Shape and topology optimization involving the eigenvalues of an elastic structure: A multi-phase-field approach

A cost function involving the eigenvalues of an elastic structure is optimized using a phase-field approach, which allows for topology changes and multiple materials.We show continuity and differentiability of simple eigenvalues in the phase-field context. Existence of global minimizers can be shown, for which first order necessary optimality conditions can be obtained in generic situations. Furthermore, a combined eigenvalue and compliance optimization is discussed.

Partially distributed outer approximation

This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression.

Learning deep linear neural networks: Riemannian gradient flows and convergence to global minimizers

We study the convergence of gradient flows related to learning deep linear neural networks (where the activation function is the identity map) from data. In this case, the composition of the network layers amounts to simply multiplying the weight matrices of all layers together, resulting in an overparameterized problem. The gradient flow with respect to these factors can be re-interpreted as a Riemannian gradient flow on the manifold of rank-$r$ matrices endowed with a suitable Riemannian metric. We show that the flow always converges to a critical point of the underlying functional. Moreover, we establish that, for almost all initializations, the flow converges to a global minimum on the manifold of rank $k$ matrices for some $k\leq r$.

DG approach to large bending plate deformations with isometry constraint

We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove [Formula: see text]-convergence of the discrete energies and discrete global minimizers. We document the flexibility and accuracy of the dG method with several numerical experiments.

Generic properties of mechanical systems on the two-torus

This chapter focuses on proving generic properties of the minimizing orbits of the slow mechanical system. It first proves Theorem 4.5 concerning non-critical but bounded energy, before proving Proposition 4.6 concerning the very high energy. The chapter then proves Proposition 4.7 concerning the critical energy. The proof of Theorem 4.5 consists of three steps. The first proves a Kupka-Smale-like theorem about non-degeneracy of periodic orbits. The second shows that a non-degenerate locally minimal orbit is always hyperbolic. The third finishes the proof by proving the finite local families obtained from the second step are “in general position,” and therefore there are at most two global minimizers for each energy.

Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime

We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic boundary data. To achieve this result, we first study minimizers subject to a physically relevant norm constraint (the Lyuksyutov constraint), and show their regularity up to the boundary. From this regularity, we rigorously derive the norm constraint from the asymptotic Lyuksyutov regime. As a consequence, isotropic melting is avoided by unconstrained minimizers in this regime, which then allows us to analyse their biaxiality sets. In the case of a nematic droplet, this also implies that the radial hedgehog is an unstable equilibrium in the same regime of parameters. Technical results of this paper will be largely employed in Dipasquale et al. (Torus-like solutions for the Landau- de Gennes model. Part II: topology of $$\mathbb {S}^1$$ S 1 -equivariant minimizers. https://arxiv.org/pdf/2008.13676.pdf; Torus-like solutions for the Landau- de Gennes model. Part III: torus solutions vs split solutions (In preparation)), where we prove that biaxiality level sets are generically finite unions of tori for smooth configurations minimizing the energy in restricted classes of axially symmetric maps satisfying a topologically nontrivial boundary condition.

Standing waves of the quintic NLS equation on the tadpole graph

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schrödinger equation with quintic power nonlinearity equipped with the Neumann–Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $$\omega \in (-\infty ,0)$$ ω ∈ ( - ∞ , 0 ) is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $$L^6$$ L 6 . The set of standing waves includes the set of ground states, which are the global minimizers of the energy at constant mass ($$L^2$$ L 2 -norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $$\omega \in (-\infty ,0)$$ ω ∈ ( - ∞ , 0 ) and correspond to a bigger interval of masses. It is proven that there exist critical frequencies $$\omega _1$$ ω 1 and $$\omega _0$$ ω 0 with $$-\infty< \omega _1< \omega _0 < 0$$ - ∞ < ω 1 < ω 0 < 0 such that the standing waves are the ground state for $$\omega \in [\omega _0,0)$$ ω ∈ [ ω 0 , 0 ) , local constrained minima of the energy for $$\omega \in (\omega _1,\omega _0)$$ ω ∈ ( ω 1 , ω 0 ) and saddle points of the energy at constant mass for $$\omega \in (-\infty ,\omega _1)$$ ω ∈ ( - ∞ , ω 1 ) . Proofs make use of the variational methods and the analytical theory for differential equations.

On the steady-states of a two-species non-local cross-diffusion model

We investigate the existence and properties of steady-state solutions to a degenerate, non-local system of partial differential equations that describe two-species segregation in homogeneous and heterogeneous environments. This is accomplished via the analysis of the existence and non-existence of global minimizers to the corresponding free energy functional. We prove that in the spatially homogeneous case global minimizers exist if and only if the mass of the potential governing the intra-species attraction is sufficiently large and the support of the potential governing the interspecies repulsion is bounded. Moreover, when they exist they are such that the two species have disjoint support, leading to complete segregation. For the heterogeneous environment we show that if a sub-additivity condition is satisfied then global minimizers exists. We provide an example of an environment that leads to the sub-additivity condition being satisfied. Finally, we explore the bounded domain case with periodic conditions through the use of numerical simulations.