feat(thermal): execute certified backward Euler histories

include/fesa/analysis/transient_heat_procedure.h

@@ -0,0 +1,39 @@
+#ifndef FESA_ANALYSIS_TRANSIENT_HEAT_PROCEDURE_H_
+#define FESA_ANALYSIS_TRANSIENT_HEAT_PROCEDURE_H_
+
+#include <vector>
+
+#include "fesa/analysis/thermal_analysis_state.h"
+#include "fesa/core/status.h"
+
+namespace fesa {
+class LinearSolver;
+class ParallelFor;
+
+/// @brief Owns complete thermal history before any result output is opened.
+struct ThermalExecutionResult {
+  ThermalAnalysisState state;
+  std::vector<Diagnostic> diagnostics;
+};
+
+/// @brief Executes fixed-increment backward Euler with checked thermal
+/// recovery.
+class TransientHeatProcedureExecutor final {
+ public:
+  /// @brief Retains caller-owned adapters, which must outlive Execute calls.
+  TransientHeatProcedureExecutor(const ParallelFor& parallel_for,
+                                 LinearSolver& linear_solver);
+  /// @brief Returns a complete independently owned state or a structured
+  /// failure.
+  /// @note Uses raw initial values; prescribed temperatures start at increment
+  /// one. No output files are opened. A failed trial is never committed to
+  /// history.
+  Result<ThermalExecutionResult> Execute(const Domain& domain) const;
+
+ private:
+  const ParallelFor& parallel_for_;
+  LinearSolver& linear_solver_;
+};
+
+}  // namespace fesa
+#endif  // FESA_ANALYSIS_TRANSIENT_HEAT_PROCEDURE_H_