Finite element method: Difference between revisions

Revision as of 09:32, 9 December 2024

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{{Differential equations}}

Finite element method (FEM) is a popular method for numerically solving [[differential equation]]s arising in engineering and [[mathematical models|mathematical modeling]]. Typical problem areas of interest include the traditional fields of [[structural analysis]], [[heat transfer]], [[fluid flow]], mass transport, and [[electromagnetic potential]]. Computers are usually used to perform the calculations required. With high-speed [[Supercomputer|supercomputerssupercomputer]]s, better solutions can be achieved and are often required to solve the largest and most complex problems.

FEM is a general [[numerical analysis|numerical method]] for solving [[partial differential equations]] in two- or three-space variables (i.e., some [[boundary value problem]]s). There are also studies about using FEM to solve high-dimensional problems.<ref>{{Cite journal

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* a set of [[ordinary differential equation]]s for [[transient state|transient]] problems.

These equation sets are element equations. They are [[linear]] if the underlying PDE is linear and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using [[numerical linear algebra|numerical linear algebraic]]ic methods. In contrast, [[ordinary differential equation]] sets that occur in the transient problems are solved by numerical integrations using standard techniques such as [[Euler's method]] or the [[Runge–Kutta methods|Runge]][[Euler–Bernoulli beam theory|–]]Kutta method.

In the second step above, a global system of equations is generated from the element equations by transforming coordinates from the subdomains' local nodes to the ___domain's global nodes. This spatial transformation includes appropriate [[Transformation matrix|orientation adjustment]]s as applied in relation to the reference [[coordinate system]]. The process is often carried out using FEM software with [[coordinates|coordinate]] data generated from the subdomains.

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Hrennikoff's work discretizes the ___domain by using a [[Lattice (group)|lattice]] analogy, while Courant's approach divides the ___domain into finite triangular sub-regions to solve [[Partial differential equation#Linear equations of second order|second-order]] [[elliptic partial differential equation]]s that arise from the problem of the [[torsion (mechanics)|torsion]] of a [[cylinder (geometry)|cylinder]]. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by [[John William Strutt, 3rd Baron Rayleigh|Lord Rayleigh]], [[Walther Ritz]], and [[Boris Galerkin]].

The application of FEM gained momentum in the 1960s and 1970s due to the developments of [[John Argyris|J. H. Argyris]] and his co-workers at the [[University of Stuttgart]]; [[Ray W. Clough|R. W. Clough]] and his co-workers at [[University of California, Berkeley|University of California Berkeley]]; [[Olgierd Zienkiewicz|O. C. Zienkiewicz]] and his co-workers [[Ernest Hinton]], [[Bruce Irons (engineer)|Bruce Irons]],<ref>{{Cite journal |last1=Hinton |first1=Ernest |last2=Irons |first2=Bruce |title=Least squares smoothing of experimental data using finite elements |journal=Strain |volume=4 |issue=3 |pages=24–27 |date=July 1968 |doi= 10.1111/j.1475-1305.1968.tb01368.x}}</ref>, and others at [[Swansea University]]; [[Philippe G. Ciarlet]] at the University of [[Pierre-and-Marie-Curie University|Paris 6]]; and [[Richard H. Gallagher|Richard Gallagher]] and his co-workers at [[Cornell University]]. During this period, additional impetus was provided by the available open-source FEM programs. NASA sponsored the original version of [[NASTRAN]]. University of California Berkeley made the finite element programs SAP IV<ref>{{cite web |title=SAP-IV Software and Manuals |url=http://nisee.berkeley.edu/elibrary/getpkg?id=SAP4 |___location=NISEE e-Library, The Earthquake Engineering Online Archive |access-date=2013-01-24 |archive-date=2013-03-09 |archive-url=https://web.archive.org/web/20130309013628/http://nisee.berkeley.edu/elibrary/getpkg?id=SAP4 |url-status=live }}</ref> and, later, [[OpenSees]] widely available. In Norway, the ship classification society Det Norske Veritas (now [[DNV GL]]) developed [[SESAM (FEM)|Sesam]] in 1969 for use in the analysis of ships.<ref>{{cite book |title=Building Trust, The history of DNV 1864-2014 |author1=Gard Paulsen |author2=Håkon With Andersen |author3=John Petter Collett |author4=Iver Tangen Stensrud |date=2014 |publisher=Dinamo Forlag A/S |isbn=978-82-8071-256-1 |___location=Lysaker, Norway |pages=121, 436}}<!-- |access-date=30 June 2015 --></ref> A rigorous mathematical basis for FEM was provided in 1973 with a publication by [[Gilbert Strang]] and [[George Fix]].<ref>{{cite book |first1=Gilbert |last1=Strang |author-link1=Gilbert Strang |first2=George |last2=Fix |author-link2=George Fix |title=An Analysis of The Finite Element Method |url=https://archive.org/details/analysisoffinite0000stra |url-access=registration |publisher=Prentice Hall |year=1973 |isbn=978-0-13-032946-2}}</ref> The method has since been generalized for the [[numerical analysis|numerical modeling]] of physical systems in a wide variety of [[engineering]] disciplines, such as [[electromagnetism]], [[heat transfer]], and [[fluid dynamics]].<ref name="ZienkiewiczTaylor2013">{{cite book |author1=Olek C Zienkiewicz |author2=Robert L Taylor |author3=J.Z. Zhu |title=The Finite Element Method: Its Basis and Fundamentals |url=https://books.google.com/books?id=7UL5Ls9hOF8C |date=31 August 2013 |publisher=Butterworth-Heinemann |isbn=978-0-08-095135-5}}</ref><ref>{{cite book |first1=K.J. |last1=Bathe |author-link1= Klaus-Jürgen Bathe |title=Finite Element Procedures |publisher= Cambridge, MA: Klaus-Jürgen Bathe |year=2006 |isbn= 978-0979004902}}</ref>

==Technical discussion==

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Another method used for approximating solutions to a partial differential equation is the [[Fast Fourier transform|Fast Fourier Transform]] (FFT), where the solution is approximated by a fourier series computed using the FFT. For approximating the mechanical response of materials under stress, FFT is often much faster,<ref>{{Cite journal |last1=Ma |first1=X |last2=Parvathaneni |first2=K |last3=Lomov |first3=S |last4=Vasiukov |first4=D |last5=Shakoor |first5=M |last6=Park |first6=C |title=Quantitative comparison between fast fourier transform and finite element method for micromechanical modeling of composite |url=https://hal.science/hal-02416258 |journal=FiBreMoD Conference|date=December 2019 }}</ref> but FEM may be more accurate.<ref name=":2">{{Cite journal |last1=Prakash |first1=A |last2=Lebensohn |first2=R A |date=2009-09-01 |title=Simulation of micromechanical behavior of polycrystals: finite elements versus fast Fourier transforms |url=https://iopscience.iop.org/article/10.1088/0965-0393/17/6/064010 |journal=Modelling and Simulation in Materials Science and Engineering |volume=17 |issue=6 |pages=064010 |doi=10.1088/0965-0393/17/6/064010 |bibcode=2009MSMSE..17f4010P |issn=0965-0393}}</ref> One example of the respective advantages of the two methods is in simulation of [[Rolling (metalworking)|rolling]] a sheet of [[Aluminium|aluminum]] (an FCC metal), and [[Wire drawing|drawing]] a wire of [[tungsten]] (a BCC metal). This simulation did not have a sophisticated shape update algorithm for the FFT method. In both cases, the FFT method was more than 10 times as fast as FEM, but in the wire drawing simulation, where there were large deformations in [[Crystallite|grains]], the FEM method was much more accurate. In the sheet rolling simulation, the results of the two methods were similar.<ref name=":2" /> FFT has a larger speed advantage in cases where the boundary conditions are given in the materials [[Strain (mechanics)|strain]], and loses some of its efficiency in cases where the [[Stress (mechanics)|stress]] is used to apply the boundary conditions, as more iterations of the method are needed.<ref>{{Cite journal |last1=Cruzado |first1=A |last2=Segurado |first2=J |last3=Hartl |first3=D J |last4=Benzerga |first4=A A |date=2021-06-01 |title=A variational fast Fourier transform method for phase-transforming materials |url=https://iopscience.iop.org/article/10.1088/1361-651X/abe4c7 |journal=Modelling and Simulation in Materials Science and Engineering |volume=29 |issue=4 |pages=045001 |doi=10.1088/1361-651X/abe4c7 |bibcode=2021MSMSE..29d5001C |issn=0965-0393}}</ref>

The FE and FFT methods can also be combined in a [[voxel]] based method (2) to simulate deformation in materials, where the FE method is used for the macroscale stress and deformation, and the FFT method is used on the microscale to deal with the effects of microscale on the mechanical response.<ref name=":3">{{Cite journal |last1=Gierden |first1=Christian |last2=Kochmann |first2=Julian |last3=Waimann |first3=Johanna |last4=Svendsen |first4=Bob |last5=Reese |first5=Stefanie |date=2022-10-01 |title=A Review of FE-FFT-Based Two-Scale Methods for Computational Modeling of Microstructure Evolution and Macroscopic Material Behavior |journal=Archives of Computational Methods in Engineering |language=en |volume=29 |issue=6 |pages=4115–4135 |doi=10.1007/s11831-022-09735-6 |issn=1886-1784|doi-access=free }}</ref> Unlike FEM, FFT methods’ similarities to image processing methods means that an actual image of the microstructure from a microscope can be input to the solver to get a more accurate stress response. Using a real image with FFT avoids meshing the microstructure, which would be required if using FEM simulation of the microstructure, and might be difficult. Because fourier approximations are inherently periodic, FFT can only be used in cases of periodic microstructure, but this is common in real materials.<ref name=":3" /> FFT can also be combined with FEM methods by using fourier components as the variational basis for approximating the fields inside an element, which can take advantage of the speed of FFT based solvers.<ref>{{Cite journal |last1=Zeman |first1=J. |last2=de Geus |first2=T. W. J. |last3=Vondřejc |first3=J. |last4=Peerlings |first4=R. H. J. |last5=Geers |first5=M. G. D. |date=2017-09-07 |title=A finite element perspective on nonlinear FFT-based micromechanical simulations: A FINITE ELEMENT PERSPECTIVE ON NONLINEAR FFT-BASED SIMULATIONS |url=https://onlinelibrary.wiley.com/doi/10.1002/nme.5481 |journal=International Journal for Numerical Methods in Engineering |language=en |volume=111 |issue=10 |pages=903–926 |doi=10.1002/nme.5481|arxiv=1601.05970 }}</ref>

==Application==