Analysis and simulations of a Viscoelastic Model of Angiogenesis

The statistical theory of the angiogenesis equations

Angiogenesis is a multiscale process by which a primary blood vessel issues secondary vessel sprouts that reach regions lacking oxygen. Angiogenesis can be a natural process of organ growth and development or a pathological induced by a cancerous tumor. A mean field approximation for a stochastic model of angiogenesis consists of partial differential equation (PDE) for the density of active tip vessels. Addition of Gaussian and jump noise terms to this equation produces a stochastic PDE that defines an infinite dimensional Lévy process and is the basis of a statistical theory of angiogenesis. …

Stochastic homogenization of a scalar viscoelastic model exhibiting stress-strain hysteresis

Motivated by rate-independent stress-strain hysteresis observed in filled rubber, this article considers a scalar viscoelastic model in which the constitutive law is random and varies on a lengthscale which is small relative to the overall size of the solid. Using stochastic two-scale convergence as introduced by Bourgeat, Mikelic and Wright, we obtain the homogenized limit of the evolution, and demonstrate that under certain hypotheses, the homogenized model exhibits hysteretic behaviour which persists under asymptotically slow loading. These results are illustrated by means of numerical simu…

Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem

In this paper, we study, from both variational and numerical points of view, a dynamic contact problem between a viscoelastic-viscoplastic piezoelectric body and a deformable obstacle. The contact is modelled using the classical normal compliance contact condition. The variational formulation is written as a nonlinear ordinary differential equation for the stress field, a nonlinear hyperbolic variational equation for the displacement field and a linear variational equation for the electric potential field. An existence and uniqueness result is proved using Gronwall's lemma, adequate auxiliary …

粘弹性相分离模型的误差分析

Error analysis for a viscoelastic phase separation model

针对粘弹性相分离模型，提出保能量耗散结构的有限元与变分时间离散方法，建立非线性稳定性及最优阶误差估计，并通过数值模拟验证。