Institute of Fundamental Technological Research

The topological sensitivity derivative of a functional expressed in terms of displacement, strain or stress fields and boundary tractions is derived for the case of an elliptical hole introduced in the plate. The derivative is specified with respect to the hole area, the length of ellipse axes and their orientation in terms of primary and adjoint state fields. The shape sensitivity derivative for a finite hole can be applied and the topological derivative with respect to the hole area is obtained in the limiting case. The transition to a plane crack occurs for vanishing length of minor axis and the topological derivative with respect to crack length is then derived from the general formulae. The results can be useful in optimal design procedures by selecting positions, shape and orientation of elliptical cutouts.

Topological sensitivity, Elliptical hole, Plate, Plane crack, Optimal design

The incremental sensitivity analysis associated with variation of structure material parameters, shape or topology variation is generally discussed by analyzing the evolution of potential and complementary energies, or arbitrary functionals of state fields. The concept of configuration and sensitivity generalized forces is used in presenting the sensitivity derivatives. The general reciprocity relations are derived for the case of potential or complementary energy variations. The topology variations in bar structures related to introduction of elements and introduction of inclusions and voids in plates are discussed, and the sensitivity forces are derived.

Sensitivity analysis, Topological derivative, Structural design, Bar structures, Plates

The concept of topological sensitivity derivative is introduced and applied to study the problem of optimal design of structures. It is assumed, that virtual topology variation is described by topological parameters. The topological derivative provides the gradients of objective functional and constraints with respect to these parameters. This derivative enables formulation of the conditions of topology transformation. In this paper formulas for the topological sensitivity derivative for bending plates are derived. Next, the topological derivative is used in the optimization process in order to formulate conditions of finite topology modifications and in order to localize positions of the modifications. In the case of plates they are related to introduction of holes and introduction of stiffeners. The theoretical considerations are illustrated by some numerical examples.

Topological sensitivity derivative, Finite topology modifications, Bending plates, Optimal topology and shape, Optimal layout of ribs

The concept of topological derivative is introduced and applied to optimal design of structural elements and to study the material microstructure evolution. For structural design the objective function and constraints provide the optimal design, for material microstructure the free energy and dissipation function generate the process of evolution such as phase transformation, crack growth or void generation. Three general modes of topology variation have been considered: generation of new elements, removing of the existing elements and a substitution of the existing elements by new elements. The cases of infinitesimal and finite topology variations have been discussed and illustrated by examples.

topological derivative, finite topology variation, structure and mate-rial evolution, optimal topology and shape

The problem of optimal design of structures with active support is analyzed in the paper. The sensitivity expressions with respect to the generalized force and the position of actuator are derived by the adjoint structure approach. Next, the optimality conditions are formulated by means of an introduced Lagrangian function. The problem of introduction of a new actuator is also considered and the condition of modification is expressed by means of the topological derivative. The obtained sensitivity formula, optimality conditions and modification conditions are applied in the optimization algorithm with respect to the number, positions and generalized forces of the actuators. Numerical examples of optimal control of beams illustrate the procedure proposed in the paper.

Smart structures, Actuators, Adjoint method, Optimal active support

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.

optimal topology, finite topology modifications, structure evolution, truss and frame structures

A heuristic algorithm for optimal design of trusses is presented with account for stress and buckling constraints. The design variables are constituted by cross-sectional areas, configuration of nodes and the number of nodes and bars. Similarly to biological growth models, it is postulated that the structure evolves with the characteristic size parameter and the “bifurcation” of topology occurs with the generation of new nodes and bars in order to minimize the cost function. The first-order sensitivity derivatives provide the necessary information on topology variation and the optimality conditions for configuration and cross-sectional parameters.

In this paper, a heuristic algorithm is presented for optimal design of trusses with varying cross-sectional parameters, configuration of nodes, and number of nodes and bars. The algorithm provides new nodes and bars at some states and for the optimal truss configuration. It is assumed that the structure evolves with the overall size parameter and a “bifurcation” of topology occurs with the generation of new nodes, in order to minimize the cost function. Both displacement and stress constraints can be introduced in the optimization procedure.

Truss and frame structures are considered at their critical states, i.e., bifurcation or limit points. When design parameters vary, the critical points evolve and the respective critical loads are modified. This paper is concerned with sensitivity analysis of critical loads with respect to cross-sectional and configuration parameters. The effect of redistribution during the prebuckling state is considered.

In this paper, it is demonstrated that incremental problems of nonlinear potential and nonpotential systems, including sensitivity problems, can be uniformly treated within the analysis of a homogeneous set of equations. Regular and critical states are considered. It is shown that rank analysis of a rectangular matrix of a homogeneous set of incremental equations reveals all possible problems associated with singularity conditions. When considering nonlinear design modification problems, it is necessary to use derivatives of the secant and tangent stiffness matrices. A direct approach to differentiation of stiffness matrices on a finite element level in sensitivity problems is also presented. Simple illustrative examples are discussed.

design sensitivity, nonlinear problems, incremental analysis

A geometrically non-linear, elastic frame structure is considered and the effect of small variation of its parameters on structure deformation response is studied. Variations of cross-sectional stiffness, member length, orientation, and also variation of node positions of a discrete structure are considered. The explicit expressions for variation of a displacement functional in terms of primary and adjoint states and of design parameter variations are provided.

An algorithm of optimal design of supports including their number, position and stiffness is proposed. The number of supports constitute topological design parameters, their positions correspond to configuration parameters. Both, elastic and rigid supports are considered and the optimization is aimed to minimize the total structure cost. The topology bifurcation points correspond to generation of new supports. The topological sensitivity derivative is used in deriving the optimality conditions

The optimization procedure provides number of supports, their position and stiffness of both supports and beam segments.