Add implicit bending stiffness for standard flex.

Standard flex (flex_interp=0) with thin-plate bending treated bending forces purely explicitly. This caused contact-induced vertex vibrations and non-physical energy injection for flat resting sheets, because the solver treated each vertex as an independent mass during contact and contact normals are orthogonal to stretch constraints.

Fix: extend the existing preconditioned CG solver to include the constant bending stiffness K_bend in the implicit operator via matrix-free mat-vec.
PiperOrigin-RevId: 914774020
Change-Id: I45e0d6749abb6f873566203bccae956514b2576b

src/engine/engine_derivative.h

@@ -43,15 +43,14 @@ MJAPI void mjd_passive_vel(const mjModel* m, mjData* d);
 // subtract (d qfrc_bias / d qvel) from qDeriv (dense version)
 MJAPI void mjd_rne_vel_dense(const mjModel* m, mjData* d);
 
-// derivative of flex_interp generalized force w.r.t position: res = (d qfrc_flexinterp / d qpos) * vec
-//  res and vec are vectors of size m->nv
-MJAPI void mjd_flexInterp_mulKD(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec, mjtNum h);
-
-// derivative of flex_interp generalized force w.r.t position (stiffness only, no damping)
-MJAPI void mjd_flexInterp_mulK(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec, mjtNum h);
-
-
+// compute res += (s1 + s2*damping) * J'*K*J * vec, for all interpolated flexes
+MJAPI void mjd_flexInterp_mul(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec,
+                              mjtNum s1, mjtNum s2);
 
+// compute res += scale * K_bend * vec for standard (non-interp) flex bending
+//   scale = s1 + s2 * flex_damping[f]  per flex
+MJAPI void mjd_flexBend_mul(const mjModel* m, mjData* d, mjtNum* res, const mjtNum* vec,
+                            mjtNum s1, mjtNum s2);
 
 
 #ifdef __cplusplus