The Finite Element Method : Introduction For studying physical phenomena or engineering problems, engineers and scientists are involved with two major tasks: Mathematical formulation of the physical problem: The behaviour of the problem is expressed or modeled by means of integro -differential equations. Such equations are quite complicated and are known as behaviour /governing equations. Numerical analysis of the mathematical model: It is very difficult or impossible to solve the complicated integro -differential governing equations by conventional methods of mathematics. Hence numerical methods which yield approximate solutions are adopted. M.B.DAVANAGERI SCEM MANGALORE

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Many numerical methods of solving the behavoiur /governing equations have been tried. But most of them have their own restrictions. The Finite Element Method (FEM) is the only numerical method which has got less restrictions of usage. In fact it can be successfully applied to almost all fields of engineering like structural engineering, thermal problems, fluid flow, electrical field, magnetism, acoustics, earth quake analysis, seepage problems, soil mechanics, etc. M.B.DAVANAGERI SCEM MANGALORE

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The basis of Finite Element Method (FEM) is ‘ Discretization ’ i.e., to represent the region (continuum) of the problem by an assemblage of finite number of standard shaped sub divisions known as ‘Elements’. Finite number of elements is a requirement of the numerical method. These elements are inter connected to each other at common points known as ‘Nodes’. The properties (geometrical and material) of the elements are first established in the form of ‘Element Equations’. M.B.DAVANAGERI SCEM MANGALORE

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They are then assembled to obtain a very large set of simultaneous equations known as ‘Global Equations’, which closely represents the behaviour of the whole problem. Boundary conditions and loads are applied to these global equations. Using a suitable matrix method the global equations are solved to obtain unknown nodal displacements. The element stresses and strains are then obtained from the evaluated nodal displacements. Hence, FEM with the help of numerical procedures produces a solution, which is close to the exact solution, Hence, FEM provides an approximate solution M.B.DAVANAGERI SCEM MANGALORE

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Boundary Value Problems: Most of the problems in engineering are Boundary Value Problems. These are problems where the values of the unknown variables are known at some region of the boundary of the problems. Example (1): In a cantilever beam problem, we intend to find the unknown variables (deflection and slope) at all points along the axis of the beam. But we know the values of deflection and slope at the fixed end which is zero each. Example (2): In a heat transfer problem of a furnace wall we intend to find the temperature distribution in the wall, i.e., temperature is the unknown variable. But we know the temperatures at the inner surface and the outside ambient surface of furnace walls. M.B.DAVANAGERI SCEM MANGALORE

? Why FEM is used widely to solve problems of engineering:

? Why FEM is used widely to solve problems of engineering In all the problems of engineering, we may require to find the value of the dependent variable at any specified point in the continuum. For this, the governing differential equations must be solved to get the value of the dependent variable. But, in actual practice, engineering problems involve complicated geometries (continuums), loadings and varying material properties. Due to this, it may be impossible to specify the boundary conditions, consider material properties and solve the governing differential equation. In such a situation, we go for numerical methods such as “Finite Element Method” (FEM) to get approximate but acceptable solutions. M.B.DAVANAGERI SCEM MANGALORE

Definition of FEM:

Definition of FEM The Finite Element Method is a numerical method for solving problems of engineering and mathematical physics where their behaviour /governing equations are expressed by integral or differential equations. The FEM formulation of the problem results in a set of simultaneous algebraic equations for solution, instead of requiring the solution of the governing differential equation. This yields approximate values of the variables at discrete points in the continuum M.B.DAVANAGERI SCEM MANGALORE

General Steps of the Finite Element Method:

General Steps of the Finite Element Method Select Element type and discretize the continuum Select a Displacement Function Define the Strain/Displacement and Stress/Strain Relationships Derive the Element Stiffness Matrix and Element Equations Assemble the Element Equations to obtain Global Equations Apply Boundary Conditions and modify the Global Equations Solve for unknown variables Solve for Element Strains and Stresses Interpret the results M.B.DAVANAGERI SCEM MANGALORE

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P 1 P 2 Finite Element Approximation Discretization Elements Data Evaluation of element stiffness matrices [K] e Assembly of element properties to get global equations {F} = [K] g {Q} Solution of the load deflection equations {Q} = [K g ] -1 { F} Apply boundary conditions Calculation of strains & stresses { } e = [B] e {q} e { } e = [D]{ } e Machine Tool Body M.B.DAVANAGERI SCEM MANGALORE

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Advantages of FEM :

Advantages of FEM Model irregularly shaped bodies quite easily. Handle general load conditions without difficulty. Model bodies composed of different materials because their element equations are evaluated individually. Handle unlimited numbers and kinds of Boundary Conditions Vary the size of the elements to make it possible to use small elements where ever necessary. Alter the FEM Model relatively easily & cheaply Include dynamic effects Handle nonlinear behaviour existing with large deformations and non linear materials M.B.DAVANAGERI SCEM MANGALORE

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Can readily handle complex geometry: The heart and power of the FEM. Can handle complex analysis types: Vibration Transients Nonlinear Heat transfer Fluids Can handle complex loading: Node-based loading (point loads). Element-based loading (pressure, thermal, inertial forces). Time or frequency dependent loading. Can handle complex restraints: Indeterminate structures can be analyzed. M.B.DAVANAGERI SCEM MANGALORE

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Can handle bodies comprised of nonhomogeneous materials: Every element in the model could be assigned a different set of material properties. Can handle bodies comprised of nonisotropic materials: Orthotropic Anisotropic Special material effects are handled: Temperature dependent properties. Plasticity Creep Swelling Special geometric effects can be modeled: Large displacements. Large rotations. Contact (gap) condition. M.B.DAVANAGERI SCEM MANGALORE

Disadvantages of FEM :

Disadvantages of FEM Solution obtained by FEM is approximate. A large number of iterations must be performed to obtain convergence of FEM solution to actual solution. Selection of proper elements (type and shape) and displacement functions depends upon the user’s knowledge, experience and skill. User of FEM should possess good modelling skills so that boundary conditions are well defined. Interpretation of FEM results is again the task of the human user. M.B.DAVANAGERI SCEM MANGALORE

Application of FEM:

Application of FEM Structural: Stress analysis of truss and frame, stress concentration problems Buckling problems Vibration analysis Non Structural: Heat Transfer Fluid flow, including seepage through porous media Distribution of electric or magnetic potential Acoustics Others: Biomedical engineering problems – analysis of human spine, skull, hip joints, jaw/gum tooth implants, heart and eyes. M.B.DAVANAGERI SCEM MANGALORE

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Example: A bracket Modeling a physical problem ...General scenario Physical Problem Mathematical Model Numerical model Does answer make sense? Refine analysis Happy YES! No! Improve mathematical model Design improvements Structural optimization Change physical problem M.B.DAVANAGERI SCEM MANGALORE

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