: Estimate Of Stiffness To Scaled Mass Ratiomentioning

confidence: 99%

“…As this equation illustrates, the additional terms in the weak formulation only affect the mass matrix. The added matrix M Γ is symmetric positive semidefinite and serves to suppress the highest eigenvalues [21,27]. The variational consistency of the added terms ensures that the true solution also satisfies Eq.…”

Section: Finite Element Formulation With Variationally Consistent Mas...mentioning

confidence: 99%

“…One potential approach for suppressing the frequencies from the lower optical branches is thus to add additional mass-scaling terms that involve jumps of higher-order derivatives. This concept is explored in an isogeometric analysis framework across patch-interfaces in [21], and on domain boundaries in [20]. However, in the current article we aim to maintain the variational consistency of the finite element formulation, which only permits us to add a penalty on the natural conditions of Eqs.…”

Section: The One-dimensional Case: a Stringmentioning

confidence: 99%

“…In the current article, we explore a mass-scaling technique based on a symmetric penalization of the natural transmission conditions across element interfaces. Such techniques are effective at reducing outlier frequencies due to reduced continuity across patch interfaces and on domain boundaries in an isogeometric analysis framework [20,21]. Penalties on the lack of continuity are also a common theme in stabilized methods, such as ghost-penalty for immersed finite element methods [22], edge-stabilization for inf-sup stability of equal-order pairs in mixed formulations [23], and discontinuous Galerkin methods in general [24].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Variationally consistent mass scaling for explicit time-integration schemes of lower- and higher-order finite element methods

Stoter¹,

Nguyen²,

Hiemstra³

et al. 2022

Preprint

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In this paper, we propose a variationally consistent technique for decreasing the maximum eigenfrequencies of structural dynamics related finite element formulations. Our approach is based on adding a symmetric positive-definite term to the mass matrix that follows from the integral of the traction jump across element boundaries. The added term is weighted by a small factor, for which we derive a suitable, and simple, element-local parameter choice. For linear problems, we show that our mass-scaling method produces no adverse effects in terms of spatial accuracy and orders of convergence. We illustrate these properties in one, two and three spatial dimension, for quadrilateral elements and triangular elements, and for up to fourth order polynomials basis functions. To extend the method to non-linear problems, we introduce a linear approximation and show that a sizeable increase in critical time-step size can be achieved while only causing minor (even beneficial) influences on the dynamic response.

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Section: Discussionmentioning

confidence: 99%

“…to the largest ξT M Γ ξ [28,21]. Based on this empirical observation, we propose the following approximation:…”

Section: Estimate Of Stiffness To Scaled Mass Ratiomentioning

confidence: 99%

Section: Finite Element Formulation With Variationally Consistent Mas...mentioning

confidence: 99%

Section: The One-dimensional Case: a Stringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Variationally consistent mass scaling for explicit time-integration schemes of lower- and higher-order finite element methods

Stoter¹,

Nguyen²,

Hiemstra³

et al. 2022

Preprint

Self Cite

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Nonlinear dynamic analysis of shear- and torsion-free rods using isogeometric discretization and outlier removal

Nguyen,

Roccia,

Hiemstra

et al. 2024

Comput Mech

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In this paper, we present a discrete formulation of nonlinear shear- and torsion-free rods introduced by Gebhardt and Romero (Acta Mechanica 232(10):3825–3847, 2021) that uses isogeometric discretization and robust time integration. Omitting the director as an independent variable field, we reduce the number of degrees of freedom and obtain discrete solutions in multiple copies of the Euclidean space $$\left( \mathbb {R}^3\right) $$ R 3 , which is larger than the corresponding multiple copies of the manifold $$\left( \mathbb {R}^3 \varvec{\times } S^2\right) $$ R 3 × S 2 obtained with standard Hermite finite elements. For implicit time integration, we choose the same integration scheme as Gebhardt and Romero in (2021) that is a hybrid form of the midpoint and the trapezoidal rules. In addition, we apply a recently introduced approach for outlier removal by Hiemstra et al. (Comput Methods Appl Mech Eng 387:114115, 2021) that reduces high-frequency content in the response without affecting the accuracy, ensuring robustness of our nonlinear discrete formulation. We illustrate the efficiency of our nonlinear discrete formulation for static and transient rods under different loading conditions, demonstrating good accuracy in space, time and the frequency domain. Our numerical example coincides with a relevant application case, the simulation of mooring lines.

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Variationally consistent mass scaling for explicit time-integration schemes of lower- and higher-order finite element methods

Stoter

Nguyen²,

Hiemstra³

et al. 2022

Computer Methods in Applied Mechanics and Engineering

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A variational approach based on perturbed eigenvalue analysis for improving spectral properties of isogeometric multipatch discretizations

Cited by 9 publications

References 26 publications

Variationally consistent mass scaling for explicit time-integration schemes of lower- and higher-order finite element methods

Variationally consistent mass scaling for explicit time-integration schemes of lower- and higher-order finite element methods

Nonlinear dynamic analysis of shear- and torsion-free rods using isogeometric discretization and outlier removal

Variationally consistent mass scaling for explicit time-integration schemes of lower- and higher-order finite element methods

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