Solving hyperelastic problems using mixed LSFEM
Solving hyperelastic problems using mixed LSFEM
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使用混合 LSFEM 解决超弹性问题
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
B. Müller
中科院分区：
文献类型：
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作者：
K. Steeger;A. Schwarz;J. Schröder;G. Starke;B. Müller
本贡献的重点是使用最小二乘有限元法 (LSFEM) 求解超弹性问题。特别是提供了混合最小二乘有限元公式并将其应用于几何非线性问题。单元公式的基础是一个 div-grad 一阶系统，由平衡条件和本构方程组成，两者都以残差形式编写。对残差采用 L2 范数，导致函数依赖于必须最小化的位移和应力。因此，两个自由变量的一阶变化必须为零。然后可以通过应用牛顿法找到解决方案。对于 W1,p 中的位移（p > 2）的连续近似，使用标准多项式。属于 Raviart-Thomas 空间的形状函数应用于应力插值。这些向量值函数确保 Sobolev 空间 H(div, Ω) 的一致离散化。最后在数值例子中测试了所提出的公式。 （© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA，魏因海姆）
The focus of this contribution is the solution of hyperelastic problems using the least‐squares finite element method (LSFEM). In particular a mixed least‐squares finite element formulation is provided and applied on geometrically nonlinear problems. The basis for the element formulation is a div‐grad first‐order system consisting of the equilibrium condition and the constitutive equation both written in a residual form. An L2‐norm is adopted on the residuals leading to a functional depending on displacements and stresses which has to be minimized. Therefore the first variations with respect to both free variables have to be zero. The solution can then be found by applying Newton's Method. For the continuous approximation of the displacements in W1,p with p > 2, standard polynomials are used. Shape functions belonging to a Raviart‐Thomas space are applied for the stress interpolation. These vector‐valued functions ensure a conforming discretization of the Sobolev space H(div, Ω). Finally the proposed formulation is tested in a numerical example. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)