完成求解器雅可比矩阵首轮优化，增加更新目录，整理了文档文件夹，增加了服务启动脚本

app/simulation/core/medium.py

@@ -1,7 +1,8 @@
 from __future__ import annotations
 
 from dataclasses import dataclass
-from typing import Protocol
+from math import isfinite
+from typing import Protocol, Sequence
 
 from app.simulation.core.errors import RecoverableTrialStateError
 from app.simulation.performance import profile_property
@@ -16,6 +17,36 @@ class ThermodynamicProperties:
     h: float
 
 
+@dataclass(frozen=True)
+class ThermodynamicPropertyTangents:
+    """Directional derivatives of a recovered thermodynamic state."""
+
+    p: tuple[float, ...]
+    T: tuple[float, ...]
+    rho: tuple[float, ...]
+    u: tuple[float, ...]
+    h: tuple[float, ...]
+
+    @property
+    def width(self) -> int:
+        return len(self.p)
+
+    @classmethod
+    def zeros(cls, width: int) -> "ThermodynamicPropertyTangents":
+        values = (0.0,) * width
+        return cls(p=values, T=values, rho=values, u=values, h=values)
+
+
+@dataclass(frozen=True)
+class ThermodynamicPropertiesLinearization:
+    """Primal properties and a validity-checked directional linearization."""
+
+    properties: ThermodynamicProperties
+    tangents: ThermodynamicPropertyTangents
+    valid: bool = True
+    reason: str | None = None
+
+
 class GasMedium(Protocol):
     """Thermodynamic contract required by pneumatic components.
 
@@ -75,6 +106,18 @@ class GasMedium(Protocol):
         V: float,
     ) -> ThermodynamicProperties: ...
 
+    def linearize_properties_from_mU(
+        self,
+        m: float,
+        U: float,
+        V: float,
+        dm: Sequence[float],
+        dU: Sequence[float],
+        dV: Sequence[float],
+        *,
+        properties: ThermodynamicProperties | None = None,
+    ) -> ThermodynamicPropertiesLinearization: ...
+
 
 @dataclass(frozen=True)
 class IdealGasMedium:
@@ -224,3 +267,100 @@ class IdealGasMedium:
         u = U / m
         h = self.specific_enthalpy(T)
         return ThermodynamicProperties(p=p, T=T, rho=rho, u=u, h=h)
+
+    def linearize_properties_from_mU(
+        self,
+        m: float,
+        U: float,
+        V: float,
+        dm: Sequence[float],
+        dU: Sequence[float],
+        dV: Sequence[float],
+        *,
+        properties: ThermodynamicProperties | None = None,
+    ) -> ThermodynamicPropertiesLinearization:
+        """Linearize properties_from_mU for several seed directions."""
+
+        dm_values = tuple(float(value) for value in dm)
+        dU_values = tuple(float(value) for value in dU)
+        dV_values = tuple(float(value) for value in dV)
+        if not (len(dm_values) == len(dU_values) == len(dV_values)):
+            raise ValueError("Thermodynamic tangent vectors must have equal lengths.")
+        props = properties or self.properties_from_mU(m, U, V)
+        width = len(dm_values)
+        expected_density = m / V
+        expected_internal_energy = U / m
+        if (
+            abs(props.rho - expected_density)
+            > 1.0e-12 * max(abs(expected_density), 1.0)
+            or abs(props.u - expected_internal_energy)
+            > 1.0e-12 * max(abs(expected_internal_energy), 1.0)
+        ):
+            return ThermodynamicPropertiesLinearization(
+                properties=props,
+                tangents=ThermodynamicPropertyTangents.zeros(width),
+                valid=False,
+                reason="properties_primal_mismatch",
+            )
+        if not all(
+            isfinite(value)
+            for values in (dm_values, dU_values, dV_values)
+            for value in values
+        ):
+            return ThermodynamicPropertiesLinearization(
+                properties=props,
+                tangents=ThermodynamicPropertyTangents.zeros(width),
+                valid=False,
+                reason="non_finite_tangent_input",
+            )
+
+        cv = self.cv_at_temperature(props.T)
+        cp = self.cp_at_temperature(props.T)
+        if not isfinite(cv) or not isfinite(cp) or cv <= 0.0 or cp <= 0.0:
+            return ThermodynamicPropertiesLinearization(
+                properties=props,
+                tangents=ThermodynamicPropertyTangents.zeros(width),
+                valid=False,
+                reason="non_positive_heat_capacity",
+            )
+
+        drho: list[float] = []
+        du: list[float] = []
+        dT: list[float] = []
+        dp: list[float] = []
+        dh: list[float] = []
+        for mass_tangent, energy_tangent, volume_tangent in zip(
+            dm_values,
+            dU_values,
+            dV_values,
+            strict=True,
+        ):
+            density_tangent = mass_tangent / V - m * volume_tangent / (V * V)
+            internal_energy_tangent = (
+                energy_tangent / m - U * mass_tangent / (m * m)
+            )
+            temperature_tangent = internal_energy_tangent / cv
+            pressure_tangent = self.R_gas * (
+                props.T * density_tangent + props.rho * temperature_tangent
+            )
+            enthalpy_tangent = cp * temperature_tangent
+            drho.append(density_tangent)
+            du.append(internal_energy_tangent)
+            dT.append(temperature_tangent)
+            dp.append(pressure_tangent)
+            dh.append(enthalpy_tangent)
+
+        tangent_values = (*drho, *du, *dT, *dp, *dh)
+        valid = all(isfinite(value) for value in tangent_values)
+        return ThermodynamicPropertiesLinearization(
+            properties=props,
+            tangents=ThermodynamicPropertyTangents(
+                p=tuple(dp),
+                T=tuple(dT),
+                rho=tuple(drho),
+                u=tuple(du),
+                h=tuple(dh),
+            ),
+            valid=valid,
+            reason=None if valid else "non_finite_property_tangent",
+        )