feat(newton): first newton solver implementation

libmeanfield/interface/operators/prepared_rotation_displacement_force.cppm

@@ -2,6 +2,7 @@ module;
 
 #include <compare>
 #include <cstdint>
+#include <expected>
 #include <optional>
 #include <vector>
 
@@ -12,6 +13,7 @@ export module mean_field:operators.prepared_rotational_displacement_force;
 export import :fem;
 export import :mapping.domain_mapper;
 export import :operators.context.rotational_displacement_force;
+export import :operators.kernels.rotational_displacement_force;
 export import :physics.rigid_rotation;
 export import :utils.blocks;
 
@@ -61,6 +63,15 @@ export namespace mean_field::operators {
             const physics::RigidRotation &rotation
         );
 
+        [[nodiscard]] std::expected<
+            PreparedRotationalDisplacementForceReport,
+            kernels::RotationalDisplacementForceRejection>
+        TryPrepare(
+            const context::rotational_displacement_force::RotationalDisplacementForceStateView &state,
+            const context::rotational_displacement_force::RotationalDisplacementForceDependencies &dependencies,
+            const physics::RigidRotation &rotation
+        );
+
         void BuildResidual(mfem::Vector &residual) const;
 
         void ApplyDensityJacobianAction(
@@ -104,7 +115,10 @@ export namespace mean_field::operators {
 
     private:
         void VerifyPrepared() const;
-        void PrepareElementData();
+        [[nodiscard]] std::expected<
+            void,
+            kernels::RotationalDisplacementForceRejection>
+        TryPrepareElementData();
         void ApplyPreparedCompleteJacobianActionTrue(
             const mfem::Vector &densityVariationTrue,
             const mfem::Vector &displacementVariationTrue,