Error estimation and adaptive meshing for vibration problems

Cook, Robert D.; Avrashi, J.

doi:10.1016/0045-7949(92)90394-f

Cited by 33 publications

(10 citation statements)

References 8 publications

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“…In the computational simulation, the mesh discretization error decreases with the decrease of the grid size, so that numerical diffusion/dispersion can be minimized by choosing the grid size appropriately [26,27,4]. When the numerical results, which are obtained from the mesh of a given grid size, agree well with the corresponding analytical solutions and experimental results, the numerical diffusion/dispersion is considered to be minimum, compared with the physical diffusion/dispersion of the problem.…”

Section: Mesh Discretization Effects Of Computational Domainsmentioning

confidence: 97%

“…To determine the dimensionless growth rate (i.e., ω) in the typical rectangular sub-domain shown in Figure 4, we need to use Equation (27) to investigate how the dimensionless growth rate varies with the dimensionless pressure gradient, p 0 axf , of the aqueous phase liquid on its left-hand-side vertical boundary. Figure 5 shows the variation of the dimensionless growth rate with the Zhao number due to different dimensionless wavenumbers in the NAPL dissolution problem.…”

Section: The Product Of ωδT Associated With the Napl Dissolution Systmentioning

confidence: 99%

“…The processes of deriving Equation (34) can be summarized as follows: we first express Equation (27) in a graphical form (in Figure 5) and then obtain Equation (34) from the consideration of Figure 5. In this regard, Equation (34) can be viewed as a simplified form of Equation (27).…”

Section: The Product Of ωδT Associated With the Napl Dissolution Systmentioning

confidence: 99%

See 2 more Smart Citations

Numerical modeling of toxic nonaqueous phase liquid removal from contaminated groundwater systems: mesh effect and discretization error estimation

Zhao

Poulet

Regenauer-Lieb

2014

Num Anal Meth Geomechanics

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SUMMARYNumerical modeling has now become an indispensable tool for investigating the fundamental mechanisms of toxic nonaqueous phase liquid (NAPL) removal from contaminated groundwater systems. Because the domain of a contaminated groundwater system may involve irregular shapes in geometry, it is necessary to use general quadrilateral elements, in which two neighbor sides are no longer perpendicular to each other. This can cause numerical errors on the computational simulation results due to mesh discretization effect. After the dimensionless governing equations of NAPL dissolution problems are briefly described, the propagation theory of the mesh discretization error associated with a NAPL dissolution system is first presented for a rectangular domain and then extended to a trapezoidal domain. This leads to the establishment of the finger-amplitude growing theory that is associated with both the corner effect that takes place just at the entrance of the flow in a trapezoidal domain and the mesh discretization effect that occurs in the whole NAPL dissolution system of the trapezoidal domain. This theory can be used to make the approximate error estimation of the corresponding computational simulation results. The related theoretical analysis and numerical results have demonstrated the following: (1) both the corner effect and the mesh discretization effect can be quantitatively viewed as a kind of small perturbation, which can grow in unstable NAPL dissolution systems, so that they can have some considerable effects on the computational results of such systems; (2) the proposed finger-amplitude growing theory associated with the corner effect at the entrance of a trapezoidal domain is useful for correctly explaining why the finger at either the top or bottom boundary grows much faster than that within the interior of the trapezoidal domain; (3) the proposed finger-amplitude growing theory associated with the mesh discretization error in the NAPL dissolution system of a trapezoidal domain can be used for quantitatively assessing the correctness of computational simulations of NAPL dissolution front instability problems in trapezoidal domains, so that we can ensure that the computational simulation results are controlled by the physics of the NAPL dissolution system, rather than by the numerical artifacts.

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Section: Mesh Discretization Effects Of Computational Domainsmentioning

confidence: 97%

Section: The Product Of ωδT Associated With the Napl Dissolution Systmentioning

confidence: 99%

See 1 more Smart Citation

Numerical modeling of toxic nonaqueous phase liquid removal from contaminated groundwater systems: mesh effect and discretization error estimation

Zhao

Poulet

Regenauer-Lieb

2014

Num Anal Meth Geomechanics

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“…Joo and Wilson [10] solved the structural dynamic problems by load dependent Ritz vector superposition in time and adaptive mesh refinement guided by a residue-type error estimator. Cook and Avrashi [11] discussed error estimation and adaptive mesh refinement for vibration problems. Belyschko and Tabbara [12] studied h-adaptive procedures for transient solid mechanics problems with emphasis on localizations due to material instability.…”

Section: Introductionmentioning

confidence: 99%

Adaptive superposition of finite element meshes in elastodynamic problems

Yue

Robbins

2005

Int. J. Numer. Meth. Engng

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SUMMARYAn adaptive finite element procedure is developed for modelling transient phenomena in elastic solids, including both wave propagation and structural dynamics. Although both temporal and spatial adaptivity are addressed, the novel feature of the formulation is the use of mesh superposition to produce spatial refinement (referred to as s-adaptivity) in transient problems. Spatial error estimation is based on superconvergent patch recovery of higher-order accurate stresses and is used to guide mesh adaptivity, while the temporal error estimation is based on the assumption of linearly varying third-order time derivatives of the displacement field and is used to adjust the time step size for the HHT-variant of the Newmark direct numerical integration method. Spatial adaptivity of the mesh is performed using a hierarchical h-refinement scheme that is efficiently implemented using a structured version of finite element mesh superposition. This particular spatial adaptivity scheme is extremely fast and consequently makes it feasible to repeatedly update both the mesh and the time increment as required in an adaptive transient analysis. This work represents the initial effort in applying this type of spatial adaptivity to transient problems. Three example problems are given to demonstrate the performance characteristics of the s-adaptive procedure.

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“…Cook and Avrashi, 1992 [21] attempt to estimate finite element model errors specifically for dynamic models, such as the models used in the case studies in this thesis. Using the strain energy distributions, the error in the natural frequency calculated by the finite element model is estimated.…”

Section: Finite Element Model Adaptive Meshmentioning

confidence: 99%

Spatial Nyquist fidelity method for structural models of opto-mechanical systems

2008

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As technology employed in complex multidisciplinary systems such as high performance telescope systems becomes ever more sophisticated, simulation is becoming more important during the initial design phases. The method of building these simulation models is often based on engineering experience with older, potentially dissimilar systems. A method is needed to measure the fidelity of simulation models so early design decisions are made based on appropriate modeling techniques.Previously, fidelity was a qualitative concept used to indicate a model's validity or accuracy. Here it is defined as the ability of a model to accurately predict chosen output figures of merit. In this thesis, a quantitative measure of fidelity, termed the Nyquist fidelity metric, is defined for commonly used structural model components. It expresses the ability of a finite element model to accurately predict structural eigenvalues based on the mesh size required by the spatial Nyquist criterion. The Nyquist fidelity method is developed which uses the fidelity metric to both assess the fidelity of existing complex models and to synthesize new multi-component models starting from architectural considerations such as geometric and material properties of the system. This method also estimates the error bound on the output figures of merit based on the fidelity levels and sensitivity analysis.The Nyquist fidelity method is applied to simple sample problems to first demonstrate the methodology and it is then applied to complex telescope models to show the accuracy and computational benefits of this method compared to current methods such as model reduction. The three high performance telescope case studies are the Modular Optical Space Telescope (MOST), the Thirty Meter Telescope, and the Stratospheric Observatory for Infrared Astronomy. It is shown in the MOST example that the Nyquist fidelity method provides a 40% improvement in computational time while assuring less than 5% modal frequency error, and less than 2.2% error in the output figure of merit.

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Error estimation and adaptive meshing for vibration problems

Cited by 33 publications

References 8 publications

Numerical modeling of toxic nonaqueous phase liquid removal from contaminated groundwater systems: mesh effect and discretization error estimation

Numerical modeling of toxic nonaqueous phase liquid removal from contaminated groundwater systems: mesh effect and discretization error estimation

Adaptive superposition of finite element meshes in elastodynamic problems

Spatial Nyquist fidelity method for structural models of opto-mechanical systems

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