Ing Ind - Inf (Mag.)(ord. 270) - MI (471) BIOMEDICAL ENGINEERING - INGEGNERIA BIOMEDICA

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050492 - COMPUTATIONAL MODELING IN ELECTRONICS AND BIOMATHEMATICS

Ing Ind - Inf (Mag.)(ord. 270) - MI (476) ELECTRONICS ENGINEERING - INGEGNERIA ELETTRONICA

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096660 - NUMERICAL METHODS IN MICROELECTRONICS

Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA

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096659 - COMPUTATIONAL MODELING IN ELECTRONICS AND BIOMATHEMATICS

Obiettivi dell'insegnamento

The course in object has a size of 8 CFUs and is associated with the following other two courses, each one sized 5 CFUs:

050492 - COMPUTATIONAL MODELING IN ELECTRONICS AND BIOMATHEMATICS

096660 - NUMERICAL METHODS IN MICROELECTRONICS

The present document defines objectives, programme and expected learning outcomes for all the above mentioned courses. Nature is by far the best Engineer worldwide and the human body is the living evidence of this statement: the dynamic interplay among skeleton, thought, breath, motion, muscles, and cardiovascular flow, is the driving force to everyday's life and a continuous inspiration for the design and development of novel materials and devices. Leveraging this unexpected connection between Life Sciences and Electronic and Biomedical Engineering, the course of Computational Modeling in Electronics and Biomathematics (CMEBM) aims at providing the student the basic instruments to construct a mathematical model of a realistic problem and the theoretical tools to develop and analyze methods and algorithms for its stable and accurate solution on a computer machine. Electronic and biological systems share significant structural similarities. Transmembrane ion flow regulating the functional response of a neuronal or cardiac cell, as well as the motion of electric charge transporting current in a nanoscale-sized transistor, obey the same phenomenological description, well known in Biology as the Nernst-Planck model and in Electronics as the Drift-Diffusion model. The goal of the course is threefold and consists of providing the students:

a unified framework for the mathematical modeling of:

1.a: cellular biology and

1.b: solid-state electronics;

the numerical methods for the simulation of:

2.a: specific cellular systems and

2.b: specific electronic devices;

a critical know-how to perform the design of complex systems in:

3.a: Life Sciences and

3.b: Electronics.

For students of the course coded 050492: goals 1.a, 2.a and 3.a.

For students of the course coded 096660: goals 1.b, 2.b and 3.b.

Risultati di apprendimento attesi

Class lectures and laboratories:

will provide students the theoretical foundations of models and numerical methods;

will allow students to verify in a quantitative manner the physical accuracy of the models and the numerical performance of the methods through the solution of exercises of increasing level of difficulty;

will enable students to gain acquaintance with the critical analysis and design of a complex system in Life Sciences and Electronics through the combined use of theoretical modeling tools and a flexible computational environment such as Matlab.

The theoretical concepts, methods and algorithms will be numerically verified during the laboratory sessions through the solution of exercises and the critical discussion of project assignments. Exercise solution and project discussion will be conducted with the interactive participation (through remote web connection) of Prof. Giovanna Guidoboni, University of Missouri, Columbia MO, USA, a worldwide expert in the development and analysis of fluid-mechanical computational models in Ophthalmology and Fluid Flow in the Cardiovascular and Lymphatic systems. Projects will be subject to evaluation and will contribute to determine the final grade of the student.

For students of the course coded 050492: the course will allow to gain acquantaince with the critical analysis and design of a complex system in Life Sciences through the combined use of theoretical modeling tools and a flexible computational environment such as Matlab.

For students of the course coded 096660: the course will allow to gain acquantaince with the critical analysis and design of a complex system in Electronics through the combined use of theoretical modeling tools and a flexible computational environment such as Matlab.

Argomenti trattati

1. Balance laws in local form. The diffusion-advection-reaction linear model problem: well-posedness analysis and numerical approximation with a stabilized Finite Element Method (FEM). Convergence analysis. Conservation properties. Numerical stability of the FEM: continuous and discrete maximum principles.

2. Introduction to cellular biology and ion electrodynamics. ODE models for transmembrane ion flow in cellular physiology. The Kirchhoff voltage and current laws: capacitive and resistive transmembrane currents. The linear resistor model; the Goldman-Hodgkin-Katz (GHK) model; the Hodgkin-Huxley (HH) model. PDE models for transmembrane ion flow in cellular physiology: the velocity-extended Poisson-Nernst-Planck (VE-PNP) system for M ionic species. Examples in cellular biology: excitable cells. Nernst potential of an ionic species; cellular homeostatis: the GHK potential. Action potential propagation: the Cable Equation model coupled with the HH ODE system. Simplified treatment of intracellular and extracellular compartments: the one-dimensional PNP model for a protein channel.

3. Multiscale structure of integrated circuits. Micro/nanoscale view: the Maxwell equation system and the quasi-static approximation. Atomic/macroscale view: charge transport in solids; Ohm's law in metal conductors; the Drift-Diffusion (DD) model in semiconductor materials. The Poisson-DD (PDD) PDE model for semiconductor device simulation at the micro/nanoscale. Model analogies: PDD = PNP with M=2. Scaling. Functional iterations: Newton's method and Gummel's map. Examples in device electronics: the p-n junction. Thermal equilibrium, reverse and forward bias. I-V curves and the ideal diode law. The full depletion approximation: analytical solution of the PDD system. 1D models for the Metal-Oxide-Semiconductor (MOS) transistor. The n-MOS capacitor. The n+ - n - n+ structure for the n-MOS channel.

For students of the course coded 050492: topics 1. and 2.

For students of the course coded 096660: topics 1. and 3.

Prerequisiti

Students are required to know the principles and methods of Calculus, Physics and Linear Algebra.

Modalità di valutazione

Student evaluation is the result of two contributions, the sum of which yields the final grade:

1. The first contribution is provided by the evaluation of a number of project assignments that must be elaborated by each student during the laboratories. Each project assignment will have a minimum grade of 0/30 and a maximum grade of 2/30, for a total of 10/30.

2. The second contribution is provided by the evaluation of a written test. Five exam sessions that will be scheduled according to the dates established by the calendar of the School of Industrial and Information Engineering. The written test consists of the solution of a number of exercises pertaining to the three main topics treated during classes and laboratories, namely: Elements of Numerical Approximation of Balance Laws in Local Form, Elements of Cellular Biology and Elements of Solid-State Electronics, and is organized according to the following structure:

a) theoretical questions on the topics covered during the course, including: statement and/or proof of a theorem or mathematical property of a method; definition of concepts such as convergence, consistency, stability of a numerical scheme; formulation and analysis of a numerical method; b) computational questions to be numerically solved through the use of built-in MATLAB functions and MATLAB functions implemented during the laboratories, including: systems of linear equations, interpolation of data and functions, nonlinear equations, numerical integration and differentiation, approximation of ordinary and partial differential equations.

The written test will have a minimum grade of 0/30 and a maximum grade of 24/30. The evaluation process will take into account the correctness and the accuracy of the provided answers, the level of critical analysis of the obtained results, the ability in the use of MATLAB functions to solve basic mathematical problems as well as problems of interest in the engineering practice.

Bibliografia

R. Sacco, G. Guidoboni, A. G. Mauri, A Comprehensive Physically Based Approach to Modeling in Bioengineering and Life Sciences, Editore: Elsevier Inc. 50 Hampshire St., Cambridge MA 02139, USA, Anno edizione: 2019
Lecture Notes of the Coursehttp://www1.mate.polimi.it/~ricsac/LectureNotesCMEBM.pdfR. Muller and T. Kamins, Device Electronics for Integrated Circuits, Editore: John Wiley and Sons, Anno edizione: 2003
J. Keener and J. Sneyd, Mathematical Physiology, Editore: Springer-Verlag, Anno edizione: 2009
I. Rubinstein, Electrodiffusion of Ions, Editore: SIAM Philadelphia, Anno edizione: 1990

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