Therefore special relaxation methods are developed to find and effectively describe highly oscillating solutions of these problems. These methods include Sufficient conditions methods, particularly the Translation method. Searching for minimization sequences, particularly for laminates of a high rank. Necessary conditions and the stable extensions. Stability of optimal designs.
Optimal structures are generally unstable.
Indeed, they concentrate the resistance capacities in certain directions to stay against a given loading, therefore they are extremely sensitive to the variation of the loadings. We are working on designs that compromise effectiveness and stability. The next picture shows the crossection of the torsioned bar with maximal rigidity. The bar is assembled from two materials with different elastic moduli, the proportion of the materials in the design is fixed. Here, the blue and the red domains denote the originally given strong and weak materials, the color reflects the volume fractions in optimal laminates, lines show their directions.