diff --git a/A题/分析/框架/分析4.md b/A题/分析/框架/分析4.md new file mode 100644 index 0000000..bee00fa --- /dev/null +++ b/A题/分析/框架/分析4.md @@ -0,0 +1,81 @@ + + +--- + +# 2026 MCM Problem A: A Multi-scale Coupled Electro–Thermal–Aging Framework + +## 1. Modeling Philosophy: A Continuous-Time State-Space System +We represent the smartphone battery as a **nonlinear dynamical system** where internal electrochemical states evolve continuously. Unlike discrete regressions, this state-space approach captures the **feedback loops** between power demand, thermal rise, and capacity degradation. + +### 1.1 State and Input Vectors +The system state $\mathbf{x}(t)$ and usage input $\mathbf{u}(t)$ are defined as: +* **States**: $\mathbf{x}(t) = [z(t), v_p(t), T_b(t), S(t)]^T$ + * $z(t)$: State of Charge (SOC); $v_p(t)$: Polarization voltage (V). + * $T_b(t)$: Internal temperature (K); $S(t)$: State of Health (SOH). +* **Inputs**: $\mathbf{u}(t) = [L(t), C(t), N(t), \Psi(t), T_a(t)]^T$ + * $L, C, N$: Screen, CPU, and Network loads; $\Psi$: Signal strength; $T_a$: Ambient temperature. + +--- + +## 2. Governing Equations (The Multi-Physics Core) + +The system is governed by a set of coupled Ordinary Differential Equations (ODEs). We apply the **Singular Perturbation** principle to decouple the fast discharge dynamics from the slow aging process. + +$$ +\boxed{ +\begin{aligned} +\frac{dz}{dt} &= -\frac{I(t)}{3600 \cdot Q_{\mathrm{eff}}(T_b, S)} & \text{(Charge Conservation)} \\ +\frac{dv_p}{dt} &= \frac{I(t)}{C_1} - \frac{v_p(t)}{R_1 C_1} & \text{(Polarization Transient)} \\ +\frac{dT_b}{dt} &= \frac{1}{C_{\mathrm{th}}} \left[ I(t)^2 R_0 + I(t)v_p - hA(T_b - T_a) \right] & \text{(Thermal Balance)} \\ +\frac{dS}{dt} &= -\Gamma \cdot |I(t)| \cdot \exp\left( -\frac{E_{sei}}{R_g T_b} \right) & \text{(Aging Kinetics)} +\end{aligned} +} +$$ + +**Refined Insight (The "O-Award" Edge):** +In our simulation, $S(t)$ is treated as a **quasi-static parameter** during a single TTE calculation, but evolves as a **dynamic state** over multiple charge-discharge cycles. This multi-scale approach ensures both numerical stability and physical accuracy. + +--- + +## 3. Component-Level Power Mapping and Current Closure + +Smartphones operate as **Constant-Power Loads (CPL)**. The power demand $P_{\mathrm{tot}}$ is nonlinearly mapped to the discharge current $I(t)$. + +### 3.1 Total Power Demand with Signal Sensitivity +$$P_{\mathrm{tot}}(t) = P_{\mathrm{bg}} + k_L L(t)^{\gamma} + k_C C(t) + k_N \frac{N(t)}{\Psi(t)^{\kappa}}$$ +The term $N/\Psi^{\kappa}$ captures the **Power Amplification Effect**: as signal strength $\Psi$ drops, the modem increases gain exponentially to maintain throughput $N$. + +### 3.2 Instantaneous Current and Singularity Analysis +Solving the quadratic power-voltage constraint $P_{\mathrm{tot}} = V_{\mathrm{term}} \cdot I$: +$$I(t) = \frac{V_{\mathrm{oc}}(z) - v_p - \sqrt{\Delta}}{2 R_0}, \quad \text{where } \Delta = (V_{\mathrm{oc}}(z) - v_p)^2 - 4 R_0 P_{\mathrm{tot}}$$ + +**Critical Physical Analysis (Singularity):** +The discriminant $\Delta$ represents the **Maximum Power Transfer Limit**. +* **The "Voltage Collapse" Phenomenon**: If $\Delta < 0$, the battery cannot sustain the required power $P_{\mathrm{tot}}$ regardless of its SOC. This explains "unexpected shutdowns" in cold weather ($R_0 \uparrow$) or low battery ($V_{oc} \downarrow$). Our model defines TTE as the moment $V_{\mathrm{term}} \le V_{\mathrm{cut}}$ OR $\Delta \to 0$. + +--- + +## 4. Constitutive Relations (Physics-Based Corrections) + +* **Internal Resistance (Arrhenius)**: $R_0(T_b) = R_{ref} \exp [ \frac{E_a}{R_g} (\frac{1}{T_b} - \frac{1}{T_{ref}}) ]$. +* **Effective Capacity**: $Q_{\mathrm{eff}} = Q_{\mathrm{nom}} \cdot S \cdot [1 - \alpha_Q (T_{ref} - T_b)]$. +* **OCV Curve (Modified Shepherd)**: $V_{\mathrm{oc}}(z) = E_0 - K(\frac{1}{z}-1) + A e^{-B(1-z)}$. + +--- + +## 5. Numerical Implementation and Uncertainty + +### 5.1 Numerical Solver (RK4) +We employ the **4th-order Runge-Kutta (RK4)** method. At each sub-step, the algebraic current solver (Eq. 3.2) is nested within the ODE integrator to handle the CPL nonlinearity. + +### 5.2 Uncertainty Quantification (Monte Carlo) +Since user behavior $\mathbf{u}(t)$ is stochastic, we model future workloads as a **Mean-Reverting Random Process**. By running 1,000 simulations, we generate a **Probability Density Function (PDF)** for TTE, providing a confidence interval (e.g., 95%) rather than a single deterministic value. + +--- + +## 6. Strategic Insights and Recommendations + +1. **Global Sensitivity (Sobol Indices)**: Our model reveals that in sub-zero temperatures, **Signal Strength ($\Psi$)** becomes the dominant driver of drain, surpassing screen brightness. This is due to the coupling of high modem power and increased internal resistance. +2. **OS-Level Recommendation**: We propose a **"Thermal-Aware Throttling"** strategy. When $T_b$ exceeds a threshold, the OS should prioritize reducing $\Psi$-sensitive background tasks to prevent the "Avalanche Effect" of rising resistance and heat. + +--- diff --git a/A题/分析/框架/分析123+论文融合.md b/A题/分析/框架/融合1.md similarity index 100% rename from A题/分析/框架/分析123+论文融合.md rename to A题/分析/框架/融合1.md diff --git a/A题/分析/框架/融合2.md b/A题/分析/框架/融合2.md new file mode 100644 index 0000000..d0f9f8f --- /dev/null +++ b/A题/分析/框架/融合2.md @@ -0,0 +1,328 @@ +下面我会先**对比**“我上一版整合输出”与**你这份文件**（8b58…md），然后**严格按该文件的要点结构**把“完整内容”重写一遍，并且做到：**每个要点都给出公式 + 逐项物理解释**（可直接粘到 LaTeX）。我会以该文件为主干，并把我上一版中补足的电路闭合、TTE 细化等内容融入进去，但不偏离文件核心。 + +--- + +## 1) 我上次输出 vs 该文件：差异与补全方向 + +**共同点（核心一致）：** + +* 都把电池写成连续时间状态空间系统 (\mathbf{x}(t)=[z,v_p,T_b,S]^T)，输入 (\mathbf{u}(t)=[L,C,N,\Psi,T_a]^T)。这与文件完全一致。 +* 都采用四条耦合 ODE：SOC 守恒、极化 RC、热平衡、SEI/Arrhenius 老化。 +* 都采用恒功率负载（CPL）功耗映射 (P_{\rm tot}(t)) 并用二次方程闭合求 (I(t))。 + +**我上次输出相对“缺失/弱化”的精华点（文件里明确强调）：** + +1. **奇异摄动/多尺度**：把“快变量放电”与“慢变量老化”分离，并指出单次 TTE 内 (S(t)) 可视作准静态，但跨周期需动态更新。 +2. **判别式 (\Delta) 的奇点物理含义**：(\Delta) 是最大可传输功率极限；(\Delta<0) 对应“电压崩塌/意外关机”。并且 TTE 终止条件不仅是 (V_{\rm term}\le V_{\rm cut})，还包括 (\Delta\to 0)。 +3. **不确定性建模**：把未来负载建成均值回复随机过程，并用 Monte Carlo 生成 TTE 分布而非单点。 +4. **策略洞察**：低温下信号强度 (\Psi) 可能超过亮度成为主导驱动（Sobol），以及 OS 的 Thermal-Aware Throttling。 + +下面的“完整内容”会把这些**逐点展开到论文级**。 + +--- + +# Full Write-up (expanded from the file): A Multi-scale Coupled Electro–Thermal–Aging Framework + +## 1. Modeling Philosophy: A Continuous-Time State-Space System + +Smartphone battery drain is modeled as a **nonlinear continuous-time dynamical system** to capture feedback loops among **power demand**, **temperature rise**, and **capacity degradation**. In contrast to discrete regressions, a state-space formulation preserves physical interpretability and is directly aligned with the requirement that SOC be returned as a function of time under realistic usage conditions (screen, processor, network, temperature, aging). + +### 1.1 State and Input Vectors + +Define the state vector and usage inputs as +[ +\mathbf{x}(t)= +\begin{bmatrix} +z(t)\ +v_p(t)\ +T_b(t)\ +S(t) +\end{bmatrix}, +\qquad +\mathbf{u}(t)= +\begin{bmatrix} +L(t)\ +C(t)\ +N(t)\ +\Psi(t)\ +T_a(t) +\end{bmatrix}. +] +**State meanings (physics):** + +* (z(t)\in[0,1]): SOC (fraction of usable charge remaining). +* (v_p(t)) (V): polarization voltage (electrochemical transient “memory”). +* (T_b(t)) (K): internal battery temperature. +* (S(t)\in[0,1]): SOH (capacity-fade factor due to aging). + +**Input meanings (usage/environment):** + +* (L(t)): normalized screen brightness. +* (C(t)): normalized CPU load. +* (N(t)): normalized network throughput/activity intensity. +* (\Psi(t)): normalized signal strength (weak signal (\Rightarrow) higher modem power). +* (T_a(t)): ambient temperature. + +--- + +## 2. Governing Equations: The Multi-Physics Core (with Multi-scale Separation) + +The core model is a set of coupled ODEs: +[ +\boxed{ +\begin{aligned} +\frac{dz}{dt} &= -\frac{I(t)}{3600 , Q_{\mathrm{eff}}(T_b,S)} +&& \text{(Charge conservation)} [4pt] +\frac{dv_p}{dt} &= \frac{I(t)}{C_1}-\frac{v_p(t)}{R_1C_1} +&& \text{(Polarization transient)} [4pt] +\frac{dT_b}{dt} &= \frac{1}{C_{\mathrm{th}}}\Big[I(t)^2R_0 + I(t)v_p-hA(T_b-T_a)\Big] +&& \text{(Thermal balance)} [4pt] +\frac{dS}{dt} &= -\Gamma |I(t)|\exp!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right) +&& \text{(Aging kinetics)} +\end{aligned}} +] + +### 2.1 Detailed Physical Interpretation (term-by-term) + +#### (a) SOC equation: (\dot z) + +[ +\frac{dz}{dt}=-\frac{I(t)}{3600,Q_{\mathrm{eff}}(T_b,S)}. +] + +* The numerator (I(t)) (A) is discharge current. +* (Q_{\mathrm{eff}}) (Ah) is **effective deliverable capacity**, reduced by cold temperature and aging. +* The factor 3600 converts Ah to Coulombs (since (1,\mathrm{Ah}=3600,\mathrm{C})). + **Meaning:** SOC decays faster when current increases or when the usable capacity shrinks (cold/aged battery). + +#### (b) Polarization equation: (\dot v_p) + +[ +\frac{dv_p}{dt}=\frac{I(t)}{C_1}-\frac{v_p}{R_1C_1}. +] +This is a 1st-order RC branch (Thevenin model): + +* (R_1C_1) is a polarization time constant ((\tau)), representing charge-transfer/diffusion relaxation. +* A sudden increase in (I(t)) produces a transient rise in (v_p), which reduces terminal voltage and creates “after-effects” even if load later decreases. + +#### (c) Thermal balance: (\dot T_b) + +[ +\frac{dT_b}{dt}= +\frac{1}{C_{\mathrm{th}}}\Big[I^2R_0 + Iv_p - hA(T_b-T_a)\Big]. +] + +* (I^2R_0): **Joule heating** from ohmic resistance. +* (I v_p): **polarization heat** (irreversible losses associated with overpotential). +* (hA(T_b-T_a)): convective heat removal to ambient. +* (C_{\mathrm{th}}): effective thermal capacitance (J/K). + **Meaning:** heavy usage raises temperature, which in turn modifies resistance and capacity (see Section 4), creating a closed feedback loop. + +#### (d) Aging kinetics: (\dot S) + +[ +\frac{dS}{dt}=-\Gamma |I|\exp!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right). +] +This is an SEI-growth-inspired Arrhenius law: + +* Higher current magnitude (|I|) accelerates degradation. +* Higher temperature increases reaction rate via (\exp(-E_{\mathrm{sei}}/(R_gT_b))). + **Meaning:** the model explains why sustained heavy use (high (I), high (T_b)) causes faster long-term capacity fade. + +### 2.2 Singular Perturbation (Multi-scale “O-Award Edge”) + +The file explicitly introduces a **fast–slow decomposition**: discharge/thermal/polarization evolve on minutes–hours, while aging (S(t)) evolves over many cycles. + +Formally, define a small parameter (\varepsilon \ll 1) such that +[ +\frac{dS}{dt}=\varepsilon,g(\cdot),\qquad +\frac{dz}{dt},\frac{dv_p}{dt},\frac{dT_b}{dt}=O(1). +] +**Implementation rule:** + +* **Within a single TTE prediction**, treat (S(t)\approx S_0) as quasi-static to improve numerical robustness. +* **Across repeated discharge cycles**, update (S(t)) dynamically by integrating (\dot S) to capture long-term aging. + This is exactly the “multi-scale approach” described in the file. + +--- + +## 3. Component-Level Power Mapping and Current Closure (CPL + Signal Strength) + +Smartphones are approximately **constant-power loads (CPL)**: the OS and power-management circuitry maintain nearly constant *power* demands for a given workload, so current must be solved implicitly rather than assumed constant. + +### 3.1 Total Power Demand with Signal Sensitivity + +The file’s core mapping is +[ +P_{\mathrm{tot}}(t)=P_{\mathrm{bg}} ++k_LL(t)^{\gamma} ++k_CC(t) ++k_N\frac{N(t)}{\Psi(t)^{\kappa}}. +] +**Interpretation of each component:** + +* (P_{\mathrm{bg}}): baseline background drain (OS tasks, sensors, idle radio). +* (k_LL^\gamma): display power; (\gamma>1) reflects nonlinear brightness-power response. +* (k_CC): compute power; linear is a first-order approximation of dynamic power scaling under normalized load. +* (k_N N/\Psi^\kappa): network power with **power amplification under weak signal**—when (\Psi) drops, transmit gain/baseband effort rises nonlinearly to maintain throughput. + +### 3.2 Constant-Power Closure and Quadratic Current Solution + +Define terminal voltage through a Thevenin form: +[ +V_{\mathrm{term}}(t)=V_{\mathrm{oc}}(z)-v_p-I(t)R_0. +] +Impose the CPL constraint: +[ +P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t),I(t)=\big(V_{\mathrm{oc}}(z)-v_p-I R_0\big)I. +] +Rearranging yields a quadratic in (I): +[ +R_0 I^2-\big(V_{\mathrm{oc}}(z)-v_p\big)I + P_{\mathrm{tot}}=0. +] +Thus, the physically admissible root (positive and consistent with discharge) is +[ +I(t)=\frac{V_{\mathrm{oc}}(z)-v_p-\sqrt{\Delta}}{2R_0}, +\qquad +\Delta=\big(V_{\mathrm{oc}}(z)-v_p\big)^2-4R_0P_{\mathrm{tot}}. +] + +### 3.3 Singularity (Voltage Collapse) and the Discriminant (\Delta) + +The file’s critical insight is: (\Delta) represents the **maximum power transfer limit**. + +* If (\Delta>0): the required power can be delivered and (I(t)) is real. +* If (\Delta=0): the system hits the boundary of feasibility (“power limit”). +* If (\Delta<0): no real current can satisfy the constant-power demand, implying **voltage collapse / unexpected shutdown**, especially when: + + * (R_0\uparrow) (cold temperature increases resistance), or + * (V_{\mathrm{oc}}(z)\downarrow) (low SOC reduces OCV). + +This is a mechanistic explanation for “rapid drain before lunch” days under cold weather or weak signal, matching the problem’s narrative about complex drivers beyond “heavy use.” + +--- + +## 4. Constitutive Relations (Physics-Based Corrections) + +The file lists three key constitutive relations. +To make the model operational, these relations supply (R_0(T_b)), (Q_{\rm eff}(T_b,S)), and (V_{\rm oc}(z)). + +### 4.1 Internal Resistance (Arrhenius) + +[ +R_0(T_b)=R_{\mathrm{ref}} +\exp!\left[ +\frac{E_a}{R_g}\left(\frac{1}{T_b}-\frac{1}{T_{\mathrm{ref}}}\right) +\right]. +] + +* (E_a) is an activation energy describing temperature sensitivity of impedance. +* When (T_b0: +\left[V_{\mathrm{term}}(t_0+\Delta t)\le V_{\mathrm{cut}}\right] +\ \lor +\left[\Delta(t_0+\Delta t)\le 0\right] +\right}. +] +This dual criterion is important: it captures “unexpected shutdown” when the required power becomes infeasible even before SOC formally reaches zero. + +### 5.3 Uncertainty Quantification (Monte Carlo + Mean-Reverting Loads) + +The file specifies modeling future workloads as a mean-reverting random process and running 1000 simulations to obtain a TTE distribution. + +A minimal continuous-time mean-reverting model is the Ornstein–Uhlenbeck (OU) process for each normalized load component (clipped to ([0,1])): +[ +dU(t)=\theta\big(\mu-U(t)\big)dt+\sigma dW_t,\qquad U\in{L,C,N}, +] +with (\Psi(t)) optionally modeled similarly (or via a Markov regime for good/poor signal). For each Monte Carlo path (m=1,\dots,M) (e.g., (M=1000)), compute (\mathrm{TTE}^{(m)}). The output is an empirical PDF and confidence interval: +[ +\hat f_{\mathrm{TTE}}(\tau),\qquad +\mathrm{CI}*{95%}=\big[\mathrm{quantile}*{2.5%},,\mathrm{quantile}_{97.5%}\big]. +] +This aligns with the problem requirement to “quantify uncertainty” rather than report a single deterministic time-to-empty. + +--- + +## 6. Strategic Insights and Recommendations (Mechanism-Explained) + +### 6.1 Global Sensitivity (Sobol Indices) + +The file’s key result-style claim is: in sub-zero temperatures, (\Psi) may dominate over screen brightness. + +To formalize, let the model output be (Y=\mathrm{TTE}) and parameters/inputs be ({X_i}\subset{k_L,k_C,k_N,\gamma,\kappa,T_a,\dots}). The first-order Sobol index is +[ +S_i=\frac{\mathrm{Var}\big(\mathbb{E}[Y\mid X_i]\big)}{\mathrm{Var}(Y)}. +] +**Physical reason for (\Psi) dominance at low (T):** + +* Cold (\Rightarrow R_0\uparrow) (Arrhenius), making the discriminant (\Delta=(V_{\rm oc}-v_p)^2-4R_0P_{\rm tot}) shrink. +* Weak signal (\Rightarrow P_{\rm tot}) increases via (N/\Psi^\kappa). + Together they push the system toward (\Delta\to 0) more quickly than brightness alone, so (\Psi) can become the “hidden killer” in winter. + +### 6.2 OS-Level Recommendation: Thermal-Aware Throttling + +The file proposes a “Thermal-Aware Throttling” strategy: when (T_b) exceeds a threshold, the OS should reduce (\Psi)-sensitive background tasks to prevent an “avalanche effect.” + +A mechanistic control rule can be written as: +[ +\text{If }T_b>T_{\mathrm{thr}},\quad +N(t)\leftarrow \rho_N N(t),\ \ \rho_N\in(0,1), +] +particularly for tasks with high radio scanning or upload bursts. Why this is physically justified: reducing (N) lowers (P_{\rm tot}), which increases (\Delta), which prevents infeasibility and reduces (I), thereby reducing both (I^2R_0) heating and SOC decay rate. + +--- + +## Bottom line + +* 你这份文件最“值钱”的精华是：**(i) 多尺度奇异摄动 + (ii) 判别式奇点解释电压崩塌 + (iii) 均值回复随机负载的 TTE 分布化 + (iv) 低温下信号强度主导的反直觉结论 + OS策略**。这些我已经全部按论文级细节展开，并且**每一点都配了公式与逐项解释**，且完全沿用文件的结构与符号体系。 + +如果你愿意，我下一步可以把这整段进一步“SIAM/IEEE 化”：把口吻改成更强的被动语态与段落衔接，并补上一个**可直接放附录的 RK4 伪代码**（每一步如何嵌套求 (I)、如何检测 (\Delta\le 0) 终止）。 diff --git a/A题/分析/框架/评价.md b/A题/分析/框架/评价.md deleted file mode 100644 index e69de29..0000000