Rank-one Convex Hull Research Articles

Hierarchical rank-one sequence convexification for the relaxation of variational problems with microstructures

This paper presents an efficient algorithm for the approximation of the rank-one convex hull in the context of nonlinear solid mechanics. It is based on hierarchical rank-one sequences and simultaneously provides first and second derivative information essential for the calculation of mechanical stresses and the computational minimisation of discretised energies. For materials, whose microstructure can be well approximated in terms of laminates and where each laminate stage achieves energetic optimality with respect to the current stage, the approximate envelope coincides with the rank-one convex envelope. Although the proposed method provides only an upper bound for the rank-one convex hull, a careful examination of the resulting constraints shows a decent applicability in mechanical problems. Various aspects of the algorithm are discussed, including the restoration of rotational invariance, microstructure reconstruction, comparisons with other semi-convexification algorithms, and mesh independency. Overall, this paper demonstrates the efficiency of the algorithm for both, well-established mathematical benchmark problems as well as nonconvex isotropic finite-strain continuum damage models in two and three dimensions. Thereby, for the first time, a feasible concurrent numerical relaxation is established for an incremental, dissipative large-strain model with relevant applications in engineering problems.

T5 configurations and hyperbolic systems

In this paper, we study the rank-one convex hull of a differential inclusion associated to entropy solutions of a hyperbolic system of conservation laws. This was introduced in [B. Kirchheim, S. Müller and V. Šverák, Studying Nonlinear PDE by Geometry in Matrix Space (Springer, 2003), Sec. 7], and many of its properties have already been shown in [A. Lorent and G. Peng, Null Lagrangian measures in subspaces, compensated compactness and conservation laws, Arch. Ration. Mech. Anal. 234(2) (2019) 857–910; A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156]. In particular, in [A. Lorent and G. Peng, On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity, Calc. Var. Partial Differential Equations 59(5) (2020) 156], it is shown that the differential inclusion does not contain any [Formula: see text] configurations. Here, we continue that study by showing that the differential inclusion does not contain [Formula: see text] configurations.

Upper semicontinuity of the lamination hull

Let K ⊆ ℝ2×2 be a compact set, let Krc be its rank-one convex hull, and let L (K) be its lamination convex hull. It is shown that the mapping K ↦ L̅(K̅) is not upper semicontinuous on the diagonal matrices in ℝ2×2, which was a problem left by Kolář. This is followed by an example of a 5-point set of 2 × 2 symmetric matrices with non-compact lamination hull. Finally, another 5-point set K is constructed, which has L (K) connected, compact and strictly smaller than Krc.

$$T_5$$ T 5 -Configurations and non-rigid sets of matrices

In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation of the rank-one convex hull for this specific set was later carried out in 2010 by W. Pompe in Calc. Var. PDE 37(3–4):461–473, (2010) by delicate geometric arguments. For more general sets of matrices a full computation of the rank-one convex hull is clearly out of reach. Therefore, in this short note we revisit the construction and propose a new, in some sense generic method for deciding whether convex integration for a given set of matrices can be carried out, which does not require the full computation of the rank-one convex hull.

Non-Laminate Microstructures in Monoclinic-I Martensite

We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of three-tuples of pairwise incompatible strains. In addition, we construct a fivedimensional continuum of T3s and show that its intersection with the boundary of the symmetrised rank-one convex hull is four-dimensional.We also show that there is another kind of monoclinic-I martensite with qualitatively different semi-convex hulls which, as far as we know, has not been experimentally observed. Our strategy is to combine understanding of the algebraic structure of symmetrised rank-one convex cones with knowledge of the faceting structure of the convex polytope formed by the strains.

On evolving deformation microstructures in non-convex partially damaged solids

The paper outlines a relaxation method based on a particular isotropic microstructure evolution and applies it to the model problem of rate independent, partially damaged solids. The method uses an incremental variational formulation for standard dissipative materials. In an incremental setting at finite time steps, the formulation defines a quasi-hyperelastic stress potential. The existence of this potential allows a typical incremental boundary value problem of damage mechanics to be expressed in terms of a principle of minimum incremental work. Mathematical existence theorems of minimizers then induce a definition of the material stability in terms of the sequential weak lower semicontinuity of the incremental functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of weak convexity notions of the stress potential. Furthermore, the variational setting opens up the possibility to analyze the development of deformation microstructures in the post-critical range of unstable inelastic materials based on energy relaxation methods. In partially damaged solids, accumulated damage may yield non-convex stress potentials which indicate instability and formation of fine-scale microstructures. These microstructures can be resolved by use of relaxation techniques associated with the construction of convex hulls. We propose a particular relaxation method for partially damaged solids and investigate it in one- and multi-dimensional settings. To this end, we introduce a new isotropic microstructure which provides a simple approximation of the multi-dimensional rank-one convex hull. The development of those isotropic microstructures is investigated for homogeneous and inhomogeneous numerical simulations.

The quasiconvex hull for the five-gradient problem

In [8 Chapter 4.3] Kirchheim and Preiss gave an example of a set K consisting of five 2 × 2 symmetric matrices without rank-one connections, for which there exists a Lipschitz mapping u satisfying \({Du \in K}\) . In the present paper we construct the rank-one convex hull of K. As a corollary we obtain that for each \({F \in {\rm int}\,K^{rc}}\) there exists a Lipschitz mapping u satisfying $$Du \in K\quad{\rm and}\quad u(x) = Fx\,{\rm for}\,x\,\in\,{\partial} \Omega \,.$$ Moreover, we show that the rank-one convex hull of K and the quasiconvex hull of K are equal.

On the gradient set of Lipschitz maps

We prove that the essential range of the gradient of planar Lipschitz maps has a connected rank-one convex hull. As a corollary, in combination with the results in [ Faraco, D., and Székelyhidi, Jr., L. , Tartar's conjecture and localization of the quasiconvex hull in R 2x2 , Acta Math., to appear.] we obtain a complete characterization of incompatible sets of gradients for planar maps in terms of rank-one convexity.

Rank-one convex hulls in $${\mathbb{R}^{2\times2}}$$

We study the rank-one convex hull of compact sets \(K\subset\mathbb{R}^{2\times2}\). We show that if K contains no two matrices whose difference has rank one, and if K contains no four matrices forming a T4 configuration, then the rank-one convex hull Krc is equal to K. Furthermore, we give a simple numerical criterion for testing for T4 configurations.

On the local structure of rank-one convex hulls

In this note we prove that ifKKis a compact set ofm×nm\times nmatrices containing an isolated pointXXwith no rank-one connection into the convex hull ofK∖{X}K\setminus \{X\}, then the rank-one convex hull separates as\[Krc=(K∖{X})rc∪{X}.K^{rc}=\bigl (K\setminus \{X\}\bigr )^{rc}\cup \{X\}.\]This is an extension of a result of P. Pedregal, which holds for2×22\times 2matrices.

Towards the efficient computation of effective properties of microstructured materials

An algorithm for partially relaxing multiwell energy densities, such as for materials undergoing martensitic phase transitions, is presented here. The detection of the rank-one convex hull, which describes effective properties of such materials, is carried out for the most prominent nontrivial case, namely the so-called T k -configurations. Despite the fact that the computation of relaxed energies (and with it effective properties) is inherently unstable, we show that the detection of these hulls ( T 4-configurations) can be carried out exactly and with high efficiency. This allows in practice for their computation to arbitrary precision. In particular, our approach to detect these hulls is not based on any approximation or grid-like discretization. This makes the approach very different from previous (unstable and computationally expensive) algorithms for the computation of rank-one convex hulls or sequential-lamination algorithms for the simulation of martensitic microstructure. It can be used to improve these algorithms. In cases where there is a strict separation of length scales, these ideas can be integrated at a sub-grid level to macroscopic finite-element computations. The algorithm presented here enables, for the first time, large numbers of tests for T 4-configurations. Stochastic experiments in several space dimensions are reported here. To cite this article: C.-F. Kreiner et al., C. R. Mecanique 332 (2004).

AN ESTIMATE ON THE APPROXIMATION OF THE RANK-ONE CONVEX HULL OF A FUNCTION

ABSTRACT We propose to approximate the rank-one convex hull of a function in terms of the maximum number of “levels” permitted in the construction and approximation of the associated laminates. Although this approximation is not new, its understanding in terms of laminates of a maximum number of levels leads to the establishment of an exponential error estimate.

On the Computation of the Rank-One Convex Hull of a Function

We report on several numerical experiments where the rank-one convexification of an energy density is computed. The explicit examples cover a whole spectrum of typical situations one may encounter. One of those is especially relevant for the computation of microstructures in crystalline solids.

Convex integration with constraints and applications to phase transitions and partial differential equations

Abstract. We study solutions of first order partial differential relations Du]K, where u:Q²ÂnMÂm is a Lipschitz map and K is a bounded set in m2n matrices, and extend Gromov's theory of convex integration in two ways. First, we allow for additional constraints on the minors of Du and second we replace Gromov's P-convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of 'wild' solutions to elliptic systems. Our work was originally motivated by questions in the analysis of crystal microstructure and we establish the existence of a wide class of solutions to the two-well problem in the theory of martensite.

Numerical Computation of Rank-One Convex Envelopes

We describe an algorithm for the numerical computation of the rank-one convex envelope of a function $f:\MM^{m\times n}\rightarrow\RR$. We prove its convergence and an error estimate in $L^\infty$.

On Functional Separately Convex Hulls

Let D be a set of vectors in R d . A function f: R d → R is called D-convex if its restriction to each line parallel to a nonzero vector of D is a convex function. For a set A⊆ R d , the functional D-convex hull of A, denoted by co D (A) , is the intersection of the zero sets of all nonnegative D -convex functions that are 0 on A . We prove some results concerning the structure of functional D -convex hulls, e.g., a Krein—Milman-type theorem and a result on separation of connected components. We give a polynomial-time algorithm for computing co D (A) for a finite point set A (in any fixed dimension) in the case of D being a basis of R d (the case of separate convexity). This research is primarily motivated by questions concerning the so-called rank-one convexity, which is a particular case of D -convexity and is important in the theory of systems of nonlinear partial differential equations and in mathematical modeling of microstructures in solids. As a direct contribution to the study of rank-one convexity, we construct a configuration of 20 symmetric 2 x 2 matrices in a general (stable) position with a nontrivial functionally rank-one convex hull (answering a question of K. Zhang on the existence of higher-dimensional nontrivial configurations of points and matrices).