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2017　Mathematical Design of Structures

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Academic unit or major
Graduate major in Urban Design and Built Environment
Instructor(s)
Kanno Yoshihiro 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Mon1-2(J231)  Thr1-2(J231)  
Group
-
Course number
UDE.S502
Credits
2
Academic year
2017
Offered quarter
2Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

Structural optimization is widely applied to explore outperforming designs of structures that have enough safety against disturbances. A key is proper mathematical modeling that captures essences both of mechanical problems and design problems. Matrix analysis and optimization, on which this course focuses, are two fundamentals for understanding diverse problems in applied mechanics.
Optimization is recognized as a useful tool in diverse fields of engineering. Matrix analysis is a fundamental of numerical analysis.

Student learning outcomes

It is desired that students will acquire proficiency in mathematical modeling of design problems of structures via optimization. Also, it is desired that they will acquire clear understandings of several concepts in engineering mechanics from the viewpoint of matrix analysis.

Keywords

Structural optimization, topology optimization, convex optimization, integer programming, contact mechanics, variational inequality

Competencies that will be developed

Specialist skills Intercultural skills Communication skills ✔ Critical thinking skills Practical and/or problem-solving skills

Class flow

A chalk talk. Visual aids are sometimes used to promote understanding.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Preliminary examination followed by its exposition. Understanding of prerequisites.
Class 2 Compatibility matrix and equilibrium matrix. Understanding of definitions.
Class 3 Rank of compatibility and equilibrium matrices: kinematical and statical determinacy (1). Understanding of rank of a matrix.
Class 4 Rank of compatibility and equilibrium matrices: kinematical and statical determinacy (2). Understanding of concepts of kinematical/statical (in)determinacy.
Class 5 Stiffness matrix and positive definiteness (1). Understanding of definitions of a stiffness matrix and a positive (semi)definite matrix.
Class 6 Stiffness matrix and positive definiteness (2). Understanding of relations between positive definiteness and eigenvalues.
Class 7 Stiffness matrix and positive definiteness (3). Understanding of the concept of positive definiteness in engineering mechanics.
Class 8 Maximum stiffness design and convex optimization (1). Understanding of the definition of the maximum stiffness design.
Class 9 Maximum stiffness design and convex optimization (2). Understanding of the concept of convexity.
Class 10 Maximum stiffness design and convex optimization (3). Understanding of algorithms for the maximum stiffness design.
Class 11 Structural optimization via integer programming (1). Understanding of fundamentals of integer programming.
Class 12 Structural optimization via integer programming (2). Understanding of usage of integer programming for structural optimization.
Class 13 Contact mechanics and convex analysis (1). Understanding of problems dealt with in contact mechanics.
Class 14 Contact mechanics and convex analysis (2). Understanding of fundamentals of convex analysis.
Class 15 Contact mechanics and convex analysis (3). Understanding of methods of contact mechanics based on the convex analysis.

Textbook(s)

There exists no specific textbook that covers all contents of the course.

Reference books, course materials, etc.

Gilbert Strang, "Linear Algebra and Its Applications", Academic Press Inc., ISBN: 978-0126736601
Yoshihiro Kanno, "Nonsmooth Mechanics and Convex Optimization", CRC Press, ISBN: 978-1420094237
Stephen Boyd, Lieven Vandenberghe, "Convex Optimization", Cambridge University Press, ISBN: 978-0521833783
Martin P. Bendsoe, Ole Sigmund, "Topology Optimization: Theory, Methods, and Applications", Springer, ISBN: 978-3642076985
Peter Wriggers: Computational Contact Mechanics (2nd ed.). Springer-Verlag, 978-3540326083

Assessment criteria and methods

Student's understanding is assessed by report(s).

Related courses

  • UDE.S406 ： Tensor Analysis for Building Structure

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed Calculus I, and Calculus II, Linear Algebra I, and Linear Algebra II or have equivalent knowledge.

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