Wolfgang Achtziger

Equivalent displacement based formulations for maximum strength Truss topology design

Abstract Maximum strength elastic truss structural design is conveniently formulated in terms of displacements and bar volumes. The resulting problem is nonconvex, and for topology design very large, as one seeks the optimal topology as a subset of a large number of potential bars connecting all nodal points of an initially chosen set. In this paper we present a number of equivalent formulations in the displacements only, taking full advantage of the structure of the optimization problem. The equivalent formulations are of min-max type or are quadratic programming problems in the displacements, reducing in some cases even to linear programming problems.

Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications

We consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints. Problems of this kind arise in various applications including optimal topology design problems of mechanical structures. We show that some standard constraint qualifications like LICQ and MFCQ usually do not hold at a local minimum of our program, whereas the Abadie constraint qualification is sometimes satisfied. We also introduce a suitable modification of the standard Abadie constraint qualification as well as a corresponding optimality condition, and show that this modified constraint qualification holds under fairly mild assumptions. We also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints.

Local stability of trusses in the context of topology optimization Part I: Exact modelling

The paper considers the problem of optimal truss topology design with respect to stress and local stability (i.e. buckling) constraints. In a context of topology optimization, the exact. management of buckling constraints is highly complex: member forces must satisfy functions which discontinuously depend on the design variables.New terminologies and an exact problem formulation are provided. It turns out that the classical constraints (equilibrium, stress) together with topological local buckling constraints do not necessarily guarantee the existence of a solution structure. We discuss a simple but typical example demonstrating this effect inherently contained in the problem. It is proved that the inclusion of slenderness constraints guarantees a solution. These additional constraints are motivated by practice and preserve the topology nature of the problem. Finally, an alternative formulation is developed serving as a basis for computational approaches. The numerical treatment is the topic of Part II.

Truss topology optimization including bar properties different for tension and compression

The problem of optimum truss topology design based on the ground structure approach is considered. It is known that any minimum weight truss design (computed subject to equilibrium of forces and stress constraints with the same yield stresses for tension and compression) is—up to a scaling—the same as a minimum compliance truss design (subject to static equilibrium and a weight constraint). This relation is generalized to the case when different properties of the bars for tension and for compression additionally are taken into account. This situation particularly covers the case when a structure is optimized which consists of rigid (heavy) elements for bars under compression, and of (light) elements which are hardly/not able to carry compression (e.g. ropes). Analogously to the case when tension and compression is handled equally, an equivalence is established and proved which relates minimum weight trusses to minimum compliance structures. It is shown how properties different for tension and compression pop up in a modified global stiffness matrix now depending on tension and compression. A numerical example is included which shows optimal truss designs for different scenarios, and which proves (once more) the big influence of bar properties (different for tension and for compression) on the optimal design.

Local stability of trusses in the context of topology optimization Part II: A numerical approach

The paper considers the problem of optimal truss topology design with respect to stress, slenderness, and local buckling constraints. An exact problem formulation is used dealing with the inherent difficulty that the local buckling constraints are discontinuous functions in the bar areas due to the topology aspect. This exact problem formulation has been derived in Part I. In this paper, a numerical approach to this nonconvex and largescale problem is proposed. First, discontinuity of constraints is erased by providing an equivalent formulation in standard form of nonlinear programming. Then a linearization concept is proposed partly preserving the given problem structure. It is proved that the resulting sequential linear programming algorithm is a descent method generating truss designs feasible for the original problem. A numerical test on a nontrivial example shows that the exact treatment of the problem leads to different designs than the usual local buckling constraints neglecting the difficulties induced by the topology aspect.

Two approaches for truss topology optimization: a comparison for practical use

This paper discusses ground structure approaches for topology optimization of trusses. These topology optimization methods select an optimal subset of bars from the set of all possible bars defined on a discrete grid. The objectives used are based either on minimum compliance or on minimum volume. Advantages and disadvantages are discussed and it is shown that constraints exist where the formulations become equivalent. The incorporation of stability constraints (buckling) into topology design is important. The influence of buckling on the optimal layout is demonstrated by a bridge design example. A second example shows the applicability of truss topology optimization to a real engineering stiffened membrane problem.

A smoothing-regularization approach to mathematical programs with vanishing constraints

We consider a numerical approach for the solution of a difficult class of optimization problems called mathematical programs with vanishing constraints. The basic idea is to reformulate the characteristic constraints of the program via a nonsmooth function and to eventually smooth it and regularize the feasible set with the aid of a certain smoothing- and regularization parameter t>0 such that the resulting problem is more tractable and coincides with the original program for t=0. We investigate the convergence behavior of a sequence of stationary points of the smooth and regularized problems by letting t tend to zero. Numerical results illustrating the performance of the approach are given. In particular, a large-scale example from topology optimization of mechanical structures with local stress constraints is investigated.

Bounds on the effect of progressive structural degradation

Abstract Problem formulations are presented for the evaluation of upper and lower bounds on the effect of progressive structural degradation. For the purposes of this study, degradation effect is measured by an increase in global structural compliance (flexibility). Thus the stated bounds are given simply by the maximum and minimum values, respectively, of the increase in compliance corresponding to a specified global interval of degradation. Solutions to these optimization problems identify the particular patterns of local degradation associated with the respective “worst case” and “least degrading” interpretations. Several formulations for extremal “loss of stiffness”, each with one or another form of model for local degradation, are compared and evaluated. An isoperimetric constraint controls the degree of loss in overall structural stiffness. Results obtained sequentially for a set of specified, increasing values for the bound in this constraint track the evolution of local degradation. While the full exposition of the paper is written specifically for trussed structures, analogues for the more useful formulations are described as well for the treatment of continuum systems. Implementation of methods for computational solution are described in detail, and computational results are given for the bound solutions corresponding to evolution from a starting structure through to its fully degraded form.

Design for maximal flexibility as a simple computational model of damage

We consider a simple model of damage, where damage is interpreted as the removal of material from a given truss structure. Occuring damage in a given scenario is modelled by a damage contribution which maximizes compliance. Assuming linear elasticity this leads to an optimization problem formulated in displacements and a set of variables which describe the damaged material of each bar in the truss. A sequence of such problems models damage as a time-dependent process, i.e. damage evolution is considered. A simple ad-hoc-method for the resulting nonconvex problems can be interpreted as a descent algorithm of feasible directions which reaches a local optimum in a finite number of steps. Some numerical examples show the use of the algorithm.

Optimization with Variable Sets of Constraints and anApplication to Truss Design

We discuss the minimization of a continuous function on a subset of Rn subject to a finite set of continuous constraints. At each point, a given set-valued map determines the subset of constraints considered at this point. Such problems arise e.g. in the design of engineering structures.After a brief discussion on the existence of solutions, the numerical treatment of the problem is considered. It is briefly motivated why standard approaches generally fail. A method is proposed approximating the original problem by a standard one depending on a parameter. It is proved that by choosing this parameter large enough, each solution to the approximating problem is a solution to the original one. In many applications, an upper bound for this parameter can be computed, thus yielding the equivalence of the original problem to a standard optimization problem.The proposed method is applied to the problem of optimally designing a loaded truss subject to local buckling conditions. To our knowledge this problem has not been solved before. A numerical example of reasonable size shows the proposed methodology to work well.