Truss Optimization

Back to Areas of Research

Topology Design

During the last decade, we devoted a major effort to the numerical treatment of problems in optimal design of trusses (bridges, electricity masts, etc.). The first question, we concentrated on, was the truss topology design problem (in short TTD); here, for an initially chosen layout of nodal points, the designer tries to find the optimal truss structure, connecting the fixed boundary and the external nodes, as a subset of an initially chosen set of connections between the nodal points. The topology of the truss is generated by varying the cross-sectional areas of the truss, allowing for zero cross-sectional areas.

The standard formulation of the TTD problem for the single-load case is a constrained nonconvex problem and thus very difficult to solve numerically, even for small numbers of nodes and bars. Therefore, our effort was devoted to finding of a reformulation of this problem, which is better suited for computation. The basic idea was to eliminate the cross-sectional areas (which are of large dimension) and keep only (the small dimensional) nodal displacement. Using some key results from convex analysis one could write the problem as an unconstrained convex problem in the displacement vector only.

For the new formulation we developed and implemented a series of algorithms which enabled us to solve numerically problems of very high dimensions; cf. Jarre, Kocvara, Zowe (1998) and Kocvara, Zibulevsky, Zowe (1998). These algorithms, based on interior-point and penalty/barrier ideas, enables us to solve problems with up to 102-103 nodes and 104-105 potential bars. This is in sharp contrast to the size of the problems discussed in the standard literature (typically not more than some 10 nodes and 100 bars).

The reformulation can also be used for (single- and multi-load) TTD problems involving unilateral contact; cf. Kocvara, Zibulevsky, Zowe (1998). This allows us to solve many real-world problems.

This substantial increase in the number of design variables adds to the problem a new dimension: Whereas the standard methods usually solve only the so-called sizing problem (where the bars are given and we ask for their optimal volumes, we are now able to solve the so-called ground structure problem (also called structural universe). It can be formulated as follows: For a set of given forces applied at given points in space, it is desired to introduce an optimal supporting pin-jointed framework whose nodes include the points of application of the forces and the given points of support. In general, the optimal layout consists of compression or tension members along curved lines of principal strain. The ground structure approach can be considered as a discrete approximation of this optimal layout. Thus the result of the topology optimization of such a ground structure contains actually an `approximate' answer also to the question of geometry optimization, where we ask also for the optimal positions of the nodes. However, the full answer to the geometry question was given in another approach, developed in Ben-Tal, Kocvara, Zowe (1993) and described the next section.

back to the top

Simultaneous Topology and Geometry Design

Most of the recent work on truss optimization is related either to the topology (volumes of bars) or to the geometry question (position of nodes). Much less effort has been devoted to the simultaneous topology and geometry optimization, because of its numerical complexity. Nevertheless, this simultaneous approach should result in the `ultimately best truss'.

The problem can be formulated by introducing in the classical truss topology problem additional design variable representing the position of the nodes. Obviously, if the original problem is hard to solve, than it is much worse with the new one, in its original form. However, by combining our effective TTD methods with appropriate tools from nonsmooth analysis, we developed a multilevel approach that proved to give by far the best results for single-load problems.

In this approach we decompose the problem into a two-level minimization problem in y (upper level) and (t,u) (lower level). The inner TTD problem can efficiently be solved by the methods mentioned above. The main part remaining is then the minimization of the so-called master function F. The number of the geometry variables y_i in the outer problem will usually be moderate. However, the master function F is generally nonconvex and nonsmooth. Hence we cannot expect to find more than local minima and we have to work with codes from nonsmooth optimization. In particular we used the BT code by Schramm and Zowe, which proved to be very efficient in this context. By this approach (which combines the topology and the geometry aspect) we were able to obtain better results (measured in the value of the objective function) in considerably shorter CPU times than with the ground structure approach mentioned at the end of the previous section. For references, see Kocvara, Zowe (1995a) and Kocvara, Zowe (1996).

We illustrate the difference between the two approaches on the following example. We consider an arch bridge which is fixed at its two end-points in both directions (a). (b) presents the geo/topo solution when the arch nodes are allowed to move in vertical direction. (c) compares this to the ground-structure/topology solution for a mesh of 11x23 potential nodes and 19500 potential bars. If we also allow vertical moves of the two end-points in (a) (i.e., only the position of the `road' is fixed), then we obtain a surprisingly better solution shown in (d).

back to the top

Topology Design with Technological Constraints

To bring our results closer to the needs of practitioners, we have included in the formulation various ``technological'' constraints. We'd like to emphasize, that it is easy to write down the problem with many constraints, but it is far more difficult to solve it numerically. All the reformulations and simplifications used above are heavily dependent on the structure of the problem. By adding just one more constraint, the structure may be destroyed and we have to look for different solution technique.

Displacement Constraints

Here we considered the minimum-compliance formulation of the truss topology problem with additional linear constraints on the displacements: the so-called displacement constraints. We proposed a new bilevel programming approach to this problem. Our primal goal (upper-level) is to satisfy the displacement constraint as well as possible - we minimize the gap between the actual and prescribed displacement. Our second goal (lower-level) is to minimize the compliance - we still want to find the stiffest structure satisfying the displacement constraints. On the lower level we solve a standard truss topology problem and hence we can solve it in the formulation suitable for the fast interior-point algorithms. The overall bilevel problem is solved by means of the so-called implicit programming approach. This approach leads to a nonsmooth optimization problem which is finally solved by a nonsmooth solver. For details, see Kocvara (1997).

Let us demonstrate the abilities of this approach by an example. We consider an 11x3 truss with all nodes connected by potential bars. For the geometry, load and boundary conditions, see Figure (a). Figure (b) shows the optimal topology for the standard TTD problem without displacement constraints. The corresponding compliance is 506.25. Now we add a constraint at the vertical displacement of the lower-middle node number 16. The original displacement (for truss (a)) at this node was u16 = -438.75 and we now want it to be u16 >= 10.0. That means we want the node 16 to move up even if the force above at node 18 pushes down. Figure (c) presents the optimal topology associated with the (locally) optimal auxiliary load g=-0.4359. The compliance for this truss is 718.21 and the constraint is satisfied: indeed the vertical displacement at the particular node is equal to 10. To satisfy this, the algorithm introduced a kind of mechanism to the structure. Finally, we want not only one node but the whole segment 13-19 to move up, i.e., u13 >= 10.0, u19 >= 10.0. The corresponding solution is shown in Figure (d): the trick is the same as in Figure (c) but, obviously, the compliance is worse now (718.21).

back to the top

Global Stability Constraints

Our goal is to formulate a rigorous mathematical model of the optimum truss design problem including constraint on the global stability of the structure. But we not only want to find the model, we also want to be able to solve numerically practical examples, that is, examples of reasonable dimension. Hence our model should be computationally tractable, i.e., solvable by a polynomial-time method of numerical optimization. In order to analyze the behaviour of a (fixed) elastic structure with respect to stability effects, one has to employ the finite elasticity model and work with a nonlinear displacement-strain relation. This leads to a nonlinear equilibrium equation and, in order to find a stable ``physical'' solution, one has to study the solution curve with respect to bifurcation and turning points. Such analysis is rather difficult, even if performed with a single parameter, usually a multiplier of the given load. We would like to control our structure with respect to a high-dimensional design variable - the bar volumes; this makes the analysis even more difficult. To keep the difficulty in the limits, we will work with the so-called linear buckling model. This model is based on an assumption that leads to simplification of the nonlinear truss analysis and that is naturally satisfied for many real-world structures. Working with this model, we can formulate the optimum design problem as a special mathematical programming problem - a semidefinite program. Convex semidefinite programs are tractable problems that can be solved by efficient interior point methods. Our problems is, however, nonconvex. Fortunately, we can use a recent algorithm introduced by Florian Jarre, generalizing primal interior point methods to nonconvex semidefinite programs. Using this algorithm, we are able to solve medium size problems in reasonable time. For details, see Kocvara 1999 and Kocvara et al. 2000.

Example. Consider the very standard example of a laced column under axial loading. Due to symmetry, we only consider one half of the column, as shown in Figure (a). The load applied at the column tip is (-1,0). Table 1 together with Figure summarize the results. Here V denotes the total volume of the truss (our objective), compl. is the compliance and Fcrit the critical force. The table also shows whether the compliance and stability constraints are active or not. Assume first that ti =1000/m, i=1,...,m (where m is the number of bars). The second line of Table 1 shows the corresponding value of compliance (0.177) and the critical force Fcrit. This eigenvalue (0.397) is smaller than one, that means the truss is unstable. The buckling mode (eigenvector) associated with this eigenvalue is shown in Figure (b); it perfectly corresponds to the notoriously known buckling mode of a solid column under axial load. The standard truss optimization without stability constraint gives a design that is twice as light as the previous one but absolutely unstable; it would collapse under any small load applied at the tip (see Figure (c)). Finally, by truss optimization with stability constraint we obtain the design shown in Figure (d). This truss is a bit heavier than the first one, it is, however, stable under the given load. To see fully the effect of the stability constraint, we chose the upper bound for the compliance 0.5, so the compliance constraint was not active.

Table 1
truss design V compl. active Fcrit active
all ti  equal 1000 0.177 n/a 0.397 n/a
optimal t without stab. constr. 509.89 0.177 yes 0 n/a
optimal t with stab. constr. 1179.61 0.239  no 1 yes

back to the top

Robust Design

Our goal is to design a truss which is as stiff as possible under the given loading and still stiff enough under incidental forces - a robust truss. One possible approach to this problem was shown by Ben-Tal and Nemirovski. In this approach one takes into account all possible incidental forces and solves a multi-load truss design problem with the original loads plus the incidental ones. This leads to a Semidefinite Programming problem of large dimension (as many load cases as the number of freedom in the discretized problem).
To make the solution process more efficient, we developed a so-called cascading technique. We first solve the original problem to get the optimal truss for the given set of loads. Then we find the most dangerous loading for this structure and add the new, incidental loads to the original ones. We solve the problem again, with the new set of loads to get an improved design, and again find the most dangerous loading for this new design. This procedure is repeated, until we get a robust design. By definition, the design is robust, if the incidental forces do not increase the compliance by more than five per cent. Typically, we obtain a robust design after 2-3 steps of the procedure and with 5-10 ``most dangerous'' incidental loads. For details, see Kocvara, Zowe, Nemirovski (2000).
back to the top
Michal Kocvara
March 5, 2001