Search

In this article we investigate a mathematical model for a retinal prosthesis made of organic polymer nanoparticles (NP) in the stationary regime. The model consists of a Drift-Diffusion system to describe free charge transport in the NP bulk; a Poisson-Nernst-Planck system to describe ion electrodiffusion in the solution surrounding the NP; and nonlinear transmission conditions at the NP-solution interface. To solve the model we use an iteration procedure for which we prove the existence and briefly comment the uniqueness of a fixed point under suitable smallness assumptions on model parameters. For system discretization we use a stabilized finite element method to prevent unphysical oscillations in the electric potential, carrier number densities and ion molar densities. Model predictions describe the amount of active chemical molecule accumulating at the neuron surface and highlight electrostatic effects induced by the sole presence of the nanoparticle. These results support the use of mathematical modeling as a virtual laboratory for the optimal design of bio-hybrid systems, whose investigation may be impervious due to experimental limits.

In this article we investigate a mathematical model for a retinal prosthesis made of organic polymer nanoparticles (NP) in the stationary regime. The model consists of a Drift-Diffusion system to describe free charge transport in the NP bulk; a Poisson-Nernst-Planck system to describe ion electrodiffusion in the solution surrounding the NP; and nonlinear transmission conditions at the NP-solution interface. To solve the model we use an iteration procedure for which we prove the existence and briefly comment the uniqueness of a fixed point under suitable smallness assumptions on model parameters. For system discretization we use a stabilized finite element method to prevent unphysical oscillations in the electric potential, carrier number densities and ion molar densities. Model predictions describe the amount of active chemical molecule accumulating at the neuron surface and highlight electrostatic effects induced by the sole presence of the nanoparticle. These results support the use of mathematical modeling as a virtual laboratory for the optimal design of bio-hybrid systems, whose investigation may be impervious due to experimental limits. [ABSTRACT FROM AUTHOR]

In this article we propose and investigate a hierarchy of mathematical models based on partial differential equations (PDE) and ordinary differential equations (ODE) for the simulation of the biophysical phenomena occurring in the electrolyte fluid that connects a biological component (a single cell or a system of cells) and a solid-state device (a single silicon transistor or an array of transistors). The three members of the hierarchy, ordered by decreasing complexity, are: (i) a 3D Poisson–Nernst–Planck (PNP) PDE system for ion concentrations and electric potential; (ii) a 2D reduced PNP system for the same dependent variables as in (i); (iii) a 2D area-contact PDE system for electric potential coupled with a system of ODEs for ion concentrations. The backward Euler method is adopted for temporal semi-discretization and a fixed-point iteration based on Gummel’s map is used to decouple system equations. Spatial discretization is performed using piecewise linear triangular finite elements stabilized via edge-based exponential fitting. Extensively conducted simulation results are in excellent agreement with existing analytical solutions of the PNP problem in radial coordinates and experimental and simulated data using simplified lumped parameter models. [ABSTRACT FROM AUTHOR]

• A novel splitting algorithm is proposed for the multiscale interface coupling of a poroelastic medium with a lumped circuit • The work is motivated by studies of tissue perfusion, but it is applicable to many engineering and life science problems • A semi-discretization in time via operator splitting ensures the balance of the physical energy at the multiscale interface • The algorithm performance is compared with two functional iteration methods inspired by domain-decomposition techniques • Unconditional stability with respect to the time discretization step for the operator splitting method is proved We consider a multiscale problem modeling the flow of a fluid through a deformable porous medium, described by a system of partial differential equations (PDEs), connected with a lumped hydraulic circuit, described by a system of ordinary differential equations (ODEs). This PDE/ODE coupled problem includes interface conditions enforcing the continuity of mass and the balance of stresses across models at different scales. In the present article, we address questions related to the solution methods of the PDE/ODE coupled problem via staggered algorithms, focusing on a detailed comparison between functional iterations and an energy-based operator splitting method and how they handle the interface conditions. We provide sufficient conditions for the convergence of functional iterations and prove that the energy-based operator splitting method is unconditionally stable with respect to the size of the time discretization step. [ABSTRACT FROM AUTHOR]

Abstract: This article deals with the analysis of the functional iteration, denoted Generalized Gummel Map (GGM), proposed in [C. de Falco, A.L. Lacaita, E. Gatti, R. Sacco, Quantum-Corrected Drift-Diffusion Models for Transport in Semiconductor Devices, J. Comp. Phys. 204 (2) (2005) 533–561] for the decoupled solution of the Quantum Drift-Diffusion (QDD) model. The solution of the problem is characterized as being a fixed point of the GGM, which permits the establishment of a close link between the theoretical existence analysis and the implementation of a numerical tool, which was lacking in previous non-constructive proofs [N.B. Abdallah, A. Unterreiter, On the stationary quantum drift-diffusion model, Z. Angew. Math. Phys. 49 (1998) 251–275, R. Pinnau, A. Unterreiter, The stationary current–voltage characteristics of the quantum drift-diffusion model, SIAM J. Numer. Anal. 37 (1) (1999) 211–245]. The finite element approximation of the GGM is illustrated, and the main properties of the numerical fixed point map (discrete maximum principle and order of convergence) are discussed. Numerical results on realistic nanoscale devices are included to support the theoretical conclusions. [Copyright &y& Elsevier]

Abstract: In this article, we deal with the numerical approximation of a quantum drift-diffusion model capable of describing tunneling effects through the thin oxide barrier in nanoscale semiconductor devices. We propose a novel reformulation of the mathematical model that allows a natural generalization of the Gummel decoupled algorithm, widely adopted in the case of the drift-diffusion system. Then, we address the finite element discretization of the linearized problems obtained after decoupling, and we prove well-posedness and a discrete maximum principle for the solution of the continuity equations. Finally, we validate the physical accuracy and the numerical stability of the proposed algorithms on the simulation of a real-life nanoscale device. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a unified framework for Quantum-corrected drift-diffusion (QCDD) models in nanoscale semiconductor device simulation. QCDD models are presented as a suitable generalization of the classical drift-diffusion (DD) system, each particular model being identified by the constitutive relation for the quantum-correction to the electric potential. We examine two special, and relevant, examples of QCDD models; the first one is the modified DD model named Schrödinger–Poisson–drift-diffusion, and the second one is the quantum-drift-diffusion (QDD) model. For the decoupled solution of the two models, we introduce a functional iteration technique that extends the classical Gummel algorithm widely used in the iterative solution of the DD system. We discuss the finite element discretization of the various differential subsystems, with special emphasis on their stability properties, and illustrate the performance of the proposed algorithms and models on the numerical simulation of nanoscale devices in two spatial dimensions. [Copyright &y& Elsevier]

We introduce a new iterative method for the computation of the minimal nonnegative solution G of the matrix equation $$X = \sum\nolimits_{i = 0}^{ + \infty } {X^i A_i } $$ , arising in the numerical solution of M/G/1 type Markov chains. The idea consists in applying a relaxation technique to customarily used functional iteration formulas. The proposed method is easy to implement and outperforms, in terms of number of iterations and execution time, the standard functional iteration techniques. [ABSTRACT FROM AUTHOR]