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A three-dimensional enriched finite element method for nonlinear transient heat transfer in functionally graded materials

机译：功能梯度材料非线性瞬态传热的三维富集有限元方法

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摘要

Nonlinear transient heat transfer in functionally graded materials is being studied more popular in present. In preliminary design, this problem can be simplified as a composite, and a three-dimensional transient heat transfer analysis is used to adjust dimensions of the considered materials. This paper is concerned with the numerical modeling of transient heat transfer in composite materials where the thermal conductivity is also dependent on the temperature; hence the problem is nonlinear. We are interested in solutions with steep boundary layers where highly refined meshes are commonly needed. Such problems can be challenging to solve with the conventional finite element method. To deal with this challenge we propose an enriched finite element formulation where the basis functions are augmented with a summation of exponential functions. First, the initial-value problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the enriched finite elements. We demonstrate through several numerical examples that the proposed approach can recover the heat transfer on coarse meshes and with much fewer degrees of freedom compared to the standard finite element method. Thus, a significant reduction in the computational requirements is achieved without compromising on the solution accuracy. The results also show the stability of the scheme when using tetrahedral unstructured grids.
机译：在功能上渐变材料中的非线性瞬态传热正在研究现在更受欢迎。在初步设计中，该问题可以简化为复合材料，并且使用三维瞬态传热分析来调节所考虑的材料的尺寸。本文涉及复合材料中瞬态传热的数值模型，导热性也取决于温度;因此，问题是非线性的。我们对具有陡峭边界层的解决方案感兴趣，其中通常需要高度精细的网格。通过传统的有限元方法解决这些问题可能是具有挑战性的。要处理这一挑战，我们提出了一个丰富的有限元制定，其中基础职能是通过指数函数的总和增强。首先，使用半隐式方案及时纳入初始值问题，然后使用丰富的有限元在空间中集成半离散问题。我们通过几个数值示例证明了所提出的方法可以在粗糙网格上恢复传热和与标准有限元方法相比的较少程度的自由度。因此，实现了计算要求的显着降低，而不会对溶液精度损害。结果还显示使用四面体非结构化网格时该方案的稳定性。

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