Nonlinear Analyzers

1. Nonlinear Analyzers

1.1 Load Control Method

The following notes describe the load control method as described by the Newton-Raphson scheme. The nonlinear equation of equilibrium can be expressed in terms of the out-of-balance force vector, as:

$\mathbf{r} = \mathbf{0}$

or in terms of external and internal force vectors:

$\mathbf{R}^{j} = \mathbf{f}^{j}$

where the superscript j indicates that the external nodal forces are applied incrementally. The external loading vector R is independent from the displacements, while the internal force vector f is nonlinearly dependent from the displacements vector u. Using the Taylor series expansion for eq.1, we get:

$\mathbf{r}(\mathbf{u}_{i}^{j}) = \mathbf{r}(\mathbf{u}_{i-1}^{j}) + \left[\frac{\partial \mathbf{r}}{\partial \mathbf{u}}\right]_{\mathbf{u}_{i-1}^{j}} \delta \mathbf{u}_{i}^{j} + ...$

where:

$\left[\frac{\partial \mathbf{r}}{\partial \mathbf{u}}\right]_{\mathbf{u}_{i-1}^{j}} = \left[\frac{\partial \mathbf{r}(\mathbf{u}_{i-1}^{j})}{\partial \mathbf{u}_{i-1}^{j}}\right]$

and the i-th iterative solution at the j-th load increment is:

$\delta \mathbf{u}_{i}^{j} = \mathbf{u}_{i}^{j} - \mathbf{u}_{i-1}^{j}$

According to the above, we obtain that for the (i-1)th iteration, the out-of-balance force vector becomes:

$\mathbf{r}_{i-1}^{j}=\mathbf{R}^{j}-\mathbf{f}_{i-1}^{j}$

Combining the above, we get:

$\mathbf{R}^{j}-\mathbf{f}_{i-1}^{j} + \left[\frac{\partial(\mathbf{R}^{j}-\mathbf{f}_{i-1}^{j})}{\partial \mathbf{u}}\right]_{\mathbf{u}_{i-1}^{j}} \delta \mathbf{u}_{i}^{j} = \mathbf{0}$

that can also be written as:

$\left[\frac{\partial(\mathbf{R}^{j}-\mathbf{f}_{i-1}^{j})}{\partial \mathbf{u}}\right]_{\mathbf{u}_{i-1}^{j}} \delta \mathbf{u}_{i}^{j}=\mathbf{R}^{j}-\mathbf{f}_{i-1}^{j}$

and takes the final incremental-iterative form:

$\mathbf{K}_{i-1}^{j}\delta \mathbf{u}_{i}^{j}=\mathbf{R}^{j}-\mathbf{f}_{i-1}^{j} \Rightarrow \mathbf{K}_{i-1}^{j}\delta\mathbf{u}_{i}^{j} = \mathbf{r}_{i-1}^{j}$

where:

$\mathbf{K}_{i-1}^{j}=\left[\frac{\partial(\mathbf{R}^{j}-\mathbf{f}_{i-1}^{j})}{\partial \mathbf{u}}\right]_{\mathbf{u}_{i-1}^{j}}$

is the tangent stiffness matrix.

Incremental-iterative implementation

The iterative solution is calculated as:

$\delta\mathbf{u}_{i}^{j} = \left[ \mathbf{K}_{i-1}^{j} \right]^{-1}\mathbf{r}_{i-1}^{j}$

and the total solution up to this point becomes:

$\mathbf{u}_{i}^{j} = \mathbf{u}_{i-1}^{j} + \delta \mathbf{u}_{i}^{j}$

Since the total displacements vector u is known, we can estimate the total internal force f, at the i-th iteration. According to the above, the updated residual force vector at this iteration can be calculated from the relation:

$\mathbf{r}_{i}^{j} = \mathbf{R}^{j} - \mathbf{f}_{i}^{j}$

Convergence has been achieved when the norm of the residual vector is small enough compared to the norm of the actual internal forces. This can be described by the following equation:

$\frac{\|\mathbf{r}_{i}^{j}\|}{\|\mathbf{R}_{i}^{j}\|} < tolerance$

where the tolerance indicates the convergence of our method and in practice it takes values between 10^6^ and 10^3^.

1.2 Displacement Control Method

The following notes describe the displacement control method for multiple prescribed DOFs as referred to BeBorst book.

Equilibrium equation is written as:

$\mathbf{K}_{i-1}^{j}\delta\mathbf{u}_{i}^{j} = \mathbf{r}_{i-1}^{j}$

where:

$\mathbf{r}_{i-1}^{j}=\mathbf{R}_{i}^{j}-\mathbf{f}_{i-1}^{j}$

is the residual force vector. The displacements vector is partitioned according to “free” and “prescribed” DOFs, as:

$\delta\mathbf{u}_{i}^{j}=[\delta\mathbf{u}_{f,i}^{j}, \delta\mathbf{u}_{p,i}^{j}]^{T}$

The partitioned stiffness matrix, external loading vector and internal forces vector can be written as:

$\mathbf{K}_{i-1}^{j} = \begin{bmatrix} \mathbf{K}_{ff,i-1}^{j} & \mathbf{K}_{fp,i-1}^{j} \\ \mathbf{K}_{pf,i-1}^{j} & \mathbf{K}_{pp,i-1}^{j} \\ \end{bmatrix}$

$\mathbf{R}_{i}^{j} = \begin{bmatrix} \mathbf{R}_{f,i}^{j} \\ \mathbf{R}_{p,i}^{j} \end{bmatrix}$

$\mathbf{f}_{i-1}^{j} = \begin{bmatrix} \mathbf{f}_{f,i-1}^{j} \\ \mathbf{f}_{p,i-1}^{j} \end{bmatrix}$

According to the above, the system of equations to be solved becomes:

$\mathbf{K}_{ff,i-1}^{j}\delta\mathbf{u}_{f,i}^{j} + \mathbf{K}_{fp,i-1}^{j}\delta\mathbf{u}_{p,i}^{j} = \mathbf{R}_{f,i}^{j} - \mathbf{f}_{f,i-1}^{j}$

$\mathbf{K}_{pf,i-1}^{j}\delta\mathbf{u}_{f,i}^{j} + \mathbf{K}_{pp,i-1}^{j}\delta\mathbf{u}_{p,i}^{j} = \mathbf{R}_{p,i}^{j} - \mathbf{f}_{p,i-1}^{j}$

The incremental displacements vector of the “free” DOFs can be calculated from the following relatioship as:

$\delta\mathbf{u}_{f,i}^{j}=(\mathbf{K}_{ff,i-1}^{j})^{-1}[\mathbf{R}_{f,i}^{j}-\mathbf{f}_{f,i-1}^{j}-\mathbf{K}_{fp,i-1}^{j}\delta\mathbf{u}_{p,i}^{j}]$

The equivalent external loading of the prescribed DOFs can be now calculated as:

$\mathbf{R}_{p,i}^{j} = \mathbf{K}_{pf,i-1}^{j}\delta\mathbf{u}_{f,i}^{j}+\mathbf{K}_{pp,i-1}^{j}\delta\mathbf{u}_{p,i}^{j}+\mathbf{f}_{p,i-1}^{j}$

Since the iterative displacements vector of the free DOFs is known, we have:

$\mathbf{R}_{p,i}^{j} = [\mathbf{K}_{pf,i-1}(\mathbf{K}_{ff,i-1}^{j})^{-1}\mathbf{K}_{fp,i-1}^{j}+\mathbf{K}_{pp,i-1}^{j}]\delta\mathbf{u}_{p,i}^{j} + \mathbf{K}_{pf,i-1}(\mathbf{K}_{ff,i-1}^{j})^{-1}(\mathbf{R}_{f,i}^{j} - \mathbf{f}_{f,i-1}^{j}) + \mathbf{f}_{p,i-1}^{j}$

The internal forces vector of the current iteration is:

$\delta\mathbf{f}_{i}^{j}=[\delta\mathbf{f}_{f,i}^{j}, \delta\mathbf{f}_{p,i}^{j}]^{T}$

While, the residual becomes:

$\mathbf{r}_{i}^{j}=\mathbf{R}_{i}^{j}-\mathbf{f}_{i}^{j}=\begin{bmatrix} \mathbf{R}_{f,i}^{j} \\ \mathbf{R}_{p,i}^{j} \\ \end{bmatrix} - \begin{bmatrix} \mathbf{f}_{f,i}^{j} \\ \mathbf{f}_{p,i}^{j} \\ \end{bmatrix}$

and he convergence criterion will be the following:

$\frac{\|\mathbf{r}_{i}^{j}\|}{\|\mathbf{R}_{i}^{j}\|} < tolerance$

where the tolerance indicates the convergence of our method and in practice it takes values between 10^6^ and 10^3^.

In the case that there are no external loadings applied and on the structure, but only displacements applied on the prescribed DOFs, we can assume that R_f,i ^j and so the external loading vector of the method becomes:

$\mathbf{R}_{i}^{j} = \begin{bmatrix} \mathbf{0} \\ \mathbf{R}_{p,i}^{j} \\ \end{bmatrix}$

Incremental-iterative implementation

At the first iteration (i=1) of the first increment (j=1) the internal forces of the “free” DOFs are zero and the initial displacements of the prescribed DOFs have a specific value, so the incremental displacements vector of the the “free” DOFs becomes:

$\delta\mathbf{u}_{f,1}=(\mathbf{K}_{ff,0})^{-1}[\mathbf{R}_{f,1}-\mathbf{K}_{fp,0}\delta\mathbf{u}_{p,1}]$

It should be noted that in δu_p,1 the boundary conditions have been taken into account, which means that in some DOFs the value is zero, but the vector’s norm is not zero due to the initial displacements that have been applied to the other prescribed DOFs.

The external forces of the “prescribed” DOFs are calculated according to the relationship:

$\mathbf{R}_{p,1} = \mathbf{K}_{pf,0}\delta\mathbf{u}_{f,1}+\mathbf{K}_{pp,0}\delta\mathbf{u}_{p,1}$

since the initial internal forces of the “prescribed” DOFs are also zero, f_p,0=0.

The external forces of the “prescribed” DOFs are the corresponding forces that occur in the “pre” DOFs as the result of the displacements of the “free” DOFs and the applied displacements of the “pre” DOFs themselves.

For i>=2, the incremental displacements of the “prescribed” DOFs must be equal to zero:

$\delta\mathbf{u}_{p,2} = \mathbf{0}$

so, the displacements vector becomes:

$\delta\mathbf{u}_{2} =\begin{bmatrix} \delta\mathbf{u}_{f,2} \\ \mathbf{0} \\ \end{bmatrix}$

where the incremental displacements of the “free” DOFs are:

$\delta\mathbf{u}_{f,2} = \mathbf{K}_{ff,1}^{-1}[-\mathbf{f}_{f,1}]$

and the external loading vector of the “prescribed” DOFs is calculated from the following equation:

$\mathbf{R}_{p,2} = \mathbf{K}_{pf,1}\delta\mathbf{u}_{f,2}+\mathbf{f}_{p,1}$

References

[1] Finite Element Procedures, K-J. Bathe, Prentice Hall, 2nd edition, 2014.

[2] Non-linear Finite Element Analysis of Solids and Structures, R. de Borst, M.A. Crisfield, J.J.C. Remmers, C.V. Verhoosel, Wiley, 2nd Edition, 2012.

NumericalAnalyzers

Library that handles static, dynamic and non-linear procedures.

1. Nonlinear Analyzers

1.1 Load Control Method

1.2 Displacement Control Method

References