## 2. Numerical algorithm

The static linear problem exposed by the model abstraction layer of B2000++ is:

{ F int u + f int = F ext u + f ext u = Rv + r L u + l = 0

Where F int u + f int is the linearised internal value of the domain (the internal force in a displacement based formulation), F ext u + f ext is the linearised natural boundary condition (the external force in a displacement based formulation), R and r is the linearised model reduction, L and l is the linearised essential boundary condition.

Using the model reduction equation, the problem becomes:

{ R T ( F int F ext ) R v = R T ( f ext f int ) + R T ( F ext F int ) r L R v = L r l

Using the Lagrange Multiplier Adjunction method for imposing the essential boundary conditions, the problem becomes:

[ R T ( F int F ext ) R R T L T L R 0 ] [ v Λ ] = [ R T ( f ext f int ) + R T ( F ext F int ) r L r l ]

This linear system is solved and the solution u is then obtained form the linearised model reduction equation.

The difference of the natural boundary condition with the internal value of the domain (this is the reaction force in a displacement based formulation) is then:

f reac = ( F int u + f int ) ( F ext u + f ext )