The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme
The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme
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DOI：
10.1007/s10915-018-0799-2
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发表时间：
2018-02
影响因子：
2.5
通讯作者：
M. Kimura;H. Notsu;Yoshimi Tanaka;H. Yamamoto
中科院分区：
文献类型：
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作者：
M. Kimura;H. Notsu;Yoshimi Tanaka;H. Yamamoto
从数学和数值的角度研究了一种带松弛参数的广义Maxwell粘弹性模型。结果表明，该模型具有关于粘弹性能量的梯度流动特性。基于梯度流结构，提出了一种保结构的时间离散模型，并证明了该模型解的存在唯一性。提出了一种保结构的P1/P0有限元格式，并利用其离散的梯度流结构证明了该格式在能量意义上的稳定性。作为典型的粘弹性现象，给出了蠕变变形和应力松弛的二维数值算例，并研究了松弛参数的影响。
An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution is proved. Moreover, a structure-preserving P1/P0 finite element scheme is presented and its stability in the sense of energy is shown by using its discrete gradient flow structure. As typical viscoelastic phenomena, two-dimensional numerical examples by the proposed scheme for a creep deformation and a stress relaxation are shown and the effects of the relaxation parameter are investigated.