, Volume 196, Issue 5, pp 1733–1748 | Cite as

Combining finite and infinite elements: Why do we use infinite idealizations in engineering?

  • Silvia De BianchiEmail author
S.I.: Infinite Idealizations in Science


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.


Infinite idealization Multiple-model idealization Engineering Finite element method Infinite element method 



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).


  1. Agrawal, R., & Hora, M. S. (2009). Coupled finite-infinite elements modeling of building frame–soil interaction system. ARPN Journal of Engineering and Applied Sciences, 4(10), 47–54.Google Scholar
  2. Albert, M., & Kliemt, H. (2017). Infinite idealizations and approximate explanations in economics. Joint Discussion Paper Series in Economics, No. 26-2017, Philipps University Marburg, School of Business and Economics, Marburg. Accessed 20 June 2018.
  3. Ardourel, V. (2018). The infinite limit as an eliminable approximation for phase transitions. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 62, 71–84.CrossRefGoogle Scholar
  4. Bangu, S. (2009). Understanding thermodynamic singularities: Phase transitions, data, and phenomena. Philosophy of Science, 76(4), 488–505.CrossRefGoogle Scholar
  5. Bangu, S. (2015). Neither weak, nor strong? Emergence and functional reduction. In B. Falkenburg & M. Morrison (Eds.), Why more is different: Philosophical issues in condensed matter physics and complex systems (pp. 153–166). Berlin: Springer.CrossRefGoogle Scholar
  6. Batterman, R. W. (2005). Critical phenomena and breaking drops: Infinite idealizations in physics. Studies in history and philosophy of science part B: Studies in history and philosophy of modern physics, 36(2), 225–244.CrossRefGoogle Scholar
  7. Batterman, R. W. (2011). Emergence, singularities, and symmetry breaking. Foundations of Physics, 41(6), 1031–1050.CrossRefGoogle Scholar
  8. Bettess, P., & Bettess, J. A. (1991a). Infinite elements for dynamic problems: Part 1. Engineering Computations, 8(2), 99–124.CrossRefGoogle Scholar
  9. Bettess, P., & Bettess, J. A. (1991b). Infinite elements for dynamic problems: Part 2. Engineering Computations, 8(2), 125–151.CrossRefGoogle Scholar
  10. Bobet, A. (2010). Numerical methods in geomechanics. The Arabian Journal for Science and Engineering, 35(1B), 27–48.Google Scholar
  11. Briaud, J. L. (2013). Geotechnical engineering: unsaturated and saturated soils. Hoboken, NJ: Wiley.CrossRefGoogle Scholar
  12. Brown, D. L., Bell, J., Estep, D., Gropp, W., Hendrickson, B., Keller-McNulty, S., Keyes, D., Oden, J. T., Petzold, L., & Wright, M. (2008). Applied mathematics at the U.S. Department of Energy: Past, present, and a view to the future. Office of Science, U.S. Department of Energy. Retrieved from
  13. Butterfield, J. (2011). Less is different: Emergence and reduction reconciled. Foundations of Physics, 41(6), 1065–1135.CrossRefGoogle Scholar
  14. Chen, W., & Wang, F. Z. (2010). A method of fundamental solutions without fictitious boundary. Engineering Analysis with Boundary Elements, 34(5), 530–532.CrossRefGoogle Scholar
  15. Chernysheva, N., & Rozin, L. (2016). Modified finite element analysis for exterior boundary problems in infinite medium. In MATEC Web of Conferences, Vol. 53. Les Ulis: EDP Sciences.
  16. Chuhan, Z., Xinfeng, C., & Guanglun, W. (1999). A coupling model of FE–BE–IE–IBE for non-linear layered soil–structure interactions. Earthquake Engineering and Structural Dynamics, 28(4), 421–441.CrossRefGoogle Scholar
  17. Damour, T. (1987). The problem of motion in Newtonian and Einsteinian gravity. In S. Hawking & W. Israel (Eds.), Three hundred years of gravitation (pp. 128–198). Cambridge: Cambridge University Press.Google Scholar
  18. Das B. M., & Sobhan, K. (2013). Principles of geotechnical engineering. Stamford, CT: Cengage Learning.Google Scholar
  19. De Bianchi, S. (2016). Which explanatory role for mathematics in scientific models? Reply to “The Explanatory Dispensability of Idealizations”. Synthese, 193(2), 387–401.CrossRefGoogle Scholar
  20. Dong, W., & Selvadurai, A. P. S. (2009). A combined finite and infinite element approach for modeling spherically symmetric transient subsurface flow. Computers & Geosciences, 35(3), 438–445.CrossRefGoogle Scholar
  21. Dutta, S. C., & Roy, R. (2002). A critical review on idealization and modeling for interaction among soil–foundation–structure system. Computers & Structures, 80(20), 1579–1594.CrossRefGoogle Scholar
  22. Gerdes, K. (2000). A review of infinite element methods for exterior Helmholtz problems. Journal of Computational Acoustics, 8(01), 43–62.CrossRefGoogle Scholar
  23. Giere, R. N. (1999). Using models to represent reality. In L. Magnani, N. J. Nersessian, & P. Thagard (Eds.), Model-based reasoning in scientific discovery (pp. 41–57). Boston, MA: Springer.CrossRefGoogle Scholar
  24. Godbole, P. N., Viladkar, M. N., & Noorzaei, J. (1990). Nonlinear soil-structure interaction analysis using coupled finite-infinite elements. Computers & Structures, 36(6), 1089–1096.CrossRefGoogle Scholar
  25. Jones, M. R., & Cartwright, N. (Eds.). (2005). Idealization XII: Correcting the model: idealization and abstraction in the sciences (Vol. 12). Amsterdam: Rodopi.Google Scholar
  26. Kavakli, E., & Loucopoulos, P. (2005). Goal modeling in requirements engineering: Analysis and critique of current methods. In J. Krogstie, T. Halpin, & K. Siau (Eds.), Information modeling methods and methodologies: Advanced topics in database research (pp. 102–124). Hershey, PA: IGI Global. Scholar
  27. Keyes, D. E., McInnes, L. C., Woodward, C., Gropp, W., Myra, E., Pernice, M., et al. (2013). Multiphysics simulations: Challenges and opportunities. The International Journal of High Performance Computing Applications, 27(1), 4–83.CrossRefGoogle Scholar
  28. Lind, M. (2005). Modeling goals and functions of control and safety systems. Nordic Nuclear Safety Research, NKS-114.Google Scholar
  29. Liu, C. (1999). Explaining the emergence of cooperative phenomena. Philosophy of Science, 66, 92–106.CrossRefGoogle Scholar
  30. Menon, T., & Callender, C. (2013). Turn and face the strange … Ch-Ch-changes: Philosophical questions raised by phase transitions. In R. Batterman (Ed.), The oxford handbook of philosophy of physics. Oxford: Oxford University Press. Scholar
  31. Morrison, M. (2012). Emergent physics and micro-ontology. Philosophy of Science, 79(1), 141–166.CrossRefGoogle Scholar
  32. Na, T. Y. (Ed.). (1979). Computational methods in engineering boundary value problems. Mathematics in science and engineering (Vol. 145). New York: Academic.Google Scholar
  33. Narens, L., & Luce, R. D. (1990). Three aspects of the effectiveness of mathematics in science. In Mathematics and science (pp. 122–135).Google Scholar
  34. Norton, J. D. (2012). Approximation and idealization: Why the difference matters. Philosophy of Science, 79(2), 207–232.CrossRefGoogle Scholar
  35. Norton, J. D. (2014). Infinite idealizations. In European philosophy of sciencephilosophy of science in Europe and the Viennese Heritage. Vienna Circle Institute Yearbook (Vol. 17, pp. 197–210). Dordrecht, Heidelberg, London, New York: Springer.Google Scholar
  36. Potochnik, A. (2017). Idealization and the Aims of Science. Chicago: University of Chicago Press.CrossRefGoogle Scholar
  37. Rice, C. (2012). Optimality explanations: A plea for an alternative approach. Biology and Philosophy, 27(5), 685–703.CrossRefGoogle Scholar
  38. Rice, C. (2015). Moving beyond causes: Optimality models and scientific explanation. Noûs, 49(3), 589–615.CrossRefGoogle Scholar
  39. Ross, D. (2016). Philosophy of economics. London: Palgrave Macmillan.Google Scholar
  40. Shech, E. (2013). What is the paradox of phase transitions? Philosophy of Science, 80(5), 1170–1181.CrossRefGoogle Scholar
  41. Silvester, P. P., & Ferrari, R. L. (1996). Finite elements for electrical engineers. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  42. Strevens, M. (2008). Depth: An account of scientific explanation. Cambridge, MA: Harvard University Press.Google Scholar
  43. Strevens, M. (2017). The structure of asymptotic idealization. Synthese. Scholar
  44. Tang, Z., et al. (2010). Infinite element method for solving open boundary field problem and its application in resistivity well-logging. In J. Zhu (Ed.), Modelling and computation in engineering (pp. 203–207). London: CRC Press.CrossRefGoogle Scholar
  45. Tegmark, M. (2015). Infinity is a beautiful concept—And it’s ruining physics. In M. J. Brockman (Ed.), This idea must die: Scientific theories that are blocking progress (pp. 48–51). New York: Harper Collins.Google Scholar
  46. Van Eck, D. (2016). The philosophy of science and engineering design. Dordrecht: Springer.Google Scholar
  47. Van Lamsweerde, A. (2001). Goal-oriented requirements engineering: A guided tour. In: Proceedings of the 5th IEEE international symposium on requirements engineering, Washington (pp. 249–262).Google Scholar
  48. Weisberg, M. (2007). Three kinds of idealization. The Journal of Philosophy, 104(12), 639–659.CrossRefGoogle Scholar
  49. West, R. P., & Pavlović, M. N. (1999). Finite-element model sensitivity in the vibration of partially embedded beams. International Journal for Numerical Methods in Engineering, 44(4), 517–533.CrossRefGoogle Scholar
  50. Wilson, M. (1992). Law along the frontier: Differential equations and their boundary conditions. In PSA: Proceedings of the Biennial meeting of the philosophy of science association (Vol. 1990, No. 2, pp. 565–575). Philosophy of Science Association.Google Scholar
  51. Woodward, J. (2003). Making things happen: A theory of causal explanation. Oxford: Oxford University Press.Google Scholar
  52. Xia, K., & Zhang, Z. (2006). Three-dimensional finite/infinite elements analysis of fluid flow in porous media. Applied Mathematical Modelling, 30(9), 904–919.CrossRefGoogle Scholar
  53. Zimmerman, W. B. (2006). Multiphysics modeling with finite element methods (Vol. 18). London: World Scientific Publishing Company.CrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Philosophy and Centre for the History of Science (CEHIC)Autonomous University of BarcelonaBarcelonaSpain

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