# Talks and presentations

I also present my works at international conferences. Below is an overview of my talks.

###### An isogeometric Petrov-Galerkin formulation with approximate dual spline functions and mass lumping for higher-order accurate explicit dynamics of shells

11th International Conference on Isogeometric Analysis (IGA2023), June 18-21, Lyon, France.

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In structural dynamics, particularly in crash and metal forming simulations, explicit methods have broad applications. Commercial codes of these computations, such as LS-DYNA, PAM-CRASH, and RADIOSS, rely on three key ingredients to achieve highly efﬁcient transient calculations: (1) low memory requirements; (2) an efﬁcient solve; and (3) relatively large critical time-step values. These ingredients are present in contemporary linear finite element codes based on mass lumping [1]. In this talk, we present a higher order accurate mass lumping technique within the context of isogeometric analysis. Our method uses compactly supported test functions that are ''approximate'' dual functionals of B-splines [2]. Because these dual functionals are linear combinations of the same B-splines, the spanning test space remains unaltered. Lumping the Galerkin mass matrix yields an identity matrix, eliminating the need for matrix inversion. We discuss two approaches that weakly and strongly enforce the essential boundary conditions without losing variational consistency and negatively affecting the accuracy. We demonstrate via numerical examples of thin-walled structures in explicit dynamics settings that using our approach retains high order accuracy. This is further supported by good spectrum properties and high efficiency of the explicit scheme.

**References**:

[1] Hughes, T. J. R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications, 2003.

[2] Chui, C. K., He, W., and Stöckler, J., Nonstationary tight wavelet frames, I: Bounded intervals. Applied and Computational Harmonic Analysis (2004) 17 (2): 141–197.

###### Nonlinear dynamic analysis of rods precluding shear and torsion with isogeometric discretizations

93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics, May 30-June 2, 2023, Dresden, Germany.

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In this work, we investigate, in the context of isogeometric analysis (IGA) [1], the recently developed formulation of nonlinear rods exhibiting only axial and bending deformations introduced in [2]. We utilize the higher-order continuity of smooth spline functions which naturally fulfill the C1 continuity required by the nonlinear formulation of [2]. The number of discrete variable fields, compared to the standard spatial discretization scheme using cubic C1 Hermite polynomials in the same reference, thus can be reduced. The resulting discrete solution belongs to R3 that is larger than the manifold (R3 × S2) of the standard scheme, however, might not preserve the same manifold structure. Inspired by [2], we employ the implicit time integration scheme that is a hybrid combination of the midpoint and trapezoidal rules. It approximately preserves the energy and exactly preserves the linear angular momentum, and thus is efficient and robust for our investigation. We demonstrate, via two- and three-dimensional numerical examples of rods, that isogeometric discretizations of the same polynomial degree and smoothness are less robust than the standard spatial discretization scheme using Hermite polynomials. Their robustness can be improved by using, for instance, the strong approach of outlier removal, or by reducing the time step. We illustrate, via an example of a swinging rod under non- and conservative, and pulsating forces, that the improved discretization scheme thus can be employed for highly nonlinear cases. We also show that the configuration-dependent mass matrix of the studied formulation behaves irregularly and thus cannot be simplified to a conﬁguration-independent one.

**References**:

[1] T. J. R. Hughes, J. A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194 (39) (2005) 4135–4195.

[2] C. G. Gebhardt, I. Romero, On a nonlinear rod exhibiting only axial and bending deformations: mathematical modeling and numerical implementation. Acta Mechanica 232 (10) (2021) 3825–3847.

###### Spectral analysis for assessing membrane locking and unlocking in isogeometric finite element formulations of the curved Euler-Bernoulli beam

10th International Conference on Isogeometric Analysis (IGA2022), November 6-9, Banff, Canada.

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In finite element discretizations of curved beam and shell models, membrane locking refers to the phenomenon of artificial bending stiffness due to the coupling of the bending response and membrane response caused by the local curvature [1,2], which negatively affects accuracy and convergence. The development of locking-preventing discretization technology has a history of more than 40 years, first within classical finite elements and then in isogeometric analysis. Given the multitude of formulations addressing membrane locking, the question arises how to best compare and assess their accuracy and effectivity. In this talk, we present spectral analysis as a tool to assess locking phenomena in finite element formulations and the effectiveness of locking-free formulations [3]. Via comparison the difference between eigenvalue and mode error curves computed on coarse meshes with "asymptotic" error curves computed on "overkill" meshes, locking can be identified and "measured". To demonstrate the intimate relation between membrane locking and spectral accuracy, we focus on the example of a circular ring discretized with isogeometric curved Euler-Bernoulli beam elements. We show that the transverse-displacement-dominating modes are locking-prone, while the circumferential-displacement-dominating modes are naturally locking-free. We use eigenvalue and mode errors to assess five isogeometric finite element formulations in terms of their locking-related efficiency: the displacement-based formulation with full and reduced integration and three locking-free formulations based on the B-bar, discrete strain gap, and Hellinger-Reissner methods. Our study shows that spectral analysis uncovers locking-related effects across the spectrum of eigenvalues and eigenmodes, rigorously characterizing membrane locking in the displacement-based formulation and unlocking in the locking-free formulations.

**References**:

[1] Stolarski, H. and Belytschko, T., Membrane locking and reduced integration for curved elements. Journal of Applied Mechanics, Transactions ASME (1982) 49 (1): 172–176.

[2] Bischoff, M., Ramm, E. and Irslinger, J., Models and finite elements for thin-walled structures. Encyclopedia of Computational Mechanics Second Edition (2018) 1–86.

[3] Nguyen, T.-H., Hiemstra, R. R. and Schillinger, D., Leveraging spectral analysis to elucidate membrane locking and unlocking in isogeometric finite element formulations of the curved Euler-Bernoulli beam. Computer Methods in Applied Mechanics and Engineering (2021) 388: 114240.

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

15th World Congress on Computational Mechanics (WCCM XV - APCOM-VIII), July 31-August 5, 2022, Yokohama, Japan.

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A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, the so-called optical branches formed by spurious outlier frequencies and modes may appear due to boundaries or reduced continuity at patch interfaces [1, 2]. The outlier frequencies are signiﬁcantly overestimated, which unnecessarily reduce the stable critical time-step size in explicit dynamics calculations. Moreover, the outlier modes behave in a spurious manner and may have a negative impact on the solution accuracy and robustness, particularly in hyperbolic problems [3]. In this talk, we present (a) a variational approach based on perturbed eigenvalue analysis to improve the spectral properties of isogeometric multipatch discretizations; and (b) a scheme for estimating optimal scaling parameters of the added perturbation term such that the outlier frequencies are effectively reduced and the accuracy in the remainder of the spectrum and modes is not negatively affected; and (c) how to cast this scheme into a pragmatic iterative procedure that can be readily implemented in any isogeometric analysis framework. We verify numerically via spectral analysis of second-and fourth-order problems that the proposed approach improves spectral properties of isogeometric multipatch discretizations in the one- and multidimensional setting. For exemplary membrane and plate structures, we confirm that our approach maintains spatial accuracy and enables a larger critical time-step size. We also demonstrate that it does not depend on the polynomial degree of spline basis functions.

**References**:

[1] Cottrell, J. A., Reali, A., Bazilevs, Y. and Hughes, T. J. R., Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering (2006) 195 (41): 5257–5296.

[2] Puzyrev V., Deng, Q. and Calo, V. Spectral approximation properties of isogeometric analysis with variable continuity. Computer Methods in Applied Mechanics and Engineering (2018) 334: 22–39.

[3] Hughes, T. J. R., Evans, J. A. and Reali, A., Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Computer Methods in Applied Mechanics and Engineering (2014) 272: 290–320.

###### Variationally consistent framework for higher-order imperfect interface models of thin layers

International Association of Applied Mathematics and Mechanics (GAMM2020@21), March 15-19, 2021, Kassel, Germany.

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In composite structures, thin films and coatings are typically used to prevent damage or to increase structure durability. Direct numerical simulation of their mechanical response requires extreme fine mesh sizes and is thus computationally expensive. Therefore, a finite-thickness interphase model is often approximated by an interface model of zero thickness, based on the reformulation of its mechanical effects as jump conditions in the relevant fields [1]. In this talk, we present a) an extension of an existing first-order into a new higher-order accurate interface model which involves higher-order differential operators in the jump formulation; and b) a variationally consistent framework combining higher-order smooth spline basis functions and cut finite element methods for its numerical approximation [2]. We demonstrate robustness and accuracy of this framework via a two-dimensional Laplace problem with a thin circular interphase.

**References**:

[1] Y. Benveniste and T. Miloh (2001): Imperfect soft and stiff interfaces in two-dimensional elasticity. Mechanics of Materials, 33:309-323, 2001.

[2] Z. Han, S.K.F. Stoter, C.T. Wu, C. Cheng, A. Mantzaflaris, S. Mogilevskaya, D. Schillinger (2019): Consistent discretization of higher-order interface models for thin layers and elastic material surfaces, enabled by isogeometric cut-cell methods. Computer Methods in Applied Mechanics and Engineering, 350:245-267, 2019.