Abstract
This contribution sheds light on the role of infinite idealization in structural analysis, by exploring how infinite elements and finite element methods are combined in civil engineering models. This combination, I claim, should be read in terms of a ‘complementarity function’ through which the representational ideal of completeness is reached in engineering model-building. Taking a cue from Weisberg’s definition of multiple-model idealization, I highlight how infinite idealizations are primarily meant to contribute to the prediction of structural behavior in Multiphysics approaches.
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Notes
To Strevens, in the case of a simple idealization a simple operation, namely, a change in the parameters of the veridical model, creates a new and partly fictional model that continues to model and to explain the target phenomenon. On the contrary, in asymptotic idealization, “a fiction is introduced by taking some sort of limit. Rather than setting a parameter to zero, for example, a limit is taken as that parameter approaches zero” (Strevens 2017, p. 3).
A recent survey of this debate has been offered by Shech (2013). For different interpretations of PT and approximation, see Menon and Callender (2013) and Norton (2014). The latter in particular shows that approximations of arbitrarily large but finite systems are often mistaken for infinite idealizations in statistical and thermal physics. For a general discussion on the use of infinite idealization in statistical mechanics, see Batterman (2011). For the distinction between approximation and idealization, see Norton (2012).
More recently, Tegmark (2015) addressed the fundamental question of the use of infinity in physics and suggested dropping this notion from the realm of physics itself.
According to Norton (2012), “an idealization is a real or fictitious system, distinct from the target system, some of whose properties provide an inexact description of some aspects of the target system”.
Weisberg (2007, pp. 649–650) considers completeness as a unique representational ideal because it directs theorists towards a goal and makes them strive to include every aspect in their representations even if this ideal can never be achieved. Therefore, he suggested that it was reminiscent of a regulative function in Kantian terms.
Zimmermann (2006, 11ff.) defined Multiphysics as any complete coupled system of differential equations that has more than one independent variable of different physical dimensions.
Safety and reliability are representational goals in civil engineering, because its models must embody them as goals. In representing the structural behavior, engineers are not simply satisfied by simulating the collapse of a bridge, rather they aim at preventing it by finding out the model that accounts for the safety and reliability of the structure. This in turn influences the kind of formalism that is used and the way in which the Multiphysics approach is employed. For instance, when designing a building and trying to meet the safety criteria, engineers have to simulate a fire and will use a Multiphysics approach considering heat flow and mass transfer, but to reach completeness they will also use other means to simulate the psychology of the people escaping from the fire, in order to find the exact points in which the fire exit signs must be put. Civil engineering models do not simply describe reality but must anticipate it. In this respect it is very different from other sciences. See Van Lamsweerde (2001), Lind (2005). On goal-oriented approaches in engineering see Kavakli and Loucopoulos (2005).
Indeed, civil engineers must give an account of complexity at a multiscale level and produce models that consider the largest number of variables, in order to obtain reliable results, see Keyes et al. 2013, p. 6. See also footnote 7 for an example.
Multiple-models idealization assumes that there is no single idealization that can fully capture all the observed behavior relevant to a phenomenon. Therefore, multiple-models idealization can imply the existence of a combination of different methods and specifically the complementarity functions of methods such as FEM and IEM.
It is possible to think of infinite idealization in terms of a meta-idealization if the system to which it applies is a new one with respect to the original target system. This concept of meta-idealization could help in unifying different types of idealization. To deal with this topic would be beyond the scope of this paper, but it is worth mentioning it here.
The mathematical techniques for BEM consist in replacing the governing differential equations valid over the entire soil mass by integral equations that consider only the boundary values. However, BEM is suitable only if the boundary surface is small compared to the volume of soil to be simulated. In other words, there must be a small boundary-to-volume ratio and this also makes the choice of BEM unsuitable in a number of cases (Bobet 2010).
However, it cannot process exterior domains which contain more than one medium, especially when the interfaces of different mediums extend to infinity (this is a typical problem solved in Multiphysics approaches).
To include vibrations in the dynamic soil-structure interaction means to follow inclusion rules, see Weisberg (2007, p. 650).
Here I mean that infinite idealization is not explanatory if taken by itself and not in connection with FEM. See also De Bianchi (2016) for the general case of idealization as being not explanatory per se.
For a review of the applications of infinite element methods, see Gerdes (2000).
Xia and Zhang (2006) had shown the superiority of merging the finite/infinite methods with respect to the use of finite element method in phenomena involving fluid flow in porous media. In numerical analysis of fluid flow problems, the most difficult task is to find the proper approach to deal with an unbounded exterior domain. In most cases, the simplest solution to such difficulties is to truncate the mesh at some large but finite distance, which results in a relative approximation to infinity. Unfortunately, this method is inaccurate and computationally inefficient. Moreover, this method cannot satisfy the real boundary conditions pertaining to the infinite boundary since displacements and pore pressures are fixed only at infinity.
According to Dong and Selvadurai (2009) this is due to the fact that the finite-infinite element coupling procedure is presented at the matrix level.
This is due to the fact that the truncated boundary \( R = b \) is a connection point between the finite computational domain and the exterior of an unbounded region. The flow potential at \( R = b \) should include the contribution of the flow potential from these two regions. For a comparison of FEM and other techniques used for the external boundary problems, see Chernysheva and Rozin (2016).
This is due to the fact that the Dirichlet condition at the truncated boundary \( R = b \) only ensures C0−continuity of the flow potential at this point. The fact that the point \( R = b \) is a physically internal boundary within the entire physical domain implies that both the flow potential and its first spatial derivative in the normal direction should be continuous across the point which leads to the set of C1-continuous conditions.
No infinite loop is generated in computation because even if the series is taken to be as infinite, still one does not count it.
I cannot deal here with this topic for reasons of space, but the analysis of idealization and infinite idealization, particularly in relationship to explanation in engineering models is fascinating. In particular, to simulate different outcomes and structural behaviors, such as crack propagation depending on the use or otherwise of infinite idealization, sheds light on many engineering models. I shall propose elsewhere this analysis but for the time being it is important to mention the potential of the application of Woodward’s (2003) approach to civil engineering models and the quest for explanation of its models. Consider the question ‘what if things had been different?’, for instance, by using or not using infinite idealization in modeling a bridge structure as it interacts with the combined action of water and wind. Which kind of model explaining vertical and horizontal vibrations would one obtain? This question relates to the reflection upon the ideal of completeness. Indeed, the latter requires that engineering models specify both (external) causal factors that affect the phenomena, as well as the difference-making properties with respect to whether or not a phenomenon occurs (for a discussion on these topics, see also van Eck 2016, p. 5).
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Acknowledgement
This research has been carried out under the Ramón y Cajal programme (RYC-2015-17289) sponsored by the Spanish Ministry of Economy, Industry and Competitiveness (MINECO).
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De Bianchi, S. Combining finite and infinite elements: Why do we use infinite idealizations in engineering?. Synthese 196, 1733–1748 (2019). https://doi.org/10.1007/s11229-018-1864-y
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DOI: https://doi.org/10.1007/s11229-018-1864-y