Course detail

# Selected Mathematical Methods in Mechanics

FSI-RMEAcad. year: 2017/2018

The course deals with the following topics: Definition of variational problems, demonstration of the equivalence of the integration of a differential equation and seeking the minimum of a suitable functional. Weak solution. Functionals and operators in the Hilbert space. Variational principles of the linear elasticity. Hashin-Shtrikman variational principles in the mechanics of composite materials. Methods of weighted residuals and direct variational methods. Method of boundary integral equations in the linear elasticity. Fundamental solution. Numerical methods for the solutions of boundary integral equations. Physical and mathematical aspects of the stability problems. Stability of elastic systems, energy criterion of stability, bifurcation and limit points. Nonlinear systems and stability criterion. Thermodynamic approach to stability. Percolation theory.

Supervisor

Learning outcomes of the course unit

Students will have an overview of modern mathematical techniques used for solution of boundary-value problems in the continuum mechanics. They realize a diversity of physical origin of stability loss and vibrations and the unity of mathematical apparatus used for solution.

Prerequisites

In the field of mechanics: Knowledge of basic concepts of the theory of elasticity (stress, principal stress, deformation, strain, general Hooke law, potential energy). Principle of virtual displacements, principle of virtual work. In the field of mathematics: Partial differential equations of 2nd order. Elements of variational calculus. Elements of functional analysis (functional spaces, Hilbert space L2).

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

W. Wunderlich, W. Pilky, Mechanics of Structures, CRC Press, 2003, Boca Raton.

P. Procházka, Základy mechaniky složených materiálů, Academia, Praha, 2001.

Satya N. Atluri: The Meshless Method (MLPG) for Domain & BIE Discretizations. CRC Press, 2004.

F. Maršík, Termodynamika kontinua, Academia Praha, 1999.

B. Bittner, J. Šejnoha, Numerické metody mechaniky, ČVUT, Praha, 1992.

K. Rektorys, Variační metody, Academia Praha,1999.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Written test examining the knowledge of basic concepts - examination paper containing 3 examples to be solved - oral discussion over examination papers with an optional additional question.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The goal of the subject is to make students familiar with selected mathematical procedures commonly used in mechanics. A particular emphasis is placed on variational methods, as well as methods based on the formulation of boundary integral equations which are of practical importance in a wide field of applications. Some basic pieces of knowledge of the mathematical theory of Hilbert spaces and partial differential equations are employed.

Specification of controlled education, way of implementation and compensation for absences

Attendance is required. One absence can be compensated by attending a seminar with another group in the same week, or by elaboration of substitute tasks. Longer absence is compensated by special tasks according to instructions of the tutor. Course-unit credit is awarded on the following conditions: - active participation in the seminars, - good results in seminar tests of basic knowledge, - solution of additional tasks in case of longer excusable absence. Seminar tutor will specify the form of these conditions in the first week of semester.

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

Historic introduction. Definition of variational problems, demonstration of the equivalence of the integration of a differential equation and seeking the minimum of a suitable functional.

Functionals and operators in the Hilbert space. Positive operators and their physical meaning. Energy space of positive definitive operators. Essential and natural boundary conditions of differential equations. Generalized or weak solution to the problem of the minimum of the energy functional.

Variational principles of the linear elasticity. Fundamental relations, extremes of functionals, classical variational principles. (Lagrange, Castiligliano, Reisssner, Hu-Washizu).

Application of the variational principles to the derivation of governing equations of selected loaded simple bodies.

Application of variational principles for the estimate of material properties of composite materials. Hashin-Shtrikman variational principles.

Methods of weighted residuals and direct variational methods. Interior and boundary trial function methods. Collocation method, min-max method, least squares method, orthogonality methods. Trefftz boundary method.

Method of boundary integral equations in the linear elasticity. Betti’s reciprocal theorems. Fundamental solution for Laplace operator.

Green tensor. Somigliani formulas. Fundamental solution of the elastostatics. Derivation of the boundary integral equations of the mixed boundary-value problem of the elastostatics.

Numerical methods for solutions of boundary integral equations.

Solution of the problems of the fracture mechanics.

Physical and mathematical aspects of the stability problems. Stability of elastic systems, energy criterion of stability, bifurcation and limit points. Eigenvalue problem and its relation to the free vibration analysis problems and stability problems.

Nonlinear systems and stability criterions. Thermodynamic approach to stability. Stability problems of materials with damage. Correlation length and the percolation theory.

Time reserve.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

Demonstrations of the energy method. Complete systems of functions. Fourier arrays. Torsion of bar with rectangle cross section.

Formulation of Ritz and Galerkin method for numerical solution of variational problems. Application of Ritz method to ordinary differential equation – bending of the beam lying on an elastic foundation.

Illustration of differences between classical Ritz method and FEM.

Illustration of extended variational principles for the formulation of hybrid version of FEM.

Hashin-Shtrikman estimates of elastic coefficients bounds of composite materials.

Demonstration of various versions of the method of weighted residuals.

Boundary integral equation method as a special case of weighted residuals method. Derivation of fundamental solution for 3D and 2D.

Illustration of the boundary integral equation method applied to the torsion of rectangular bar.

Calculation of singular and hypersingular integrals.

Demonstration of mathematical methods applied for the solution of fracture problems.

Application of the 1st and 2nd order theory in the examination of stability problems. Application of variational methods in stability problems. Solution of eigenvalue problems.

Examples of localization (bifurcation) phenomenon in materials with damage.

Awarding course-unit credits.