Professor Andrej Cherkaev

(Андрей Всеволодович Черкаев)
Department of Mathematics 
University of Utah, 

 email: cherk@math.utah.edu
URL:
http://www.math.utah.edu/~cherk

 

Предварительные задания по курсу

Новые мат. методы для новых материалов
New math for new materials

URL: www.math.utah.edu/~cherk/spb07

ИПМ РАН, cПб,  Большой 61, В.О. Ноябрь-Декабрь 2007.

 

             С вопросами обращайтесь по email: cherk@math.utah.edu.  Пожалуйста, используйте латиницу.

 

Prerequisites: Before the class starts, the participants are kindly requested to work on the topics, listed below. This would allow for quick start of the short intensive course. The problems highlight the concerts that will be discussed. They are not easy, and it is not expected that you will solve  all of them  completely. However,  it is expected that you think  about them  and suggest  ideas for the solution.

 

=============  1  =============

 

Dynamic of unstable materials.

 

Dynamic homogenization, waves of damage and phase transitions.

 

 

Phenomenon. Consider a train of standing domino pieces. Assume that a wave of transition is initialized by a distant falling piece that collides with the next piece and make it fall, and so on. Falling domino pieces form a wave of transition.

 

Problems

 

a.     Formulate the mathematical model(s) of falling dominos train. Suggest simplifications of the model. Formulate the impact conditions. Derive the equations of the wave motion. 

b.    Compute (or estimate) the steady speed of this wave if size and mass of a piece and the distance between then are known and assuming an inelastic collisions. Compare with experiment. 

 

Advanced problems

 

c.     Formulate a continuum description of the transitional wave.

d.    Assume that the distances between domino pieces are not exactly the same, but are randomly perturbed. Reformulate the problem; find the average speed of the wave and condition of its propagation.

e.     Assume that the neighbor domino pieces are linked by a linear spring. Modify the model.

f.     Suggest a two-dimensional extension of the domino problem.


Related problem:  
Slinky - compute the speed of walking slinkypicture of slinky


===============  2  ===============

 

Optimal design of a structure

 

Consider an infinite strip made from a laminate of materials with the given conductivities k1 and k2. The difference of potentials at the ends of the strip is equal to one. Determine the proportion of the materials in the laminate and the angle of its inclination to the boundary of the strip that maximizes the component of the current along the strip.


Hint: the current across the field gradient (along the strip) is due to anisotropy of the laminate. The anisotropic conductivity depends on the orientation of laminates and the fractions of materials within.

 

===============  3  ===============

 

Structural optimization and the finite-dimensional models.

Consider a square region filled with a conducting composite. The conductivity of isotropic components equals to k1 and k2, respectively. 

 

Problem

Build a finite (NxN) square lattice that models the conductivity of heterogeneous material (Figure below). Enforce the isotropy of components.

 









 

 

 

 

 

 

 

 

 

 

 

 

 


Assume that the unit density current propagates through the upper and lower sides of the square and two other sites are insulated (Left figure).

Next Problems

1.     Find an optimal materials’ layout in the domain.  The aim is to maximize the conductivity of the domain. Assume that the fractions of materials are given. Is the solution unique?

 

2.     Formulate a similar optimal layout problem for the lattice that mimics the composite.  Derive and analyze the optimality conditions.

 

 

============= 4  ==============

 

 

Search for new, hot, and exciting research topics

 

1. Familiarize yourself with popular and demanded areas contemporary mechanics/applied math and learn the searching strategy. Find the most popular and demanded areas in contemporary mechanics/applied math. Describe the recent advances in the chosen area (you may choose nanomaterials, biomaterials, smart materials, MEM, flexible materials and folding, multiscale models, etc) based on listed below websites. Analyze the most discussed topics. Why are these topics popular?

 

·       Find websites of major conferences: SES 2007,  USNCCM9,  IUTAM Congresses, SES Annual Technical Meetings, Euromech. Analyze the most discussed topics. Why are these topics popular?

·       Read the table of contents of most important journals in the area: Nature, Science News, JMPS, Applied mechanics, SIAM review, J of Solid and Structures, J of Structural and Multidisciplinary optimization, Theory of Elasticity, SIAM Journals.

·       Investigate websites of major USA grant agencies: NSF (National Science Foundation), Department of Energy (DOE), National Institute for Health (NIH), DARPA: AFOSR (Air Force Office of Scientific Research), and Office of Naval Research.

·       Look through webpages of leading specialists in the field and faculty of departments of Mechanics, Mechanical Engineering, Airspace Engineering, and Applied Math in the leading Universities.

 

2. Write a review about your findings. Emphasize the areas of your personal interest. Focus on the needed mathematical technique. Emphasize the unknown terminology, methods, and topics. Select the most exciting and interesting applications.

3. Exchange information with your fellows and start an informal junior seminar “Contemporary problems in mechanics” discussing problems outside of the current area of expertise of your group.


Выберете ваш(и) проект(ы) из списка

Source research:
Используя Интернет, подберите информацию об одном из следующих "горячих" направлениях. Подготовьте доклад на 10-15 мин.

1. Optimal structures of composites
2. Optimal design
3. Phase transition and shape memory alloys
4. Smart materials
5. Nanostructures (nanotubes, nanocoils, C60, etc.)
6. Granular materials, flow.
7. Biostructures (bone mechanics, soft tissues)
8. Protein structures
9. Flexible materials: design and optimization
10. Wool, spaghetti, hairball: How to describe them?

Информация включает ответы на вопросы:

a. Почему люди изучают эту проблему? Почему именно сейчас?
b. Что решено и что еще нет? Сформулируйте нерешенные задачи.

and also:
c. Какие конференции по этой теме были и будут?
d. Где работают на проблемой (Университеты, Лаборатории, etc.)?
e. Проводятся ли летние школы по теме? Как туда подать заявку?
f. Какие организации финансируют исследования? Что эти организации хотят?
g. Где публикуются результаты? Есть ли монографии по теме?
h. Имена. Контактные группы. Веб-сайты.



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