339-0918/01 – Numerical Methods (NM)

Gurantor departmentDepartment of Mechanics of MaterialsCredits10
Subject guarantorprof. Ing. Jiří Lenert, CSc.Subject version guarantorprof. Ing. Jiří Lenert, CSc.
Study levelpostgraduateRequirementChoice-compulsory
YearSemesterwinter + summer
Study languageCzech
Year of introduction1999/2000Year of cancellation2012/2013
Intended for the facultiesFSIntended for study typesDoctoral
Instruction secured by
LoginNameTuitorTeacher giving lectures
LEN30 prof. Ing. Jiří Lenert, CSc.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 25+0
Combined Credit and Examination 25+0

Subject aims expressed by acquired skills and competences

Teach a students derive benefit from the newest knowledge of subject with possibility the knowledge further evolve and apply for complicated problems.

Teaching methods

Individual consultations
Project work

Summary

Finite Element Method Matrix Algebra. Definitions, additions and subtractions of matrices, matrix multiplication, determinants, matrix inverse, solution of simultaneous equations, integration and differentiation of matrices. Basic Structural Concepts and Energy Theorems. Stiffness and flexibility, stiffness matrix, the principle of virtual work, the principle of complementary virtual work, method of minimum potential, complementary energy theorem. The Discrete System. Electrical networks, fluid flow networks. The Application of the Principle of Virtual Work. Ritz method, application of Ritz method for bending of beams, application for tension and pressure, the solution of rotate-symmetric problems (rotate disc of constant thickness), complementary variant of Ritz method. Static Analysis of Pin-jointed Trusses. To obtain of stiffness matrix for a rod element, to obtain of stiffness matrix for a two-dimensional rod element, to obtain global stiffness matrix for plane pin-jointed trusses. Finite Element Analysis. Derivation of element stiffness matrices, virtual work approach, rod element, Hermite element, beam element, grid element, in-plane triangular element, in-plane quadrilateral element. Isotropic element, four- nod quadrilateral element, other higher-order elements. Finite Element Aanalysis. Global stiffness matrix, solution of simultaneous equations systems. Boundary Element Method Global System of Differential Equations. Formulation of the system of equations, introduction of boundary conditions, reactions, variable transformations. Numerical Procedures. Numerical integration, one-dimensional numerical integration (Gauss method), numerical integration in two dimensions (bi- directional methods), numerical integration in three dimensions. Fundamental Solutions. The point load (Kelvin) solution, reciprocal work theorem (Betti), Somigliana integral identity for displacements. Two Dimensional Potential Problems. Normal loading of the plane (Flamants solution), uniformly distributed load. The Direct Boundary Element Method. Influence coefficients, creation of equation system, the fundamental solution.

Compulsory literature:

DHATT,G.-TOUZOT,G.: The Finite Element Method Displeyd, John Wiley and Sons, New York 1984 BEER,G.-WATSON,J.O.: Introduction to Finite and Boundary Elemetn Methods for Engineers, John Wiley & Sons,1992

Recommended literature:

BORESI,A.P.-SCHMIDT,R.J.-SIDEBOTTOM,O.M.: Advanced Mechanics of Materials, John Wiley & Sons,Inc., 1993 CANDRUPATLA,T.R.-BELEGUNDU,A.D.: Introduction to Finite Elements in Engineering, Prentice-Hall International, Inc., 1991

Way of continuous check of knowledge in the course of semester

Příprava zadané problematiky v písemné formě.

E-learning

Další požadavky na studenta

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

Předmět rozšiřuje teoretické základy MKP a MHP získané v bakalářském a magisterském studiu. Numerické metody jsou v současné době v široké formě využívány pro řešení numerické analýzy mechanických vlastností konstrukcí. Konkrétně se zabývá dále uvedenými problémy: Metoda konečných prvků. Základní principy a energetické teorémy. Tuhost a poddajnost, matice tuhosti, princip virtuálních prací, princip komplementární virtuální práce, princip minima potenciální energie systému, komplementární energetický teorém. Diskrétní systémy. Systém elektrické sítě. Potrubní kapalinový systém. Aplikace principu virtuálních prácí. Ritzova metoda, aplikace Ritzovy metody u ohybu nosníků, aplikace pro namáhání v tahu a tlaku, řešení rotačně symetrických problémů (rotující tenký kruhový disk konstantní tloušťky). Varianta Ritzovy metody za použití komplementární potenciální energie. Statická analýza prutových soustav. Matice tuhosti pro tyčový prvek, matice tuhosti pro tyčový prvek v dvourozměrném prostoru, globální matice tuhosti pro prutovou soustavu. Odvození matice tuhosti elementu pomocí principu virtuální práce, tyčový element, Hermitovský element, nosníkový element, roštový element, rovinný trojúhelníkový element, čtyřúhelníkový element, isotropický element. Analýza konstrukce. Sestavení globální matice tuhosti. Metody řešení soustavy lineárních rovnic. Metoda hraničních prvků. Sestavení soustavy diferenciálních rovnic elastického problému. Formulace soustavy diferenciálních rovnic, zavedení okrajových a hraničních podmínek, reakce, transformace proměnných. Numerické procedury. Numerická integrace, jednodimenzionální numerická integrace (Gaussova metoda), numerická integrace v dvojrozměrném systému, numerická integrace v trojrozměrném systému. Fundamentální řešení. Kelvinova úloha bodového zatížení roviny, Bettiho teorie vzájemnosti posuvů, Somiglianův integrál identity pro posunutí. Dvoudimenzionální potenciální problém. Normálové zatížení poloroviny (Flamantova úloha), spojité zatížení. Přímá metoda hraničních prvků. Koeficienty vlivu, vytvoření systému rovnic, fundamentální řešení.

Conditions for subject completion

Combined form (validity from: 1960/1961 Summer semester, validity until: 2012/2013 Summer semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (145) 51
        Examination Examination 100  0
        Exercises evaluation Credit 45  0
Mandatory attendence parzicipation:

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Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty
2012/2013 (P2346) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2012/2013 (P2346) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2012/2013 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2012/2013 (P2301) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2011/2012 (P2346) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2011/2012 (P2346) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2011/2012 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2011/2012 (P2301) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2010/2011 (P2301) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2010/2011 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2009/2010 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2009/2010 (P2301) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2008/2009 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2008/2009 (P2301) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2007/2008 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2007/2008 (P2301) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2006/2007 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2006/2007 (P2301) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2005/2006 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava Choice-compulsory study plan
2005/2006 (P2301) Mechanical Engineering (3901V003) Applied Mechanics K Czech Ostrava Choice-compulsory study plan
2004/2005 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan
2001/2002 (P2301) Mechanical Engineering (3901V003) Applied Mechanics P Czech Ostrava 1 Choice-compulsory study plan

Occurrence in special blocks

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