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Fundamentals of Finite Element Analysis.
- 1. FUNDAMENTALS OF FINITE ELEMENT ANALYSIS BY Dr.VINAYAK H.KHATAWATE Associate Professor, SVKM’s Dwarkadas J. Sanghvi College of Engineering, Mumbai.
- 2. INTRODUCTION • METHODS TO SOLVE ANY ENGINEERING PROBLEM 1. ANALYTICAL METHOD 2. NUMERICAL METHOD 3. EXPERIMENTAL METHOD
- 3. Analytical Method Numerical Method Experimental Method - Classical approach - 100 % accurate results - Closed form solution - Applicable only for simple problems like cantilever & simply supported beams etc. - Complete in itself -Mathematical representation - Approximate, assumptions made - Applicable even if physical prototype not available (initial design phase) - Real life complicated problems - Results can not be believed blindly & must be verified by experimental method or hand calculation for knowing the range of results - Actual measurement - Time consuming & needs expensive set up - Applicable only if physical prototype is available - Results can not be believed blindly & min. 3 to 5 prototypes must be tested Though analytical methods could also give approximate results if the solution is not closed form, but in general and broad sense, analytical methods are considered as closed form solutions i.e. 100 % accurate. Finite Element Method: Linear, Nonlinear, Buckling, Thermal, Dynamic & Fatigue analysis Boundary Element Method: Acoustics / NVH( noise vibration and harshness) Finite Volume Method: CFD (Computational Fluid Dynamics) & Computational Electromagnetics Finite Difference Method: Thermal & Fluid flow analysis (in combination with FVM) - Strain gauge - Photo elasticity - Vibration measurements - Sensors for temp. & pressure etc. - Fatigue test
- 4. Procedure for solving any analytical or Numerical problem There are two steps Step 1) Writing governing equation - Problem Definition or in other words formulating the problem in the form of a mathematical equation. Step 2) Mathematical solution of governing equation. Final result is summation of step 1 & step 2. Result will be 100 % accurate when there is no approximation at either of the steps (Analytical method).Numerical methods make approximation at step 1 as well as at step 2 & hence all the numerical methods are approximate.
- 5. Analytical Approximation" Numericall Approximation Step 1 Little or no approximation Approximate Step 2 Little or no approximation Approximate Accuracy High accuracy Approximate results
- 6. THE BASIC CONCEPT OF THE FINITE ELEMENT METHOD • The most distinctive feature of the finite element method that separates it from others is the division of a given domain into a set of simple subdomains, called finite elements. • Here the basic ideas underlying the finite element method are introduced via following simple example. • Determination of the circumference of a circle using a finite number of line segments
- 7. Approximation of the Circumference of a Circle Consider the problem of determining the perimeter of a circle of radius R . a) Circle of radius R b) Uniform and nonuniform meshes used to represent the circumference of the circle. c) A typical element.
- 8. STEPS IN FEA 1. Finite element descretization 2. Element equations. 3. Assembly of element equations and solutions. 4. Convergence and error estimate.
- 9. 1.Finite element discretization. • First, the domain (i.e., the circumference of the circle) is represented as a collection of a finite number (n) of subdomains, namely, line segments. This is called discretization of the domain. • Each subdomain (i.e., line segment) is called an element. • The collection of elements is called the finite element mesh. • The elements are connected to each other at points called nodes. • In the present case, we discretize the circumference into a mesh of five (n = 5) line segments. • Mesh may be either uniform or nonuniform mesh (see Fig )
- 10. 2.Element equations. • A typical element (i.e., line segment, he ) is isolated and its required properties, i.e., length, are computed by some appropriate means. Let he be the length of element he in the mesh. For a typical element he, he is given by (see Fig. ) • he = 2R sin (1/2) θe • where R is the radius of the circle and θe <π is the angle subtended by the line segment. The above equations are called element equations.
- 11. 3.Assembly of element equations and solution. • The approximate value of the circumference (or perimeter) of the circle is obtained by putting together the element properties in a meaningful way; this process is called the assembly of the element equations. It is based, in the present case, on the simple idea that the total perimeter of the polygon (assembled elements) is equal to the sum of the lengths of individual elements: n • Pn =∑ he e=1 • Then Pn represents an approximation to the actual perimeter, p. If the mesh is uniform, or he is the same for each of the elements in the mesh, then θe = 2 π /n, and we have • Pn = n( 2R sin [π/n] )
- 12. 4.Convergence and error estimate. • For this simple problem, we know the exact solution: p = 2 π R. • We can estimate the error in the approximation and show that the approximate solution Pn converges to the exact p in the limit as n tends to infinite. • Consider the typical element Ωe. The error in the approximation is equal to the difference between the length of the sector and that of the line segment. i.e.,Ee=Se-he. (see Fig. C): • where Se = R.θe is the length of the sector
- 14. 5.conclusion • it is shown that the circumference of a circle can be approximated as closely as we wish by a finite number of piecewise- linear functions. • As the number of elements is increased, the approximation improves, i.e., the error in the approximation decreases.
- 15. 6.Advantages of the Finite Element Method • 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.
- 16. Advantages of the Finite Element Method (cont.) • 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.
- 17. 7.Disadvantages of the Finite Element Method • A specific numerical result is obtained for a specific problem. A general closed-form solution, which would permit one to examine system response to changes in various parameters, is not produced. • The FEM is applied to an approximation of the mathematical model of a system (the source of so-called inherited errors.) • Experience and judgment are needed in order to construct a good finite element model. • A powerful computer and reliable FEM software are essential. • Input and output data may be large and tedious to prepare and interpret.
- 18. Disadvantages of the Finite Element Method (cont.) • Numerical problems: • Computers only carry a finite number of significant digits. • Round off and error accumulation. • Can help the situation by not attaching stiff (small) elements to flexible (large) elements. • Susceptible to user-introduced modeling errors: • Poor choice of element types. • Distorted elements. • Geometry not adequately modeled. • Certain effects not automatically included: • Buckling • Large deflections and rotations. • Material nonlinearities . • Other nonlinearities.
- 19. 8.FEA Application Areas • Automotive industry • Static analyses • Modal analyses • Transient dynamics • Heat transfer • Mechanisms • Fracture mechanics • Metal forming • Crashworthiness • Aerospace industry » Static analyses » Modal analyses » Aerodynamics » Transient dynamics » Heat transfer » Fracture mechanics » Creep and plasticity analyses » Composite materials » Aeroelasticity » Metal forming » Crashworthiness • Architectural » Soil mechanics » Rock mechanics » Hydraulics » Fracture mechanics » Hydroelasticity
- 20. • Primary Variable (PV) • Secondary Variable (SV) 9.Types of Variables in FEA:
- 21. 21 Field Variables: Primary Variables (PV) These are the Dependent Variables y = f(x) • These are the Variables derived from PVs • Mostly, derivatives of PVs: y`, y`` etc. • Less accurate than PVs • Obtained only after finding out PVs Secondary Variables (SV)
- 22. 22 10.Domain: A Domain is a Field with end boundary conditions u(0) L u(L) u x u = f(x) Scalar/Vector Field with End Boundary Conditions L—Extent of Domain
- 23. 23 Definition of Boundary Conditions: Specified values of field variables (or related variables such as derivatives of field variables) on the boundaries of the Field or Domain.
- 24. 24 u(0)=1 u(L)=2 Essential Non-Homogeneous BCs u(0)=0 u`(L)=1 Mixed Non-Homogeneous BCs u(0)=0 u(L)=0 Essential Homogeneous BCs u`(0)=0 u`(L)=0 Natural Homogeneous BCs Homogeneous BCs (BCs = 0) u(0)=0 u`(2)=1 BCs: PV-SV Mixed BCs u`(0)=0 u`(2)=1 BCs: SV-SV Natural BCs (Neumann BCs) u(0)=0 u(4)=2 BCs: PV-PV Essential BCs (Dirichlet BCs) Classifications of Boundary Conditions (BCs): Non-homogeneous BCs (one or both BCs 0)
- 25. 11. Process Flow in a Typical FEM Analysis Start Problem Definition Pre-processor • Creates geometry of nodes and elements (ex: ANSYS, Hypermesh, etc.) • Assigns material property data. • Assigns boundary conditions (loads and constraints.) Processor • Generates element shape functions • Calculates master element equations • Calculates transformation matrices • Maps element equations into global system • Assembles element equations • Introduces boundary conditions • Performs solution procedures Post-processor • Prints or plots contours of stress components. • Prints or plots contours of displacements. • Evaluates and prints error bounds. Analysis and design decisions Stop Step 1, Step 4 Step 6 Steps 2, 3, 5
- 26. 12.Information Available from Various Types of FEM Analysis • Static analysis • Deflection • Stresses • Strains • Forces • Energies • Dynamic analysis • Frequencies • Deflection (mode shape) • Stresses • Strains • Forces • Energies • Heat transfer analysis »Temperature » Heat fluxes » Thermal gradients » Heat flow from convection faces • Fluid analysis » Pressures » Gas temperatures » Convection coefficients » Velocities
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