Bounds on the extremal eigenvalues of the finite element stiffness and mass matrices and their spectral condition number

Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.

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Implementing model reduction to the JuliaFEM platform

Rakenteiden Mekaniikka ◽

10.23998/rm.69026 ◽

2018 ◽

Vol 51 (1) ◽

pp. 36-54 ◽

Cited By ~ 1

Author(s):

Marja Liisa Rapo ◽

Jukka Aho ◽

Hannu Koivurova ◽

Tero Frondelius

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Model Reduction ◽

Open Source ◽

Degrees Of Freedom ◽

Mass Matrices ◽

Dynamic Condensation ◽

Dynamic Analyses ◽

Reduction Methods ◽

Large Model

JuliaFEM is an open source finite element method solver written in the Julia language. This paper presents an implementation of two common model reduction methods: the Guyan reduction and the Craig-Bampton method. The goal was to implement these algorithms to the JuliaFEM platform and demonstrate that the code works correctly. This paper first describes the JuliaFEM concept briefly after which it presents the theory of model reduction, and finally, it demonstrates the implemented functions in an example model. This paper includes instructions for using the implemented algorithms, and reference the code itself in GitHub. The reduced stiness and mass matrices give the same results in both static and dynamic analyses as the original matrices, which proves that the code works correctly. The code runs smoothly on relatively large model of 12.6 million degrees of freedom. In future, damping could be included in the dynamic condensation now that it has been shown to work.

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Solving Topology Optimization Problems Using Wavelet Approximations

Volume 2: 24th Design Automation Conference ◽

10.1115/detc98/dac-5618 ◽

1998 ◽

Author(s):

Giuseppe C. A. DeRose ◽

Alejandro R. Díaz

Keyword(s):

Finite Element ◽

Topology Optimization ◽

Finite Element Methods ◽

Condition Number ◽

Optimization Problems ◽

Mesh Size ◽

Elasticity Problem ◽

New Method ◽

Galerkin Scheme

Abstract A new method to solve topology optimization problems is discussed. This method is based on the use of a Wavelet-Galerkin scheme to solve the elasticity problem associated with each iteration of the topology optimization sequence. Typically, finite element methods are used for this analysis. However, as the mesh size grows, the computational requirements necessary to solve the finite element equations increase beyond the capacity of current desk top computers. This problem is inherent to finite element methods, as the condition number of finite element matrices increases with mesh size. Wavelet-Galerkin techniques are used to replace standard finite element methods in an attempt to alleviate this problem. Examples demonstrating the performance of the new methodology are presented.

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On the condition number of high order finite element methods: Influence of p-refinement and mesh distortion

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2020.05.012 ◽

2020 ◽

Vol 80 (11) ◽

pp. 2289-2339

Author(s):

S. Eisenträger ◽

E. Atroshchenko ◽

R. Makvandi

Keyword(s):

Finite Element ◽

Finite Element Methods ◽

Condition Number ◽

High Order ◽

Mesh Distortion

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On a Finite Element Formulation of Dynamical Torsion of Beams Including Warping Inertia and Shear Deformation Due to Nonuniform Warping

Journal of Vibration and Acoustics ◽

10.1115/1.3269131 ◽

1983 ◽

Vol 105 (4) ◽

pp. 476-483

Author(s):

A. Potiron ◽

D. Gay

Keyword(s):

Finite Element ◽

Shear Deformation ◽

Cross Section ◽

Experimental Results ◽

Finite Element Formulation ◽

Element Formulation ◽

Secondary Effects ◽

Mass Matrices ◽

Warping Displacement ◽

Thin Walled

We start from the energetical expressions of dynamical torsion of beams in terms of angular and warping displacement and velocity. We derive the stiffness and two mass matrices including both secondary effects for torsion: the shear deformation due to nonuniform warping and the warping inertia. The suitability of these matrices for evaluation modified torsional frequencies is investigated in the case of thick, as well as thin-walled, cross section beams by comparison with analytical and experimental results.

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Extension of the FGM Beam Finite Element by Warping Torsion

Strojnícky casopis – Journal of Mechanical Engineering ◽

10.2478/scjme-2019-0017 ◽

2019 ◽

Vol 69 (2) ◽

pp. 57-76

Author(s):

Murín Justín ◽

Hrabovský Juraj ◽

Aminbaghai Mehdi ◽

Kutiš Vladimír ◽

Paulech Juraj ◽

...

Keyword(s):

Finite Element ◽

Cantilever Beam ◽

Cross Section ◽

Timoshenko Beam ◽

Material Properties ◽

Torsion Effect ◽

Mass Matrices ◽

Effective Material Properties ◽

Fgm Beam ◽

Torsional Analysis

AbstractIn the contribution, our 3D FGM Timoshenko beam finite element with 12x12 stiffness and mass matrices for doubly symmetric open and closed cross-section [1] is extended by warping torsion effect (non-uniform torsion) to 14x14 finite element matrices. A longitudinal continuous variation of effective material properties is considered by the finite element equations derivation, which can be obtained by homogenization of the spatial varying material properties in real FGM beam. Results of numerical elastostatic non-uniform torsional analysis of the FGM cantilever beam of I-cross-section are presented and the accuracy and effectiveness of the new FGM beam finite element is discussed and evaluated.

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