18.4 Comments on the matrices

  • The stiffness matrix

    • \(K\) is \(N \times N\) where \(N\) is the number of DOF

    • \(K\) is symmetric

    • \(K\) is sparse (has many zeros)

    • \(K\) is singular until appropriate constraints are applied (ie it can float in space, non-unique solution)

    • \(K\) typically has a “band” of non-zero values.

  • Boundary conditions are enforced on the nodes

    • Force (general) boundary conditions are added to the loading vector.

    • Displacement boundary conditions are enforced on the nodal displacement vector.

      • “Elimination method/Matrix Partitioning”

      • “Penalty method”

  • The “cost” of solving the problem scales exponentially with \(N\) (and bandwidth)

    • Reduce \(N\) or reduce bandwidth to reduce cost significantly