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Lecture on introduction to finite element methods & its contents

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## Lecture on introduction to finite element methods & its contents

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**Finite Element Method (FEM)**Module Code: Lecture on Introduction to FEM, 15 September 2019 By: Dr. Mesay Alemu Tolcha Faculty of Mechanical Engineering, JiT, Ethiopia. 1/21**Content I**1. Introduction • What is the Finite Element Method? • Common Application Areas • Modeling and Finite Element Methods • FEM Procedures 2. Direct approach to solution of one-dimensional problem • Spring Element and Truss Element • Analysis of Truss Structure • Heat Transfer in a composite wall • Pipe flow element 3. Finite Element Methods Formulation • Introduction • Weighted Residual Methods • Galerkin’s Method • Galerkin’s Finite Element Method/ Dicretization Principle • Variational Method 2/21**Content II**4. Interpolation Functions and Isoparametric Elements • One-Dimensional Elements • Two-Dimensional Elements • Triangular Elements • Area Coordinates • Rectangular Elements • Natural Coordinates • Three-Dimensional Elements • Isoparametric Formulation • Axisymmetric Elements 5. Numerical Integration • Transformation of integral from global to local coordinates • Jacobian Matrix • Gaussian quadrature, Trapezoidal rule, Simpson’s rule 6. Plain Stress and Strain Analysis by FEM • Constitutive Equations 3/21**Outline III**• Plane stress and Plane Analysis using plate Elements • Axis Symmetric problems 7. Plane Steady State heat Conduction Analysis • Element Thermal Stiffness Matrix and Load Vector Derivation • Assembly of Element Matrix • Boundary Conditions • Imposition of Prefixed Temperature • Imposition of Heat Flux • Imposition of Convective Heat Transfer 8. Transient Heat Conduction Analysis • Element Thermal Stiffness and Capacitance Matrix Derivation • Assembly of element matrix • Imposition of boundary conditions 9. Dynamic Analysis of a Beam by FEM • Element Mass, stiffness matrices and load vectors • Modal analysis 4/21**...Cont**10. Software Practice • Python • ABAQUS/ANSYS Assessment Strategy: • Exercises: 15% • Project 1: 15% • Project 2: 20% • Final exam: 50% 5/21**Application area of FEM**What do you expect from this course? • The finite element method is now widely used for analysis of structural engineering problems. ◦ In civil, aeronautical, mechanical, ocean, mining, nuclear, biomechanical,... engineering • Since the first applications two decades ago, • Now we see applications in linear, nonlinear, static and dynamic analysis • Various computer programs are available and in significant use • Therefore, how do we cover the entire course within the stipulated time? 6/21**Engineering Analyzing Tools**Experimental design & test Theoretical justification Analytical methods Constitutive equations Math models New Exist Governing models Physical problem FEM solution process Establish FEM models Refine the models Solve the models/simulate Interpret the results No Results OK? Yes Extract the results Experimental validation 7/21**Methods of Engineering Analysis...**There are three approaches usually followed to undertake any engineering analysis: • Experimental methods ◦ Accurate but it needs man power and materials. So, it is time consuming and cost to process. • Analytical methods ◦ Quick and closed form solution but for simple geometries and simple loading conditions. • FEM or approximate methods ◦ Approximate but acceptable solution for problems involving complex material properties and loading. 8/21**Numerical Methods**The common four numerical methods are: • Finite Difference Method ◦ For heat transfer, fluid and structural mechanics. This method is difficult to use when regions have curved or irregular boundaries • Boundary element method (BEM) and Finite volume method (FVM)... to solve thermal and computational fluid dynamics • Finite Element Method ◦ This method is a popular numerical technique that used to determine the approximated solution for a partial differential equation (PDE) • Functional approximation/Method implemented in FEM ◦ Rayleigh-Ritz (variation approach) for complex structural ◦ Weighted residual method for solving non-structural 9/21**What is the finite element method ?**Finite element method is a numerical method for solving problems of Engineering and Mathematical Physics. • So, Finite element method can be viewed simply as a method of finding approximate solutions for partial differential equations. • Or as a tool to transform partial differential equations into algebraic equations, which are then easily solved. In this method the body is considered as an assemblage of elements connected at a finite number of Nodes. • On other the hand, the FEM reduces the degree of freedom from infinite to finite with the help of discretion or meshing (nodes and elements) 10/21**Introduction to FEM more...**Structural/Stress Analysis Figure 1: Application area of FEM 11/21**Introduction to FEM more...**b) a) d) c) Figure 2: Multiphyics application. a) Car crush, b)Thermal stress analysis, c) Electromagnetic analysis, d) Aerodynamics analysis 12/21**History of Finite Element Methods**• Hrenikoff (1941), proposed framework method • Courant (1943), used principle of stationary potential energy • 1953, -Stiffness equations were written and solved using digital computers by several research • Clough (1960), made up the name "finite element method" • 1970s, -FEA carried on "mainframe" computers • 1980, -FEM code run on PCs • 2000s, -Parallel implementation of FEM (large-scale analysis, virtual design • There are improvement every times in feature also..... 13/21**Major FEM Analysis Steps**• Discretization of the domain into a finite number of subdomains (elements) • Selection of interpolation functions • Development of the element matrix for the subdomain (element) • Assembly of the element matrices of each subdomain to obtain the global matrix for the entire domain, • Imposition of the boundary conditions • Solution of equations • Additional computations (if desired). 14/21**Simple Example,...Approximation the area of a circle**Element Node Edge Figure 3: Discretisation of a Continuum Deiscretization modelling is dividing a body it into a equivalent system of finite elements interconnected at finite number points on each element called node. 15/21**Finite Element Method Continued**• Assume a trial Solution that satisfies the boundary condition • The domain residual or error is calculated while satisfying the differential equation • The weighted sum of the domain residual computed over the entire domain is rendered zero • The accuracy of the assumed trial solution can be improved by taking additional higher order terms but computations becomes tedious. • Therefore, it is not a trivial task to choose a single trial function over the entire domain satisfying the boundary condition. • It is preferred to discretize the domain in to several elements and use several piece wise continuous trial functions, each valid with in a segment. 16/21**Common Types of Elements**1. One-Dimensional Elements (1D) • Line, Rods, Beams, Trusses, Frames 2. Two-Dimensional Elements (2D) • Triangular, Quadrilateral Plates, Shells, 2D Continua 3. Three-Dimensional Elements (3D) • Tetrahedral, Rectangular, Prism (Brick), 3-D Continua If this the case, finite element method only makes calculations at a limited (finite) number of points and then interpolated the results for the entire domain Therefore, any continuous solution field such as stress, displacement, temperature, pressure, etc. can be approximated by a discrete model composed of a set of piecewise continuous functions defined over a finite number of sub-domains. 17/21**Two-Dimensional Discretization Refinement**Node With 228 elements Piecewise linear representation Triangular element Figure 4: A finite element partition To understand the physics of such representation, mathematical equation must be developed. The finite element equation can be derived by either of the following methods: 1. Direct equilibrium method 2. Variation method /or Rayleigh-Ritz method 3. Weighted Residual method /or Galerkin method 18/21**FE Equation Derivation**• The direct method is easy to understand but difficult for programming while the variation and weighted residual methods are difficult to understand but easy from a programming. • The Galerkin Weighted Residual formulation is the most popular from the finite element point of view. Weighted Residual Method • For any problem where the differential equation of the phenomenon can be easily formulated, Weighted residual becomes very useful. But, • For structural problems, potential energy function can be easily formed, Rayleigh-Ritz method is used. 19/21**FE Equation Derivation...In Direct Method**Direct approach to solution of one-dimensional problem including the following points: • Spring Element • Truss Element • Analysis of Truss Structure • Element Transformation • Heat Transfer in a composite wall • Pipe flow element Direct method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. 20/21**FE Equation Derivation...**There are many types of weighted residual methods, among of them, three are very popular. 1. Point collocation method 2. Least square method 3. Galerkin’s method • Among these three, the Galerkin approach has the widest choice and is used in FEMs. • In Galerkin’s, the trial function itself is considered as the weighting function. • In point collocation, residuals are set to zero at n different locations 21/21

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