Static Analysis

In static nonlinear finite element problems we seek a solution (\(\boldsymbol{u}\), \(\lambda\)) to the nonlinear vector function

\[ \boldsymbol{r}(\boldsymbol{u}, \lambda) = \lambda \boldsymbol{p}_f - \boldsymbol{p}_r(\boldsymbol{u}) = \boldsymbol{0} %\label{staticGenForm} \]

The most widely used technique for solving the non-linear finite element equation, equation femGenForm, is to use an incremental scheme. In the incremental formulation, a solution to the equation is sought at successive incremental steps.

\[ \boldsymbol{r}(\boldsymbol{u}_{n}, \lambda_n) = \lambda_n \boldsymbol{p}_f - \boldsymbol{p}_{\sigma}(\boldsymbol{u}_{n}) \]

The solution of this equation is typically obtained using an iterative procedure, in which a sequence of approximations (\(\boldsymbol{u}_{n}^{(i)}\), \(\lambda_n^{(i)}\)), \(i=1,2, ..\) is obtained which converges to the solution (\(\boldsymbol{u}_n\), \(\lambda_n)\). The most frequently used iterative schemes, such as Newton-Raphson, modified Newton, and quasi Newton schemes, are based on a Taylor expansion of equation staticIncForm about (\(\boldsymbol{u}_{n}\), \(\lambda_n\)):

\[ \boldsymbol{r}(\boldsymbol{u}, \lambda)(\boldsymbol{u}_{n},\lambda_n) = \lambda_n^{(i)} \boldsymbol{p}_f - \boldsymbol{p}_{\sigma}\left(\boldsymbol{u}_{n}^{(i)} \right) - \left[ \begin{array}{cc} \boldsymbol{K}_n^{(i)} & -\boldsymbol{p}_f \\ \end{array} \right] \left\{ \begin{array}{c} \boldsymbol{u}_{n} - \boldsymbol{u}_{n}^{(i)} \\ \lambda_n - \lambda_n^{(i)} \end{array} \right\} %\label{staticFormTaylor} \]

which a system of of \(N\) equations with (\(N+1\)) unknowns. Two solve this, an additional equation is required, the constraint equation. The constraint equation used depends on the static integration scheme, of which there are a number, for example load control, arc length, and displacement control.

The following are the aggregates of such an analysis type:

StaticIntegrator - an algorithmic class which provides methods which are invoked by the FE_Element to determine their current tangent and residual matrices; that is this is the class that sets up the system of equations. It also provides the update() method which is invoked to set up the appropriate dof response values once the solution algorithm has formed and solved the system of equations.
- Load Control – Specifies the incremental load factor to be applied to the loads in the domain
- Displacement Control – Specifies the incremental displacement at a specified DOF in the domain
- Minimum Unbalanced Displacement Norm – Specifies the incremental load factor such that the residual displacement norm in minimized
- Arc Length – Specifies the incremental arc-length of the load-displacement path
EquiSolnAlgo - an algorithmic class specifying the sequence of operations to be performed in setting up and solving the finite element equation which can be represented by the equation \[ \boldsymbol{K}_{\mathrm{eff}}(\boldsymbol{u}) \boldsymbol{u} = \boldsymbol{p}(\boldsymbol{u}) \]
ConstraintHandler - a class which creates the DOF_Group and FE_Element objects, the type of objects created depending on how the specified constraints in the domain are to be handled.
DOF_Numberer - a class responsible for providing equation numbers to the individual degrees of freedom in each DOF_Group object.
LinearSOE - a numeric class responsible for the creation and subsequent solution of large systems of linear equations of the form \(Ax = b\), where \(A\) is a matrix and \(x\) and \(b\) are vectors.

	`-det`	`-iterScale`	`-exp`
`MinUnbalDisp`	Fix
`ArcLength`	Add	Add	Add
`LoadControl`	Add
`DispControl`	Add

	`-det`
`Umfpack`
`FullGen`
`BandGen`
`ProfileSPD`
`Band SPD`
`SparseGen`
`SparseSym`
`UmfGen`
`Conjugate Gradient`