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1. Introduction to Finite Element Method By: Zerihun G . Arba Minch University , Arba Minch Institute of Technology , Faculty of Mechanical Engineering Course: - Introduction to Finite Element Method ( MEng 5 283) March, 202 4

2. CH 1 Introduction to FEM, Computational Modeling

3. What is finite element method (FEM)? FEM: is a numerical technique used for finding an approximate solutions to value problems (field for partial differential boundary problems) equations It is a method for dividing up a very complicated problem into small elements that can be solved in relation to each other CH 1 Introduction How does FEM work? It uses a complex system of points called nodes which make a grid called a mesh The mesh is programmed to contain the material and structural properties which define how the structure will react to certain loading conditions Nodes are assigned at a certain density throughout the material depending on the anticipated stress level of a particular area It is useful of problems with complicated geometries, loadings and material properties where analytical solutions can not be obtained 1

4. How does FEM work? Regions which will receive large amounts of stress usually have a higher node density than those regions tested which experience or no stress. These include: fracture point of material, fillets, corners, previously complex detail and high stress area. The mesh acts like a spider web in that from each node, there extends a mesh element to the neighboring nodes. This web od is what carries the material to the object creating many each of vectors properties elements. How can FEM Help a Mechanical Engineer? The FEM has many advantages to the Mechanical engineer To solve complex , irregular- shaped objects composed of several different materials and having complex boundary conditions Applicable to steady state, time dependent and eigenvalue problems Applicable to linear and nonlinear problems One method can solve a wide variety of problems, including problems in solid mechanics, fluid mechanics, chemical reactions , electromagnetics, biomechanics, heat transfer and acoustics to name a few 2 CH 1 Introduction

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7. Advantages of FEM Can handle complex geometries Can handle complex analysis types including vibration, heat transfer, fluid etc. Can handle complex loading Node based loading (point load) Element based loading (pressure, thermal, inertial forces) Time or frequency dependent loading Can handle complex restraints Disadvantages of FEM A specific numerical result is obtained for a specific problem The FEM is applied to an approximation of the mathematical model of a system (which is usually open for inherent 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 Numerical errors such as the limitation of the number of significant digits, rounding- off occur very often Fluid elements with boundaries at infinity can be computed and treated by suing boundary element method CH 1 Introduction 5

8. Discretization Selection of approximation of functions Formulation of elemental stiffness matrix Formulation of total (global) stiffness matrix Formulation of element loading matrix Formulation of overall (global) loading matrix There are two general direct approaches traditionally associated with the finite element method as applied to structural mechanics problems. One approach, called the force, or flexibility , method, uses internal forces as the unknowns of the problem. The second approach, called the displacement, or stiffness , method, assumes the displacements of the nodes as the unknowns of the problem. For computational purposes, the displacement (or stiffness) method is more desirable because its formulation is simpler for most structural analysis problems. Basic Steps and Phases Involved in FEM Direct Stiffness Method Formulation of overall equilibrium equation Implementation of boundary conditions Calculation of unknown nodal displacements Calculation of stress and strains CH 1 Introduction 6

9. Phases in FEM Pre-processing It is a stage where finite element mesh is developed to divide the given geometry into subdomains for mathematical analysis and the material properties are applied and also the boundary conditions Solution In this phase , governing matrix equations are derived and the solution for the primary quantities is generated Post-processing In this phase, checking of the validity of the solution generated, examination of the values of primary quantities. CH 1 Introduction 7

10. Types of Elements: The primary line elements consist of bar (or truss) and beam elements (1D). They have a cross- sectional area but are usually represented by line segments. In general, the cross- sectional area within the element can vary, but it will be considered to be constant for the sake of this section. These elements are often used to model trusses and frame structures. The simplest line element (called a linear element) has two nodes , one at each end. The basic two- dimensional (or plane) elements are loaded by forces in their own plane (plane stress or plane strain conditions). They are triangular or quadrilateral elements. The simplest two- dimensional elements have corner nodes only (linear elements) with straight sides or boundaries The elements can have variable thicknesses throughout or be constant. 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. CH 1 Introduction 8

11. BASIC ELEMENTS The shapes, sizes, number, and configurations of the elements have to be chosen carefully such that the original body or domain is simulated as closely increasing as possible without the computational effort needed for the solution. Mostly the choice of the type of element is dictated by the geometry number of necessary to of the body and the independent coordinates describe the system. If the geometry, material properties, and the field variable of the problem can be described in terms of only one spatial coordinate, we can use the one- dimensional or line elements shown in Figure 2.1(a). Figure 2.1 9 (a) CH 1 Introduction

12. BASIC ELEMENTS The temperature distribution in a rod (or fin), the pressure distribution in a pipe flow. and the deformation of a bar under axial load, for example, can be determined using these elements. Although these elements have cross- are generally as a line element sectional area, they shown schematically (Figure 2.1(b)). In some cases, the cross- area of the element may uniform. sectional be non For a simple analysis, one- dimensional elements are assumed to have two nodes, one at each end, with the corresponding value of the field variable chosen as the unknown (degree of freedom). However, for the analysis of beams, the values of the field variable (transverse displacement) and its derivative (slope) are chosen as the unknowns (degrees of freedom) at each node as shown in Figure 2.1(c) Figure 2.1 10 CH 1 Introduction (a)

13. and other BASIC ELEMENTS When the configuration details of the problem can be described in terms of two independent spatial coordinates, we can use the two- dimensional elements shown in Figure 2.2. The basic element useful for two- dimensional analysis is the triangular element. Although a quadrilateral (or its special forms, rectangle and parallelogram) element can be obtained by assembling two or four triangular elements, as shown in Figure 2.3, in some cases the use of quadrilateral (or rectangle or parallelogram) elements proves to be advantageous. For the bending analysis of plates, multiple degrees of freedom (transverse displacement and its derivatives) are used at each node. CH 1 Introduction 11

14. BASIC ELEMENTS If the geometry, material properties, and other parameters of the body can be described by three independent spatial coordinates, we can idealize the body by using the three dimensional in Figure 2.4. elements shown For the discretization of problems involving curved geometries, finite elements with curved sides are useful. Typical elements having curved boundaries 2.5. The ability to model curved boundaries has by the addition of mid side nodes. Finite elements with straight sides are are shown in Figure been made possible known as linear elements, whereas those with curved sides are called higher order elements Figure 2.5. Finite Elements with Curved Boundaries 12 CH 1 Introduction

15. Ch 2 Fundamentals for Finite Element Method

16. OUTLINES OF CH-2 & CH-3 : Principle of minimum potential energy. Raleigh’s Ritz method. Direct approach for stiffness matrix formulation of bar element. 3. Deriving an element stiffness matrix 16

17. Principle of minimum potential energy: Total potential energy: The total potential energy of the elastic body is defined as the sum of strain energy due to internal stresses produced and the work potential due to the external force. Total Potential energy (PE) = Strain Energy (SE) + Work Potential (WP) PE (π) =SE+WP 17

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22. Find ( i ) the deflection of the spring by using MPE ii ) the total PE at equilibrium K=1000 N/m, m=200kg, g=10 m/s Assume a linear spring   22

23.       The total PE of the system in equilibrium is indeed minimum. Thus the PE in Equilibrium is stationary and minimum 23

24. Variational Method: 24

25. Rayleigh Ritz method: The Rayleigh Ritz method is a classical approximate method to find the displacement function of an object such that, it is in equilibrium with the externally applied loads. The Rayleigh Ritz method relies on the principle of minimum potential energy for conservative systems. The method involves assuming a form or a shape for the unknown displacement functions, and thus, the displacement functions would have a few unknown parameters. 25

26.     In practice , the displacement function y(x) can be expressed in terms of polynomial series or trigonometric series such as Substituting the displacement function in Eq.1.27 gives 26

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44. Direct Stiffness Method For Linear springs The direct equilibrium approach, will be used to derive the stiffness matrix for a one- dimensional linear spring— that is, a spring that obeys Hooke’s law and resists forces only in the direction of the spring. Reference points 1 and 2 are located at the ends of the element. These reference points are called the nodes of the spring element. The local nodal forces are f1x and f2x for the spring element associated with the local axis x. The local axis acts in the direction of the spring so that we can directly measure displacements and forces along the spring. The local nodal displacements are d1x and d2x for the spring element. These nodal displacements are called the degrees of freedom at each node. Positive directions for the forces and displacements at each node are taken in the positive x direction as shown from node 1 to node 2 in the figure. The symbol k is called the spring constant or stiffness of the spring. Ch 2 Fundamentals 1 4

45. Direct Stiffness Method For Linear springs We know that a prismatic uniaxial bar has a spring constant 𝒌 = 𝑨𝑬 / 𝑳 , where A represents the cross- sectional area of the bar, E is the modulus of elasticity, and L is the bar length. Similarly, we know that a prismatic circular cross- section bar in torsion has a spring constant 𝒌 = where J is the polar moment of inertia and G is the shear modulus of the material. 𝑱𝑮 / 𝑳 , For one- dimensional heat conduction, 𝑘 = 𝐴𝑘𝑥𝑥 / 𝐿 , where Kxx is the thermal conductivity of the material. And for one- dimensional fluid flow through a porous medium, the stiffness 𝒌 = 𝑨𝒌𝒙𝒙 / 𝑳 , where Kxx is the permeability coefficient of the material. Observe that the stiffness method can be applied to nonstructural problems, such as heat transfer, fluid flow, and electrical networks, as well as structural problems by simply applying the proper constitutive law (such as Hooke’s law for structural problems, Fourier’s law for heat transfer, Darcy’s law for fluid flow and Ohm’s law for electrical networks) and a conservation principle such as nodal equilibrium or conservation of energy. Ch 2 Fundamentals 1 4

46. The Linear Spring FEM Select the element type: Consider the linear spring element (which can be an element in a system of springs) subjected to resulting nodal tensile forces T (which may result from the action of adjacent springs) directed along the spring axial direction x as shown in Figure below, so as to be in equilibrium. The local x axis is directed from node 1 to node 2 . We represent the spring by labeling nodes at each end and by labeling the element number. The original distance between nodes before deformation is denoted by L. The material property ( spring constant ) of the element is k . Fig. Linear spring subjected to tensile forces Ch 2 Fundamentals 1 5

47. Select a Displacement Function: We must choose in advance the mathematical function to represent the deformed shape of the spring element under loading. Because it is difficult, if not impossible at times, to obtain a closed form or exact solution, We assume a solution shape or distribution of displacement within the element by using an appropriate mathematical function. The most common functions used are polynomials . Because the spring element resists axial loading only with the local degrees of freedom for the element being displacements d1x and d2x along the x direction, we choose a displacement function u to represent the axial displacement throughout the element. Here a linear displacement variation along the x axis of the spring is assumed , because a linear function with specified endpoints has a unique path. 𝒖 = 𝒂 𝟏 + 𝒂 𝟏 𝒙 Therefore, , In general, the total number of coefficients a is equal to the total number of degrees of freedom associated with the element. Here the total number of degrees of freedom is two— an axial displacement at each of the two nodes of the element In matrix form, becomes: Ch 2 Fundamentals 1 6

48. Select a Displacement Function: We now want to express u as a function of the nodal displacements d1x and d2x. This will allow us to apply the physical boundary conditions on nodal displacements directly as indicated in Step 3 and to then relate the nodal displacements to the nodal forces in Step 4. We achieve this by evaluating u at each node and solving for a1 and a2 . Ch 2 Fundamentals Back substituting the values: In matrix form: Where : 1 7

49. 04- Oct- 21 18 Select a Displacement Function: N1 and N2 are called the shape functions because the Ni’s express the shape of the assumed displacement function over the domain (x coordinate) of the element when the ith element degree of freedom has unit value and all other degrees of freedom are zero. In this case, N1 and N2 are linear functions that have the properties that N1= 1 at node 1 and N1 = at node 2, whereas N2 = 1 at node 2 and N2 =0 at node 1. Also, N1 + N2 =1 for any axial coordinate along the bar. In addition, the Ni’s are often called interpolation functions because we are interpolating to find the value of a function between given nodal values. Ch 2 Fundamentals Fig. ( a) Spring element showing plots of (b) displacement function u and Shape functions (c) N1 and (d) N2 over domain of element

50. Define the Strain/Displacement and Stress/Strain Relationships: The tensile forces T produce a total elongation (deformation) d of the spring. Here d1x is a negative value because the direction of displacement is opposite to the positive x direction, whereas d2x is a positive value. The deformation of the spring is then represented by: For a spring element, we can relate the force in the spring directly to the strain/displacement relationship is not necessary here. deformation. Therefore, the The stress/strain relationship can be expressed in terms of the force/deformation relationship instead as: Ch 2 Fundamentals 19

51. 4: Derive the Element Stiffness Matrix and Equations We now derive the spring element stiffness matrix. By the sign convention for nodal forces and equilibrium, we have: or In a single matrix equation yields: This relationship holds for the spring along the Y axis. From our basic definition of a stiffness matrix and application. Here k is called the local stiffness matrix for the element. We observe from that k is a symmetric (that is, kij = kji square matrix (the number of rows equals the number of columns in k). The main diagonal of the stiffness matrix is always positive. Otherwise, a positive nodal force Fi could produce a negative displacement di— a behavior contrary to the physical behavior of any actual structure. Ch 2 Fundamentals 20

52. 5 : Assemble the Element Equations to Obtain the Global Equations and Introduce Boundary Conditions The global stiffness matrix and global force matrix are assembled using nodal force equilibrium equations, force/deformation and compatibility equations, and the direct stiffness method described. This step applies for structures composed of more than one element such that where k and f are now element stiffness and force matrices expressed in a global reference frame. (the ∑ sign used in this context does not imply a simple summation of element matrices but rather denotes that these element matrices must be assembled properly according to the direct stiffness method. 6: Solve for the Nodal Displacements The displacements are then determined by imposing boundary conditions, such as support conditions, and solving a system of equations, F = Kd, simultaneously. 7 :Solve for the Element Forces Finally, the element forces are determined by back- substitution, applied to each element, into equations Ch 2 Fundamentals 21

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69. INTERPOLATION The basic idea of the finite element method is piecewise approximation that is, the solution of a complicated problem is obtained by dividing the region of interest into small regions (finite elements) and approximating the solution over each sub region by a simple function. Thus, a necessary and important step is that of choosing a simple function for the solution in each element. The functions used to represent the behavior of the solution within an element are called interpolation functions or approximating functions or interpolation models . Polynomial- type interpolation functions have been most widely used in the literature due to the following reasons: It is easier to formulate and computerize the finite element equations with polynomial- type interpolation functions. Specifically. it is easier to perform differentiation or integration with polynomials. It is possible to improve the accuracy of the results by increasing the order of the polynomial, as shown in in the figure below. Theoretically. a polynomial of infinite order corresponds to the exact solution. But in practice we use polynomials of finite order only as an approximation. Ch 2 Fundamentals 38

70. INTERPOLATION Ch 2 Fundamentals When the interpolation polynomial is of order one. the element is termed a linear element. A linear element is called a simplex element if the number of nodes in the element is 2, 3. and 4 in 1, 2, and 3 dimensions, respectively . If the interpolation polynomial is of order two or more. the element is known as a higher order element. In higher order elements, some secondary (mid- side and/or interior) nodes are introduced in addition to the primary (corner) nodes in order to match the number of nodal degrees of freedom with the number of constants (generalized coordinates) in the interpolation polynomial. 39

71. INTERPOLATION Problems involving curved boundaries cannot be modeled satisfactorily by using straight- sided elements. The family of elements known as "iso- parametric" elements has been developed for this purpose. The basic idea underlying the iso- parametric elements is to use the same interpolation functions to define the element shape or geometry as well as the variation of the field variable within the element. To derive the iso- parametric element equations, we first introduce a local or natural coordinate system for each element shape. Then the interpolation or shape functions are expressed in terms of the natural coordinates. The representation of geometry in terms of (nonlinear) shape functions can be considered as a mapping procedure that transforms a regular shape, such as a straight- sided triangle or rectangle in the local coordinate system, into a distorted shape. such as a curved- sided triangle or rectangle in the global Cartesian coordinate system. Ch 2 Fundamentals 40

72. INTERPOLATION Polynomial Form of Interpolation Function If a polynomial type of variation is assumed for the field variable Q(x) in a one- dimensional element, r can be expressed as Similarly, in two- and three- dimensional finite elements the polynomial form of interpolation functions can be expressed as Ch 2 Fundamentals 41

73. INTERPOLATION Polynomial Form of Interpolation Function In most of the practical applications the order of the polynomial in the interpolation functions is taken as one. two, or three. Ch 2 Fundamentals 42

74. INTERPOLATION 1. Domain Discretization Consider the temperature distribution along the one- dimensional fin A one- dimensional continuous temperature distribution with an infinite number of unknowns is shown in (a). The fin is discretized in (b) – i.e. divided into 4 subdomains (or elements). The nodes are numbered consecutively from left to right, as are the elements. The elements are first order elements; the interpolation scheme between the nodes is therefore linear. Note that there are only 5 nodes for this system, since the internal nodes are shared between the elements. Since we are only solving for temperature, there are only 5 degrees of freedom in this model of the continuous system. It should be clear that a better approximation for T(x) would be obtained if the number of elements was increased (i.e. if the element lengths were reduced). It is also apparent that the nodes should be placed closer together in regions where the temperature (or any other unknown solution) changes rapidly. Ch 2 Fundamentals 43

75. INTERPOLATION 1. Domain Discretization It is useful also to place a node wherever a step change in numerical value of the temperature is needed. temperature is expected and where a It is good practice to continue to increase the number of nodes until a converged solution is reached Ch 2 Fundamentals 44

76. INTERPOLATION In (c), the fin has been divided into two subdomains – elements 1 and 2. However, in this instance we have chosen to use a second order (quadratic) element. These elements contain ‘mid- side’ nodes as shown, and the interpolation between the nodes is quadratic which permits a much closer approximation to the real system. For this model system there are still just 5 degrees of freedom. However, the analysis takes longer for (c) than it does for (b) because the quadratic interpolation (which calculates the temperature at locations between the nodes) is more demanding than the corresponding linear case. Ch 2 Fundamentals 45

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86. DIRECT FORMULATION Ch 2 Fundamentals 𝑝 55

87. DIRECT FORMULATION 1. Discretize the solution domain into finite elements 1. Discretize the solution domain into finite elements Dividing the model by Five nodes and Four elements having a uniform cross section. The cross- sectional area of each element is represented by an average area of the cross sections at the nodes that make up (define) the element as Ch 2 Fundamentals 56

88. DIRECT FORMULATION 2. Assume a solution that approximates the behavior of the element In order to study the behavior of a typical element, let’s consider the deflection of a solid member with a uniform cross section A that has a length 𝑙 when force F , The average stress 𝜎 in the member is given by subjected to a Ch 2 Fundamentals 57

89. DIRECT FORMULATION 2. Assume a solution that approximates the behavior of the element Ch 2 Fundamentals 58

90. DIRECT FORMULATION 2. Assume a solution that approximates the behavior of the element Ch 2 Fundamentals 59

91. DIRECT FORMULATION 2. Assume a solution that approximates the behavior of the element Ch 2 Fundamentals 60

92. DIRECT FORMULATION 2. Assume a solution that approximates the behavior of the element Ch 2 Fundamentals 61

93. DIRECT FORMULATION 3. Develop an equation for an element Ch 2 Fundamentals 62

94. Ch 2 Fundamentals DIRECT FORMULATION 4. Assemble the elements to present the entire problem 63

95. Ch 2 Fundamentals DIRECT FORMULATION 4. Assemble the elements to present the entire problem 64

96. Ch 2 Fundamentals DIRECT FORMULATION 5.Apply boundary conditions and loads 65

97. Ch 2 Fundamentals DIRECT FORMULATION 6. Solve a system of algebraic equations simultaneously (solution stage) 66

98. Ch 2 Fundamentals DIRECT FORMULATION 6. Solve a system of algebraic equations simultaneously (solution stage) 67

99. Ch 2 Fundamentals DIRECT FORMULATION 7. Obtain other information (post processing stage) 68

100. Ch 2 Fundamentals DIRECT FORMULATION 7. Obtain other information (post processing stage) 69

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