Solution of engineering problems
Download as PPTX, PDF6 likes2,741 views
This document discusses numerical methods for solving engineering problems. It introduces analytical methods, numerical methods, and experimental methods. It then describes various numerical methods in more detail, including the finite element method (FEM), finite difference method (FDM), finite volume method (FVM), and boundary element method (BEM). It provides examples of the types of problems each method can be applied to and notes the advantages and limitations of each approach.
Solution of engineering problems
- 1. INTRODUCTION
- 2. Structural Problems ◦ Stress analysis of beam, Fatigue analysis of mechanical component, Modal analysis of beams, etc. Thermal problems ◦ Heat transfer analysis of plate, temperature distribution in fin, Heat flux distribution in a object Computational Fluid Dynamics Problems - High resolution predictions that hold across a large range of flow conditions, motion for the dispersed phase, shock capturing, simulation of turbulent flows 2 ME 6603 FEA Erode Sengunthar Engineering College
- 3. Analytical Method Classical approach 100 % accurate Closed form solution Complete in itself Numerical Method • Mathematical representation • Approximate assumptions • Results must be verified experimentally Experimental method • Correct procedure • Time consuming • Expensive • Physical prototype required (3-5 protype reqd.) 3 ME 6603 FEA Erode Sengunthar Engineering College
- 4. Analytical Method • Provides Closed form solution • Example: Theory of Bending Numerical Method • Examples - FAM - FEM - FDM - FVM - BEM Experimental method • Examples - Strain gauge measurements - Photoelasticity analysis - Vibration measurement - Sensors for temperature measurement - Fatigue test 4 ME 6603 FEA Erode Sengunthar Engineering College
- 5. An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally accepted set. A closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well- known" operations (e.g., + − × ÷), and functions but e.g. usually no limit. 5 ME 6603 FEA Erode Sengunthar Engineering College
- 6. Continuum Modeling Discrete Modelling 6 ME 6603 FEA Erode Sengunthar Engineering College
- 7. Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the particular material studied is added through constitutive relations. 7 ME 6603 FEA Erode Sengunthar Engineering College
- 8. Model obtained using finite number of well defined components Number of elements is very large Overcomes the intractability of realistic types of continuum problems 8 ME 6603 FEA Erode Sengunthar Engineering College
- 9. Types - Functional Approximation method (FAM) - Finite Element Method (FEM) - Finite Difference Method (FDM) - Boundary Element Method (BEM) - Finite Volume Method (FVM) 9 ME 6603 FEA Erode Sengunthar Engineering College
- 10. It can be applied for linear, non linear and continuum problems Problems mentioned in terms of differential equations or mathematical expressions are solved The whole system called the domain is expressed by differential equations Examples : Rayleigh Ritz Methods, Weighted Residual Method 10 ME 6603 FEA Erode Sengunthar Engineering College
- 11. Nature of problems - Equilibrium problem or boundary value problems - Eigen Value Problem - Propagation problems or initial value problems In Equilibrium Problems, the domain is closed and the boundary conditions are prescribed around the entire boundary. They are expressed as elliptic equations 11 ME 6603 FEA Erode Sengunthar Engineering College
- 12. Finding a function to describe the temperature of this idealised 2D rod is a boundary value problem 12 ME 6603 FEA Erode Sengunthar Engineering College
- 13. Eigen Value Problem are the special problems where solution exists only for a special values of a parameter of the problem. - Mechanical vibration problems are eigen value problems 13 ME 6603 FEA Erode Sengunthar Engineering College
- 14. The Eigen values provide the natural frequencies of the system. The eigenvectors represent the mode shapes of the system. The solution of an Eigenvalue problem can be quite cumbersome (especially for problems with many degrees of freedom), but fortunately most math analysis programs have Eigen value routines 14 ME 6603 FEA Erode Sengunthar Engineering College
- 15. Propagation problems are intial value problems in open domain in which the solution in the domain of interest. Propagation problems are governed by parabolic or hyperbolic PDE’s Propagation problems are to predict the subsequent stresses or deformation states of a system under the time-varying loading and deformation states. It is called initialvalue problems in mathematics or disturbance transmissions in wave propagation. 15 ME 6603 FEA Erode Sengunthar Engineering College
- 16. 16 ME 6603 FEA Erode Sengunthar Engineering College
- 17. Numerical method with unknown functions of the problem domain is approximated by piecewise defined functions Complex regions defining the domain is divided into smaller elements called finite elements Physical properties like shape, dimensions and other boundary conditions are imposed The elements are assembled in a proper way and the solution for the entire system can be revealed. 17 ME 6603 FEA Erode Sengunthar Engineering College
- 18. Approximated the derivatives in the governing differential equation using difference equations FDM replaces derivative terms in the differential equations by the difference equivalents Used for solving heat transfer and fluid mechanics problems Method cannot be used effectively for regions having curved and irregular boundaries 18 ME 6603 FEA Erode Sengunthar Engineering College
- 19. Unit Volume is considered as finite volume Variable properties such as Pressure, Velocity, Area, Mass, etc. can be assessed. Based on Navier Stokes Equation (Mass, Momentum and energy conservation equilibrium equations) Computational fluid Dynamics (CFD) problems are based on FVM 19 ME 6603 FEA Erode Sengunthar Engineering College
- 20. The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations The boundary element method is often more efficient than other methods, including finite elements, in terms of computational resources for problems where there is a small surface/volume ratio Solves Acoustics or Noise vibration Harshness problems Solving the problem faster Reduces the dimensionality of the problem 20 ME 6603 FEA Erode Sengunthar Engineering College
- 21. 1906 – Civil engineering problems for structure analysed for 1 D problems 1909 – Ritz Variational method (FAM) 1915 – Galerikin Weighted Residual methods (FAM) 1940 – Courant , Pragger and Synge Mathematical foundation for present form of FEA 1941 – Hreinkoff solution for elasticity problems “Frame work method” 1943 – Piecewise polynomial interpolation over triangular elements (FEA) 21 ME 6603 FEA Erode Sengunthar Engineering College
- 22. 1950’s – Argyris, Kelsery, Turner: Direct Continuum elements, Aerospace industry engineers formulated Stiffness problems 1956 – Turner derived stiffness matrix for beam, truss and other elements 1960 – FAM used for stress analysis, fluid flow, heat transfer problems and other areas 1967 – First FEA book published by Zienkiewicz and Chung 1972 – Oden’s book on non-linear problems (Development of main frames appeared) 1980 – Graphical and computational development 1990’s – Emergence of low cost powerful PC workstations and FEA adopted by mid and small scale industries 22 ME 6603 FEA Erode Sengunthar Engineering College
- 23. 23 ME 6603 FEA Erode Sengunthar Engineering College
- 24. ME 6603 FEA Erode Sengunthar Engineering College 24
- 25. ME 6603 FEA Erode Sengunthar Engineering College 25
- 26. ME 6603 FEA Erode Sengunthar Engineering College 26