数据驱动多孔介质问题的热力学一致解的存在性
Existence of thermodynamically consistent solutions for data-driven porous media problems
- Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
- International Centre for Numerical Methods in Engineering（国际数值方法工程中心）
- University of Bergen（卑尔根大学）
- Geophysical Institute and Bergen Offshore Wind Centre（地球物理研究所与卑尔根海上风电中心）
- California Institute of Technology（加州理工学院）
- Division of Engineering and Applied Science（工程与应用科学系）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
针对数据驱动多孔介质问题，提出变分DDCM框架将热力学第二定律作为硬约束，通过补偿紧性论证确立极小解存在性，分析拉格朗日乘子与惩罚方案，数值实验验证其可恢复热力学一致性。
中文摘要 AI 辅助
数据驱动计算力学（DDCM）用实验或细尺度模拟得到的局部材料状态数据直接重构边值问题，替代传统的现象学本构模型。标准DDCM公式仅满足平衡与相容性，无法固有保证符合热力学第二定律，当输入材料数据集存在噪声或局部物理不可容许时，该问题尤为突出。本研究针对扩散-反应问题提出一种变分DDCM框架，利用梯度-通量系统中热力学约束的简洁性，提出增广公式，在能量极小化问题中显式将热力学第二定律作为硬约束。尽管热力学可容许状态集非凸，且在环境相空间中不弱闭，本研究通过补偿紧性论证，证明可容许集与相容且平衡的场子空间的交集弱序列闭，从而确立极小解的存在性。为实现实用计算，本研究分析拉格朗日乘子公式与惩罚方案，证明惩罚泛函Γ收敛至精确约束问题，并建立结合空间有限元离散与经验数据集近似的全离散收敛框架。数值实验证实，所提惩罚方案即使在材料数据严重损坏时，也能有效恢复热力学一致性。
英文摘要
Data-Driven Computational Mechanics (DDCM) replaces traditional phenomenological constitutive models by directly reformulating boundary-value problems in terms of local material state data obtained from experiments or fine-scale simulations. Standard DDCM formulations, only enforcing equilibrium and compatibility, do not inherently guarantee compliance with the second law of thermodynamics. This breakdown occurs particularly when input material data sets are subject to noise or local physical non-admissibility. In this work, we present a variational DDCM framework specifically tailored to diffusion--reaction problems. Taking advantage of the simplicity of the thermodynamic constraint in gradient-flux systems, we propose an augmented formulation that explicitly enforces the second law of thermodynamics as a hard constraint within the energy-minimization problem. Although the set of thermodynamically admissible states is non-convex and fails to be weakly closed in the ambient phase space, we establish existence of minimizers by proving that the intersection of the admissible set with the subspace of fields that are compatible and in equilibrium is weakly sequentially closed via a compensated compactness argument. To enable practical computations, we analyze both a Lagrange multiplier formulation and a penalization scheme. We prove the $Γ$-convergence of the penalized functionals to the exact constrained problem and establish a fully discrete convergence framework incorporating spatial finite-element discretization and empirical data-set approximations. Numerical experiments confirm that the proposed penalty scheme effectively restores thermodynamic consistency even in the presence of severely corrupted material data.
