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Christian Haase

Publications and source records attributed to Christian Haase.

At least 37 records · Page 2Linked to original sources
arXiv2019

Thermo-micro-mechanical simulation of bulk metal forming processes

The newly proposed microstructural constitutive model for polycrystal viscoplasticity in cold and warm regimes (Motaman and Prahl, 2019), is implemented as a microstructural solver via user-defined material subroutine in a finite element (FE) software. Addition of the microstructural solver to the default thermal and mechanical solvers of a standard FE package enabled coupled thermo-micro-mechanical or thermal-microstructural-mechanical (TMM) simulation of cold and warm bulk metal forming processes. The microstructural solver, which incrementally calculates the evolution of microstructural state variables (MSVs) and their correlation to the thermal and mechanical variables, is implemented based on the constitutive theory of isotropic hypoelasto-viscoplastic (HEVP) finite (large) strain/deformation. The numerical integration and algorithmic procedure of the FE implementation are explained in detail. Then, the viability of this approach is shown for (TMM-) FE simulation of an industrial multistep warm forging.

physics.comp-ph↗
arXiv2018

Ehrhart-equivalent $\boldsymbol 3$-polytopes are equidecomposable

We show that if two lattice $3$-polytopes $P$ and $P'$ have the same Ehrhart function then they are $\operatorname{GL}_3({\mathbb Z})$-equidecomposable; that is, they can be partitioned into relatively open simplices $U_1,\dots, U_k$ and $U'_1,\dots,U'_k$ such that $U_i$ and $U'_i$ are unimodularly equivalent, for each $i$.

math.CO↗
arXiv2018

Smooth centrally symmetric polytopes in dimension 3 are IDP

In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric $3$-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.

math.CO↗
arXiv2018

Levelness of Order Polytopes

Since their introduction by Stanley~\cite{StanleyOrderPoly} order polytopes have been intriguing mathematicians as their geometry can be used to examine (algebraic) properties of finite posets. In this paper, we follow this route to examine the levelness property of order polytopes. The levelness property was also introduced by Stanley~\cite{Stanley-CM-complexes} and it generalizes the Gorenstein property. This property has been recently characterized by Miyazaki~\cite{Miyazaki} for the case of order polytopes. We provide an alternative characterization using weighted digraphs. Using this characterization, we give a new infinite family of level posets and show that determining levelness is in $\operatorname{co-NP}$. This family can be used to create infinitely many examples illustrating that the levelness property can not be characterized by the $h^{\ast}$-vector. We then turn to the more general family of alcoved polytopes. We give a characterization for levelness of alcoved polytopes using the Minkowski sum. Then we study several cases when the product of two polytopes is level. In particular, we provide an example where the product of two level polytopes is not level.

math.CO↗
arXiv2017

Existence of unimodular triangulations - positive results

Unimodular triangulations of lattice polytopes arise in algebraic geometry, commutative algebra, integer programming and, of course, combinatorics. In this article, we review several classes of polytopes that do have unimodular triangulations and constructions that preserve their existence. We include, in particular, the first effective proof of the classical result by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation that admits a unimodular triangulation. Our proof yields an explicit (although doubly exponential) bound for the dilation factor.

math.CO↗
arXiv2017

Influence of deformation and annealing twinning on the microstructure and texture evolution of face-centered cubic high-entropy alloys

The influence of the physical mechanisms activated during deformation and annealing on the microstructure and texture evolution as well as on the mechanical properties in the equiatomic CoCrFeMnNi high-entropy alloy (HEA) were investigated. A combination of cold rolling and annealing was used to investigate the HEA in the deformed, recovered, partially recrystallized, and fully recrystallized states. Detailed microstructure and texture analysis was performed by electron backscatter diffraction, transmission electron microscopy, and X-ray diffraction. The mechanical properties were evaluated using uniaxial tensile testing. A specific focus of this investigation was put on studying the influence of deformation and annealing twinning on the material behavior. It was substantiated that during cold rolling deformation, twinning facilitates the transition from the Cu-type to the Brass-type texture, whereas annealing-twinning leads to a strong modification of the texture formed during recrystallization. The formation of specific twin orientations and the randomization of the recrystallization texture were proven by experiments as well as by cellular automaton simulations. During tension of the cold-rolled and annealed material high work-hardenability was observed. We attribute this behavior primarily to the dominance of planar dislocation slip and reduced tendency for dynamic recovery, since deformation twinning was observed to activate only in the non-recrystallized grains and thus, contributed minimally to the overall plasticity. The correlation between deformation/annealing twinning and the material behavior was discussed in detail.

cond-mat.mtrl-sci↗
arXiv2017

Mixed Ehrhart polynomials

For lattice polytopes $P_1,\ldots, P_k \subseteq \mathbb{R}^d$, Bihan (2014) introduced the discrete mixed volume $\mathrm{DMV}(P_1,\dots,P_k)$ in analogy to the classical mixed volume. In this note we initiate the study of the associated mixed Ehrhart polynomial $\mathrm{ME}_{P_1,\dots,P_k}(n) = \mathrm{DMV}(nP_1,\dots,nP_k)$. We study properties of this polynomial and we give interpretations for some of its coefficients in terms of (discrete) mixed volumes. Bihan (2014) showed that the discrete mixed volume is always non-negative. Our investigations yield simpler proofs for certain special cases. We also introduce and study the associated mixed $h^*$-vector. We show that for large enough dilates $r P_1, \ldots, rP_k$ the corresponding mixed $h^*$-polynomial has only real roots and as a consequence the mixed $h^*$-vector becomes non-negative.

math.CO↗
arXiv2016

Discrete Mixed Volume and Hodge-Deligne Numbers

Generalizing the famous Bernstein-Kushnirenko Theorem, Khovanskii proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskii in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

math.CO↗
arXiv2015

Face-subgroups of permutation polytopes

In [Baumeister, H., Nill, Paffenholz, On permutation polytopes, Adv. Math. 222 (2009), 431-452 / arXiv:0709.1615] we conjectured a characterization of subgroups H of a permutation group G so that, on the level of permutation polytopes, P(H) is a face of P(G). Here we present the embarrassingly simple proof of this conjecture.

math.CO↗
arXiv2014

Convex-normal (pairs of) polytopes

In 2012 Gubeladze (Adv.\ Math.\ 2012) introduced the notion of k-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+1) have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between k- and (k+1)-convex-normality (for k >= 3) and improve the bound to 2d(d+1). In the second part we extend the definition to pairs of polytopes and show that for rational polytopes P and Q, where the normal fan of P is a refinement of the normal fan of Q, if every edge e_P of P is at least d times as long as the corresponding edge e_Q of Q, then (P+Q) \cap \Z^d = (P\cap \Z^d) + (Q \cap \Z^d).

math.CO↗
arXiv2013

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.

math.AG↗
arXiv2013

Integer decomposition property of dilated polytopes

Let $\mathcal{P} \subset \mathbb{R}^N$ be an integral convex polytope of dimension $d$ and write $k \mathcal{P}$, where $k = 1, 2, \ldots$, for dilations of $\mathcal{P}$. We say that $\mathcal{P}$ possesses the integer decomposition property if, for any integer $k = 1, 2, \ldots$ and for any $α\in k \mathcal{P} \cap \mathbb{Z}^N$, there exist $α_{1}, \ldots, α_k$ belonging to $\mathcal{P} \cap \mathbb{Z}^N$ such that $α= α_1 + \cdots + α_k$. A fundamental question is to determine the integers $k > 0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.

math.CO↗
arXiv2012

Polytopes associated to Dihedral Groups

In this note we investigate the convex hull of those $n \times n$-permutation matrices that correspond to symmetries of a regular $n$-gon. We give the complete facet description. As an application, we show that this yields a Gorenstein polytope, and we determine the Ehrhart $h^*$-vector.

math.CO↗
arXiv2012

Polyhedral adjunction theory

In this paper we give a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we define two convex-geometric notions: the Q-codegree and the nef value of a rational polytope P. We define the adjoint polytope P^(s) as the set of those points in P, whose lattice distance to every facet of P is at least s. We prove a structure theorem for lattice polytopes P with high Q-codegree. If P^(s) is empty for some s < 2/(dim(P)+2), then the lattice polytope P has lattice width one. This has consequences in Ehrhart theory and on polarized toric varieties with dual defect. Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.

math.CO↗
arXiv2011

Permutation Polytopes of Cyclic Groups

We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. We give formulas for their dimension and vertex degree. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets.

math.CO↗
arXiv2009

Linear Systems on Tropical Curves

A tropical curve Γis a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system |D| of a divisor D on a tropical curve Γanalogously to the classical counterpart. We investigate the structure of |D| as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, |D| defines a map from Γto a tropical projective space, and the image can be extended to a tropical curve of degree equal to °(D). The tropical convex hull of the image realizes the linear system |D| as a polyhedral complex. We show that curves for which the canonical divisor is not very ample are hyperelliptic. We also show that the Picard group of a \Q-tropical curve is a direct limit of critical groups of finite graphs converging to the curve.

math.AG↗
arXiv2009

Grid graphs, Gorenstein polytopes, and domino stackings

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniques use results from graph theory, polyhedral geometry, and enumerative combinatorics.

math.CO↗