A fundamental new approach of deep drawing processes for uniform wall thickness is suggested and proved experimentally. It is based on imposition of a back-up fluid pressure which is prescheduled to vary with respect to the punch position during the drawing stroke in the hydroforming process. The formula for scheduling this fluid pressure path is generated via the plastic limit analysis and presented in terms of the geometry of the product, the work-hardening of the material, and the friction coefficient. It leads systematically to better results than empirical procedures reported in the current literature and industrial technical reports. Experiments in hydroforming sheets to cups and hemispheres performed on different materials (Al, Cu, and stainless steel) substantiate the improvement suggested. The associated overall punch load is overestimated (at most by 15 percent) with the unique feature of including Coulomb friction into the rigorous upper bound analysis.

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This work considers the concentration of measures for low-density parity-check (LDPC) code ensembles. The two results derived in this paper follow from Azuma's inequality for Doob martingales with bounded differences. The first result is a tightened concentration inequality for the conditional entropy (originally derived by Méasson et al.), and the second result is a concentration inequality for the cardinality of the fundamental systems of cycles of a bipartite graph from the ensemble.