Course detail

# Selected Mathematical Methods in Mechanics

FSI-RMEAcad. year: 2015/2016

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.

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Learning outcomes of the course unit

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Co-requisites

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

Course curriculum

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Aims

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

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Prerequisites and corequisites

Basic literature

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

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

Recommended reading

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

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

#### Type of course unit

Lecture

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Syllabus

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.

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Exercise

Teacher / Lecturer

Syllabus

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.