A Study on the Effect of Multi-Axial Forging Type on the Deformation Heterogeneity of AA1100 Using Finite Element Analysis

The effect of 3 forging routes (Route A - 1~12 passes by plane forging (PF) and reverse-plane forging (R-PF), Route B – 1~6 passes by PF and R-PF, 7~12 passes by diagonal forging (DF) and reversediagonal forging (R-DF), Route C – 1~12 passes by DF and R-DF) on maximum load to produce the workpiece, deformation heterogeneity and hydrostatic pressure distribution in AA1100 was theoretically investigated using finite element analysis (FEA). The maximum load per pass required to complete 1 cycle of the SPD process was different depending on the forging routes. Route A was relatively higher than Route B and C. From the results of effective strain, the deformation heterogeneity was predicted at the center, edge, and corner regions of the AA1100 workpiece produced by Route A and B. However, the distribution of effective strain in Route C was relatively more homogeneous than Route A and B. The average hydrostatic pressure, which is closely related to the suppression of crack formation in the workpiece under multi-axial forging, was predicted to be relatively bigger in Route C than Route A and B.

Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions

International audience The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.

Hook formulas for skew shapes

International audience The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.

Greene–Kleitman Invariants for Sulzgruber Insertion

R. Sulzgruber's rim hook insertion and the Hillman–Grassl correspondence are two distinct bijections between the reverse plane partitions of a fixed partition shape and multisets of rim-hooks of the same partition shape. It is known that Hillman–Grassl may be equivalently defined using the Robinson–Schensted–Knuth correspondence, and we show the analogous result for Sulzgruber's insertion. We refer to our description of Sulzgruber's insertion as diagonal RSK. As a consequence of this equivalence, we show that Sulzgruber's map from multisets of rim hooks to reverse plane partitions can be expressed in terms of Greene–Kleitman invariants.

Refined Dual Stable Grothendieck Polynomials and Generalized Bender-Knuth Involutions

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the $K$-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries $1$ and $2$.

Enumeration of Cylindric Plane Partitions

International audience Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. As in the reverse plane partition case, the right hand side of this identity admits a simple factorization form in terms of the "hook lengths'' of the individual boxes in the underlying shape. The first result of this paper is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result of this paper is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog in terms of the non-intersecting lattice path model for cylindric plane partitions. Les partitions planes cylindriques sont une généralisation naturelle des partitions planes renversées. Une série génératrice pour énumération des partitions planes cylindriques a été donnée récemment par Borodin. Comme dans le cas des partitions planes renversées, la partie droite de cette identité peut être factoriser en terme de "longueur d’équerres'' des carrés dans la forme sous-jacente. Le premier résultat de cet article est une nouvelle preuve bijective de l'identité de Borodin qui utilise le cadre de "diagramme de croissance'' de Fomin pour la correspondance de RSK généralisée. Le deuxième résultat de cette article est une $(q,t)$-déformation d'identité de Borodin qui généralise un résultat de Okada dans le cas des partitions planes renversées. Le troisième résultat de cet article est une formule combinatoire explicite pour le poids de Macdonald qui utilise le modèle des chemins non-intersectant pour les partitions planes cylindriques.

Plane Reverse Based on Coordinate Measuring

Plane is a basic element to constitute the part and a benchmark to assess shape and position error. It’s precision directly influence to components processing flatness and shape position error's evaluation precision. Therefore carries on measuring and reversing in the manufacturing process is very meaningful to improve manufacturing precision. Using the smallest region principle to reverse plane compiles a system of measuring and reversing based on machining center, it can not only raise the machining center use efficiency, but also can increase the online working accuracy.