Critical assessment of different mass lumping schemes for higher order serendipity finite elements

Duczek, Sascha; Gravenkamp, Hauke

doi:10.1016/j.cma.2019.03.028

Cited by 39 publications

(38 citation statements)

References 34 publications

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“…1. These findings are discussed in detail in forthcoming publications by the second author concerning mass lumping in the context of serendipity finite elements and SEM [4,5]. The visualized mode shapes correspond to the data points within the grey domain.…”

Section: Resultsmentioning

confidence: 64%

“…The results are shown in Fig. In the context of the FEM, a lumped mass matrix results in a deteriorated convergence [4]. On the left hand side the convergence behavior of the 10th eigenfrequency is compared, while on the right hand side the eigenmodes are illustrated.…”

Section: Resultsmentioning

confidence: 99%

“…In this contribution, the advantages of highorder finite element methods, such as a possibly exponential rates of convergence, will be exploited. These findings are discussed in detail in forthcoming publications by the second author concerning mass lumping in the context of serendipity finite elements and SEM [4,5]. Consequently, the computational time required for the analysis can be notably reduced for a given error threshold.…”

mentioning

confidence: 87%

“…These tools are typically based on conventional low-order finite element methods (h-FEM) for solving the governing partial differential equations (PDEs) of the problem. In the context of the FEM, a lumped mass matrix results in a deteriorated convergence [4]. Based on this approach a similar accuracy compared to classical h-FEM simulations using significantly less degrees of freedom (dof) can be achieved.…”

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confidence: 96%

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High‐order shape functions for interior acoustics

Duvigneau

Duczek

2019

Proc Appl Math and Mech

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The acoustic behavior of structures has lately been in the focus of industrial applications, due to the fact that the acoustic emission of a product is one major parameter which significantly influences the customer perception with respect to comfort and functionality. In this context, the numerical simulation of vibroacoustic problems needs to provide reliable information in order to be able to evaluate the acoustic behavior of new products already in an early stage of the product development cycle. With the help of a suitable simulation model expensive experimental studies can be reduced and acoustically improved designs can be developed. However, there are already commercial software tools available which offer the opportunity to solve coupled vibroacoustic problems. These tools are typically based on conventional low-order finite element methods (h-FEM) for solving the governing partial differential equations (PDEs) of the problem. In this contribution, the advantages of highorder finite element methods, such as a possibly exponential rates of convergence, will be exploited. Based on this approach a similar accuracy compared to classical h-FEM simulations using significantly less degrees of freedom (dof) can be achieved. Consequently, the computational time required for the analysis can be notably reduced for a given error threshold. Reducing the computational effort is of special importance when investigating complex structures of practical relevance. Even today, the overall efficiency of the simulation process is still an issue, despite the ever growing computational power. To exploit the described advantages, high-order FEMs have to be extended to acoustical problems. In this contribution the developed high-order simulation approach is discussed and the obtained results are compared to commercial finite element solutions focussing on the accuracy and the computational effort of the different approaches. High-order finite element methods for acousticsIn this contribution different high-order finite element methods are implemented and compared for interior acoustic problems. Namely we distinguished between three methods: the p-FEM (hierarchical shape functions based on Legendre polynomials), the conventional FEM (Lagrange polynomials with equidistant nodal distributions) and the spectral element method (Lagrange polynomials with with Gauß-Lobatto-Legendre or Gauss-Lobatto-Chebyshev nodal distributions). For further information interested readers are referred to the following monographs [1][2][3]. In contrast to previous works, these methods are applied to the Helmholtz equation which is derived from the acoustic wave equation by a transformation into the frequency domain. The acoustic problem can now be solved for each frequency component assuming a harmonic excitation. ResultsTo demonstrate the capabilities of the different methods an academic benchmark problem is investigated: the modal analysis of a two-dimensional rectangular domain. This article focusses on interior acoustics, consequently, no Sommerfel...

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Section: Resultsmentioning

confidence: 64%

Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 87%

mentioning

confidence: 96%

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High‐order shape functions for interior acoustics

Duvigneau

Duczek

2019

Proc Appl Math and Mech

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“…Mass matrix lumping is one of the important techniques making explicit finite element transient analysis practical . A lumped mass matrix makes the solutions of the discretized momentum equations as efficient as a division.…”

Section: Introductionmentioning

confidence: 99%

A general mass lumping scheme for the variants of the extended finite element method

Asareh

Song

Mullen

et al. 2020

Numerical Meth Engineering

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Summary A new strategy for the mass matrix lumping of enriched elements for explicit transient analysis is presented. It is shown that to satisfy the kinetic energy conservation, the use of zero or negative masses for enriched degrees of freedom of lumped mass matrix may be necessary. For a completely cracked element, by lumping the mass of each side of the interface into the finite element nodes located at the same side and assigning zero masses to the enriched degrees of freedom, the kinetic energy for rigid body translations is conserved without transferring spurious energy across the interface. The time integration is performed by adopting an explicit‐implicit technique, where the regular and enriched degrees of freedom are treated explicitly and implicitly, respectively. The proposed method can be viewed as a general mass lumping scheme for the variants of the extended finite element methods because it can be used irrespective of the enrichment method. It also preserves the optimal critical time step of an intact finite element by treating the enriched degrees of freedom implicitly. The accuracy and efficiency of the proposed mass matrix are validated with several benchmark examples.

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Partitioned formulation of contact‐impact problems with stabilized contact constraints and reciprocal mass matrices

González

Kopačka

Kolman

et al. 2021

Numerical Meth Engineering

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This work presents an efficient and accuracy‐improved time explicit solution methodology for the simulation of contact‐impact problems with finite elements. The proposed solution process combines four different existent techniques. First, the contact constraints are modeled by a bipenalty contact‐impact formulation that incorporates stiffness and mass penalties preserving the stability limit of contact‐free problems for efficient explicit time integration. Second, a method of localized Lagrange multipliers is employed, which facilitates the partitioned governing equations for each substructure along with the completely localized contact penalty forces pertaining to each free substructure. Third, a method for the direct construction of sparse inverse mass matrices of the free bodies in contact is combined with the localized Lagrange multipliers approach. Finally, an element‐by‐element mass matrix scaling technique that allows the extension of the time integration step is adopted to improve the overall performance of the algorithm. A judicious synthesis of the four numerical techniques has resulted in an increased stable explicit step‐size that boosts the performance of the bipenalty method for contact problems. Classical contact‐impact numerical examples are used to demonstrate the effectiveness of the proposed methodology.

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Critical assessment of different mass lumping schemes for higher order serendipity finite elements

Cited by 39 publications

References 34 publications

High‐order shape functions for interior acoustics

High‐order shape functions for interior acoustics

A general mass lumping scheme for the variants of the extended finite element method

Partitioned formulation of contact‐impact problems with stabilized contact constraints and reciprocal mass matrices

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