FEM-Introduction to Finite Element Method.pptx

FEM Intro

1. ME- 6202 Finite Element Analysis Lect. # 1 Introduction to FEA Dr. Nazeer Ahmad Anjum Mechanical Engineering Program University of Engineering Taxila

2. Introduction to Finite Element Analysis Introduction to Matrix Algebra Introduction to the stiffness (displacement) Method Theory of Elasticity Intro to Truss & Beam Problems solving by Direct Approach Higher Order and Iso -Parametric Elements Development of Plane stress and Plane strain Stiffness equation Variational and Weighted Residual Formulations General Approach to Structural Analysis (Elasticity, -) Parameter Functions; C O -Continuous Shape Functions Force & Stress Analysis Heat Transfer and Mass transport Fluid flow in porous Media Analysis of Thermal Stresses Course Contents 3 9/19/2019

3. A First Course in the Finite Element Method by D. Logan, Fifth Edition. Applied finite Element Analysis for Engineers by Stasa Fundamentals of the FEM for Heat and Fluid Flow by Roland W. Lewis The Finite Element Method in Engineering by S.S. Rao . An Introduction to the Finite Element Method by J.N. Reddy, Tata McGraw-Hill Edition . Recommended Books 4 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

4. Why you need to study Finite Element Analysis? Introduction 5 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

5. Analysis is the Key to effective design. Introduction 6 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

6. introduction Definition Basic Terms Coordinate System Discretization History Brief Description General Steps Advantages FEM Today’s Topics 7 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

7. Many problems in Engineering and applied sciences are governed by differential or integral equations . The solutions to these equations would provide an exact , closed-form solution to the particular problem being studied. However, complexities in the geometry , properties and in the boundary conditions that are seen in most real-world problems usually means that an exact solution cannot be obtained or obtained in a reasonable amount of time . Introduction 8 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

8. Current product design cycle times imply that engineers must obtain design solutions in a ‘short’ amount of time . They are content to obtain approximate solutions that can be readily obtained in a reasonable time frame , and with reasonable effort . The FEA is one such approximation solution technique . Introduction 9 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

9. It is a numerical method for solving problems of engineering and mathematical physics . Useful for problems with complicated geometries , loadings , and material properties where analytical solutions can not be obtained. FEM Definition 10 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

10. Node “ The Point of attachment of element to an other element is called NODE .” Node, DOF, Coordinate System 11 9/19/2019 Degrees of freedom “ The number of independent coordinates necessary to specify the configuration of a system.” Coordinate System “Node point locations are specified relative to a particular coordinate system. The global coordinate system is usually the Cartesian coordinate system with axes labeled X,Y,Z. The three axis are mutually perpendicular. Dr. Nazeer A. Anjum, MED, UET, Taxila

11. Base Function “A function describing the distribution of deformation parameters (displacements) inside the element (between nodes).” Shape Function “A function describing the distribution of strains inside the element , it represents a derivative of the base function .” Mesh density “It influences the time consumption and accuracy of the solution.” Matrixes “These are created by summarization of contributions of the individual elements of displacements , stiffness , base function Node, DOF, Coordinate System 12 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

12. “T he stiffness K is defined as the force necessary to produce a unit displacement” or “The ratio of force to displacement .” or “I s the resistance of an elastic body to deflection by an applied force. The stiffness k of a body that deflects a distance δ under an applied force F is:” “ The structural stiffness matrix is a square , symmetric matrix with dimension equal to the number of degrees of freedom .” Stiffness Matrix 13 9/19/2019 k =F/  Dr. Nazeer A. Anjum, MED, UET, Taxila

13. “A method of representing points in a space of given dimensions by coordinates.” Local Coordinate System “Measurement indicates into a local coordinate system or a local Coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc. Global Coordinate System “ This system is usually the traditional Cartesian coordinate system.” Coordinate System 14 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

14. X Y Z x y Global Coordinate System Local Coordinate System Coordinate System 15 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

15. Elements 16 9/19/2019 In FEA, a complex region defining a continuum is discretized into simple geometric shapes called element. 1 2 3 Dr. Nazeer A. Anjum, MED, UET, Taxila

16. Nodes 17 9/19/2019 The properties and governing relationships are assumed over elements and expressed mathematically in terms of unknown values at specific points in the elements called nodes . Dr. Nazeer A. Anjum, MED, UET, Taxila 3 1 2 5 4

17. 3 1 2 5 4 1 2 3 x y X Y Z Elements, Nodes & Coordinate System 18 9/19/2019 An assembly process is used to link the individual elements to the given system . When the effects of loads and boundary conditions are considered, a set of linear or nonlinear algebraic equations is usually obtained. Solution of these equations gives the approximate behavior of the continuum or system Dr. Nazeer A. Anjum, MED, UET, Taxila

18. Introduction 19 9/19/2019 The continuum has an infinite number of DOF , while the discretized model has a finite number of DOF. This is the origin of the Finite Element Method . The number of equations is usually rather large for most real-world applications of the FEM, and requires the computational power of the digital computer . The FEM has little practical value if the digital computer were not available. Advances in and ready availability of computers and software has brought the FEA within reach of engineers working in small industries , and even studies. Dr. Nazeer A. Anjum, MED, UET, Taxila

19. Introduction 20 9/19/2019 Two features of the FEA are worth noting. The piecewise approximation of the physical field (continuum) on finite element provides good precision even with simple approximating functions. Simply increasing the number of elements can achieve increasing precision. The locality of the approximation leads to sparse equation systems for a discretized problem. This helps to ease the solution of problems having very large numbers of nodal unknowns . It is not uncommon today to solve systems containing a million primary unknowns. Dr. Nazeer A. Anjum, MED, UET, Taxila

20. There are two types of frame of references Local frame of reference “When one of the axis is always inclined at an angle.” T hat is only expected to function over a small region or a restricted region of space or spacetime . Global frame of reference “This type of reference is assembled with the whole structure. It is not necessarily to align with the axis. When the nodes are joined continuously in an order, it is called GFR.” Frame of References 21 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

21. The Finite E lement A nalysis ( FEA), is a computational technique used to obtain approximate solutions of boundary value problems in engineering. Boundary value problems are also called field problems . The field is the domain of interest and most often represents a physical structure . The field variables are the dependent variables of interest governed by the differential equation. The boundary conditions are the specified values of the field variables (or related variables such as derivatives) on the boundaries of the field. Principles of FEA 22 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

22. “The sub-division of a continuum or a large element into a large number of small discrete elements .” Model body by dividing it into an equivalent system of smaller bodies or units (finite elements) interconnected at points common to two or more elements (nodes or nodal points) and/or boundary lines and/or surfaces. Discretizations 23 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

23. One-Dimensional Elements Line Rods, Beams, Trusses, Frames Two-Dimensional Elements Triangular, Quadrilateral Plates, Shells, 2-D Continua Three-Dimensional Elements Tetrahedral, Rectangular Prism (Brick) 3-D Continua Common Types of Elements 24 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

24. One-Dimensional Frame Elements Two-Dimensional Triangular Elements Three-Dimensional Brick Elements Discretizations Examples 25 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

25. It is difficult to document the exact origin of the FEM, because the basic concepts have evolved over a period of 150 or more years. Basic ideas of the finite element method originated from advances in aircraft structural analysis in 1940 . In 1941 , Hrenikoff presented a solution of elasticity problems using the “frame work method.” Courant’s paper, which used piecewise polynomial interpolation over triangular subregions to model torsion problems, appeared in 1943 . A book by Argyris in 1955 on energy theorems and matrix methods laid a foundation for further developments in finite element studies. Turner et al. derived stiffness matrices for truss , beam , and other elements and presented their findings in 1956 . Historical Background 26 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

26. The term “ finite element ” was first coined and used by Clough in 1960 . In the early 1960s , engineers used the method for approximate solution of problems in stress analysis , fluid flow , heat transfer , and other areas. The first book on finite elements by Zienkiewicz and Chung was published in 1967 . In the late 1960s and early 1970s , finite element analysis was applied to nonlinear problems and large deformations . Historical Background 27 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

27. The 1970s marked advances in mathematical treatments , including the development of new elements , and convergence studies. Most commercial FEM software packages originated in the 1970s (ABAQUS, ADINA, ANSYS, MARK, PAFEC) and 1980s (FENRIS, LARSTRAN ‘80, SESAM ‘80.) Today , developments in distributed or multi-node computers and availability of powerful microcomputers have brought this method within reach of students and engineers working in small industries . The FEM is one of the most important developments in computational methods to occur in the 20th century . In just a few decades , the method has evolved from one with applications in structural engineering to a widely utilized and richly varied computational approach for many scientific and technological areas . Historical Background 28 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

28. Grew out of aerospace industry . Post-WW II jets , missiles , space flight Need for light weight structures Required accurate stress analysis Paralleled growth of computers Structural/Stress Analysis Fluid Flow Heat Transfer Electro-Magnetic Fields Soil Mechanics Acoustics Application of FEM 29 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

29. Obtain a set of algebraic equations to solve for unknown nodal quantity (displacement). Secondary quantities (stresses and strains) are expressed in terms of nodal values of primary quantity Feature 30 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

30. F = The external force is positive to the right and negative to the left. u ( ) = The displacement is positive to the right and negative to the left. Sign Convention F = Ku or (  ) 31 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

31. Easy input - preprocessor. Solves many types of problems Modular design - fluids, dynamics, heat, etc. Can run on PC’s now. Relatively low cost. High development costs. Less efficient than smaller programs, Often proprietary. User access to code limited. Advantages of General Purpose Programs 32 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

32. Step 1 - Discretization: The problem domain is discretized into a collection of simple shapes, or elements. Step 2 - Develop Element Equations : Element equations are developed be applying laws of physics related to the problem, and typically Galerkin’s Method or variational principles. Step 3 - Assembly: The element equations for each element in the FEM mesh are assembled into a set of global equations that model the properties of the entire system. Step 4 - Application of Boundary Conditions: Solution cannot be obtained unless boundary conditions are applied. They reflect the known values for certain primary unknowns. Imposing the boundary conditions modifies the global equations. Step 5 - Solve for Primary Unknowns: The modified global equations are solved for the primary unknowns at the nodes. Step 6 - Calculate Derived Variables: Calculated using the nodal values of the primary variables. Finite Element Method Steps 33 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

33. Primary line elements consist of bar (or truss) and beam elements. They have a cross-sectional area but are usually represented by line segments. The simplest line element (called a linear element ) has two nodes, one at each end, although higher-order elements having three nodes or more (called quadratic , cubic , etc. elements ) also exist. Step 1 - Discretize and Select Element Types 34 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

34. The most common three-dimensional elements are tetrahedral and hexahedral (or brick ) elements ; they are used when it becomes necessary to perform a three- dimensional stress analysis. The basic three dimensional elements have corner nodes only and straight sides , whereas higher-order elements with mid-edge nodes (and possible mid-face nodes) have curved surfaces for their sides Step 1 - Discretize and Select Element Types 35 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

35. Step 1 - Discretize and Select Element Types 36 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

36. Step 1 - Discretize and Select Element Types 37 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

37. The axisymmetric element is developed by rotating a triangle or quadrilateral about a fixed axis located in the plane of the element through 360°. This element can be used when the geometry and loading of the problem are axisymmetric . Step 1 - Discretize and Select Element Types 38 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

38. Step 1 - Discretize and Select Element Types 39 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

39. Step 2 involves choosing a displacement function within each element . The function is defined within the element using the nodal values of the element. Linear , quadratic , and cubic polynomials are frequently used functions because they are simple to work with in finite element formulation. Step 2 - Select a Displacement Function 40 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

40. The functions are expressed in terms of the nodal unknowns (in the two-dimensional problem, in terms of an x and a y component). Hence, the finite element method is one in which a continuous quantity , such as the displacement throughout the body, is approximated by a discrete model composed of a set of piecewise-continuous functions defined within each finite domain or finite element. Step 2 - Select a Displacement Function 41 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

41. Strain/displacement and stress/strain relationships are necessary for deriving the equations for each finite element. For one-dimensional small deformation, say in the ‘x’ direction, we have strain  x , related to displacement ‘u’ by Stresses must be related to the strains through the stress/strain law (generally called the constitutive law ). The simplest of stress/strain laws, Hooke’s law , often used in stress analysis, is given by: Step 3 – Define the strain/Displacement & Stress/Strain Relationships 42 9/19/2019  x  E  x Dr. Nazeer A. Anjum, MED, UET, Taxila

42. Direct Equilibrium Method - According to this method, the stiffness matrix and element equations relating nodal forces to nodal displacements are obtained using force equilibrium conditions for a basic element, along with force/deformation relationships. This method is most easily adaptable to line or one-dimensional elements ( spring , bar , and beam elements) Step 4 -Derive the Element Stiffness Matrix and Equations 43 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

43. Work or Energy Methods - To develop the stiffness matrix and equations for two and three-dimensional elements, it is much easier to apply a work or energy method. The principle of virtual work (using virtual displacements), the principle of minimum potential energy , and Castigliano’s theorem are methods frequently used for the purpose of derivation of element equations. We will present the principle of minimum potential energy (probably the most well known of the three energy methods mentioned here) Step 4 -Derive the Element Stiffness Matrix and Equations 44 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

44. Methods of Weighted Residuals - The methods of weighted residuals are useful for developing the element equations (particularly popular is Galerkin’s method ). These methods yield the same results as the energy methods , wherever the energy methods are applicable. They are particularly useful when a functional such as potential energy is not readily available. The weighted residual methods allow the finite element method to be applied directly to any differential equation Step 4 -Derive the Element Stiffness Matrix and Equations 45 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

45. The individual element equations generated in Step 4 can now be added together using a method of superposition (called the direct stiffness method ) whose basis is nodal force equilibrium (to obtain the global equations for the whole structure). Implicit in the direct stiffness method is the concept of continuity , or compatibility , which requires that the structure remain together and that no tears occur anywhere in the structure. The final assembled or global equation written in matrix form  F    K   d  Step 5 – Assemble the Element Equations and Introduce Boundary Conditions 46 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

46. Once the element equations are assembled and modified to account for the boundary conditions , a set of simultaneous algebraic equations that can be written in expanded matrix form as : where n is the structure total number of unknown nodal degrees of freedom. These equations can be solved for the d’s by using an elimination method (such as Gauss’s method ) or an iterative method (such as Gauss Seidel’s method) Step 6 – Solve for the Unknown Degrees of Freedom (or Generalized Displacements) 47 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

47. For the structural stress-analysis problem, important secondary quantities of strain and stress (or moment and shear force ) can be obtained in terms of the displacements determined in Step 6. Step 7 – Solve for the Element Strains and Stresses 48 9/19/2019 The final goal is to interpret and analyze the results for use in the design/analysis process. Determination of locations in the structure where large deformations and large stresses occur is generally important in making design/analysis decisions . Step 8 – Interpret the Results Dr. Nazeer A. Anjum, MED, UET, Taxila

48. Start Problem Definition Pre-processor Reads or generates nodes and elements (ex: ANSYS) Reads or generates material property data. Reads or generates 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 Process Flow in a Typical FEA Analysis 49 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

49. airfoil geometry (from CAD program) mesh generator surface model ET,1,SOLID45 N, 1, 183.894081 , -.770218637 , 5.30522740 N, 2, 183.893935 , -.838009645 , 5.29452965 . . TYPE, 1 E, 1, 2, 80, 79, 4, 5, 83, 82 E, 2, 3, 81, 80, 5, 6, 84, 83 . . . meshed model Step 1: Discretization - Mesh Generation 50 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

50. Displacements : DOF constraints usually specified at model boundaries to define rigid supports. Forces and Moments: Concentrated loads on nodes usually specified on the model exterior. Pressures: Surface loads usually specified on the model exterior. Temperatures: Input at nodes to study the effect of thermal expansion or contraction. Inertia Loads: Loads that affect the entire structure (ex: acceleration, rotation). Step 4: Boundary Conditions 51 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

51. Temp mapper Nodes from FE Modeler Thermal Soln Files bf, 1,temp, 149.77 bf, 2,temp, 149.78 . . . bf, 1637,temp, 303.64 bf, 1638,temp, 303.63 Step 4: Boundary Conditions (Thermal Loads) 52 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

52. Speed, temperature and hub fixity applied to sample problem. FE Modeler used to apply speed and hub constraint. antype,static omega,10400*3.1416/30 d,1,all,0,0,57,1 Boundary Conditions (Other Loads) 53 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

53. A single degree of freedom translational spring will be used to introduce the force-displacement equation that is continuously encountered in the finite element method. In addition, the spring will also be used to introduce the concept of stiffness and degrees of freedom. The Basic Finite Element Equation ( Examle ) 54 9/19/2019 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

54. 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila Force Displacement Relationship For Single Degree of freedom Linear Spring Force Displacement Relationship For Single Degree of freedom Non-Linear Spring The Basic Finite Element Equation ( Examle ) 55 9/19/2019

55. Examples of Linear Spring Stiffness 56 9/19/2019 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

56. The figure below shows a transnational spring which only needs one coordinate to specify its position. The displacement u defines the horizontal movement (translation) of point P due to a horizontal external force F applied at that point. Therefore, the spring system shown has a single (one) degree of freedom. Force-Displacement Relationship for Single Degree of Freedom (SDOF) Linear Spring 57 9/19/2019 9/19/2019 Dr. Nazeer A. Anjum, MED, UET, Taxila

57. The finite element method has been applied to numerous problems, both structural and non-structural. This method has a number of advantages that have made it very popular. Advantages of the Finite Element Method 58 9/19/2019 Model irregularly shaped bodies quite easily Handle general load conditions without difficulty Model bodies composed of several different materials because the 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 necessary Alter the finite element model relatively easily and cheaply Include dynamic effects Handle nonlinear behavior existing with large deformations and nonlinear materials . Dr. Nazeer A. Anjum, MED, UET, Taxila

58. The systematic generality of FEA procedure makes it a powerful and versatile tool for a wide range of problems. FEA is simple , compact and result oriented and hence widely popular among engineering community. FEA can be easily coupled with CAD programs in various streams of engineering . Availability of large number of Computer software packages and literature makes FEA a versatile and powerful numerical method Advantages of the Finite Element Method 59 9/19/2019 The FEA of structural analysis enables the designer to detect stress , vibration , and thermal problems during the design process and to evaluate design changes before the construction of a possible prototype . Dr. Nazeer A. Anjum, MED, UET, Taxila