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methods described above. For example, one CV may represent a region which is not very homogeneous, but nevertheless described by one set of state variables. By decreasing the size of the CVs, the error can be decreased systematically [243]. The strategy followed in this work is to choose the CV size according to the expected gradients, see also [243]. Nevertheless, as long as the CV size is larger than the length scale of the “smallest” physical effect, the error will not vanish. Convergence errors occur in all iterative algorithms. Except for trivial cases, the solver will never produce an exact solution of the system of equations. Instead, the solution is improved until the difference between two consecutive iterations is smaller than a certain tolerance. Typically one defines an absolute and relative upper bound- ary, which are a direct measure of the solution’s uncertainty. To improve the quality of the solution, the tolerance can be decreased, which in turn increases computation time. In order to limit the computational cost of finding a solution which satisfies the tolerance requirements, either all variables in the solution vector y should be scaled to the range [0:1] or the tolerance should be set independently for each variable in the solution vector. Finally, modeling errors are errors which are present in the design or implemen- tation of the model. While this includes plain mistakes, most errors of this type are introduced on purpose in order to reduce the complexity of the model, e.g. the as- sumption of dilute solutions or 100% activity of dissolved species as discussed above. Another important source of errors is the neglect of certain effects. What is tricky about modeling errors is that there is no way to systematically check for or even de- crease them. Instead, assumptions and the errors associated with them can only be justified by additional studies or calculations. Even though all errors present in modeling studies are systematical errors, it makes sense to discriminate between precision and accuracy of the results [240, chap. 1.1]. While convergence and discretization errors only affect the precision and can be re- duced by increasing the computational effort, there is no systematic strategy to avoid modeling errors, which typically affect the accuracy of the results. The strategy fol- lowed in this work is to increase the precision of a given simulation until the results do not change significantly anymore. That way, a benchmark for the uncertainty is available. It is, however, by no means a measure for the true accuracy, i.e. the devia- tion of the simulation results from “reality” (however one may define this). It would be a daunting task (probably worth its own thesis) to define and analyze the effect of all modeling errors in this or any other reasonably complex model. Finally therefore, it has to stated that, even though the model was carefully de- signed, calibrated, and partly validated, there is no hard evidence that it actually describes something “true”. Confidence in the model must be built in the context of 89PDF Image | Lithium-Sulfur Battery: Design, Characterization, and Physically-based Modeling
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