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Lubrication in Counterformal Contacts—Elastohydrodynamic Lubrication (EHL)
Published in Q. Jane Wang, Dong Zhu, Interfacial Mechanics, 2019
It is well known that, for any computational analysis based on numerical discretization, the higher the mesh density, the smaller the discretization error but the longer the computing time, if the model and the numerical algorithms remain the same. Specifically, for EHL problems, when mesh density is doubled, the change of converged film thickness solution between two subsequent meshes reduces by a factor of 4, but the computing time required for each iteration is increased approximately by the same factor of 4, if a second-order discretization scheme is used. Therefore, it is always needed in practice to find a reasonably good compromise between the numerical accuracy and the computation efficiency. The usage of extremely dense meshes should be avoided unless absolutely necessary.
Numerical Differentiation
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
Numerical differentiation is an ill-conditioned problem in the sense that small perturbations in the input can lead to significant differences in the outcome. It is, however important, since many problems require approximations to derivatives. It introduces the concept of discretization error, which is the error occurring when a continuous function is approximated by a set of discrete values. This is also often known as the local truncation error. It is different from the rounding error, which is the error due to the inherent nature of floating point representation. For a given function f(x) : ℝ → ℝ the derivative f′(x) is defined as f′(x)=limh→0f(x+h)−f(x)h.
The finite element method
Published in Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee, Numerical Methods in Mechanics of Materials, 2017
Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee
Discretization error is due to the approximation of the domain with a finite number of elements of fixed geometry. For instance, consider the analysis of a rectangular plate with a centrally located hole, Figure 4.1a. Due to symmetry, it is sufficient to model only one-fourth of the plate. If the region is subdivided into triangular elements (a triangular mesh or grid), the circular hole is approximated by a series of straight lines. If a few large triangles are used in a coarse mesh (Figure 4.1b), a greater discretization error results than if a large number of small elements is used in a fine mesh (Figure 4.1c). Other geometric shapes may be chosen. For example, with quadrilateral elements that can represent curved sides, the circular hole is more accurately approximated (Figure 4.1d). Hence, discretization error may be reduced by grid refinement. The grid can be refined by using more elements of the same type but of smaller size (h-refinement; Cook et al., 2007) or by using elements of different type (p-refinement).
Validation of Pronghorn Pressure Drop Correlations Against Pebble Bed Experiments
Published in Nuclear Technology, 2022
Jieun Lee, Paolo Balestra, Yassin A. Hassan, Robert Muyshondt, Duy Thien Nguyen, Richard Skifton
The simulation uncertainty results from three different sources: numerical, input, and model uncertainties.31 The numerical uncertainty arises from discretization error, assuming that the double precision round-off error is negligible and an iterative convergence tolerance set to sufficiently converges the solutions.32 The input uncertainty comes from uncertain input parameters, such as the porosity and fluid flow rate. The model formulation based on specific assumptions causes the model uncertainty , which is considered as a completely epistemic uncertainty due to the lack of the inferable knowledge in the model.33 Assuming all the errors are independent, the overall simulation uncertainty is determined to be34
Verification of numerical solutions of thermal radiation problems in participating and nonparticipating media
Published in Numerical Heat Transfer, Part B: Fundamentals, 2023
Antonio Carlos Foltran, Carlos Henrique Marchi, Luís Mauro Moura
If the analytical solution of the mathematical model is not yet available, it can be created for a specific problem similar to the one intended to be solved and the asymptotic order can be verified by numerical experiments in grids of successive grid refinement. This technique is called Order Verification Via the Manufactured Solution Procedure (OVMSP) and it is detailed in [1] and [4]. Interestingly, Eq. (2) can be used in cases where the analytical solution is not available. In those cases, it is possible to estimate the discretization error, where represents the estimated analytical solution.
CFD modelling of pump-around jet mixing tanks: a reliable model for overall mixing time prediction
Published in Journal of the Chinese Institute of Engineers, 2019
Eakarach Bumrungthaichaichan, Santi Wattananusorn
According to the previous works, it can be seen that the CFD models can be used to predict the overall mixing times instead of the mixing time correlations obtained by experiments. However, the numerical errors of these CFD models were not assessed. Generally, there are three different components of numerical error, including round-off error, iterative error, and discretization error (Roache 2009). Eça and Hoekstra (2009, 2014) stated that the contribution of round-off and iterative errors to the numerical error is negligible as compared to the discretization error. Furthermore, the suitable numerical methods for pump-around jet mixing tank simulation were not clearly studied and represented.