Theory and practice of finite elements
Theory and practice of finite elements
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DOI：
10.1007/978-1-4757-4355-5
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
A. Ern;J. Guermond
中科院分区：
文献类型：
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作者：
A. Ern;J. Guermond
有限元法的起源可以追溯到20世纪50年代，当时工程师们开始解决航空中的数值结构力学问题。从那时起，应用领域稳步扩大，现在包括非线性固体力学，流体/结构相互作用，工业或地球物理环境中的流动，多组分反应湍流，多孔介质中的传质，医学科学中的粘弹性流动，电磁学，波散射问题和期权定价（仅举几个例子）。多年来，已经开发了许多基于有限元法的商业和学术代码。该方法在求解偏微分方程（PDE）方面非常成功，以至于现在的术语“有限元法”不仅指插值技术，而且还指偏微分方程和逼近技术的模糊集。有限元方法的效率依赖于两个不同的要素：有限元的插值能力（在本书中称为逼近性属性）和用户在适当的数学设置中逼近其模型（主要是一组偏微分方程）的能力（因此保证连续性、稳定性和一致性）。经验表明，如果不能以可接受的精度得出近似解，几乎总是与偏离数学基础有关。典型的例子包括非物理振荡、寄生模式和锁定效应。在大多数情况下，如果适当地建立数学框架，就可以设计出补救措施。
The origins of the finite element method can be traced back to the 1950s when engineers started to solve numerically structural mechanics problems in aeronautics. Since then, the field of applications has widened steadily and nowadays encompasses nonlinear solid mechanics, fluid/structure interactions, flows in industrial or geophysical settings, multicomponent reactive turbulent flows, mass transfer in porous media, viscoelastic flows in medical sciences, electromagnetism, wave scattering problems, and option pricing (to cite a few examples). Numerous commercial and academic codes based on the finite element method have been developed over the years. The method has been so successful to solve Partial Differential Equations (PDEs) that the term" Finite Element Method" nowadays refers not only to the mere interpolation technique it is, but also to a fuzzy set of PDEs and approximation techniques. The efficiency of the finite element method relies on two distinct ingredi ents: the interpolation capability of finite elements (referred to as the approx imability property in this book) and the ability of the user to approximate his model (mostly a set of PDEs) in a proper mathematical setting (thus guar anteeing continuity, stability, and consistency properties). Experience shows that failure to produce an approximate solution with an acceptable accuracy is almost invariably linked to departure from the mathematical foundations. Typical examples include non-physical oscillations, spurious modes, and lock ing effects. In most cases, a remedy can be designed if the mathematical framework is properly set up.