Novel higher order mass matrices for isogeometric structural vibration analysis

Wang, Dongdong; Liu, Wei; Zhang, Hanjie

doi:10.1016/j.cma.2013.03.011

Cited by 73 publications

(38 citation statements)

References 45 publications

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“…We should mention that we have never seen in the literature such a significant improvement of the order of the dispersion error for the conventional finite elements, the spectral elements or the isogeometric elements. For example, for the linear and high-order finite and isogeometric elements (e.g., see [2,22,20,11,14]), the order of the dispersion error has been improved from the order 2p to the order 2p + 2 and this leads to a significant decrease in the computation time at a given accuracy. The increase in accuracy for the new isogeometric elements with the order 4p of the dispersion error is much higher compared with the known approaches.…”

Section: Discussionmentioning

confidence: 99%

“…(21) with the help of Eq. (20) we get that a 1 = 131/210. Next, let us assume that the displacement is the same for the entire domain.…”

Section: A New Approach For the Quadratic Isogeometric Elements With mentioning

confidence: 99%

“…(3), respectively (at this point, these elemental matrices are not defined). For example, for the conventional quadratic isogeometric elements, the coefficients of the stencil equation (10) are (e.g., see [10,6,20,21]):…”

Section: A New Approach For the Quadratic Isogeometric Elements With mentioning

confidence: 99%

“…It has been shown in these papers that the isogeometric elements yield more accurate numerical results for wave propagation problems compared with the high-order finite elements. The modification of the non-diagonal mass matrix in [20,21] for the isogeometric elements allows the increase in the order of the dispersion error from order 2p to order 2p + 2 in the 1-D case and for one specific direction of harmonic waves in the 2-D case (these techniques do not improve the order of the dispersion error in the general 2-D case). It is also necessary to mention that in contrast to the high order finite and spectral elements, the stencil equation for the isogeometric elements on uniform meshes is the same for all internal degrees of freedom (for the high order finite and spectral elements there are several stencil equations with different structures depending on the location of nodes).…”

Section: Introductionmentioning

confidence: 99%

“…(2) contains the numerical dispersion error. Usually the analysis of the numerical dispersion error and its improvement for many space-discretization techniques such as the finite elements, spectral elements, isogeometric elements and others starts with the analysis and modifications of the elemental mass and stiffness matrices; see [10,1,2,7,8,15,16,17,18,22,11,14,20,21]. For example, one simple and effective finite-element technique for acoustic and elastic wave propagation problems is based on the calculation of the mass matrix M M M in Eq.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

Idesman¹

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. Based on the optimal coefficients of the stencil equation, a numerical technique for the reduction of the numerical dispersion error has been suggested. New isogeometric elements with the reduced numerical dispersion error for wave propagation problems have been developed with the suggested approach. By the minimization of the order of the dispersion error of the stencil equation, the order of the dispersion error is improved from order 2p (the conventional isogeometric elements) to order 4p (the isogeometric elements with reduced dispersion) where p is the order of the polynomial approximations. Because all coefficients of the stencil equation are obtained from the minimization procedure, the obtained accuracy is maximum possible. The corresponding elemental mass and stiffness matrices of the isogeometric elements with reduced dispersion are calculated with help of the optimal coefficients of the stencil equation. The analysis of the dispersion error of the isogeometric elements with the lumped mass matrix has also shown that independent of the procedures for the calculation of the lumped mass matrix, the second order of the dispersion error cannot be improved with the conventional stiffness matrix. However, the dispersion error with the lumped mass matrix can be improved from the second order to order 2p by the modification of the stiffness matrix. The numerical examples confirm the computational efficiency of the isogeometric elements with reduced dispersion that significantly reduce the computation time at a given accuracy. The numerical results obtained by the new and conventional isogeometric elements may include spurious oscillations due to the dispersion error. These oscillations can be quantified and filtered by the two-stage timeintegration technique developed recently. The approach developed can be directly applied to other space-discretization techniques with similar stencil equations. 946Available online at www.eccomasproceedia.org Eccomas Proceedia COMPDYN (2017) 946-954

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Section: Discussionmentioning

confidence: 99%

“…(21) with the help of Eq. (20) we get that a 1 = 131/210. Next, let us assume that the displacement is the same for the entire domain.…”

Section: A New Approach For the Quadratic Isogeometric Elements With mentioning

confidence: 99%

Section: A New Approach For the Quadratic Isogeometric Elements With mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

Idesman¹

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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The simulation of transport processes in cementitious materials with embedded healing systems

Freeman

Jefferson

2019

Num Anal Meth Geomechanics

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A new model for simulating the transport of healing agents in self-healing (SH) cementitious materials is presented. The model is applicable to autonomic SH material systems in which embedded channels, or vascular networks, are used to supply healing agents to damaged zones. The essential numerical components of the model are a crack flow model, based on the Navier-Stokes equations, which is coupled to the mass balance equation for simulating unsaturated matrix flow. The driving forces for the crack flow are the capillary meniscus force and the force derived from an external (or internal) pressure applied to the liquid healing agent. The crack flow model component applies to non-uniform cracks and allows for the dynamic variation of the meniscus contact angle, as well as accounting for inertial terms. Particular attention is paid to the effects of curing on the flow characteristics. In this regard, a kinetic reaction model is presented for simulating the curing of the healing agent and a set of relationships established for representing the variation of rheological properties with the degree of cure. Data obtained in a linked experimental programme of work is employed to justif the hoi e a d fo of the o stituti e elatio ships, as ell as to ali ate the odel s evolution functions. Finally, a series of validation examples are presented that include the analysis of a series of concrete beam specimens with an embedded vascular network. These examples demonstrate the ability of the model to capture the transport behaviour of this type of SH cementitious material system.

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Implicit dynamic analysis using an isogeometric Reissner–Mindlin shell formulation

Sobota

Dornisch

Müller

et al. 2016

Numerical Meth Engineering

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SummaryIn isogeometric analysis, identical basis functions are used for geometrical representation and analysis. In this work, non‐uniform rational basis splines basis functions are applied in an isoparametric approach. An isogeometric Reissner–Mindlin shell formulation for implicit dynamic calculations using the Galerkin method is presented. A consistent as well as a lumped matrix formulation is implemented. The suitability of the developed shell formulation for natural frequency analysis is demonstrated by a numerical example. In a second set of examples, transient problems of plane and curved geometries undergoing large deformations in combination with nonlinear material behavior are investigated. Via a zero‐thickness stress algorithm for arbitrary material models, a J2‐plasticity constitutive law is implemented. In the numerical examples, the effectiveness, robustness, and superior accuracy of a continuous interpolation method of the shell director vector is compared with experimental results and alternative numerical approaches. Copyright © 2016 John Wiley & Sons, Ltd.

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Novel higher order mass matrices for isogeometric structural vibration analysis

Cited by 73 publications

References 45 publications

Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

The simulation of transport processes in cementitious materials with embedded healing systems

Implicit dynamic analysis using an isogeometric Reissner–Mindlin shell formulation

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