EFGM求解传热边界形状辨识和相变问题及自适应EFGM计算
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摘要
传热边界形状辨识和相变传热问题的数值模拟具有重要的工程应用背景与理论探讨价值。由于这两个问题都涉及到边界变化的问题,因此,如何避免网格重构造成的计算困难成为相关数值模拟中所关注的一个重要问题。无网格伽辽金法(Element-free Galerkin method, EFGM)基于点的近似,不需要网格的生成和重构,具有良好的精度、稳定性和收敛性。因此,本文以EFGM为基础,探讨了传热边界形状辨识和相变问题的数值求解技术。
自适应EFGM只对计算精度不满足要求的区域进行模型改进,和均匀加密相比,可在提高精度的同时,显著减少所需的节点数量,降低计算量,具有重要的工程应用价值。因此,本文还对EFGM的自适应计算进行了研究。
主要的研究工作包括：
1.将EFG和水平集方法(Level set method, LSM)相结合,提出了一个求解传热边界形状辨识问题的数值模型。EFG求解边界变化的的温度场,不需网格重构；LSM可在固定网格下方便和准确地描述形状演化。通过一个曲线边界的识别问题对所提模型进行了数值验证,考虑了初值、EFG节点/LSM有限差分网格和测量误差的影响,计算结果令人满意。
2.将EFG与函数光滑化技术相结合,提出了一个求解相变传热问题的光滑等效热容数值模型。借助Sigmoid函数得到光滑连续的等效热容曲线,有效避免了等效热容阶跃变化引起的计算误差,不需使用传统的平均近似,对集中/非集中质量热容阵均适用。采用EFGM/有限差分法进行空间/时域离散。通过平板/角域凝固的数值算例对所提模型进行了数值验证,考虑了EFG节点、光滑函数参数的影响,并与平均近似方法对比,计算结果令人满意。
3.建立了一个基于节点误差估计的自适应EFG数值模型,并应用到弹性力学问题中。基于节点的误差估计可减小估计误差在节点间的虚假振荡。和Chung-Belytschko误差估计的自适应相比,可在引入较少节点数量的条件下取得较高的精度。
4.提出了一个偶应力正问题的h自适应EFG数值模型。EFG避免了有限元求解中因C1连续性要求造成的困难,自适应计算提高了计算精度,与均匀加密相比,显著降低了计算量。本文还建立了一个多宗量偶应力反问题的EFG和Gauss-Newton数值模型,可有效地对区域非均质的本构参数／载荷进行单一／组合反演,具有良好的抗噪性。
论文工作,有望为无网格方法的进一步研究／应用及相关问题的数值求解提供有价值的参考。
Identification of boundary configurations and phase change in heat transfer problems have important application background in engineering and research value in theory. As these two problems involve changing boundaries, how to avoid the difficulties because of remeshing is a very important problem in numerical simulation. The Element-Free Galerkin Method (EFGM) builds the approximation using nodes only, thus it can avoid the generation and remeshing for mesh with good accuracy, stability and convergence. Hence, this thesis studies the identification of boundary configurations and phase change in heat transfer problems based on EFGM solutions.
The adaptive EFGM computing can only refine the region where accuracy is not satisfied, so in comparision with uniform refinement, it can improve the accuracy and reduce the number of nodes needed, which achieve a significant decrease of computation cost. Hence, besides above two heat transfer problems, the adaptive EFGM computing is also researched in this thesis.
The contribution of this thesis includes:
1. Presenting a numerical model for the identification of boundary configurations in heat transfer problems combining EFG and Level set method (LSM). The remeshing is avoided in solving temperature field with changing boundary; LSM is an easy way to discrible shape evolution convieniently and accurately on a fixed grid. The proposed numerical model is verified via an identification of a curvilinear boundary, and the effects of initial guess, number of probing points, measurement error, and density of EFGM nodes and the LSM FD grid are considered. Satisfactory results are obtained.
2. Presenting a smoothed effective heat capacity numerical model for phase change heat transfer problems combining EFG and function smoothing technique. The Sigmoid function is used to achieve a discontinuous and smooth effective heat capacity, which avoids the error caused by the discontinuous step-jump of the effective heat capacity, and it is efficient for either lumped or non-lumped heat capacity matrix. The proposed numerical model is verified via solidifications of slab and coner region examples, and the EFG nodes, the parameter relevant with Sigmoid function and the comparison with traditional averaging method are considered. Satisfactory results are obtained.
3. Presenting an adaptive EFGM numerical model based on a node-based error estimator, this model is applied in elasticity problems. This node-based error estimator effectively reduces the spurious oscillation of estimated error between nodes. In comparison with the adaptive computing using Chung-Belytschko error estimator, the proposed model can achieve higher accuracy with fewer nodes introduced.
4. Presenting an adaptive EFGM model for direct couple-stress problems, in which the EFG avoids the inconvenience that may be caused by C1 continuity requirement in the implementation of FEM, and the adaptive refinement increases computational accuracy with significantly reducing the computational cost compared with uniform refinement. This thesis also builds a numerical model for solving multi-variables inverse couple-stress problems based on EFGM and Gauss-Newton algorithm. This model is effective for single/combined identification for material parameters/load in case of regional inhomogeneity.
This thesis provides a valuable reference for further research/application of meshless and other numerical solutions for relevant problems.
自适应EFGM只对计算精度不满足要求的区域进行模型改进,和均匀加密相比,可在提高精度的同时,显著减少所需的节点数量,降低计算量,具有重要的工程应用价值。因此,本文还对EFGM的自适应计算进行了研究。
主要的研究工作包括：
1.将EFG和水平集方法(Level set method, LSM)相结合,提出了一个求解传热边界形状辨识问题的数值模型。EFG求解边界变化的的温度场,不需网格重构；LSM可在固定网格下方便和准确地描述形状演化。通过一个曲线边界的识别问题对所提模型进行了数值验证,考虑了初值、EFG节点/LSM有限差分网格和测量误差的影响,计算结果令人满意。
2.将EFG与函数光滑化技术相结合,提出了一个求解相变传热问题的光滑等效热容数值模型。借助Sigmoid函数得到光滑连续的等效热容曲线,有效避免了等效热容阶跃变化引起的计算误差,不需使用传统的平均近似,对集中/非集中质量热容阵均适用。采用EFGM/有限差分法进行空间/时域离散。通过平板/角域凝固的数值算例对所提模型进行了数值验证,考虑了EFG节点、光滑函数参数的影响,并与平均近似方法对比,计算结果令人满意。
3.建立了一个基于节点误差估计的自适应EFG数值模型,并应用到弹性力学问题中。基于节点的误差估计可减小估计误差在节点间的虚假振荡。和Chung-Belytschko误差估计的自适应相比,可在引入较少节点数量的条件下取得较高的精度。
4.提出了一个偶应力正问题的h自适应EFG数值模型。EFG避免了有限元求解中因C1连续性要求造成的困难,自适应计算提高了计算精度,与均匀加密相比,显著降低了计算量。本文还建立了一个多宗量偶应力反问题的EFG和Gauss-Newton数值模型,可有效地对区域非均质的本构参数／载荷进行单一／组合反演,具有良好的抗噪性。
论文工作,有望为无网格方法的进一步研究／应用及相关问题的数值求解提供有价值的参考。
Identification of boundary configurations and phase change in heat transfer problems have important application background in engineering and research value in theory. As these two problems involve changing boundaries, how to avoid the difficulties because of remeshing is a very important problem in numerical simulation. The Element-Free Galerkin Method (EFGM) builds the approximation using nodes only, thus it can avoid the generation and remeshing for mesh with good accuracy, stability and convergence. Hence, this thesis studies the identification of boundary configurations and phase change in heat transfer problems based on EFGM solutions.
The adaptive EFGM computing can only refine the region where accuracy is not satisfied, so in comparision with uniform refinement, it can improve the accuracy and reduce the number of nodes needed, which achieve a significant decrease of computation cost. Hence, besides above two heat transfer problems, the adaptive EFGM computing is also researched in this thesis.
The contribution of this thesis includes:
1. Presenting a numerical model for the identification of boundary configurations in heat transfer problems combining EFG and Level set method (LSM). The remeshing is avoided in solving temperature field with changing boundary; LSM is an easy way to discrible shape evolution convieniently and accurately on a fixed grid. The proposed numerical model is verified via an identification of a curvilinear boundary, and the effects of initial guess, number of probing points, measurement error, and density of EFGM nodes and the LSM FD grid are considered. Satisfactory results are obtained.
2. Presenting a smoothed effective heat capacity numerical model for phase change heat transfer problems combining EFG and function smoothing technique. The Sigmoid function is used to achieve a discontinuous and smooth effective heat capacity, which avoids the error caused by the discontinuous step-jump of the effective heat capacity, and it is efficient for either lumped or non-lumped heat capacity matrix. The proposed numerical model is verified via solidifications of slab and coner region examples, and the EFG nodes, the parameter relevant with Sigmoid function and the comparison with traditional averaging method are considered. Satisfactory results are obtained.
3. Presenting an adaptive EFGM numerical model based on a node-based error estimator, this model is applied in elasticity problems. This node-based error estimator effectively reduces the spurious oscillation of estimated error between nodes. In comparison with the adaptive computing using Chung-Belytschko error estimator, the proposed model can achieve higher accuracy with fewer nodes introduced.
4. Presenting an adaptive EFGM model for direct couple-stress problems, in which the EFG avoids the inconvenience that may be caused by C1 continuity requirement in the implementation of FEM, and the adaptive refinement increases computational accuracy with significantly reducing the computational cost compared with uniform refinement. This thesis also builds a numerical model for solving multi-variables inverse couple-stress problems based on EFGM and Gauss-Newton algorithm. This model is effective for single/combined identification for material parameters/load in case of regional inhomogeneity.
This thesis provides a valuable reference for further research/application of meshless and other numerical solutions for relevant problems.
引文
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