Fast evaluation of finite element weak forms using python tensor contraction packages

Cimrman, Robert

doi:10.1016/j.advengsoft.2021.103033

Cited by 6 publications

(4 citation statements)

References 23 publications

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“…The sensor has the following properties: All results reported here were obtained using the Open Source finite element software SfePy (Cimrman et al, 2019;Cimrman, 2021). The results for the non-symmetric Nitsche's method without penalty term (β = 0), our first choice, are illustrated in Fig.…”

Section: Numerical Convergence Resultsmentioning

confidence: 99%

Weak Imposition of Potential Dirichlet Boundary Conditions in Piezoelasticity: Numerical Study

Cimrman

2024

Engineering Mechanics

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Detailed modelling of piezoelastic structures often includes also an external circuit. This is tied with the need for a weak imposition of electric potential Dirichlet boundary conditions and questions of simulation accuracy arise. Several methods allowing the weak imposition, namely the non-symmetric Nitsche's method, penalty method and their combinations, are evaluated and their numerical convergence w.r.t. uniform mesh refinement for various finite element approximation orders is reported. The numerical study is performed on a geometry corresponding to a piezoelectric sensor.

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Section: Numerical Convergence Resultsmentioning

confidence: 99%

Weak Imposition of Potential Dirichlet Boundary Conditions in Piezoelasticity: Numerical Study

Cimrman

2024

Engineering Mechanics

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“…12: end for All computations were performed using the Open Source finite element package SfePy. 37,38 The numerous calculations were launched and the results assembled with the help of the package soops. 39…”

Section: Algorithmmentioning

confidence: 99%

Higher‐order inverse mass matrices for the explicit transient analysis of heterogeneous solids

Cimrman,

Kolman,

González

et al. 2024

Numerical Meth Engineering

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New methods are presented for the direct computation of higher‐order inverse mass matrices (also called reciprocal mass matrices) that are used for explicit transient finite element analysis. The motivation of this work lies in the need of having appropriate sparse inverse mass matrices, which present the same structure as the consistent mass matrix, preserve the total mass, predict suitable frequency spectrum and dictate sufficiently large critical time step sizes. For an efficient evaluation of the reciprocal mass matrix, the projection matrix should be diagonal. This condition can be satisfied by adopting dual shape functions for the momentum field, generated from the same shape functions used for the displacement field. A theoretically consistent derivation of the inverse mass matrix is based on the three‐field Hamilton principle and requires the projection matrix to be evaluated from the integral of these shape functions. Unfortunately, for higher‐order FE shape functions and serendipity FE elements, the projection matrix is not positive definitive and can not be employed. Therefore, we study several lumping procedures for higher order reciprocal mass matrices considering their effect on total‐mass preserving, frequency spectra and accuracy in explicit transient simulations. The article closes with several numerical examples showing suitability of the direct inverse mass matrix in dynamics.

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“…All results reported here were obtained using the Open Source finite element software SfePy (Cimrman et al, 2019;Cimrman, 2021). The initial state was a result of a static calculation with the gravity load by 280 g weight attached in the point W .…”

Section: Numerical Examplementioning

confidence: 99%

Dynamics of a cantilever beam with piezoelectric sensor: Finite element modeling

Cimrman,

Kolman,

Musil

et al. 2023

Engineering Mechanics

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An elastodynamical model of a cantilever beam coupled with a piezoelectric sensor is introduced and its discretization using the finite element method is presented. The mathematical model includes additional terms that enforce the floating potential boundary condition for keeping a constant charge on an electrode of the sensor. The behaviour of the model is illustrated using a numerical example corresponding to an experimental setup, where vibrations of the beam and the potential on the sensor are measured.

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Fast evaluation of finite element weak forms using python tensor contraction packages

Cited by 6 publications

References 23 publications

Weak Imposition of Potential Dirichlet Boundary Conditions in Piezoelasticity: Numerical Study

Weak Imposition of Potential Dirichlet Boundary Conditions in Piezoelasticity: Numerical Study

Higher‐order inverse mass matrices for the explicit transient analysis of heterogeneous solids

Dynamics of a cantilever beam with piezoelectric sensor: Finite element modeling

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