We consider the optimal distribution of several elastic materials in a fixed working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the description of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steepest descent algorithm. The shape gradient obtained for a sharp interface involves jump of discontinuous quantities at the interface which are difficult to numerically evaluate. Therefore we suggest an alternative smoothed interface approach which yields more convenient shape derivatives. We rely on the signed distance function and we enforce a fixed width of the transition layer around the interface (a crucial property in order to avoid increasing “grey” regions of fictitious materials). It turns out that the optimization of a diffuse interface has its own interest in material science, for example to optimize functionally graded materials. Several 2-d examples of compliance minimization are numerically tested which allow us to compare the shape derivatives obtained in the sharp or smoothed interface cases.

This paper presents a new method to analyze frictionless grasp immobility based on defined surface-to-surface signed distance function. Distance function's differential properties are analyzed and its second-order Taylor expansion with respect to differential motion is deduced. Based on the non-negative condition of the signed distance function, the first- and second-order free motions are defined and the corresponding conditions for immobility of frictionless grasp are derived. As one benefit of the proposed method, the second-order immobility check can be formulated as a nonlinear programming problem. Numerical examples are used to verify the proposed method.