Subsections

• Formulation
• Performance

15 Elastic-Plastic Torsion

Determine the stress potential in an infinitely long cylinder when torsion is applied.

Formulation

The elastic-plastic torsion problem [15, pages 41-46] can be formulated in terms of the cross-section ${\cal D}$ of the cylinder, and the torsion angle $c$ per unit length. The stress potential $u$ minimizes the quadratic $q:K \mapsto \mbox{${\mathbb{R}}$}$,

$$q(v)= \int_{{\cal D}} \left\{ \frac{1}{2} \Vert\nabla v(x)\Vert^2 - c \, v(x) \right\} dx ,$$

over the convex set $K$, where

$$K=\{v \in H_0^1 ({\cal D}):\vert v\vert \leq dist(x,\partial {\cal D}), \ \ x \in {{\cal D}}\} ,$$

$dist(x,\partial {{\cal D}})$ is the distance from $x$ to the boundary of ${{ \cal D}}$, and $H_0^1 ({{\cal D}})$ is the space of functions with gradients in $L^2 ( {\cal D} )$ that vanish on the boundary of $\cal {D}$.

A finite element approximation to the elastic-plastic torsion problem is obtained by triangulating ${{\cal D}}$ and minimizing $q$ over the space of piecewise linear functions with values $v_{i,j}$ at the vertices of the triangulation. We follow [15,3] by choosing ${\cal D} = [0,1] \times [0,1]$, and using a triangulation with, respectively, $n_x$ and $n_y$ internal grid points in the coordinate directions. Data for this problem appears in Table 15.1.

Table 15.1: Elastic-plastic torsion problem data

Variables	$n_x n_y$
Constraints	0
Bounds	$n_x n_y$
Linear equality constraints	0
Linear inequality constraints	0
Nonlinear equality constraints	0
Nonlinear inequality constraints	0
Nonzeros in $\nabla ^2 f(x)$	$5n_x n_y -2(n_x+n_y)$
Nonzeros in $c'(x)$	0

Performance

We provide results for the AMPL formulation with $c = 5$ in Table 15.2. For these results we fix $n_x = 50$ and vary $n_y$. The starting guess is the function $dist(x,\partial {{\cal D}})$ evaluated at the grid nodes. Figure 15.1 shows the potential in the torsion problem with $c = 5$. The number of active constraints in this problem increases with $c$. Also

$$\lim _{ c \to \infty} v_c(x) = dist (x,\partial {{\cal D}}),$$

where $v_c$ is the potential as a function of $c$.

Table 15.2: Performance on elastic-plastic torsion problem

Solver	$n_y = 25$	$n_y = 50$	$n_y = 75$	$n_y = 100$
LANCELOT	3.01 s	7.1 s	11.85 s	17.19 s
$f$	-4.17510e-01	-4.18087e-01	-4.18199e-01	-4.18239e-01
$c$ violation	0.00000e+00	0.00000e+00	0.00000e+00	0.00000e+00
iterations	14	18	19	21
LOQO	2.99 s	6.94 s	11.55 s	15.86 s
$f$	-4.17510e-01	-4.18087e-01	-4.18199e-01	-4.18239e-01
$c$ violation	1.9e-15	1.7e-14	3.3e-15	3.7e-15
iterations	19	19	21	21
MINOS	108.31 s	830.16 s	2758.52 s	$\ddagger$
$f$	-4.17510e-01	-4.18087e-01	-4.18199e-01	$\ddagger$
$c$ violation	0.0e+00	0.0e+00	0.0e+00	$\ddagger$
iterations	1	1	1	$\ddagger$
SNOPT	125.58 s	1207.62 s	$\ddagger$	$\ddagger$
$f$	-4.17510e-01	-4.18087e-01	$\ddagger$	$\ddagger$
$c$ violation	0.0e+00	0.0e+00	$\ddagger$	$\ddagger$
iterations	65	110	$\ddagger$	$\ddagger$
$\dagger$ Errors or warnings. $\ddagger$ Timed out.

Figure 15.1: Elastic-plastic torsion problem with $c=5$