# Special Numerical Techniques to Joint Design

• Andreas Öchsner
Living reference work entry

## Abstract

The aim of this chapter is to introduce special numerical techniques. The first part covers special finite element techniques which reduce the size of the computational models. In the case of the substructuring technique, internal nodes as parts of a finite element mesh can be condensed out so that they do not contribute to the size of the global stiffness matrix. A postcomputational step allows to determine the unknowns of the condensed nodes. In the case of the submodel technique, the results of a finite element computation based on a coarse mesh are used as input, i.e., boundary conditions, for a refined submodel. The second part of this chapter introduces alternative approximation methods to solve the partial differential equations which describe the problem. The boundary element method is characterized by the fact that the problem is shifted to the boundary of the domain and as a result, the dimensionality of the problem is reduced by one. In the case of the finite difference method, the differential equation and the boundary conditions are represented by finite difference equations. Both methods are introduced based on a simple one-dimensional problem in order to demonstrate the major idea of each method. Furthermore, advantages and disadvantages of each alternative approximation methods are given in the light of the classical finite element simulation. Whenever possible, examples of application of the techniques in the context of adhesive joints are given.

## Keywords

Numerical method Finite element method Boundary element method Finite difference method Substructure Submodel Static condensation Superelement Internal forces Enforced displacement Weighted residual method Weighting function Strong formulation Weak formulation Inverse formulation Residuum Partial differential equation Fundamental solution Representation formula Load point

## References

1. Banerjee PK (1994) Boundary element methods in engineering. McGraw-Hill, LondonGoogle Scholar
2. Bathe KJ (1996) Finite element procedures. Prentice-Hall, Upper Saddle River
3. Beer G, Smith I, Duenser C (2008) The boundary element method with programming. Springer, Wien
4. Brebbia CA, Felles JCF, Wrobel JCF (1984) Boundary element techniques: theory and applications. Springer, Berlin
5. Bushnell D, Almroth BO, Brogan F (1971) Finite-difference energy method for nonlinear shell analysis. Comput Struct 1:361–387
6. Carrara P, Ferretti D (2013) A finite-difference model with mixed interface laws for shear tests of FRP plates bonded to concrete. Compos Part B Eng 54:329–342
7. Collatz L (1966) The numerical treatment of differential equations. Springer, BerlinGoogle Scholar
8. Cook RD, Malkus DS, Plesha ME, Witt RJ (2002) Concepts and applications of finite element analysis. Wiley, New YorkGoogle Scholar
9. Crocombe AD, Bigwood DA (1992) Development of a full elasto-plastic adhesive joint design analysis. J Strain Anal Eng Des 27(4):211–218
10. da Silva LFM, Campilho RDSG (2012) Advances in numerical modelling of adhesive joints. Springer, Heidelberg
11. de G Allen DN (1955) Relaxation methods. McGraw-Hill, New YorkGoogle Scholar
12. Dohrmann CR, Key SW, Heinstein MW (2000) A method for connecting dissimilar finite element meshes in two dimesions. Int J Numer Meth Eng 48:655–678
13. Fish J, Belytschko T (2007) A first course in finite elements. Wiley, Chichester
14. Forsythe GE, Wasow WR (1960) Finite-difference methods for partial differential equations. Wiley, New York
15. Gaul L, Kögl M, Wagner M (2003) Boundary element methods for engineers and scientists. Springer, Berlin
16. Gmür TC, Kauten RH (1993) Three-dimensional solid-to-beam transition elements for structural dynamics analysis. Int J Numer Meth Eng 36:1429–1444
17. Gonçalves DJS, Campilho RDSG, Silva LFMD, Fernandes JLM (2014) The use of the boundary element method in the analysis of single lap joints. J Adhes 90:50–64
18. Groth HL (1986) Calculation of stresses in bonded joints using the substructuring technique. Int J Adhes Adhes 6(1):31–35
19. IMSL Math/Library Manual (1997) Chapter 5: differential equations. Visual Numeric Inc, HoustonGoogle Scholar
20. Javanbakht Z, Öchsner A (2017) Advanced finite element simulation with MSC Marc: application of user subroutines. Springer, Cham
21. Knight NF, Ransom JB, Griffin OH, Thompson DM (1991) Global/local methods research using a common structural analysis framework. Finite Elem Anal Des 9:91–112
22. Lee SS (1998) Boundary element analysis of the stress singularity at the interface corner of viscoelastic adhesive layers. Int J Solids Struct 35(13):1385–1394
23. Lourenço F, Aguirre IF, Alburquerque EL, Sollero P (2003) Boundary element analysis of panels reinforced by adhesive plates. In: Pesce CP (ed) Proceedings of the 17th international congress of mechanical engineering – COBEM 2003Google Scholar
24. Love AEH (1944) A treatise on the mathematical theory of elasticity. Dover Publications, Mineola
25. Mackerle J (1995a) Fastening and joining: finite element and boundary element analyses – a bibliography (1992–1994). Finite Elem Anal Des 20:205–215
26. Mackerle J (1995b) Some remarks on progress with finite elements. Comput Struct 55:1101–1106
27. Mitchell AR, Griffiths DF (1980) The finite difference method in partial differential equations. Wiley, New York
28. Öchsner A (2016) Computational statics and dynamics: an introduction based on the finite element method. Springer, Singapore
29. Öchsner A, Merkel M (2013) One-dimensional finite elements: an introduction to the FE method. Springer, Berlin
30. Paroissien E, Gaubert F, Veiga AD, Lachaud F (2013) Elasto-plastic analysis of bonded joints with macro-elements. J Adhes Sci Technol 27(13):1464–1498
31. Pisa CD, Aliabadi MH (2015) Boundary element analysis of stiffened panels with repair patches. Eng Anal Bound Elem 56:162–175
32. Raamachandran J (2000) Boundary and finite elements – theory and problems. Alpha Science International, PangbourneGoogle Scholar
33. Raghu ES (2009) Finite element modeling techiques in MSC.NASRAN and LS/DYNA. Arup, LondonGoogle Scholar
34. Reddy JN (2006) An introduction to the finite element method. McGraw Hill, SingaporeGoogle Scholar
35. Sokolnikoff I (1956) Mathematical theory of elasticity. McGraw-Hill, New York
36. Southwell RV (1946) Relaxation methods in theoretical physics. Clarendon Press, Oxford
37. Stratford T, Cadei J (2006) Elastic analysis of adhesion stresses for the design of a strengthening plate bonded to a beam. Constr Build Mater 20(1–2):34–45
39. Wahab MMA, Ashcroft IA, Crocombe AD (2004) A comparison of failure prediction methods for an adhesively bonded composite beam. J Strain Anal 39(2):173–185
40. Wang RX, Cuia J, Sinclair AN, Spelt JK (2003) Strength of adhesive joints with adherend yielding: I. Analytical model. J Adhes 79(1):23–48
41. Wang ZY, Wang L, Guo W, Deng H, Tong JW, Aymerich F (2009) An investigation on strain/stress distribution around the overlap end of laminated composite single-lap joints. Compos Struct 89:589–595
42. Wen PH, Aliabadi MH, Young A (2002) Boundary element analysis of flat cracked panels with adhesively bonded patches. Eng Fract Mech 69:2129–2146
43. Wen PH, Aliabadi MH, Young A (2003) Boundary element analysis of curved cracked panels with adhesively bonded patches. Int J Numer Methods Eng 58:43–61
44. Xu W, Li G (2010) Finite difference three-dimensional solution of stresses in adhesively bonded composite tubular joint subjected to torsion. Int J Adhes Adhes 30:191–199
45. Young A, Rooke DP (1992) Analaysis of patched and stiffened cracked panels using the boundary element method. Int J Solids Struct 29(17):2201–2216
46. Zhao C, Hobbs BE, Mühlhaus HB, Ord A (1999) A consistent point-searching algorithm for solution interpolation in unstructured meshes consisting of 4-node bilinear quadrilateral elements. Int J Numer Methods Eng 45:1509–1526
47. Zienkiewicz OC, Taylor RL (2000) The finite element method – volume 1: the basis. Butterworth-Heinemann, Oxford