# Finite Element Method

Hrennikoff’s

##### Discretization

A function in {\displaystyle H_{0}^{1},} with zero values at the endpoints (blue), and a piecewise linear approximation (red) P1 and P2 are ready to be discretized which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem:

##### where we define {\displaystyle x_{0}=0} and {\displaystyle x_{n+1}=1}. Observe that functions in {\displaystyle V} are not differentiable according to the elementary definition of calculus. Indeed, if {\displaystyle v\in V} then the derivative is typically not defined at any {\displaystyle x=x_{k}}, {\displaystyle k=1,\ldots ,n}. However, the derivative exists at every other value of {\displaystyle x} and one can use this derivative for the purpose of integration by parts.

A piecewise linear function in two dimensions
##### Choosing a basis
Interpolation of a Bessel function
16 scaled and shifted triangular basis functions (colors) used to reconstruct a zeroeth order Bessel function J0 (black).
The linear combination of basis functions (yellow) reproduces J0(blue) to any desired accuracy.
##### Small support of the basis

Solving the two-dimensional problem {\displaystyle u_{xx}+u_{yy}=-4} in the disk centred at the origin and radius 1, with zero boundary conditions.
(a) The triangulation.

(b) The sparse matrix L of the discretized linear system

(c) The computed solution, {\displaystyle u(x,y)=1-x^{2}-y^{2}.}

##### Application

Visualization of how a car deforms in an asymmetrical crash using finite element analysis.[1]

##### FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modelling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.

Finite Element Model of a human knee joint. [18]

## Reference

1. Daryl L. Logan (2011). A first course in the finite element method. Cengage Learning. ISBN 978-0495668251.
2. Reddy, J.N. (2006). An Introduction to the Finite Element Method (Third ed.). McGraw-Hill. ISBN 9780071267618.
3. “Finite Elements Analysis (FEA)”http://www.manortool.com. Retrieved 2017-07-28.
4.  Hrennikoff, Alexander (1941). “Solution of problems of elasticity by the framework method”. Journal of applied mechanics8.4: 169–175.
5.  Courant, R. (1943). “Variational methods for the solution of problems of equilibrium and vibrations”. Bulletin of the American Mathematical Society49: 1–23. doi:10.1090/s0002-9904-1943-07818-4.
6.  “СПб ЭМИ РАН”emi.nw.ru. Retrieved 17 March 2018.
7.  Hinton, Ernest; Irons, Bruce (July 1968). “Least squares smoothing of experimental data using finite elements”. Strain4: 24–27. doi:10.1111/j.1475-1305.1968.tb01368.x.
8.  “SAP-IV Software and Manuals”. NISEE e-Library, The Earthquake Engineering Online Archive.
9.  Gard Paulsen; Håkon With Andersen; John Petter Collett; Iver Tangen Stensrud (2014). Building Trust, The history of DNV 1864-2014. Lysaker, Norway: Dinamo Forlag A/S. pp. 121, 436. ISBN 978-82-8071-256-1.
10.  Strang, GilbertFix, George (1973). An Analysis of The Finite Element Method. Prentice Hall. ISBN 0-13-032946-0.
11. Zienkiewicz, O.C.; Taylor, R.L.; Zhu, J.Z. (2005). The Finite Element Method: Its Basis and Fundamentals (Sixth ed.). Butterworth-Heinemann. ISBN 0750663200.
12.  Bathe, K.J. (2006). Finite Element Procedures. Cambridge, MA: Klaus-Jürgen Bathe. ISBN 097900490X.
13.  Babuška, Ivo; Banerjee, Uday; Osborn, John E. (June 2004). “Generalized Finite Element Methods: Main Ideas, Results, and Perspective”. International Journal of Computational Methods1 (1): 67–103. doi:10.1142/S0219876204000083.
14. P. Solin, K. Segeth, I. Dolezel: Higher-Order Finite Element Methods, Chapman & Hall/CRC Press, 2003
15. “Spectral Element Methods”State Key Laboratory of Scientific and Engineering Computing. Retrieved 2017-07-28.
16. “What’s The Difference Between FEM, FDM, and FVM?”Machine Design. 2016-04-18. Retrieved 2017-07-28.
17. Template:Kiritsis, D.; Eemmanouilidis, Ch.; Koronios, A.; Mathew, J. (2009). “Engineering Asset Management” Proceedings of the 4th World Congress on Engineering Asset Management (WCEAM), 591-592.
18. Naghibi Beidokhti, Hamid; Janssen, Dennis; Khoshgoftar, Mehdi; Sprengers, Andre; Perdahcioglu, Emin Semih; Boogaard, Ton Van den; Verdonschot, Nico. “A comparison between dynamic implicit and explicit finite element simulations of the native knee joint”Medical Engineering & Physics38 (10): 1123–1130. doi:10.1016/j.medengphy.2016.06.001.
19. Hastings, J. K., Juds, M. A., Brauer, J. R., Accuracy and Economy of Finite Element Magnetic Analysis, 33rd Annual National Relay Conference, April 1985.
20. McLaren-Mercedes (2006). “McLaren Mercedes: Feature – Stress to impress”. Archived from the original on 2006-10-30. Retrieved 2006-10-03.
21.  Peng Long; Wang Jinliang; Zhu Qiding (19 May 1995). “Methods with high accuracy for finite element probability computing”. Journal of Computational and Applied Mathematics59(2): 181–189. doi:10.1016/0377-0427(94)00027-X.
22. Haldar, Achintya; Mahadevan, Sankaran (2000). Reliability Assessment Using Stochastic Finite Element Analysis. John Wiley & Sons. ISBN 978-0471369615.
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