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Precursors of Isogeometric Analysis pp 559–579Cite as

Review and Epilogue

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Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 256))

Abstract

In this chapter, we summarize the most important issues which we expect that the reader should have learned from the previous chapters. We repeat the six most important CAGD interpolations, and then we sketch a draft picture on the evolution of the computational methods which are based on the first five of them (older than isoGeometric Analysis). As the author happened to have been involved in these older CAD-based methodologies since 1982 and has somehow contributed in a large part of the whole CAE spectrum (FEM, BEM, and global collocation), the short history of this research is given from his point of view; a full list of the sixty papers, properly classified (in FEM, BEM, and collocation categories), is given in Appendix. In some places, the text is inspired by self-criticism. We believe that the compact information provided in this chapter, as well as the details in the whole book, will strengthen and broaden the horizon of researchers and postgraduate students in the field of Computational Mechanics. In more detail, we anticipate that the gap between CAD and CAE approaches and communities will be further reduced, bridging the older with the contemporary ideas.

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Author information

Authors and Affiliations

  1. School of Mechanical Engineering, National Technical University of Athens, Athens, Greece

    Christopher G. Provatidis

Authors
  1. Christopher G. Provatidis
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Corresponding author

Correspondence to Christopher G. Provatidis .

Appendix 1: Author’s Papers Related with CAD/CAE Integration

Appendix 1: Author’s Papers Related with CAD/CAE Integration

The overall output, including the first paper of the NTUA Group, is shown below. The prefix “J” refers to Journal whereas “C” stands for Conference. Numbering is according to the chronological date, and the existing gaps are due to author’s work in different fields, which may be found at:

http://users.ntua.gr/cprovat/index_en.htm

  1. I.

    Galerkin Method

Journal Papers

  1. J0.

    Kanarachos A, Deriziotis D (1989) On the solution of Laplace and wave propagation problems using ‘C-elements’. Finite Elem Anal Des 5:97–109

  2. J17.

    Provatidis C, Kanarachos A (2001) Performance of a macro-FEM approach using global interpolation (Coons’) functions in axisymmetric potential problems. Comput Struct 79:1769–1779

  3. J19.

    Provatidis CG (2003) Analysis of axisymmetric structures using Coons’ interpolation. Finite Elem Anal Des 39:535–558

  4. J24.

    Provatidis CG (2002) Coons-patch macroelements in potential Robin problems. Forsch Ingenieurwes 67:19–26

  5. J28.

    Provatidis CG (2004a) Coons-patch macroelements in two-dimensional eigenvalue and scalar wave propagation problems. Comput Struct 82(4–5):383–395

  6. J29.

    Provatidis CG (2004b) Solution of two-dimensional Poisson problems in quadrilateral domains using transfinite Coons interpolation. Commun Numer Methods Eng 20(7):521–533

  7. J36.

    Provatidis CG (2005) Three-dimensional Coons macroelements in Laplace and acoustic problems. Comput Struct 83(4–5):1572–1583

  8. J37.

    Provatidis CG (2005) Analysis of box-like structures using 3-D Coons’ interpolation. Commun Numer Methods Eng 21(8):443–456

  9. J41.

    Provatidis CG (2006a) Coons-patch macroelements in two-dimensional parabolic problems. Appl Math Model 30(4):319–351

  10. J42.

    Provatidis CG (2006b) Three-dimensional Coons macroelements: application to eigenvalue and scalar wave propagation problems. Int J Numer Methods Eng 65:111–134

  11. J44.

    Provatidis CG (2006c) Free vibration analysis of two-dimensional structures using Coons-patch macroelements. Finite Elem Anal Des 42(6):518–531

  12. J47.

    Provatidis CG (2006d) Transient elastodynamic analysis of two-dimensional structures using Coons-patch macroelements. Int J Solids Struct 43(22–23):6688–6706

  13. J69.

    Provatidis CG (2009) Eigenanalysis of two-dimensional acoustic cavities using transfinite interpolation. J Algorithms Comput Technol 3(4):477–502

  14. J92.

    Provatidis CG (2011) Equivalent finite element formulations for the calculation of eigenvalues using higher-order polynomials. Appl Math 1(1):13–23. (https://doi.org/10.5923/j.am.20110101.02). Open access. Available online at: http://article.sapub.org/10.5923.j.am.20110101.02.html

  15. J93.

    Provatidis CG (2012) Two-dimensional elastostatic analysis using Coons-Gordon interpolation. Meccanica 47(4):951–967

  16. J108.

    Provatidis CG (2013) A review on attempts towards CAD/CAE integration using macroelements. Comput Res 1(3):61–84. Open access. Online at: http://www.hrpub.org/download/20131107/CR2-13801513.pdf

  17. J111.

    Provatidis CG, Angelidis D (2014) Performance of Coons’ macroelements in plate bending analysis. Int J Comput Methods Eng Sci Mech 15:110–125

  18. J114.

    Provatidis CG (2014) Bézier versus Lagrange polynomials-based finite element analysis of 2-D potential problems. Adv Eng Softw 73:22–34

  19. J119.

    Provatidis CG (2014) Finite element analysis of structures using C1-continuous cubic B-splines or equivalent Hermite elements. J Struct 2014(754561):9 pp. Online at: http://www.hindawi.com/journals/jstruc/2014/754561/

  20. J135.

    Provatidis CG (2018) Engineering analysis with CAD-based macroelements. Arch Appl Mech 88(1–2):121–140. Free online access at: http://rdcu.be/uoQh

  • Conference Papers

  1. C4.

    Kanarachos A, Grekas D, Provatidis C (1995) Generalized formulation of Coons’ interpolation (chap 7). In: Kaklis P, Sapidis N (eds) Computer aided geometric design: from theory to practice. NTUA, pp 65–76. ISBN: 960-254-068-0

  2. C12.

    Kanarachos A, Provatidis C, Deriziotis D, Photeas N (1999) A new approach of the FEM analysis of two-dimensional elastic structures using global (Coons’) interpolation functions. In: Wunderlich W (ed) ECCM’99—CD proceedings European conference on computational mechanics (ECCM’99): CD proceedings, Munich, Germany, 28 Aug–3 Sept 1999, 763.pdf: pp 1–19. Available online at: http://users.ntua.gr/cprovat/yliko/763.pdf

  3. C19.

    Provatidis C, Kanarachos A (2000) On the use of Coons’ interpolation in CAD/CAE systems. MCME 2000: 2nd international conference on mathematics and computers in mechanical engineering, Vouliagmeni, Greece, 10–15 July 2000. In: Mastorakis N (ed) Systems and control: theory and applications. World Scientific and Engineering Society Press, pp 343–348. ISBN: 960-8052-11-4. Available online at: http://www.wseas.us/e-library/conferences/athens2000/Papers2000/547.pdf

  4. C22.

    Provatidis CG (2001) A global approximation technique for noise-control analysis. In: Tsahalis DT (ed) Proceedings of EuroNoise-2001, 4th European conference on noise control, 14–17 Jan 2001, vol 1, pp 23–31. University of Patras, Greece. Available online at: http://users.ntua.gr/cprovat/yliko/C22.pdf

  5. C25.

    Provatidis C (2001) Acoustic analysis of two-dimensional mufflers using large finite elements derived from Coons’ interpolation. Proceedings of ASME—Greek section, Patras, Greece, 17–20 Sept 2001. In: Drakatos PA (ed) CD proceedings, paper ANG1/P029, pp 1–6. Available online at: http://users.ntua.gr/cprovat/yliko/C25.pdf

  6. C40.

    Provatidis CG (2002) ‘ACOUSYS’: a computer code for the prediction of acoustical behavior in cavities and sound radiation from elastic structures. In: Trohidis A (ed) Proceedings acoustics 2002, Patras, Greece, 30 Sept–1 Oct 2002, paper no. AcP049, pp 333–338. ISBN: 960-7620-25-5

  7. C48.

    Provatidis C (2003) Frequency analysis and transient response of two-dimensional structures using Coons-patch macroelements. In: Brennan MJ et al (eds) Proceedings VIII international conference on recent advances in structural dynamics, organized by the Institute of Sound & Vibration Research University of Southampton, 14–16 July 2003, paper no. 24. ISBN: 0-854-32-7894

  8. C53.

    Provatidis CG (2003) Free vibrations of two-dimensional structures using Coons-patch macroelements, FEM and BEM. In: Atluri SN (ed) Proceedings ICCES’ 03, Corfu, Greece, 24–29 July 2003

  9. C57.

    Provatidis CG (2003) Coons-patch macroelements for boundary-value problems governed by fourth-order partial differential equation. In: Lipitakis EA (ed) Proceedings Hellenic-European conference on computational mathematics (HERCMA) 2003, Athens, Greece, 25–27 Sept 2003, pp 634–643. ISBN: 960-87275-1-0

  10. C78.

    Provatidis CG (2005) Performance of large Gordon-Coons finite elements in 2-D potential problems. In: Georgiou G, Papanastasiou P, Papadrakakis M (eds) Proceedings GRACM 05, Limassol, Cyprus, 29 June–1 July 2005, vol 2, pp 545–552. ISBN: 9963-644-52-Xv.2

  11. C89.

    Provatidis CG (2006) A global method in biomechanical modeling: application to dental mechanics. In: Middleton J (ed) Proceedings 7th international symposium on computer methods in biomechanics and biomedical engineering, Antibes, Juan Les Pins, Côte d’Azur, France, 22–25 Mar 2006.

  12. C96.

    Provatidis CG, Vossou CG, Theodorou EG (2006) On the CAD/CAE integration using Coons interpolation. In: Tsahalis D (ed) Proceedings 2nd international conference “From scientific computing to computational engineering”, Athens, Greece, 5–8 July 2006, vol III, pp 991–998. ISBN: 960-530-083-4

  13. C108.

    Provatidis CG (2006) Transient elastodynamic analysis of plane structures using Coons-patch macroelements and modal superposition. In: The eighth international conference on computational structures technology, Las Palmas de Gran Canaria—Spain, 12–15 Sept 2006. (https://doi.org/10.4203/ccp.83.197. You can find paper webpage in the repository by following the link http://dx.doi.org/10.4203/ccp.83.197)

  14. C136.

    Provatidis C (2008) Plate bending analysis using transfinite interpolation. In: Talaslidis D, Manolis G (eds) Proceedings 6th GRACM international congress on computational mechanics, Thessaloniki, Greece, 19–21 June 2008

  15. C173.

    Provatidis CG (2009) Higher order Galerkin/Ritz approximations in 1-D eigenvalue problems. In: Proceedings 3rd international conference on experiments/process/system modeling/simulation & optimization (3rd IC-EpsMsO), Athens, 8–11 July 2009

  16. C191.

    Provatidis CG (2011) Some issues on CAD/CAE integration: global interpolation using isoparametric and isogeometric techniques. In: Proceedings 7th GRACM international congress on computational mechanics, Athens, Greece, 30 June–2 July 2011

  17. C192.

    Provatidis CG (2011) Global versus local interpolation in the FEM free vibration analysis of prismatic bars. In: Proceedings 7th GRACM international congress on computational mechanics, Athens, Greece, 30 June–2 July 2011

  18. C236.

    Provatidis C (2016) Engineering analysis using CAD-based macroelements. In: Katsikadelis T, Stavroulakis GE (eds) Recent advances in mechanics. Book of abstracts, 9th German-Greek-Polish symposium, Orthodox Academy of Crete, Kolympari, Greece, 4–9 Sept 2016. Publisher: Technical University of Crete, Chania, Greece, 2016 (e-book), pp 51–52. ISBN: 978-960-8475-25-0

  1. II.

    Boundary Element Method

  • Conference Papers

  1. C26.

    Provatidis C (2001) Stress analysis of 3D solid structures using large boundary elements derived from 2D-Coons’ interpolation. Proceedings of ASME—Greek section, Patras, Greece, 17–20 Sept 2001. In: Drakatos PA (ed) CD proceedings, paper ANG1/P129, pp 1–6. Available online at: http://users.ntua.gr/cprovat/yliko/C26_asme_3d_elasticity.pdf

  2. C35.

    Provatidis CG (2002) A comparative study between Coons-patch macroelements and Boundary elements in two-dimensional potential problems. In: Tsahalis DT (ed) Proceedings 4th GRACM congress on computational mechanics, Patras, 27–29 June 2002 (CD proceedings); also: vol I, pp 43–50. Available online at: http://users.ntua.gr/cprovat/yliko/C35.pdf

  3. C36.

    Provatidis CG (2002) Analysis of three-dimensional sound radiation problems using trimmed patch boundary elements. In: 4th GRACM congress on computational mechanics, Patras, 27–29 June 2002 (CD proceedings); also: vol I, pp 402–409. Available online at: http://users.ntua.gr/cprovat/yliko/C36.pdf

  4. C55.

    Provatidis CG, Zafiropoulos NK (2003) Free-vibration analysis of three-dimensional solids using Coons-patch boundary superelements. In: Fifth European solids mechanics conference, Thessaloniki, Greece, 17–22 Aug 2003. Extended version, online at: http://users.ntua.gr/cprovat/yliko/C55.pdf

  5. C56.

    Provatidis CG, Zafiropoulos NK (2003) Determination of eigenfrequencies in three-dimensional acoustic cavities using Coons-patch boundary superelements. In: Fifth European solids mechanics conference, Thessaloniki, Greece, 17–22 Aug 2003. Extended version, online at: http://users.ntua.gr/cprovat/yliko/C56.pdf

  • Global Collocation Method

  • Journal Papers

  1. J59.

    Provatidis CG (2008) Free vibration analysis of elastic rods using global collocation. Arch Appl Mech 78(4):241–250

  2. J60.

    Provatidis CG (2008) Global collocation method for 2-D rectangular domains. J Mech Mater Struct 3(1):185–194

  3. J63.

    Provatidis CG (2008) Time- and frequency-domain analysis using lumped mass global collocation. Arch Appl Mech 78(11):909–920

  4. J67.

    Provatidis CG (2009) Integration-free Coons macroelements for the solution of 2-D Poisson problems. Int J Numer Meth Eng 77:536–557

  5. J70.

    Provatidis CG, Ioannou KS (2010) Static analysis of two-dimensional elastic structures using global collocation. Arch Appl Mech 80(4):389–400

  6. J99.

    Provatidis CG, Isidorou SK (2012) Solution of one-dimensional hyperbolic problems using cubic B-splines collocation. Int J Comput Sci Appl (IJCSA) 1(1):12–18. Open access. Available online: http://dpi-journals.com/index.php/IJCSA/article/view/795

  7. J117.

    Provatidis CG (2014) B-splines collocation eigenanalysis of 2D acoustic problems. J Mech Mater Struct 9(3):259–285

  8. J121.

    Provatidis CG (2014) Lumped mass acoustic and membrane eigenanalysis using the global collocation method. Cogent Eng 1(1):981366 (http://www.tandfonline.com/doi/abs/10.1080/23311916.2014.981366#.VIKnImc-eSo)

  9. J125.

    Provatidis C, Fillipatos A (2015) Numerical determination of eigenfrequencies in two-dimensional acoustic cavities using a global collocation method. Rom J Acoust Vibr XII(2):111–115. Open access. Available online: http://sra.ro/Arhiva/2015/nr2/Paper_4_page132-138.pdf

  10. J126.

    Provatidis C (2015) Lumped mass collocation method for 2D elastodynamic analysis. Int J Adv Eng Technol Comput Sci 2(2):1–15. http://www.irosss.org/ojs/index.php/IJAETCS/article/view/676/207

  11. J128.

    Provatidis CG (2017) CAD-based collocation eigenanalysis of 2-D elastic structures. Comput Struct 182:55–73

  12. J129.

    Provatidis CG (2017) B-splines collocation for plate bending eigenanalysis. J Mech Mater Struct 12(4):353–371

  • Conference Papers

  1. C122.

    Provatidis CG (2007) Performance of a Lagrange based global finite element collocation method for eigenvalue structural analysis. In: Proceedings 8th Hellenic society of theoretical and applied mechanics (HSTAM) congress, Patras, Greece, 12–14 July 2007, pp 153–160

  2. C135.

    Provatidis C (2008) Analysis of one-dimensional wave propagation problems using global collocation. In: Proceedings 6th GRACM international congress on computational mechanics, Thessaloniki, Greece, 19–21 June 2008

  3. C172.

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Provatidis, C.G. (2019). Review and Epilogue. In: Precursors of Isogeometric Analysis. Solid Mechanics and Its Applications, vol 256. Springer, Cham. https://doi.org/10.1007/978-3-030-03889-2_14

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