Abstract
The present work addresses two-dimensional structures with stiff inclusions embedded in a soft matrix. An isotropic damage model has been chosen for the interface and matrix softening failure whereas inclusions have been assumed to be elastic for the time being. Interface and matrix failure are described by an eXtended Finite Element Method (X-FEM) applying a cohesive zone concept and levelsets for the description of the inclusion geometry as well as the enrichment functions. After reaching a crack initiation criterion the cracks propagate element by element. To enable the integration of finite elements with local discontinuities a triangulation is applied that incorporates cracks and interface contours. The aim of this work is maximizing the overall structural ductility by variation of the local geometrical layout of the inclusions for a prescribed inclusion mass. The ductility is defined as the integral of the strain energies over a specified displacement range. In the gradient based structural optimization process shape and location parameters of the inclusions are taken as design variables. If the inclusions are specified as ellipses the design variables are the semi-axes, the angle and the coordinates of the ellipses center. The variation of these parameters is restricted by a prescribed minimum distance to the boundaries of the design space and a minimum distance of the inclusions with respect to each other. Although the geometry of the inclusions in the matrix is permanently changing during optimization, a fixed structural mesh is used to avoid continuous adaptation to a material conforming mesh. The Optimality Criteria (OC) method or the Method of Moving Asymptotes (MMA) is used as optimization algorithm. Necessary gradients with respect to a variation of the optimization parameters are determined by an analytical sensitivity analysis. Using gradients of the objective function "maximization of ductility" and of the constraint "constant inclusion mass" the design of the inclusions is changed automatically in an iterative process. The calculation of sensitivities is an important issue of this work. In order to derive the sensitivities of the objective "maximization of ductility" the gradients of the state variables, i.e. the nodal displacements, have to be determined in advance. This is necessary because of the implicit dependence of the displacement field on the optimization variables. In order to determine the gradients of the nodal displacements, the derivatives of the equilibrium equations with respect to the optimization variables are computed. Here, the influence of the material interfaces, the related enrichment functions of the X-FEM, as well as the kinematic and constitutive relations have to be taken into account. The potential of the gradient based structural optimization with repect to the maximization of the overall structural ductility is demonstrated by various numerical examples.
Users
Please
log in to take part in the discussion (add own reviews or comments).
