Initial Attempts on CAD/CAE Integration

  • Christopher G. ProvatidisEmail author
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 256)


This chapter discusses the meaning of the conventional “integrated CAD/CAE systems,” which is contradicted from the “CAD/ CAE integration ” (under the umbrella of isogeometric analysis) adopted throughout this book. The history of several important CAD interpolations since 1964 is outlined. Five precursors of the NURBS-based isogeometric analysis are discussed. The general boundary value problem is posed. In order to solve it, three computational methods, i.e., the finite element method, the Boundary Element Method, and the collocation method are presented in brief. The implementation of Coons and Gordon interpolation formulas in mesh generation is discussed. Moreover, the utilization of the closely related transfinite elements in engineering analysis in conjunction with the aforementioned three major computational methods is discussed.


CAD/CAE integration Finite element Boundary element Collocation method Coons interpolation Transfinite element 


  1. 1.
    Argyris JH (1955) Energy theorems and structural analysis: a generalized discourse with applications on energy principles of structural analysis including the effects of temperature and non-linear stress strain relations. Aircr Eng 26(1):347–356Google Scholar
  2. 2.
    Auricchio F, Beirão da Veiga L, Hughes TJR, Reali A, Sangalli G (2010) Isogeometric collocation methods. Math Models Methods Appl Sci 20(11):2075–2107MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bajaj C, Chen J, Xu G (1995) Modeling with cubic A-patches. ACM Trans Graph 14:103–133CrossRefGoogle Scholar
  4. 4.
    Barnhill RE, Birkhoff G, Gordon WJ (1973) Smooth interpolation on triangles. J Approx Theory 8:114–128MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Barnhill RE, Mansfield L (1974) Error bounds for smooth interpolation on triangles. J Approx Theory 11:306–318MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Barnhill RE (1985) A survey of patch methods. NASA Report. Available at:
  7. 7.
    Barnhill RE, Gregory JA (1975) Compatible smooth interpolation on triangles. J Approx Theory 15:214–225MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bathe KJ (1996) Finite element procedures. Prentice-Hall, Upper Saddle River, New JerseyzbMATHGoogle Scholar
  9. 9.
    Beer G, Watson JO (2002) Introduction to finite and boundary element methods for engineers. Wiley, Chichester, Chapter 11, pp 357–377Google Scholar
  10. 10.
    Bercovier M, Shilat E (1993) Enhancement of Gordon-Coons interpolations by ‘bubble functions’. Comput Aided Des 10:253–265MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Βezier P (1966) Définition numérique des courbes et surfaces I. Automatisme 11:625–632Google Scholar
  12. 12.
    Βezier P (1967) Définition numérique des courbes et surfaces II. Automatisme 12:17–21Google Scholar
  13. 13.
    Βezier P (1968) Procédé de définition numérique des courbes et surfaces non mathématiques. Automatisme 13(5):189–196Google Scholar
  14. 14.
    Βezier P (1968b) How Renault uses numerical control for car body design and tooling. SAE Paper 680010Google Scholar
  15. 15.
    Bezier PE (1971) Example of an existing system in motor industry: the UNISURF system. Proc R Soc Lond Ser A 321:207–218Google Scholar
  16. 16.
    Bezier P (1971) Numerical control and foundries. FOND-FR 26(299):77Google Scholar
  17. 17.
    Bezier P (1972) Numerical control: mathematics and applications. Wiley (translated by A. R. Forrest)Google Scholar
  18. 18.
    Bezier P (1973) UNISURF system: principles, program, language. In: Harvany J (ed)Proceedings of 1973 PROLAMAT conference, Budapest, North Holland Publ. Co., AmsterdamGoogle Scholar
  19. 19.
    Bezier P (1974) Mathematical and practical possibilities of UNISURF. In: Barnhill R, Riesenfeld R (eds) Computer aided geometric design. Academic Press, CambridgeGoogle Scholar
  20. 20.
    Βezier P (1977) Essay de définition numérique des courbes et des surfaces expérimentales. Ph.D. thesis, University of Paris VIGoogle Scholar
  21. 21.
    Βezier P (1978) General distortion of an ensemble of biparametric surfaces. Comput Aided Des 10(2):116–120CrossRefGoogle Scholar
  22. 22.
    Bezier PE (1981) A view of CAD-CAM. Comput Aided Des 13(4):207–209CrossRefGoogle Scholar
  23. 23.
    Bezier PE, Sioussiou S (1983) Semi-automatic system for defining free-form curves and surfaces. Comput Aided Des 15(2):65–72CrossRefGoogle Scholar
  24. 24.
    Bezier PE (1983) UNISURF, from styling to tool-shop. Comput Ind 4(2):115–126CrossRefGoogle Scholar
  25. 25.
    Bezier PE (1984) CADCAM—past, requirements, trends. Comput Aided Des 16(2):102CrossRefGoogle Scholar
  26. 26.
    Bezier P (1986) The mathematical basis of the UNISURF CAD system. Butterworths, LondonGoogle Scholar
  27. 27.
    Bezier P (1989) 1st steps of CAD. Comput Aided Des 21(5):259–261CrossRefGoogle Scholar
  28. 28.
    Bezier P (1990) Style, mathematics and NC. Comput Aided Des 22(9):524–526CrossRefGoogle Scholar
  29. 29.
    Bezier P (1998) A view of the CAD/CAM development period. IEEE Ann Hist Comput 20(2):37–40Google Scholar
  30. 30.
    Boyse JW, Rosen JM (1981) GMSOLID—a system for interactive design and analysis of solids. SAE Trans, 90, Section 1: 810010-810234, pp 847–857Google Scholar
  31. 31.
    Brebbia CA (1982) Finite element systems: a handbook, 2nd edn. Springer, BerlinzbMATHCrossRefGoogle Scholar
  32. 32.
    Brebbia CA, Dominguez J (1992) Boundary elements: an introductory course. Computational Mechanics Publications, McGraw-Hill Book Company, SouthamptonzbMATHGoogle Scholar
  33. 33.
    Casale MS (1989) Integration of geometric analysis and structural analysis using trimmed patches. Ph.D. thesis, University of California, IrvineGoogle Scholar
  34. 34.
    Casale MS, Bobrow JE (1989) The analysis of solids without mesh generation using trimmed patch boundary elements. Eng Comput 5:249–257CrossRefGoogle Scholar
  35. 35.
    Casale MS, Bobrow JE, Underwood R (1992) Trimmed-patch boundary elements: bridging the gap between solid modeling and engineering analysis. Comput Aided Des 24:193–199zbMATHCrossRefGoogle Scholar
  36. 36.
    Cavendish JC, Gordon WJ, Hall CA (1976) Ritz-Galerkin approximations in blending function spaces. Numer Math 26:155–178MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Cavendish JC, Gordon WJ, Hall CA (1977) Substructured macro elements based on locally blending interpolation. Int J Numer Meth Eng 11:1405–1421zbMATHCrossRefGoogle Scholar
  38. 38.
    Cavendish JC, Hall CA (1984) A new class of transitional blended finite elements for the analysis of solid structures. Int J Numer Meth Eng 28:241–253zbMATHCrossRefGoogle Scholar
  39. 39.
    Charlesworth WW, Cox JJ, Anderson DC (1994) The domain decomposition method applied to Poisson’s equation in two dimensions. Int J Solids Struct 37(18):3093–3115zbMATHGoogle Scholar
  40. 40.
    Clark BW, Anderson DC (2003) The penalty boundary method for combining meshes and solid models in finite element analysis. Eng Comput 20(4):344–365zbMATHCrossRefGoogle Scholar
  41. 41.
    Clark BW, Anderson DC (2003) The penalty boundary method. Finite Elem Anal Des 39:387–401CrossRefGoogle Scholar
  42. 42.
    Collatz L (1960) The numerical treatment of differential equations, 3rd edn. Springer-Verlag, BerlinzbMATHCrossRefGoogle Scholar
  43. 43.
    Cook WA (1974) Body oriented (natural) co-ordinates for generating three-dimensional meshes. Int J Numer Meth Eng 8:27–43zbMATHCrossRefGoogle Scholar
  44. 44.
    Coons SA (1964) Surfaces for computer aided design of space form, Project MAC, MIT (1964), revised for MAC-TR-41 (1967), Springfield, VA 22161, USA. Available as AD 663 504 from the National Technical Information Service (CFSTI), Sills Building, 5285 Port Royal Road. Now, online available at:
  45. 45.
    Coons S (1968) Rational bibubic surface patches. Technical report, MIT, Project MACGoogle Scholar
  46. 46.
    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: towards integration of CAD and FEA. Wiley, ChichesterzbMATHCrossRefGoogle Scholar
  47. 47.
    Cox MG (1972) The numerical evaluation of B-splines. J Inst Math Its Appl 10:134–149MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    De Boor C (1972) On calculating with B-splines. J Approx Theory 6:50–62MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    De Casteljau P (1959) Courbes à poles. National Industrial Property Institute (INPI, France)Google Scholar
  50. 50.
    De Casteljau PF (1999) De Casteljau’s autobiography: my life at Citroën. Comput Aided Geom Des 16:583–586Google Scholar
  51. 51.
    Dimitriou V (2004) Adaptive finite elements and related meshes, Ph.D. Dissertation (advisor: Prof. Andreas E. Kanarachos), National Technical University of Athens, School of Mechanical Engineering, Athens, August, 2004Google Scholar
  52. 52.
    El-Zafrany A, Cookson RA (1986) Derivation of Lagrangian and Hermitian shape functions for triangular elements. Int J Numer Meth Eng 23(2):275–285MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    El-Zafrany A, Cookson RA (1986) Derivation of Lagrangian and Hermitian shape functions for quadrilateral elements. Int J Numer Meth Eng 23(10):1939–1958zbMATHCrossRefGoogle Scholar
  54. 54.
    Farin G (1990) Curves and surfaces for computer aided geometric design: a practical guide. Academic Press, BostonzbMATHGoogle Scholar
  55. 55.
    Farin G, Hoschek J, Kim MS (2002) Handbook of computer aided geometric design. Elsevier, North-HollandzbMATHGoogle Scholar
  56. 56.
    Forrest AR (1968) Curves and surfaces for computer-aided design, Ph.D. dissertation, Cambridge University, Cambridge, UKGoogle Scholar
  57. 57.
    Gordon WJ (1969) Spline-blended surface interpolation through curve networks. Indiana Univ Math J 18, 10, pp 931–952. Online available at:
  58. 58.
    Gordon WJ (1971) Blending functions methods of bivariate and multivariate interpolation and approximation. SIAM J Numer Anal 8:158–177MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Gordon WJ (1983) An operator calculus for surface and volume modeling. IEEE Comput Graph Appl 3:18–22CrossRefGoogle Scholar
  60. 60.
    Gordon WJ, Hall CA (1973) Construction of curvilinear co-ordinate systems and application to mesh generation. Int J Numer Meth Eng 7:461–477MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Gordon WJ, Hall CA (1973) Transfinite element methods: Blending-function interpolation over arbitrary curved element domains. Numer Math 21:109–112MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Gordon WJ, Thiel LC (1982) Transfinite mappings and their application to grid generation. In: Thompson JP (ed) Numerical grid generation. Applied mathematics and computation, vol 11–12. Elsevier, pp. 171–233Google Scholar
  63. 63.
    Gregory JA (1983) N-sided surface patches. In: Gregory JA (ed) Math Surf. Clarendon Press, Oxford, pp 217–232Google Scholar
  64. 64.
    Haber RB, Shephard MS, Abel JF, Gallagher RH, Greensberg DP (1981) A general three-dimensional graphical finite element preprocessor utilizing discrete transfinite mappings. Int J Numer Meth Eng 17:1015–1044zbMATHCrossRefGoogle Scholar
  65. 65.
    Haber RB, Abel JF (1982) Discrete transfinite mappings for the description and meshing of three-dimensional surfaces using interactive computer graphics. Int J Numer Meth Eng 18:41–66zbMATHCrossRefGoogle Scholar
  66. 66.
    Hartmann S, Benson D, Nagy A (2016) Isogeometric analysis with LS-DYNA. J Phys Conf Ser 734:032125. CrossRefGoogle Scholar
  67. 67.
    Higuchi F, Gofuku S, Maekawa T, Mukundan H, Patrikalakis NM (2007) Approximation of involute curves for CAD-system processing. Eng Comput 23(3):207–214CrossRefGoogle Scholar
  68. 68.
    Höllig K (2002) Finite element approximation with splines. In: Farin G, Hoschek J, Kim MS (eds) Handbook of computer aided geometric design, North-Holland, Amsterdam, Chapter 11, pp 283–307CrossRefGoogle Scholar
  69. 69.
    Höllig K (2003) Finite element methods with B-splines. SIAM, PhiladelphiazbMATHCrossRefGoogle Scholar
  70. 70.
    Hughes TJR (2000), The finite element method: Linear static and dynamic finite element analysis. Dover Publications, Mineola, NYGoogle Scholar
  71. 71.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Inoue K, Kikuchi Y, Masuyama T (2005) A NURBS finite element method for product shape design. J Eng Des 16(2):157–174CrossRefGoogle Scholar
  73. 73.
    Kagan P, Fischer A (2000) Integrated mechanical based CAE systems using B-spline based finite elements. Comput Aided Des 32(8–9):539–552zbMATHCrossRefGoogle Scholar
  74. 74.
    Kanarachos A, Röper O (1979) Rechnerunterstützte Netzgenerierung mit Hilfe der Coonsschen Abbildung. VDI-Z 121:297–303Google Scholar
  75. 75.
    Kanarachos A, Provatidis Ch (1987) Performance of mass matrices for the BEM dynamic analysis of wave propagation problems. Comput Methods Appl Mech Eng 63:155–165zbMATHCrossRefGoogle Scholar
  76. 76.
    Kanarachos AE, Spentzas CN (1988) Comparison of four finite element solutions of self-adjoint and non-self-adjoint problems governed by differential equations with predominant lower order derivatives. Eng Comput 5:71–74CrossRefGoogle Scholar
  77. 77.
    Kanarachos A, Deriziotis D (1989) On the solution of Laplace and wave propagation problems using ‘C-elements’. Finite Elem Anal Des 5:97–109zbMATHCrossRefGoogle Scholar
  78. 78.
    Kanarachos A, Provatidis C, Deriziotis D, Foteas N (1999),A new approach of the FEM analysis of two-dimensional elastic structures using global (Coons’s) interpolation functions. In: Wunderlich J (ed) CD proceedings first european conference on computational mechanics, München-Germany, August–September, 1999Google Scholar
  79. 79.
    Lanczos C (1938) Trigonometric interpolation of empirical and analytical functions. Stud Appl Math 17(1–4):123–199zbMATHGoogle Scholar
  80. 80.
    Lee K (1999) Principles of CAD/CAM/CAE systems. Addison-Wesley, Reading, MAGoogle Scholar
  81. 81.
    Liu B, Xing Y, Wang Z, Lu X, Sun H (2017) Non-uniform rational Lagrange functions and its applications to isogeometric analysis of in-plane and flexural vibration of thin plates. Comput Methods Appl Mech Eng 321:173–208MathSciNetCrossRefGoogle Scholar
  82. 82.
    Loop CT, DeRose TD (1989) A multisided generalization of Bezier surfaces. ACM Trans Graph 8:204–234zbMATHCrossRefGoogle Scholar
  83. 83.
    Martin W, Cohen E (2001) Representation and extraction of volumetric attributes using trivariate splines: a mathematical framework. In: Proceedings of the solid modelling symposium (
  84. 84.
    Natekar D, Zhang X, Subbarayan G (2004) Constructive solid analysis: a hierarchical, geometry-based meshless analysis procedure for integrated design and analysis. Comput Aided Des 36(5):473–486CrossRefGoogle Scholar
  85. 85.
    Park S, Lee K (1997) High-performance trivariate NURBS representation for analyzing and visualizing fluid flow data. Comput Graph 21:473–482CrossRefGoogle Scholar
  86. 86.
    Provatidis C (1987) On the application of the boundary element method in the analysis of field and dynamic problems, Doctoral Dissertation, National Technical University of Athens, Mechanical Engineering Department, Greece, November 1987 (in Greek)Google Scholar
  87. 87.
    Provatidis CG, Kanarachos AE (1995) Further research on the performance of consistent mass matrices using BEM for symmetric/nonsymmetric formulations. Comput Mech 16:197–207MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Provatidis C, Kanarachos A (2001) Performance of a macro-FEM approach using global interpolation (Coons’) functions in axisymmetric potential problems. Comput Struct 79(19):1769–1779CrossRefGoogle Scholar
  89. 89.
    Provatidis C (2001b) A global approximation technique in noise-control. In: Tsahalis DT (ed) Proceedings 4th European conference on noise control (EuroNoise-2001, 14–17 January 2001, Patras, Greece), vol I, pp 23–31Google Scholar
  90. 90.
    Provatidis C (2001c) Acoustic analysis of two-dimensional mufflers using large finite elements derived from Coons’ interpolation. In: Drakatos PA (ed) CD proceedings of ASME—Greek Section, September 17–20, 2001, Patras, Greece (Paper ANG1/P099, pages 1–6)Google Scholar
  91. 91.
    Provatidis C (2001d) Stress analysis of 3D solid structures using large boundary elements derived from 2D-Coons’ interpolation. In: Drakatos PA (ed) CD proceedings of ASME—Greek Section, September 17–20, 2001, Patras, Greece (Paper ANG1/P129, pages 1–6)Google Scholar
  92. 92.
    Provatidis C (2002) Coons-patch macroelements in potential Robin problems. Forsch Ingenieurwes 67(1):19–26CrossRefGoogle Scholar
  93. 93.
    Provatidis C (2002b) 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, 27–29 June, 2002, Patras, Greece, pp 43–50Google Scholar
  94. 94.
    Provatidis C (2002c) Analysis of three-dimensional sound radiation problems using trimmed patch boundary elements. In: Tsahalis DT (ed) Proceedings 4th GRACM Congress on Computational Mechanics, 27–29 June, 2002, Patras, Greece, pp. 402-409Google Scholar
  95. 95.
    Provatidis CG (2002d) CAD-FEA integration using Coons interpolation. Technical Report MD&CS 01–2002, National Technical University of Athens, Mechanical Engineering Department; also submitted to “Engineering with Computers” (EWC03-019, Sept 12, 2003)Google Scholar
  96. 96.
    Provatidis C (2003) Analysis of axisymmetric structures using Coons’ interpolation. Finite Elem Anal Des 39:535–558CrossRefGoogle Scholar
  97. 97.
    Provatidis C (2003b) Frequency analysis and transient response of two-dimensional structures using Coons-patch macroelements. In: Brennan MJ, Ferman MA, Petersson BAT, Rizzi SA, Wentz K (eds) Proceedings of the VIII international conference on recent advances in structural dynamics, 14–16 July 2003, Southampton, UK (paper No. 24)Google Scholar
  98. 98.
    Provatidis C (2003c) Free vibrations of two-dimensional structures using Coons-patch macroelements, FEM and BEM. In: Atluri SN, Beskos DE, Polyzos D (eds) Proceedings from ICCES 2003, advances in computational & experimental engineering & sciences, 24–29 July 2003, Corfu, Greece, Chapter 5, Paper No. 249Google Scholar
  99. 99.
    Provatidis C, Zafiropoulos N (2003d) Free-vibration analysis of three-dimensional solids using Coons-patch Boundary Superelements. In: Aifantis E (ed) Proceedings 5th European solids mechanics conference, Thessaloniki, 17–22 August 2003Google Scholar
  100. 100.
    Provatidis C, Zafiropoulos N (2003e) Determination of eigenfrequencies in three-dimensional acoustic cavities using Coons-patch Boundary Superelements. In: Aifantis E (ed) Proceedings 5th European solids mechanics conference, Thessaloniki, 17–22 August 2003Google Scholar
  101. 101.
    Provatidis CG (2004) Coons-patch macroelements in two-dimensional eigenvalue and scalar wave propagation problems. Comput Struct 82(4–5):383–395MathSciNetCrossRefGoogle Scholar
  102. 102.
    Provatidis CG (2004) On DR/BEM for eigenvalue analysis of 2-D acoustics. Comput Mech 35:41–53zbMATHCrossRefGoogle Scholar
  103. 103.
    Provatidis CG (2004) Solution of two-dimensional Poisson problems in quadrilateral domains using transfinite Coons interpolation. Commun Numer Methods Eng 20(7):521–533MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Provatidis CG (2005) Performance of large Gordon-Coons finite elements in 2-D potential problems. In: Proceedings GRACM 05, Limassol, Cyprus, 29 June–1 July, 2005Google Scholar
  105. 105.
    Provatidis CG, Vossou CG, Theodorou EG (2006) On the CAD/CAE integration using Coons interpolation. In: Proceedings 2nd international conference “from scientific computing to computational engineering”, Athens, Greece, 5–8 July, 2006Google Scholar
  106. 106.
    Provatidis CG (2007) Final report for the project: PROTAGORAS (DIDYMO: Investigation on CAD/CAE integration), EDEIL Contract No. 65/1388-June 2004, February, 2007Google Scholar
  107. 107.
    Provatidis C, Kanarachos A (2000) On the use of Coons’ interpolation in CAD/CAE systems. In: Mastorakis N (ed) Systems and control: theory and applications. World Scientific and Engineering Society Press, pp 343–348. Online available on:
  108. 108.
    Provatidis CG (2005) Three-dimensional Coons macroelements in Laplace and acoustic problems. Comput Struct 83:1572–1583MathSciNetCrossRefGoogle Scholar
  109. 109.
    Provatidis CG (2005) Analysis of box-like structures using 3-D Coons’ interpolation. Commun Numer Methods Eng 21:443–456zbMATHCrossRefGoogle Scholar
  110. 110.
    Provatidis CG (2005c) Performance of several RBFs in DR/BEM eigenvalue analysis of 2-D structures. GRACM 05, Limassol, Cyprus, 29 June–1 July 2005Google Scholar
  111. 111.
    Provatidis CG (2006) Coons-patch macroelements in two-dimensional parabolic problems. Appl Math Model 30:319–351zbMATHCrossRefGoogle Scholar
  112. 112.
    Provatidis CG (2006) Free vibration analysis of two-dimensional structures using Coons-patch macroelements. Finite Elem Anal Des 42(6):18–531zbMATHCrossRefGoogle Scholar
  113. 113.
    Provatidis CG (2006) Transient elastodynamic analysis of two-dimensional structures using Coons-patch macroelements. Int J Solids Struct 43(22–23):6688–6706zbMATHCrossRefGoogle Scholar
  114. 114.
    Provatidis CG (2006) Three-dimensional Coons macroelements: application to eigenvalue and scalar wave propagation problems. Int J Numer Meth Eng 65:111–134zbMATHCrossRefGoogle Scholar
  115. 115.
    Provatidis CG (2007) Performance of a Lagrange based global finite element collocation method for eigenvalue structural analysis. In: Proceedings 8th HSTAM international congress on mechanics, Patras, 12–14 July, 2007Google Scholar
  116. 116.
    Provatidis CG (2008) Free vibration analysis of elastic rods using global collocation. Arch Appl Mech 78(4):241–250zbMATHCrossRefGoogle Scholar
  117. 117.
    Provatidis CG (2008) Time- and frequency-domain analysis using lumped mass global collocation. Arch Appl Mech 78(11):909–920zbMATHCrossRefGoogle Scholar
  118. 118.
    Provatidis CG (2008) Global collocation method for 2-D rectangular domains. J Mech Mater Struct 3(1):185–194CrossRefGoogle Scholar
  119. 119.
    Provatidis CG (2009) Integration-free Coons macroelements for the solution of 2-D Poisson problems. Int J Numer Meth Eng 77:536–557MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Provatidis CG (2009) Eigenanalysis of two-dimensional acoustic cavities using transfinite interpolation. J Algorithms Comput Technol 3(4):477–502MathSciNetCrossRefGoogle Scholar
  121. 121.
    Provatidis CG (2009c) Higher order Galerkin/Ritz approximations in 1-D eigenval problems. In: Tsahalis D (ed) Proceedings 3rd international conference on experiments/process/system modeling/simulation & optimization (3rd IC-EpsMsO), Athens, 8–11 July, 2009Google Scholar
  122. 122.
    Provatidis CG, Isidorou S (2009) Comparison of advanced collocation methods for the solution of ordinary differential equations. In: Tsahalis D (ed) Proceedings 3rd international conference on experiments/process/system modeling/simulation & optimization (3rd IC-EpsMsO), Athens, 8–11 July, 2009Google Scholar
  123. 123.
    Provatidis CG, Ioannou KS (2010) Static analysis of two-dimensional elastic structures using global collocation. Arch Appl Mech 80(4):389–400zbMATHCrossRefGoogle Scholar
  124. 124.
    Provatidis CG (2011) Equivalent finite element formulations for the calculation of eigenvalues using higher-order polynomials. Appl Math 1(1):13–23MathSciNetCrossRefGoogle Scholar
  125. 125.
    Provatidis CG (2011b) Global versus local interpolation in the FEM free vibration analysis of prismatic bars. In: Proceedings 7th GRACM international congress in mechanics, Athens, 30 June–2 July, 2011Google Scholar
  126. 126.
    Provatidis CG (2011c) Some issues on CAD/CAE integration: global interpolation using isoparametric and isogeometric techniques. In: Proceedings 7th GRACM international congress in mechanics, Athens, 30 June-2 July, 2011Google Scholar
  127. 127.
    Provatidis CG, Isidorou SK (2011) B-splines collocation eigenvalue analysis of 1-D problems. In: Proceedings 7th GRACM international congress in mechanics, Athens, 30 June–2 July, 2011Google Scholar
  128. 128.
    Provatidis CG (2012) Two-dimensional elastostatic analysis using Coons-Gordon interpolation. Meccanica 47(4):951–967MathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    Provatidis C (2013) A review on attempts towards CAD/CAE integration using macroelements. Comput Res 1(3):61–84Google Scholar
  130. 130.
    Provatidis CG (2014) Bezier versus Lagrange polynomials-based finite element analysis of 2-D potential problems. Adv Eng Softw 73:22–34CrossRefGoogle Scholar
  131. 131.
    Provatidis CG (2018) Engineering analysis with CAD-based macroelements. Arch Appl Mech 88:121–140CrossRefGoogle Scholar
  132. 132.
    Rabut C (2002) On Pierre Bezier’s life and motivations. Comput Aided Des 34(7):493–510zbMATHCrossRefGoogle Scholar
  133. 133.
    Renken FP, Subbarayan G (2000) NURBS-based solution to inverse boundary problems in droplet shape prediction. Comput Methods Appl Mech Eng 190(11):1391–1406zbMATHCrossRefGoogle Scholar
  134. 134.
    Röper O (1978) Ein Geometrieprozessor für die rechnerunterstützte Auslegung von Maschinenbauteilen mit Hilfe der Methode der Finite Elemente, Dissertation, Ruhr-Universität Bochum, Mai 1978Google Scholar
  135. 135.
    Schillinger D, Ruthala PK, Nguyen LH (2016) Lagrange extraction and projection for NURBS basis functions: a direct link between isogeometric and standard nodal finite element formulations. Int J Numer Meth Eng 108(6):515–534MathSciNetCrossRefGoogle Scholar
  136. 136.
    Schoenberg IJ (1946) Contributions to the problem of approximation of equidistant data by analytic functions. Q Appl Math 4:45–99CrossRefGoogle Scholar
  137. 137.
    Schramm U, Pilkey WD (1993) The coupling of geometric descriptions and finite element using NURBS—a study in shape optimization. Finite Elem Anal Des 15:11–34zbMATHCrossRefGoogle Scholar
  138. 138.
    Schramm U, Pilkey WW (1994) Higher order boundary elements for shape optimization using rational B-splines. Eng Anal Boundary Elem 14(3):255–266CrossRefGoogle Scholar
  139. 139.
    Szabó B, Babuška I (1991) Finite element analysis. Wiley, New YorkzbMATHGoogle Scholar
  140. 140.
    Turner MJ, Clough RW, Martin HC, Topp LT (1956) Stiffness and deflection analysis of complex structures. Journal of the Aeronautical Sciences 23 (9):805–823Google Scholar
  141. 141.
    Van Blerk JJ, Botha JF (1993) Numerical solution of partial differential equations on curved domains by collocation. Numer Methods Partial Differ Equat 9:357–371MathSciNetzbMATHCrossRefGoogle Scholar
  142. 142.
    Versprille KJ (1975) Computer-aided design applications of the rational B-spline approximation form, Ph.D. Dissertation, Syracuse University, USA, 1975, (advisor: Prof. S. A. Coons)
  143. 143.
    Wu SC, Abel JF (1979) Representation and discretization of arbitrary surfaces for finite element shell analysis. Int J Numer Meth Eng 14:813–836zbMATHCrossRefGoogle Scholar
  144. 144.
    Yildir YB, Wexler A (1983) MANDAP—a FEM/BEM preparation package. IEEE Trans Magn 19:2562–2565CrossRefGoogle Scholar
  145. 145.
    Zhaobei Z, Zhiqiang X (1987) Coons’ surface method for formulation of finite element of plates and shells. Comput Struct 27(1):79–88zbMATHCrossRefGoogle Scholar
  146. 146.
    Zhang YJ (2017) Integrating CAD with Abaqus: a practical isogeometric analysis software platform for industrial applications. In: IACM 19th international conference on finite elements in flow problems—FEF 2017, 5–7 April 2017, Rome, Italy (paper: a79.pdf)Google Scholar
  147. 147.
    Zienkiewicz OC (1977) The finite element method, 3rd edn. McGraw-Hill, LondonzbMATHGoogle Scholar
  148. 148.
    Zienkiewicz OC, Taylor RL (1988) The finite element method, 4rd edn, vol 1: Basic formulation and linear problems, and vol 2: Solid and fluid mechanics, dynamics and non-linearity. McGraw-Hill, LondonGoogle Scholar
  149. 149.
    Zienkiewicz OC, Taylor RL (2000) The finite element method, 5rd edn, 3 volume set. Butterworth-Heinemann, Elsevier, Kidlington, OxfordGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mechanical EngineeringNational Technical University of AthensAthensGreece

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