二维共轭传热的隐式伴随有限体积拓扑优化
Implicit-adjoint finite-volume topology optimization of two-dimensional conjugate heat transfer
AI总结：
该研究提出二维有限体积共轭传热拓扑优化格式，采用隐函数定理计算伴随，经验证后优化四个热流体设计基准，为可验证的共轭热流体拓扑优化提供确定性参考栈。
AI中文摘要：
设计紧凑、高效的热架构需要在高度受限的物理空间内协调固体导热、流体对流和流动阻力之间的竞争需求。基于密度的拓扑优化为合成这些耦合布局提供了系统框架，但所得设计的可重复性和数值稳定性关键取决于底层离散求解器和伴随灵敏度机制。本研究在交错标记与单元网格上提出了一种透明、自包含的二维有限体积格式，用于受设计相关能量输运、耦合斯托克斯-布林克曼（Stokes-Brinkman）或达西（Darcy）流动（固体体积固定）控制的共轭传热。为防止多孔弱可压缩布林克曼域中出现虚假人工热源，离散平流算子被构造为逐单元满足等式$\boldsymbol{u}\nabla T = \nabla \boldsymbol{u}T - T(\nabla \boldsymbol{u})$，确保即使在连续性条件不精确满足时，均匀温度场仍保持精确的离散零空间。反向模式导数通过隐函数定理而非展开迭代循环计算，得到具有有界内存需求的精确离散伴随。离散算子通过方法制造解和方向泰勒余项测试进行系统验证。采用带$\beta$-延拓的投影梯度方案对四个代表性热流体设计基准进行优化，其中候选迭代仅在满足严格、预先声明的残差收敛、质量守恒、体积可行性和数值有限性的准则时才被接受和发布。所得格式为可验证的共轭热流体拓扑优化提供了可检查、确定性的参考栈。
英文摘要：
Designing compact, high-efficiency thermal architectures requires resolving the competing demands of solid conduction, fluid convection, and flow resistance within highly constrained physical envelopes. Density-based topology optimization provides a systematic framework for synthesizing these coupled layouts, yet the reproducibility and numerical stability of the resulting designs depend critically on the underlying discrete solvers and adjoint sensitivity mechanics. In this work, we present a transparent, self-contained two-dimensional finite-volume formulation on a staggered Marker-and-Cell grid for conjugate heat transfer governed by design-dependent energy transport coupled to Stokes--Brinkman or Darcy flow at fixed solid volume. To prevent spurious artificial thermal sources in porous, weakly compressible Brinkman domains, the discrete advection operator is constructed to satisfy the identity $\mathbf{u}\cdot\nabla T=\nabla\cdot(\mathbf{u}T)-T(\nabla\cdot\mathbf{u})$ cellwise, ensuring that uniform temperature fields remain exact discrete nullspaces even under inexact continuity satisfaction. Reverse-mode derivatives are evaluated via the implicit function theorem rather than unrolled iterative loops, yielding exact discrete adjoints with bounded memory requirements. The discrete operators are systematically validated through the method of manufactured solutions and directional Taylor remainder tests. Four representative thermofluid design benchmarks are optimized using a projected-gradient scheme with $β$-continuation, wherein candidate iterates are accepted and published only upon satisfying rigorous, predeclared gates on residual convergence, mass conservation, volume feasibility, and numerical finiteness. The resulting formulation provides an inspectable, deterministic reference stack for verifiable conjugate thermofluidic topology optimization.
