Think of a bonded tie as enforcing a constraint of  C*u = 0

MPC (multi-point constraints) 

• Bonded–MPC turns 𝐶 𝑢 = 0 Cu=0 into elimination equations: slave DOFs are expressed as linear combinations of master DOFs and removed from the system.
• This requires the constraint rows in $C$ to be independent.
  If you close a loop with several MPC bonds (e.g., arm↔bracket, bolt threads↔bracket holes, plus other rigid attachments), you create dependent or conflicting constraints. Algebraically, the transformation matrix becomes rank-deficient → singular/near-singular stiffness, tiny pivots, “CE/MPC loop” warnings, and non-convergence

Pure Penalty

• Penalty does not eliminate DOFs. It adds a very stiff spring across the constraint, which yields a linear system.

Causes 2 things:

1. There are no CE (elimination) equations, so constraint loops do not cause rank deficiency. Redundant bonds simply add stiffness “in parallel”.

2. The global matrix stays symmetric positive-definite. So your “MPC over-constraint” disappears when you switch the bonds on that bracket from MPC to Pure Penalty.

Trade-Off: the interface is not mathematically exact; the displacement jump is auto chosen from local $E$, facet size, and contact depth, so the jump is typically microscopic.

You can only use MPC bonded when there is no closed loop of bonds. 

Use Pure Penalty when many bonds on the same body