Solvabilty of generalized Hammerstein integral equations on unbounded domains, with sign-changing kernels

In this work we study an Hammerstein generalized integral equation u(t)=∫_{-∞}^{+∞}k(t,s) f(s,u(s),u′(s),...,u^{(m)}(s))ds, where k:ℝ²→ℝ is a W^{m,∞}(ℝ²), m∈ℕ, kernel function and f:ℝ^{m+2}→ℝ is a L¹-Carathéodory function. To the best of our knowledge, this paper is the first one to consider discontinuous nonlinearities with derivatives dependence, without monotone or asymptotic assumptions, on the whole real line. Our method is applied to a fourth order nonlinear boundary value problem, which models moderately large deflections of infinite nonlinear beams resting on elastic foundations under localized external loads.

A symbolic approach to nonlinearly perturbed heat equation

We consider a system described by the linear heat equation with adiabatic boundary conditions which is perturbed periodicaly. This perturbation is nonlinear and is characterized by a one-parameter family of quadratic maps. The system, depending on the parameters, presents very complex behaviour. We introduce a symbolic framework to analyze the system and resume its most important features.

A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement

We propose an alternative crack propagation algo- rithm which effectively circumvents the variable transfer procedure adopted with classical mesh adaptation algo- rithms. The present alternative consists of two stages: a mesh-creation stage where a local damage model is employed with the objective of defining a crack-conforming mesh and a subsequent analysis stage with a localization limiter in the form of a modified screened Poisson equation which is exempt of crack path calculations. In the second stage, the crack naturally occurs within the refined region. A staggered scheme for standard equilibrium and screened Poisson equa- tions is used in this second stage. Element subdivision is based on edge split operations using a constitutive quantity (damage). To assess the robustness and accuracy of this algo- rithm, we use five quasi-brittle benchmarks, all successfully solved.

Damage and fracture algorithm using the screened Poisson equation and local remeshing

We propose a crack propagation algorithm which is independent of particular constitutive laws and specific element technology. It consists of a localization limiter in the form of the screened Poisson equation with local mesh refinement. This combination allows the cap- turing of strain localization with good resolution, even in the absence of a sufficiently fine initial mesh. In addition, crack paths are implicitly defined from the localized region, cir- cumventing the need for a specific direction criterion. Observed phenomena such as mul- tiple crack growth and shielding emerge naturally from the algorithm. In contrast with alternative regularization algorithms, curved cracks are correctly represented. A staggered scheme for standard equilibrium and screened equations is used. Element subdivision is based on edge split operations using a given constitutive quantity (either damage or void fraction). To assess the robustness and accuracy of this algorithm, we use both quasi-brittle benchmarks and ductile tests.