Sparse direct factorization is suited to declared structural matrices and can be reused when the effective matrix is unchanged.
01 Model and scope
A direct solver factors the assembled matrix and reuses that factorization only while the operator is unchanged. Residual and memory checks distinguish a successful factorization from an accurate, affordable solution.
Route assumptions and calculation checks
- Audit output distinguishes requested, selected, and executed method.
- An enforced memory budget can refuse a direct factorization above its estimated cost.
- Factor reuse is recorded and does not apply across changed stiffness, damping, or time step.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- Linear system.
- Algebraic residual.
- ||r||/scale
- Reported relative residual with a scale floor.
- factor reuse
- Reuse valid only for an identical effective matrix.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Form the linear system that is actually solved
After supports, multipoint constraints, and effective dynamic terms are applied, the direct route receives a specific sparse matrix A and right-hand side b. Its solver eligibility and factorization budget apply to that effective operator, not just the unmodified structural stiffness.
2 · Factor once and solve by triangular substitutions
Sparse direct methods reorder and factor the matrix to reduce fill, then solve the resulting triangular systems. A later right-hand side can reuse the factor only if the effective matrix and its ordering remain unchanged.
3 · Audit residual, memory, and executed backend
Compute the residual in the original system and report its scale, factor memory estimate, selected backend, actual backend, reuse decision, and termination status. A memory guard may reject a solve before factorization; that is an explicit policy result.
4 · Do not generalize factor reuse
Changing stiffness, damping, boundary conditions, load-dependent tangent, or time step changes A and invalidates reuse. A low residual is an algebraic check only; it does not validate model quality or make an over-budget direct solve affordable.
04 Algorithm and implementation
Read the linked versioned source for the actual assembly and solver behavior. The derivation above explains the mathematics and is not a verbatim source excerpt.
05 Verification and limitations
STANDARD SCIPY ROUTE · factor reuse is route-specific
The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.