Multimaterial design of physically nonlinear structures

A theoretical consideration of optimization problems for physically nonlinear hyperelastic structures is carried out. The structures are subjected to a single static “dead” loading and a multimaterial design approach is analysed. Structural materials are assumed to be isotropic, with stress-strain relations being weakly concave. The problems considered are mass minimization with prescribed structural stiffness, stiffness maximization with prescribed structural mass, and mass minimization with constrained stresses. Optimality conditions for the problems are analysed. Generalizations of Maxwell’s and Michell’s theorems for the considered structures are proved. Some regularities inherent in the third problem are analysed using the analytical example of a three-rod physically nonlinear truss made of two materials. An algorithm for compliance decreasing in the case of prescribed structural mass is proposed. The monotonicity property of the algorithm is proved. Numerical examples (bi-material beam and airframe) are presented and corresponding results are analysed on a basis of the theoretical approaches developed. Received November 3, 1999     

On optimal geometrically non-linear trusses

The paper is devoted to the investigation of regularities inherent to optimal geometrically non-linear trusses. The single static loading case is considered, a single structural material is used (except specially indicated cases) and buckling effects are neglected. The so-called small strains and large rotations case is investigated. Some regularities inherent to the kinematic and static variational principles for geometrically non-linear trusses are considered. Then the strain compatibility conditions resulting from the static variational principle are obtained and explored. It is shown that 1) the conditions are linear with respect to subcomponents of rod Green strains such as rotations and geometrically linear strains, 2) strains (in particular, rotations and geometrically linear strains) within rods which are not members of the so-called basic structure are fully determined by geometrically linear strains in rods of the basic structure.Extensions of some theorems (Maxwells theorem, Michells theorem, theorems on the stiffness properties of equally-stressed structures, etc.) known for geometrically linear structures are proved.Conditions assuring better or worse quality of equally-stressed geometrically non-linear truss as compared to geometrically linear ones are obtained.It is shown that in numerical optimization of geometrically non-linear trusses in the case of negligible rotations of compressed rods some updated analytical optimization algorithms (derived earlier for geometrically linear case) are monotonic. A simple numerical example confirming the features is presented.     

Some general results for optimal structures

This paper considers some approaches to structural optimization that meet real design requirements to a greater extent than has previously been achieved. In all of the problems considered, the structures are statically loaded, strains and displacements are small, and buckling effects are neglected.Topology optimization problems for multi-material (both linear and non-linear) elastic trusses, in particular, bi-material ones, are considered. Several general theorems are proven. Well-known classical results are generalized to minimal mass structures with restricted stresses or stiffness values. Some general theorems for trusses of given geometry are also proven. Moreover, it is proven that in the case of restricted stresses, the minimum mass non-linear-elastic truss of a given geometry is not heavier than the corresponding optimal linear-elastic one.     

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.     

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.     

Sizing and layout design of truss structures under dynamic and static constraints with an integrated particle swarm optimization algorithm

Natural frequencies offer useful knowledge on the dynamic response of the structures. It is possible to avoid from the destructive effects of dynamic loads on the structures by optimizing layout and size of their subject to constraints on natural frequencies. Since optimization problems including frequency constraints are highly nonlinear, this kind of problems forms a challenging area to test the performance of the different optimization techniques. This study tests the performance of an integrated particle swarm optimization algorithm (iPSO), a new particle swarm optimizer integrated with the improved fly-back mechanism and the weighted particle concept, in four weight minimization of truss structures with sizing and layout variables under multiple frequency constraints. Optimization results demonstrate that the new algorithm is competitive with other state-of-the-art metaheuristic algorithms in dynamic and static structural optimization 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.     

On the validity of Prager's example of nonunique Michell structures

Prager (1974) demonstrated through an example that the optimal layout of least-weight trusses for a stress constraint — termed also Michell trusses—can be nonunique. The strain field for the above example, however, seems to violate Michell's optimality criteria. It is shown in this note that the above example of nonuniqueness is completely correct if we restrict the truss members to a smaller subset of the plane.     

On design derivatives for optimization with a critical point constraint

Design optimization of geometrically nonlinear structures with a critical point constraint is considered. A staggered scheme is applied to the optimization problem and the reduced optimization problem is solved at the critical point. Derivatives of the objective function and constraints are defined consistently with the algorithmic steps of the staggered scheme.It is noticed that different schemes require different design derivatives of the objective function and constraints. It is stressed that a distinction must be made between the derivative of displacements at the critical load and the derivative of critical displacements. For the sake of simplicity a nonlinear two-bar truss structure is used to show that their properties are quite different; while the first one grows to infinity when approaching the critical point and thus does not exist, the other exists at the critical point and is equal to zero.Subsequently, two methods of computing the design derivative of critical loads are analysed, and it is demonstrated, for the truss example, that both methods yield correct results. Then, two optimization problems, i.e. the minimum volume problem and the maximum critical load problem, are formulated. Both problems are solved for the two-bar truss, and yield results that compare favourably with those obtained analytically. The solution scheme is shown to be insensitive to initial errors in the determination of the critical point.     

On design derivatives for optimization with a critical point constraint

Design optimization of geometrically nonlinear structures with a critical point constraint is considered. A staggered scheme is applied to the optimization problem and the reduced optimization problem is solved at the critical point. Derivatives of the objective function and constraints are defined consistently with the algorithmic steps of the staggered scheme. It is noticed that different schemes require different design derivatives of the objective function and constraints. It is stressed that a distinction must be made between the derivative of displacements at the critical load and the derivative of critical displacements. For the sake of simplicity a nonlinear two-bar truss structure is used to show that their properties are quite different; while the first one grows to infinity when approaching the critical point and thus does not exist, the other exists at the critical point and is equal to zero. Subsequently, two methods of computing the design derivative of critical loads are analysed, and it is demonstrated, for the truss example, that both methods yield correct results. Then, two optimization problems, i.e. the minimum volume problem and the maximum critical load problem, are formulated. Both problems are solved for the two-bar truss, and yield results that compare favourably with those obtained analytically. The solution scheme is shown to be insensitive to initial errors in the determination of the critical point.     

Exact optimal structural layouts for non-self-adjoint problems

Using optimality criteria for trusses with displacement constraints, a truss layout is optimized for a point load and a given displacement in different directions, with a view to demonstrating some unexpected features of non-self-adjoint problems.On leave from the Institute of Structural Mechanics, Warsaw University of Technology, supported by the Humboldt Foundation     

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.     

Truss optimization with natural frequency constraints using generalized normal distribution optimization

The newly proposed Generalized Normal Distribution Optimization (GNDO) algorithm is used to design the truss structures with optimal weight. All trusses optimized have frequency constraints, which make them very challenging optimization problems. A large number of locally optimal solutions and non-convexity of search space make them difficult to solve, therefore, they are suitable for testing the performance of optimization algorithm. This work investigates whether the proposed algorithm is capable of coping with such problems. To evaluate the GNDO algorithm, three benchmark truss optimization problems are considered with frequency constraints. Numerical data show GNDO’s reliability, stability, and efficiency for structural optimization problems than other meta-heuristic algorithms. We thoroughly analyse and investigate the performance of GNDO in this engineering area for the first time in the literature.

     

Michell-like 2D layouts generated by genetic ESO

The theory of least-weight trusses for one load condition and a stress constraint was established by Michell (Philos Mag 8:589–597, 1904). His work was largely ignored for about 50 years, but from then on, a lot of research effort has been devoted to construct optimal topologies satisfying Michell’s optimality criteria. But the exact analytical Michell layout is still not known for most boundary conditions. It is therefore useful to develop numerical methods for generating approximate Michell-like topologies. We show in this paper that genetic ESO (GESO) methods are suitable for constructing Michell-like layouts. To illustrate the validity of GESO, seven different load cases for a plane structure with two point supports are considered. The Michell-like layouts are generated for vertical or horizontal loads applied at different heights and different angles. The results are compared and classified and a new Michell truss is proposed on the basis of the GESO results. Plane structures under two point loads are also considered, and the GESO results are compared with Melchers (Struct Multidisc Optim 29:85–92, 2005) solutions.     

Hybridizing electromagnetism-like mechanism algorithm with migration strategy for layout and size optimization of truss structures with frequency constraints

The optimal design of a truss structure with dynamic frequency constraints is a highly nonlinear optimization problem with several local optimums in its search space. In this type of structural optimization problems, the optimization methods should have a high capability to escape from the traps of the local optimums in the search space. This paper presents hybrid electromagnetism-like mechanism algorithm and migration strategy (EM–MS) for layout and size optimization of truss structures with multiple frequency constraints. The electromagnetism-like mechanism (EM) algorithm simulates the attraction and repulsion mechanism between the charged particles in the field of the electromagnetism to find optimal solutions, in which each particle is a solution candidate for the optimization problem. In the proposed EM–MS algorithm, two mechanisms are utilized to update the position of particles: modified EM algorithm and a new migration strategy. The modified EM algorithm is proposed to effectively guide the particles toward the region of the global optimum in the search space, and a new migration strategy is used to provide efficient exploitation between the particles. In order to test the performance of the proposed algorithm, this study utilizes five benchmark truss design examples with frequency constraints. The numerical results show that the EM–MS algorithm is an alternative and competitive optimizer for size and layout optimization of truss structures with frequency constraints.     

On using genetic algorithms for optimum damper placement in space trusses

Although similar in some ways to the design of aircraft and other lightweight structures, the optimal design of space structures has several unique challenges. A flexible optimization system allowing for multiple analysis techniques and including continuous and discrete design variables is desired. Although not as computationally efficient as traditional optimization techniques, genetic algorithms meet this requirement.The present investigation used a genetic algorithm to place passive viscous dampers in space trusses. The flexibility of the system was demonstrated through the use of fixed and free boundary conditions. The results showed that four dampers are generally sufficient to suppress bending motion in a seventy-two-bar fixed truss and a seventy-eight-bar free truss. The results were intuitive, demonstrating the suitability of the genetic algorithm to this class of problem.     

Force density method for simultaneous optimization of geometry and topology of trusses

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.     

A swift technique for damage detection of determinate truss structures (2)

This paper introduces a novel and robust probable statistical approach for the applied damage detection of determinate truss structures. This technique involves two steps; the first is called most probable damaged element identification step and the second is called probable damage severity prediction step. In the first step, a new index based on modal residual forces plays a major role to independently identify damage-suspected elements for each considered mode. Then among them, the elements, the most probable to damage, are extracted. In the second step, the probable damage severity for each most probable damaged element is individually predicted using a novel statistical approach. Finally, to justify the validity and robustness of the technique, three commonly used bridge trusses including a 29-bar Pratt truss, a 29-bar Warren truss, and finally, a 37-bar K truss under different damage scenarios are thoroughly studied while their modal parameters are corrupted by noise. The obtained results indicate that the method is innovatively capable of swiftly predicting, for determinate truss structures, not only damaged elements but also their damage severities by carrying out solely few structural analyses.

     

Optimal strengthening of elasto-plastic trusses with plastic deformation and stability constraints

In this paper a design method for the optimal strengthening of trusses is presented. Trusses with given configuration composed of linearly elastic-perfectly plastic material are considered and it is assumed that because of the increase or change of the loads their strength and/or stiffness is not satisfactory. The problem is to strengthen the truss by the application of additional elastoplastic bars and/or supports, such that the truss be strong enough to carry the loads, does not undertake excessive plastic deformations, and the cost of the strengthened truss be a minimum. For the solution of the highly nonlinear problem an iterative method is presented. Besides the constraints on overall plastic deformations, the stability effects are also taken into consideration during iteration. The application is illustrated by the solution of numerical examples.     

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.