Topology optimization of trusses by growing ground structure method

A new method called the growing ground structure method is proposed for truss topology optimization, which effectively expands or reduces the ground structure by iteratively adding or removing bars and nodes. The method uses five growth strategies, which are based on mechanical properties, to determine the bars and nodes to be added or removed. Hence, the method can optimize the initial ground structures such that the modified, or grown, ground structures can generate the optimal solution for the given set of nodes. The structural data of trusses are manipulated using C++ standard template library and the Boost Graph Library, which help alleviate the programming efforts for implementing the method. Three kinds of topology optimization problems are considered. The first problem is a compliance minimization problem with cross-sectional areas as variables. The second problem is a minimum compliance problem with the nodal coordinates also as variables. The third problem is a minimum volume problem with stress constraints under multiple load cases. Six numerical examples corresponding to these three problems are solved to demonstrate the performance of the proposed method.     

Geometry and topology optimization of plane frames for compliance minimization using force density method for geometry model

A new method is proposed for simultaneous optimization of shape, topology and cross section of plane frames. Compliance against specified loads is minimized under constraint on structural volume. Difficulties caused by the melting nodes can be alleviated to some extent by introducing force density as design variables for defining the geometry, where the side constraints are assigned for force density to indirectly avoid the existence of extremely short members. Force density method is applied to an auxiliary cable-net model with different boundary and loading conditions so that the regularity of force density matrix is ensured by positive force densities. Sensitivity coefficients of the objective and constraint functions with respect to the design variables are also explicitly calculated. After the optimal geometry of the frame is obtained, the topology is further improved by removing the thin members and combining closely spaced nodes. It is demonstrated in the numerical examples of three types of frames that rational geometry and topology can be achieved using the proposed method, and the effect of bending moment on the optimal solution is also discussed.

     

Discrete optimal design of elasto-plastic trusses using compliance and stability constraints

In this paper two discrete optimization methods are presented for the minimum volume design of elasto-plastic trusses with given geometry. The design is based on given sets of discrete cross-sectional sizes. Both methods enable the use of the plastic reserve of the truss; the plastic deformations, however, are controlled by compliance constraints on plastic deformations. In the second solution method, the shakedown of the truss is also taken into consideration. The stability of the bars is also controlled by using permissible stresses for compression. The methods are based on continuous optimal elasto-plastic design methods and on the discrete optimization method of elastic trusses using segmental approach. By the iterative application of these methods, solution procedures that use standard linear programming have been developed.Dedicated to Prof. F. Ziegler, for his 60-th birthday; extended version of a paper presented at the Second World congress of Structural and Multidisciplinary Optimization (WCSMO-2), in Zakopane, Poland, May 1997     

The exact weight of discretized Michell trusses for a central point load

A discretized optimal structure is derived in a closed analytical form based on Michell truss. The result shows that the discretized optimal structure is most similar to Michell truss in topology and shape. The difference in volume, displacement and strain energy between the discretized optimal structure and Michell truss decreased sharply as the number of members increased in discretized structure. A discretized optimal structure may be obtained from Michell truss by using finite members. This work is meaningful for studying discretized optimal topology based on Michell truss. This result is useful for engineering structural design.     

Finite topology variations in optimal design of structures

The method of optimal design of structures by finite topology modification is presented in the paper. This approach is similar to growth models of biological structures, but in the present case, topology modification is described by the finite variation of a topological parameter. The conditions for introducing topology modification and the method for determining finite values of topological parameters characterizing the modified structure are specified. The present approach is applied to the optimal design of truss, beam, and frame structures. For trusses, the heuristic algorithm of bar exchange is proposed for minimizing the global compliance subject to a material volume constraint and it is extended to volume minimization with stress and buckling constraints. The optimal design problem for beam and frame structures with elastic or rigid supports, aimed at minimizing the structure cost for a specified global compliance, is also considered.     

Analytical benchmarks for topological optimization IV: square-shaped line support

Michell’s problem of optimizing truss topology for stress or compliance constraints under a single load condition is solved analytically for plane trusses having a square-shaped line support. Geometrical characteristics of the Hencky nets giving the truss layout are expressed in terms of Lommel functions. Analytically derived truss volumes for the above problem are compared with those of trusses supported along circles of equivalent area. Some general implications of the results are also discussed.     

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.     

Geometry and sizing optimisation of discrete structure using the genetic programming method

This paper presents a structural optimisation method using the genetic programming (GP) technique. This method applied linear GP to derive optimum geometry and sizing of discrete structure from an arbitrary initial design space. The linear GP was used to find out the optimum nodal locations and member sizing of the structure through a linear sequence of programming instructions. The nodal locations and member cross-sectional areas of the structure were used as the design variable for these instructions, with the optimal geometry and sizing obtained by evolving a population of GP individuals satisfying the optimisation design objective. The approach was applied to the benchmark example of ten-bar planar truss for verification. Other truss examples, including 18-bar planar truss and 25-bar space truss, were also used to demonstrate the effectiveness of this method. The optimum results obtained demonstrate the practicability and generality of using the proposed method in geometry and sizing optimisation problems.     

A displacement based optimization method for geometrically nonlinear frame structures

An extension of the displacement based optimization method to frames with geometrically nonlinear response is presented. This method, when applied to small-scale trusses with linear and nonlinear response, appeared to be efficient providing the same solutions as the classical optimization method. The efficiency of the method is due to the elimination of numerous finite element analyses that are required in using the traditional optimization approach. However, as opposed to trusses, frame problems have typically a larger number of degrees of freedom than cross sectional area design variables. This leads to difficulties in the implementation of the method compared to the truss implementation. A scheme that relaxes the nodal equilibrium equations is introduced, and the method is validated using test examples. The optimal designs obtained by using the displacement based optimization and the classical approaches are compared to validate the application to frame structures. The characteristics and limitations of the optimization in the displacement space for sizing problems, based on the current formulation, are discussed.     

Simultaneous material selection and geometry design of statically determinate trusses using continuous optimization

In this work, we explore simultaneous geometry design and material selection for statically determinate trusses by posing it as a continuous optimization problem. The underlying principles of our approach are structural optimization and Ashby’s procedure for material selection from a database. For simplicity and ease of initial implementation, only static loads are considered in this work with the intent of maximum stiffness, minimum weight/cost, and safety against failure. Safety of tensile and compression members in the truss is treated differently to prevent yield and buckling failures, respectively. Geometry variables such as lengths and orientations of members are taken to be the design variables in an assumed layout. Areas of cross-section of the members are determined to satisfy the failure constraints in each member. Along the lines of Ashby’s material indices, a new design index is derived for trusses. The design index helps in choosing the most suitable material for any geometry of the truss. Using the design index, both the design space and the material database are searched simultaneously using gradient-based optimization algorithms. The important feature of our approach is that the formulated optimization problem is continuous, although the material selection from a database is an inherently discrete problem. A few illustrative examples are included. It is observed that the method is capable of determining the optimal topology in addition to optimal geometry when the assumed layout contains more links than are necessary for optimality.     

An improved CSS for damage detection of truss structures using changes in natural frequencies and mode shapes

Non-destructive structural damage identification can be carried out using the difference between a structure’s characteristics before and after a catastrophic event. An approach is to formulate the problem as an inverse optimization problem, in which the amounts of damage to each element are considered as the optimization variables. The objective is to set these variables such that the characteristics of the model correspond to the experimentally measured characteristics of the actual damaged structure. Since the structures are usually symmetric, this is an optimization problem with several global optimal solutions each representing a probable state of damage, where unlike many other optimization problems, it is not enough to merely find one of these optimal solutions; it is important to capture all such possible states and to compare them. In this paper, structural damage detection of planar and spatial trusses using the changes in structures’ natural frequencies and mode shapes is addressed. An improved Charged System Search algorithm is developed and utilized to tackle the problem of finding as many global optimal solutions as possible in a single run. A 10-bar planar truss and a 72-bar spatial truss are considered as numerical examples. Experimental results show that it is important to incorporate mode shapes in order to determine the actual damage scenario among other possibilities.     

A ground-structure-based representation with an element-removal algorithm for truss topology optimization

In this study, a new ground-structure-based representation for truss topology optimization is proposed. The proposed representation employs an algorithm that removes unwanted elements from trusses to obtain the final trusses. These unwanted elements include kinematically unstable elements and useless zero-force elements. Since the element-removal algorithm is used in the translation of representation codes into corresponding trusses, this results in more representation codes in the search space that are mapped into kinematically stable and efficient trusses. Since more representation codes in the search space represent stable and efficient trusses, the strategy increases meaningful competition among representation codes. This remapping strategy alleviates the problem of having large search spaces using ground structures, and encourages faster convergences. To test the effectiveness of the proposed representation, it is used with a simple multi-population particle swarm optimization algorithm to solve several truss topology optimization problems. It is found that the proposed representation can significantly improve the performance of the optimization process.     

An efficient method for optimal structural design by substructuring

An optimal structural design technique incorporating the concept of substructuring in its formulation is presented. The method is developed using functional analysis techniques and the state space formulation of the optimal design problem. Design sensitivity analysis for the problem is discussed and an integrated computational algorithm for optimal design is presented. As an application of the method, optimal design of general trusses is presented. Numerical results for two standard truss structures of 25 and 200 members are obtained, using substructuring, and are compared with results obtained without substructuring. It is shown that the algorithm with substructuring is up to 66% more efficient.     

Why multi-load topology designs based on orthogonal microstructures are in general non-optimal

It is well established that for a compliance constraint, the optimal topology of perforated plates under plane stress tends to that for least-weight trusses (Michell structures) as the volume fraction (i.e. the ratio material volume/available volume) approaches zero. It is shown in this note that for two loading conditions the optimal bar orientations for Michell structures are in general non-orthogonal and hence the assumption of orthogonal microstructures in multi-load plate topology optimization must lead to erroneous results.     

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.     

An iterative solution for geometrically nonlinear trusses

Successive applications of the stiffness method were used to analyze geometrically nonlinear space trusses. Each cycle consists of the computation of corrective displacements using a revised stiffness matrix subjected to a residule load vector. The stiffness matrix is based upon the displaced geometry of the structure. The residule load vector is obtained by computing the differences between the applied loads and the components of bar forces at each joint. These force components are based upon the displaced joint locations.A computer program based upon this method was developed and applied to several small plane trusses and a large transmission tower space truss. The method was found to converge rapidly in the cases investigated. Comparisons with the piece-wise linear method are presented, indicating some important advantages to the iterative approach.     

A well-posed non-selfadjoint layout problem: Least-weight plane truss for one load condition and two displacement constraints

Exact optimal plane truss layouts are derived for a vertical support and a concentrated load with two displacement constraints. The latter are imposed at the point of application of the load, in the direction of the load and in another direction. It is shown that for the above class of problems the optimal solution always consists of two symmetrically positioned bars. These solutions are derived analytically by two independent methods: (i) in the first one a two-bar topology is assumed and then the orientations and cross-sectional areas of the bars are optimized; (ii) in the second one, the same optimal solutions are derived from general optimality criteria, which show that the optimum is valid even when we consider all possible topologies. The paper demonstrates the power and versatility of continuum-type optimality criteria and also shows that for two displacement constraints at a loaded point the problem is non-selfadjoint but always well-posed, having a stationary optimum with a finite structural weight. The exact layout solutions given in this paper can be used as test examples for numerical methods in topology optimization.On leave from the Institute of Structural Mechanics, Warsaw University of Technology     

Effect of horizontal diaphragm flexibility on the P-delta analysis

This paper investigates the importance of including the effects of the flexibility of the horizontal diaphragms when using the P-delta method of analysis, especially when considering the loads applied to intermediate frames on trusses that are not part of the lateral force resisting system. Analyses were conducted for structural systems with a variable number of stories, number of bays and diaphragm stiffnesses and supported by rigid jointed plane frames or vertical trusses.     

An efficient approximate analysis method based on an exact univariate model for the element loads

Approximate analysis modules for structural design are usually based on a linear Taylor expansion of the nodal displacements in terms of the reciprocals of the design variables. Direct approximations of the member forces have received lesser attention. This paper describes an approach for the direct calculation of the member forces in a truss as a function of the design variables. It is based on the exact expression of the member forces if only one design variable is allowed to vary at a time. In the case of an arbitrary move in the design space the method gives approximate results of a very good quality. This is obtained by enforcing zero order homogeneity of the element loads and by refining the results via a virtual work equation. The method is illustrated with numerical results on previously published test cases for elastic trusses. Preliminary results for an elastic frame are also presented. This new approximate force model is shown to yield excellent results.