Linear constraints connect selected DOFs through a visible reduced-system transformation.
01 Model and scope
Linear multipoint constraints C u = d restrict admissible displacement combinations. Writing u = Tq + u₀ eliminates constrained components before solving; reaction recovery must still account for constraint forces and preserve global balance.
Route assumptions and calculation checks
- Recorded mixed route: connected conforming TET4/WEDGE6/HEX8 with homogeneous zero-offset translational constraints.
- This does not imply modal, Newmark, harmonic, contact, nonlinear, or mixed-distributed MPC support.
- Reaction recovery must preserve original force balance and constraint work.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- Constraint matrix.
- Prescribed constraint offsets.
- Null-space or reduced-coordinate map.
- Particular displacement satisfying the offsets.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Express relations as a linear constraint system
A multipoint constraint couples selected global degrees of freedom through rows of C. The vector d contains prescribed offsets; homogeneous constraints have d=0. Constraint equations must be independent enough to avoid redundant rows or an underdetermined relation set.
2 · Build admissible coordinates
Find one particular solution u₀ with C u₀=d and a basis T for the null space of C. Every admissible displacement is then u=u₀+Tq. This projection approach eliminates constrained combinations before the structural solve.
3 · Project equilibrium and reconstruct the field
Project the full stiffness and load into the admissible subspace, solve for q, and map back to all original DOFs. This route preserves the constraints by construction, subject to the numerical quality of the basis and the declared constraint convention.
4 · Audit forces, work, and route scope
Verify C u−d, free residual, total force/moment balance, and the work done by constraint reactions. Current mixed evidence is limited to connected conforming TET4/WEDGE6/HEX8 in a linear-static translational route; it does not transfer to modal, transient, contact, nonlinear, or distributed MPC cases.
04 Algorithm and implementation
This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.
05 Verification and limitations
BOUNDED · homogeneous translational mixed route
The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.