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.     

A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints

The present paper investigates problems of truss topology optimization under local buckling constraints. A new approach for the solution of singular problems caused by stress and local buckling constraints is proposed. At first, a second order smooth-extended technique is used to make the disjoint feasible domains connect, then the so-called ε-relaxed method is applied to eliminate the singular optima from problem formulation. By means of this approach, the singular optimum of the original problem caused by stress and local buckling constraints can be searched approximately by employing the algorithms developed for sizing optimization problems with high accuracy. Therefore, the numerical problem resulting from stress and local buckling constraints can be solved in an elegant way. The applications of the proposed approach and its effectiveness are illustrated with several numerical examples. Received May 2, 2000     

Some aspects of truss topology optimization

The present paper studies some aspects of formulations of truss topology optimization problems. The ground structure approach-based formulations of three types of truss topology optimization problems, namely the problems of minimum weight design for a given compliance, of minimum weight design with stress constraints and of minimum weight design with stress constraints and local buckling constraints are examined. The common difficulties with the formulations of the three problems are discussed. Since the continuity of the constraint or/and objective function is an important factor for the determination of the mathematical structure of optimization problems, the issue of the continuity of stress, displacement and compliance functions in terms of the cross-sectional areas at zero area is studied. It is shown that the bar stress function has discontinuity at zero crosssectional area, and the structural displacement and compliance are continuous functions of the cross-sectional area. Based on the discontinuity of the stress function we point out the features of the feasible domain and global optimum for optimization problems with stress and/or local buckling constraints, and conclude that they are mathematical programming with discontinuous constraint functions and that they are essentially discrete optimization problems. The difference between topology optimization with global constraints such as structural compliance and that with local constraints on stress or/and local buckling is notable and has important consequences for the solution approach.     

Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming

The advance in digital fabrication technologies and additive manufacturing allows for the fabrication of complex truss structure designs but at the same time posing challenging structural optimization problems to capitalize on this new design freedom. In response to this, an iterative approach in which Sequential Linear Programming (SLP) is used to simultaneously solve a size and shape optimization sub-problem subject to local stress and Euler buckling constraints is proposed in this work. To accomplish this, a first order Taylor expansion for the nodal movement and the buckling constraint is derived to conform to the SLP problem formulation. At each iteration a post-processing step is initiated to map a design vector to the exact buckling constraint boundary in order to facilitate the overall efficiency. The method is verified against an exact non-linear optimization problem formulation on a range of benchmark examples obtained from the literature. The results show that the proposed method produces optimized designs that are either close or identical to the solutions obtained by the non-linear problem formulation while significantly decreasing the computational time. This enables more efficient size and shape optimization of truss structures considering practical engineering constraints.     

Topology optimization of trusses with stress and local constraints on nodal stability and member intersection

A truss topology optimization problem under stress constraints is formulated as a Mixed Integer Programming (MIP) problem with variables indicating the existence of nodes and members. The local constraints on nodal stability and intersection of members are considered, and a moderately large lower bound is given for the cross-sectional area of an existing member. A lower-bound objective value is found by neglecting the compatibility conditions, where linear programming problems are successively solved based on a branch-and-bound method. An upper-bound solution is obtained as a solution of a Nonlinear Programming (NLP) problem for the topology satisfying the local constraints. It is shown in the examples that upper- and lower-bound solutions with a small gap in the objective value can be found by the branch-and-bound method, and the computational cost can be reduced by using the local constraints.     

Optimum design of truss topology under buckling constraints

The present paper studies the optimum design of truss topology under buckling constraints based on a new formulation of the problem. Through the incorporation of a global system stability constraint into the problem formulation, isolated compressive bars are eliminated from the final optimal topology. Furthermore, by including overlapping bars in the initial ground structure, the difficulty caused by hinge cancellation as pointed out by Rozvany (1996) can be overcome. Also, the importance of inclusion of compatibility conditions in the problem formulation is demonstrated. Finally, several numerical examples are presented for demonstration of the effectiveness of the proposed approach.     

Strength-based topology optimisation of plastic isotropic von Mises materials

Conventionally, topology optimisation is formulated as a non-linear optimisation problem, where the material is distributed in a manner which maximises the stiffness of the structure. Due to the nature of non-linear, non-convex optimisation problems, a multitude of local optima will exist and the solution will depend on the starting point. Moreover, while stress is an essential consideration in topology optimisation, accounting for the stress locally requires a large number of constraints to be considered in the optimisation problem; therefore, global methods are often deployed to alleviate this with less control of the stress field as a consequence. In the present work, a strength-based formulation with stress-based elements is introduced for plastic isotropic von Mises materials. The formulation results in a convex optimisation problem which ensures that any local optimum is the global optimum, and the problems can be solved efficiently using interior point methods. Four plane stress elements are introduced and several examples illustrate the strength of the convex stress-based formulation including mesh independence, rapid convergence and near-linear time complexity.

     

Shape equilibrium constraint: a strategy for stress-constrained structural topology optimization

In topology optimization of a continuum, it is important to consider stress-related objective or constraints, from both theoretical and application perspectives. It is known that the problem is challenging. Although remarkable achievements have been made with the SIMP (Solid Isotropic Material with Penalization) framework, a number of critical issues are yet to be fully resolved. In the paper, we present an approach of a shape equilibrium constraint strategy with the level-set/X-FEM framework. We formulate the topology optimization problem under (spatially-distributed) stress constraints into a shape equilibrium problem of active stress constraint. This formulation allows us to effectively handle the stress constraint, and the intrinsic non-differentiability introduced by local stress constraints is removed. The optimization problem is made into one of continuous shape-sensitivity and it is solved by evolving a coherent interface of the shape equilibrium concurrently with shape variation in the structural boundary during a level-set evolution process. Several numerical examples in two dimensions are provided as a benchmark test of the proposed shape equilibrium constraint strategy for minimum-weight and fully-stressed designs and for designs with stress constraint satisfaction.     

Optimum design of steel frames with stability constraints

Optimum design algorithms based on the optimality criteria approach are proven to be efficient and general. They have the flexibility of accomodating variety of design constraints such as displacement, stress, stability and frequency in the design problem. The design methods developed recently, although considering one or more of these constraints, lack the necessity of referring to any relevant design code. The algorithm presented for the optimum design of street frames implements the displacement and combined stress limitations according to AISC. The recursive relationship for design variables in the case of dominant displacement constraints is obtained by the optimality criteria approach. The combined stress inequalities which include in-plane and lateral buckling of members are reduced into nonlinear equations of design variables. The solution of these equations gives the values of bounds for the variables in the case where the stress constraints are dominant in the design problem. The use of effective length in the combined stress constraints makes it possible to study the effect of the end rigidities on the final designs. The design procedure is simple and easy to program which makes it particularly suitable for microcomputers. A number of design examples are considered to demonstrate the practical applicability of the method. It is also shown that the design procedure can be employed in selecting the optimum topology of steel frames.     

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.     

Including global stability in truss layout optimization for the conceptual design of large-scale applications

Including stability in truss topology optimization is critical to avoid unstable optimized designs in practical applications. While prior research addresses this challenge by implementing local buckling and linear prebuckling, numerical difficulties remain due to the global stability singularity phenomenon. Therefore, the goal of this paper is to develop an optimization formulation for truss topology optimization including global stability without numerical singularities, within the framework of the preliminary design of large-scale structures. This task is performed by considering an appropriate simultaneous analysis and design formulation, in which the use of a disaggregated form for the equilibrium equations alleviates the singularities inherent to global stability. By implementing a local buckling criterion for hollow truss elements, the resulting formulation is well-suited for the preliminary design of large-scale trusses in civil engineering applications. Three applications illustrate the efficiency of the proposed approach, including a benchmark truss structure and the preliminary design of a footbridge and a dome. The results demonstrate that including local buckling and global stability can considerably affect the optimized design, while offering a systematic means of avoiding unstable solutions. It is also shown that the proposed approach is in a good agreement with linear prebuckling assumptions.     

Stress isolation through topology optimization

This paper introduces a problem of stress isolation in structural design and presents an approach to the problem through topology optimization. We model the stress isolation problem as a topology optimization problem with multiple stress constraints in different regions. The shape equilibrium constraint approach is employed to effectively control the local stress constraints. The level set based structural optimization is implemented with the extended finite element method (X-FEM) for providing an adequately accurate stress analysis. Numerical examples of stress isolation design in two dimensions are investigated as a benchmark test of the proposed method. The results, from the force transmittance point of view, suggest that the guard “grooves” obtained can change the force path to successfully realize the stress isolation in the structure.     

Constrained optimization of structures with displacement constraints under various loading conditions

Optimization problems often involve constraints and restrictions which must be considered in order to obtain an optimum result and the resultant solution should not deviate from any of the imposed constraints. These constraints and restrictions are imposed either on the design variables or on the algebraic relations between them. Constraints of allowable stress, minimum size and buckling of members in the absence of allowable displacement constraint are the most important factors in optimization of the cross-sectional area of structural elements. When the allowable displacement constraint is included in the problem as a determinant parameter, since the specifications of most of elements affect the displacement rate, the way of imposing and considering this constraint requires special care. In this research the way of simultaneous imposition of multi displacement constraints for optimum design of truss structures in several load cases is described. In this method various constraints for different load cases are divided into active and passive constraints. The mathematical formulation is based on the classical method of Lagrange Multipliers. Overall, this simple method can be employed along with other constraints such as buckling, allowable stress and minimum size of members for imposing the displacement constraint in various load cases.     

Finite prism method based topology optimization of beam cross section for buckling load maximization

The use of the finite element method (FEM) for buckling topology optimization of a beam cross section requires large numerical cost due to the discretization in the length direction of the beam. This investigation employs the finite prism method (FPM) as a tool for linear buckling analysis, reducing degrees of freedom of three-dimensional nodes of FEM to those of two-dimensional nodes with the help of harmonic basis functions in the length direction. The optimization problem is defined as the maximization problem of the lowest eigenvalue, for which a bound variable is introduced and set as the design objective to treat mode switching phenomena of multiple eigenvalues. The use of the bound formulation also helps the proposed optimization to treat beams having local plate buckling modes as the fundamental modes as well as beams having global buckling modes. The axial stress is calculated according to the distribution of material modulus which is interpolated using the SIMP approach. Optimization problems finding cross-section layouts from rectangular, L-shaped and generally-shaped design domains are solved for various beam lengths to ascertain the effectiveness of the proposed method.     

Truss topology optimization considering local buckling constraints and restrictions on intersection and overlap of bar members

This paper illustrates the application of a two-level approximation method for truss topology optimization with local member buckling constraints and restrictions on member intersections and overlaps. Previously developed for truss topology optimization with stress and displacement constraints, that method is achieved by starting from an initial ground structure, and, combined with genetic algorithm (GA), it can handle both discrete and continuous variables, which denote the existence and cross-sectional areas of bar members respectively in the ground structure. In this work, this method is improved and extended to consider member buckling constraints and restrict intersection and overlap of members for truss topology optimization. The temporary deletion technique is adopted to temporarily remove buckling constraints when related bar members are deleted, and in order to avoid unstable designs, the validity check for truss topology configuration is conducted. By using GA to search in each possible design subset, the singularity encountered in buckling-constrained problems is remedied, and meanwhile, as the required structural analysis is replaced with explicit approximation functions in the process of executing GA, the computational cost is significantly saved. Moreover, for the consideration of restrictions on member intersecting and overlapping, the definition of such phenomena and mathematical expressions to recognize them are presented, and a new fitness function is developed to include such considerations. Numerical examples are presented to show the efficacy of the proposed techniques.     

On compliance and buckling objective functions in topology optimization of snap-through problems

This paper deals with topology optimization of static geometrically nonlinear structures experiencing snap-through behaviour. Different compliance and buckling criterion functions are studied and applied for topology optimization of a point loaded curved beam problem with the aim of maximizing the snap-through buckling load. The response of the optimized structures obtained using the considered objective functions are evaluated and compared. Due to the intrinsic nonlinear nature of the problem, the load level at which the objective function is evaluated has a tremendous effect on the resulting optimized design. A well-known issue in buckling topology optimization is artificial buckling modes in low density regions. The typical remedy applied for linear buckling does not have a natural extension to nonlinear problems, and we propose an alternative approach. Some possible negative implications of using symmetry to reduce the model size are highlighted and it is demonstrated how an initial symmetric buckling response may change to an asymmetric buckling response during the optimization process. This problem may partly be avoided by not exploiting symmetry, however special requirements are needed of the analysis method and optimization formulation. We apply a nonlinear path tracing algorithm capable of detecting different types of stability points and an optimization formulation that handles possible mode switching. This is an extension into the topology optimization realm of a method developed, and used for, fiber angle optimization in laminated composite structures. We finally discuss and pinpoint some of the issues related to buckling topology optimization that remains unsolved and demands further research.     

Stress-based topology optimization

Previous research on topology optimization focussed primarily on global structural behaviour such as stiffness and frequencies. However, to obtain a true optimum design of a vehicle structure, stresses must be considered. The major difficulties in stress based topology optimization problems are two-fold. First, a large number of constraints must be considered, since unlike stiffness, stress is a local quantity. This problem increases the computational complexity of both the optimization and sensitivity analysis associated with the conventional topology optimization problem. The other difficulty is that since stress is highly nonlinear with respect to design variables, the move limit is essential for convergence in the optimization process. In this research, global stress functions are used to approximate local stresses. The density method is employed for solving the topology optimization problems. Three numerical examples are used for this investigation. The results show that a minimum stress design can be achieved and that a maximum stiffness design is not necessarily equivalent to a minimum stress design.     

A lower-bound formulation for the geometry and topology optimization of truss structures under multiple loading

In this contribution, we propose an effective formulation to address the stress-based minimum volume problem of truss structures. Starting from the lower-bound formulation in topology optimization, the problem is further expanded to geometry optimization and multiple loading scenarios, and systematically reformulated to alleviate numerical difficulties related to the melting node effect and stress singularities. The subsequent simultaneous analysis and design (SAND) formulation is well suited for a direct treatment by introducing a barrier function. Using exact second derivatives, this difficult class of problem is solved by sequential quadratic programming with trust regions. These building blocks result into an integrated design process. Two examples–including a large-scale application–illustrate the robustness of the proposed formulation.     

Topology optimization of continuum structures with local and global stress constraints

Topology structural optimization problems have been usually stated in terms of a maximum stiffness (minimum compliance) approach. The objective of this type of approach is to distribute a given amount of material in a certain domain, so that the stiffness of the resulting structure is maximized (that is, the compliance, or energy of deformation, is minimized) for a given load case. Thus, the material mass is restricted to a predefined percentage of the maximum possible mass, while no stress or displacement constraints are taken into account. This paper presents a different strategy to deal with topology optimization: a minimum weight with stress constraints Finite Element formulation for the topology optimization of continuum structures. We propose two different approaches in order to take into account stress constraints in the optimization formulation. The local approach of the stress constraints imposes stress constraints at predefined points of the domain (i.e. at the central point of each element). On the contrary, the global approach only imposes one global constraint that gathers the effect of all the local constraints by means of a certain so-called aggregation function. Finally, some application examples are solved with both formulations in order to compare the obtained solutions.     

Optimum design of stiffened cylindrical shells with natural frequency constraints

The design optimization of axially loaded, simply supported stiffened cylindrical shells for minimum mass is considered. The design variables are thickness of shell wall, thicknesses and depths of rings and stringers, number/spacing of rings and stringers. Natural frequency, local and overall buckling strengths and direct stress constraints are considered in the design problems. Three different combinations of stiffeners are considered. In each case, the independent effects of behaviour constraints are also studied. The optimum designs are achieved with one of the standard nonlinear constrained optimization techniques (Davidon-Fletcher-Powell method with interior penalty function formulation) and few optimal solutions are checked for the satisfaction of Kuhn-Tucker conditions.