Finite element method: Difference between revisions

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== Finite element and fast fourier transform (FFT) methods ==

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 FFtFFT 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>