Approximated decompositions for computational continuum mechanics
Approximated decompositions for computational continuum mechanics
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DOI：
10.1016/j.jcp.2023.112576
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发表时间：
2023-10-23
影响因子：
4.1
通讯作者：
Shu，Chi-Wang
中科院分区：
文献类型：
--
作者：
Borges，Rafael B. deR.;Colman，Flavio C.;Shu，Chi-Wang
固体和流体力学问题经常涉及特征值和特征向量的计算。一个是欧拉方程和理想气体状态方程的简短代数表达式。然而，固体力学模型的代数表达式可以变得很重要。用代数方法计算不同模型的本征值和本征向量会变得很麻烦。此外，较大的公式可能导致增加的总计算时间。内能在模拟介质中起着重要的作用，热力学一致性是必不可少的。模型应正确捕获热传递、机械响应和流体流动，例如，机械热量和骨重建（BR）问题。在机械热问题中，由于加载和卸载应力而发生温度变化。在BR问题中，开发了几种数学模型来模拟其行为。测试不同模型的灵活性是一项基本要求。因此，我们的目标是提出和测试近似特征系统和奇异值（SVD）分解连续介质力学问题。我们在基准CFD问题中测试了我们的命题的分辨率和准确性。在固体力学方面，解决了几个经典问题，验证了其精度，解决了力热试验和拉伸杆件。虽然更昂贵，近似特征系统分解是更灵活的。它是高阶和高分辨率的，可以捕获热传递，流体流动，不连续性和机械响应，并可以模拟真实世界的应用。奇异值分解是一种可行的方法，可以用来调整严重情况下的计算时间，但它需要改进。
Solid and fluid mechanics problems often involve the computation of eigenvalues and eigenvectors. One has short algebraic expressions for the Euler equations and the perfect gas equation of state. However, algebraic expressions for solid mechanics models can become substantial. Computing the eigenvalues and eigenvectors algebraically for different models can become cumbersome. Also, larger formulae may result in increased overall computational time. Internal energy plays a significant role in modeling the medium, and thermodynamic consistency is indispensable. The models should correctly capture heat transfer, mechanical response, and fluid flow, e.g., in mechanocaloric and bone remodeling (BR) problems. In mechanocaloric problems, temperature variation occurs because of load and unload stresses. In BR problems, several mathematical models were developed for modeling its behavior. Flexibility for testing different models is an essential requirement. Hence, our goal is to propose and test approximated eigensystem and singular value (SVD) decompositions for continuum mechanics problems. We tested the resolution and accuracy of our propositions in benchmarking CFD problems. For the solid mechanics, we solved a few classical problems, verified the accuracy, and solved mechanocaloric tests and bar under tension. Although more expensive, the approximated eigensystem decomposition is more flexible. It is high-order and high-resolution, can capture heat transfer, fluid flow, discontinuities, and mechanical response, and can model real-world applications. The SVD is a viable possibility and can be used to adjust the computational time of severe situations, but it needs to be improved.