Introduction to finite element method
- 1. Finite Element Method By Dr. Sohail Iqbal
- 2. Modeling & Simulation • Modeling and simulation is the foundation of predictive and functional design • Idea of modeling and simulation is inherent in the basic concept of science • To quantify the physical observations, physical laws are expressed in the form of mathematical equations • Since physical laws generally change in space and time , the equations describing the physical systems are generally differential equations • Analytical solution of such equations are possible on fast digital computers(simulations) • Numerical solutions are approximate solutions which can approach the exact solution with better numerical algorithms and sufficient computing power • The finite element method (FEM) provides a powerful numerical simulation technique to simulate complicated scientific and engineering models
- 3. Finite Difference Method Based on Taylor series expansion of the function leading to numerical approximation of derivatives. The differential equations are then changed to finite difference equations, which are then solved at discrete mesh points subject to specified initial and boundary conditions. Finite Volume Method Based on integrating the differential equations over a control volume and then applying the finite difference techniques The integral approach ensures the conservation of physical quantities
- 4. Numerical Simulation Methods DETERMINISTIC METHODS • Based on equation • Generally discretize a continuum represented by differential equations into finite domains • Represented by discrete equations STOCHASTIC METHODS • Based on probabilities
- 5. Formulations Variational Formulation • Uses energy minimization based on Rayleigh- Ritz method of variational formulation Weighted Residual Formulation • Uses error minimization based on methods such as Galerkin method of the weighted residual formulation
- 6. Variational Method • Based on minimization and maximization of functionals subject to specified constraints for the numerical approximation of an ordinary or partial differential equation. • Equation is put into equivalent weighted integral form and then an approximate solution over the entire domain is assumed to be a linear combination of the type Ū(x)= 𝑖=1 𝑛 Ci øi(x) U(x) represents an approximate solution in terms of approximately chosen functions øi and undetermined coefficients ci. • Creation of approximate solution or trial function u(x) involves creating a function øi that 1. satisfies boundary conditions 2. contains arbitrary adjustable constants Ci • These coefficient are determined such that the integral equation equivalent to original equation is satisfied
- 7. Variational Method Total P.E can be written as a functional in the form of a definite integral that has an integrand involving a single independent variable, x, a function u(x), and possibly a derivative of u(x) w.r.t x 𝑥1 𝑥2 𝑓 𝑥, 𝑢 𝑥 , 𝑢′ 𝑥 𝑑𝑥
- 8. Variational Method • Different approximation methods i.e Rayleigh- Ritz, Galerkin, and least square methods differ from each other in the choice of integral form, weight function and approximation function. • The essence of the approach is to find the functional in the form of total potential energy of the system and to invoke the stationarity of potential energy subject to specified constraints to ensure equilibrium. • The variational method suffer from the disadvantage that the approximation functions for problems with arbitrary domains are difficult to construct.
- 9. The Rayleigh Ritz Method • If variational form of the problem exist i.e we have a functional for the problem, an approximate solution for the dependent variable can be made by creating a solution function that minimizes the functional. The selection of the solution function involves assuming a function that 1. satisfies boundary conditions 2. contain arbitrary constants that can be adjusted. • The Rayleigh Ritz Method assumes a trial solution function of the form Ū(x)= 𝑖=1 𝑛 Ciøi(x) Substituting into function 𝑥1 𝑥2 𝑓 𝑥, 𝑖=1 𝑛 Ciøi(x) , 𝑖=1 𝑛 Ciøi′(x) 𝑑𝑥 • The functional now involves n arbitrary constants, Ci which are determined by finding the extreme of the functional w.r.t each constant. The resulting Ū(x) provides the approximate solution
- 10. Principle of Minimum Potential Energy For stable equilibrium a body must have minimum total potential energy Total P.E = U+V Strain P.E U= 𝐸𝐴 2 𝑥 𝜀2 𝑑𝑥 Force P.E V= -Fu The true configuration of the deformed elastic continuum yields a minimum value of total P.E. This is basics of the development of approximation techniques for the equilibrium elasticity problem.
- 11. The Principle of Minimum Potential Energy and the strong formulation are exactly equivalent statements of the same problem. The exact solution (uexact) that satisfies the strong form, renders the potential energy of the system a minimum. So, why use the Principle of Minimum Potential Energy? The short answer is that it is much less demanding than the strong formulation. The long answer is, it 1. requires only the first derivative to be finite 2. incorporates the force boundary condition automatically. The admissible displacement (which is the function that you need to choose) needs to satisfy only the displacement boundary condition
- 12. Weak Formulations • Weak formulations are tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. • In a weak formulation, an equation is no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". • Note : For details please visit http://math.stackexchange.com/questions/408615/conceptual-difference- between-strong-and-weak-formulations
- 13. Step 1. Assume a solution Rayleigh-Ritz Principle The minimization of the potential energy is difficult to perform exactly. The Rayleigh-Ritz principle is an approximate way of doing this. Step 2. Plug the approximate solution into the potential energy Step 3. Obtain the coefficients ao, a1, etc by setting boundary conditions
- 14. FE Modification of the Rayleigh-Ritz Method • In the Rayleigh-Ritz method o A single trial function is applied throughout the entire region o Trial functions of increasing complexity are required to model all but the simplest problems • The FE approach o uses comparatively simple trial functions that are applied piece-wise to parts of the region o These subsections of the region are then the finite elements 14
- 15. Galerkin Method • Galerkin method incorporates differential equations in their weak form.
- 16. Finite Element Method (FEM) Definition: • Finite element method (FEM) is a numerical technique for finding approximate solutions using the given boundry conditions. Description: • FEM gives approximate solution of the given problem. • It is done by dividing the given model into smaller bodies and then perform analysis. This is called Discretization. • Connecting points of these bodies are called Nodes. • The accuracy of solution depends upon the size of discreted body. Fine discretization would give accurate results. • Primary unknowns are displacements from which we find secondary unknowns which are stresses. 16
- 17. FEM METHODS S.No. Direct Approach Variational Approach (Rayleigh-Ritz) Weighted Residual Approach (Galerkin) 1 This approach can be used only for relatively simple problems This approach can also be used for non-linear problems Solution to non-linear and non-structural problems was achieved through the method of weighted residuals (MWR) 2 In direct approach we can directly write down the elemental equations through physical reasoning Variational approach relies on the calculus of variation and involves extremizing a functional It does not rely on a variational statement. This approach is required in order to generate the necessary and sufficient number of simultaneous equations to solve for approximate solution
- 18. FEM METHODS S.No. Direct Approach Variational Approach (Rayleigh-Ritz) Weighted Residual Approach (Galerkin) 3 Direct method is used for one dimensional cases (straight elements, like straight bar and straight beam) It can be used for 3D problems of complex shapes It can be used for 3D problems which can not even be solved by Variational approach 4 One dimensional elements are simple enough that the characteristic matrix (structural stiffness matrix) can usually be formulated by the “direct method” For complex geometries and curved surfaces this characteristic matrix cannot be obtained using direct method. Therefore, Variational approach is used For advanced types of problems, where the application of variational principle is limited, the MWR was found to provide the ideal theoretical basis for a much wider basis of problems as opposed to the Rayleigh- Ritz method (RR)
- 19. FEM METHODS S.No. Direct Approach Variational Approach (Rayleigh-Ritz) Weighted Residual Approach (Galerkin) 5 Direct method is used to determine the element behavior through matrix algebra It is a systematic procedure for producing more general FE approximations. The procedure requires only that a functional* be available Functional needed for a variational approach cannot be written for every Degree equations. Galerkin method (GM) is a way of formulating an approximate solution when one knows the DE but not the functional
- 20. Idealization Mathematical Models • “A model is a symbolic device built to simulate and predict aspects of behavior of a system.” • Abstraction of physical reality Implicit vs. Explicit Modelling • Implicit modelling consists of using existent pieces of abstraction and fitting them into the particular situation (e.g. Using general purpose FEM programs) • Explicit modelling consists of building the model from scratch
- 21. Dicretization 1. Finite Difference Discretization • The solution is discretized • Stability Problems • Loss of physical meaning 2. Finite Element Discretization • The problem is discretized • Physical meaning is conserved on elements • Interpretation and Control is easier
- 22. Solution 1. Linear System Solution Algorithms • Gaussian Elimination • Fast Fourier Transform • Relaxation Techniques 2. Error Estimation and Convergence Analysis
- 23. Interpretations Physical Interpretation: The continous physical model is divided into finite pieces called elements and laws of nature are applied on the generic element. The results are then recombined to represent the continuum. Mathematical Interpretation: The differetional equation reppresenting the system is converted into a variational form, which is approximated by the linear combination of a finite set of trial functions.
- 24. Notation Elements are defined by the following properties: 1. Dimensionality 2. Nodal Points 3. Geometry 4. Degrees of Freedom 5. Nodal Forces
- 25. Element Types
- 27. Chronicle of Finite Element Method Year Scholar Theory 1941 Hrennikoff Presented a solution of elasticity problem using one-dimensional elements. 1943 McHenry Same as above. 1943 Courant Introduced shape functions over triangular subregions to model the whole region. 1947 Levy Developed the force (flexibility) method for structure problem. 1953 Levy Developed the displacement (stiffness) method for structure problem. 1954 Argyris & Kelsey Developed matrix structural analysis methods using energy principles. 1956 Turner, Clough, Martin, Topp Derived stiffness matrices for truss, beam and 2D plane stress elements. Direct stiffness method. 1960 Clough Introduced the phrase finite element . 1960 Turner et. al Large deflection and thermal analysis. 1961 Melosh Developed plate bending element stiffness matrix. 1961 Martin Developed the tetrahedral stiffness matrix for 3D problems. 1962 Gallagher et al Material nonlinearity.
- 28. Year Scholar Theory 1963 Grafton, Strome Developed curved-shell bending element stiffness matrix. 1963 Melosh Applied variational formulation to solve nonstructural problems. 1965 Clough et. al 3D elements of axisymmetric solids. 1967 Zienkiewicz et. Published the first book on finite element. 1968 Zienkiewicz et. Visco-elasticity problems. 1969 Szabo & Lee Adapted weighted residual methods in structural analysis. 1972 Oden Book on nonlinear continua. 1976 Belytschko Large-displacement nonlinear dynamic behavior. ~1997 New element development, convergence studies, the developments of supercomputers, the availability of powerful microcomputers, the development of user-friendly general- purpose finite element software packages. Chronicle of Finite Element Method
- 29. Finite Element Analysis Steps Classifying the problem: The First step is to identify the problem that what physical conditions are given and what is required to be found. Creating a Model: The analysis is applied to an appropriate model representing the original physical model. Analyzing a model is much easier than dealing with the original problem. Discretization of the Model The model is divided into a finite continuous mesh to be analysed using FEA. Defining relations Appropriate relations/ Stiffness matrices are formed to find out primary unknowns ( Displacements ). Solving for unknowns Primary unknowns are then used to find secondary unknowns which are stresses etc. Interpreting the results Results are then interpreted on the original physical problem. 30
- 30. Finite Element Analysis Steps used in FEA (cont) : 31
- 31. Finite Element Analysis Advantages: • FEA is applicable to any field problem. • There is no restriction of shape or size of the model. • Boundry conditions or loading conditions can be taken anywhere on the model according to the required unknowns. • Can also deal with composite materials. • Mesh size can easily be changed according to the desired accuracy. 32
- 32. Finite Element Analysis Applications It is used to analyze structural and non structural problems – Stress analysis – Buckling – Vibration analysis – Fluid flow – Heat transfer – Distribution of electric and magnetic potentials etc 33
- 33. Finite Element Analysis Software’s • Includes: – NASTRAN/PATRAN, CREO, ADAMS, INTELLI-SUITE, ANSYS, COSMOS, NISA, ALGOR, CATIA etc • Involves: – Pre-processing – Numerical analysis – Post-processing 34
- 34. 1D elements & computations procedure • 1D elements include – Straight bar loaded axially – Straight beam loaded laterally – Bar conducts heat or electricity etc • Degree of freedom (DOF) is number of parameters that may vary independently • Total DOF in structure = number of nodes * DOF at each node • Order of structural stiffness = Total DOF in structure * number of nodes in structure • i.e spring element with 2 nodes has order of structural stiffness of 2*2 35
- 35. References • Concepts & applications of finite element analysis 4th edition, by Robert D. cook • Fundamentals of Finite element analysis 1st edition, by David V. Hutton • Class notes • Wikipedia • Web Links 36