CHAPTER 5 FINITE ELEMENT METHOD 5.1 Introduction to Finite Element Method Finite element analysis is a computer based numerical method to deduce engineering structures strength and behaviour. Its use can be calculation of buckling behaviour, vibration, stress, deflection & many other phenomena. It can analyze elastic or out of shape deformation. For applications requiring astronomical numbers for analysis purpose, computer is required. Modern computers low cost & power has made FEA implementable to many industries & disciplines. Quick technological advancements & complication in designing is also increasing. This scenario demands from engineers to design problems speedily, efficiently optimally. For better output and developmental pace, the engineer today resorting to numerical methods. For complicated boundary conditions problems, complex shapes & material properties, it is cumbersome & interactive to obtain solution in many. Approximate but acceptable solutions to such problems may be provided by numerical methods. Finite element analysis is one of such numerical procedure for analysis & solution of engineering problems which are complex and wide ranging & whose satisfactory solutions are difficult to obtain if we use classical methods. Intervention of computer is the backbone of the procedure since it involves a number of algebraic and simultaneous equations. Computers can solve these problems easily. Actually, origination of FEA came from stress analysis method. But today the applications are numerous. Now days, Finite Element Analysis is the main computational method for developing any kind of design. The numerous applications include the fields of magnetic and electric fields, lubrication, fluid flow heat transfer, seepage and other problems related to flow. Numerous applications areas contain design of buildings and bridges, electric motors, space crafts, aircraft structures heat engines etc. With the advancements of Interactive CAD/CAM systems, modeling of complex problems has become quite easy.
Simple geometrical shapes are achieved after discretization of complex region by this method. Name given to these discretized shapes are finite elements.. The governing relationships and the material properties are considered over these elements and expressed in terms of unknown values. After consideration of constraints & loading, set of equations are achieved during process of assembly.. The approximate behavior of continuum is obtained with the help of these equations' solutions. Getting solution of complex problems using FEM and to replace it with a simpler one is the basic idea of it. Instead of actual solution, approximate solution can be obtained because replacement of actual problem. To get solutions exactly, existing mathematical tools is not enough and sometimes even the approximate solutions in all uneven conditions and by spending some computational effort the approximate solutions can be improved. The key step in Finite Element \Method is discretization of the model and then modeling it. Hence the Finite Element method is a method of piece wise approximation in which we form, connecting simple functions, an approximating function Φ. Each defined in a small region (i.e. discritized element), and the approximating function Φ is one which represents any of the several physical quantities varies smoothly in the structure. A Finite element model typically yields a piece wise smooth representation of Φ. The power of Finite Element method resides principally in its versatility. Support conditions, loads and arbitrary shape can be obtained from the body analyzed. Elements of different physical properties, shapes & types can be obtained by meshing them. This versatile nature of meshing can be achieved with the help of a computer program. User prepared Input data prepared by user controls the selection of elements problem type selection, boundary conditions geometry and so on. Another feature of FEA is that the finite element model resembles closely with the actual structural model. Description of virtually every phenomenon in nature, whether mechanical, biological or geological, may be done with the help of physics laws, in terms of algebraic, differential, or integral equation relating various quantities of interest. Determining the stress distribution in a pressure vessel with oddly shaped holes and numerous stiffeners and subjected to aerodynamic,
mechanical or/and thermal loads, determination of pollutants' concentration in sea water or in the atmosphere & simulating weather to predict & understand the mechanics of formation of tornadoes and thunderstorms are a few example of many important practical problems. While the derivation of the governing equations for these problems is not unduly difficult, their solution exact methods of analysis is a formidable task. In such cases alternatives of getting solutions are provided by approximate methods of analysis. Among these the finite difference methods and the variation methods such as the Rayleigh Ritz and Galerkin methods are more frequently used. The derivatives in a differential equation are substituted by difference quotients. At a domain's discrete mesh points the solution's values are used. By imposition of boundary conditions, the solution of the resulting discrete equations is obtained. This is done at the mesh points for the solution values. Even though the method of finite difference is conceptually easy, it carries numerous disadvantages. The most visible disadvantages are: It is difficult to represent geometrically complex domains accurately, Along non-straight boundaries, it is difficult to impose boundary conditions, The inability to employ non-uniform and non-rectangular meshes. In the variation solution of differential equation, the differential equation is put into an equivalent variation form, and then the approximate solution is assumed to be a combination (Zc1) of given approximation functions j. Determination of parameter is done with the help of variation form. The variation methods suffer from the disadvantage that it is difficult to construct, with arbitrary domains, the approximation functions for problems. Variation methods difficulties are overcome by finite element method because it provides a systematic procedure for the derivation of the approximation functions. Two basic features are endowed in this method. This feature is superior to other methods in competition. Firstly, representation of a collection of geometrically simple sub domains, called finite elements is achieved in terms of geometrically complex domain. Secondly, derivation of approximation functions over each finite element is done and any continuous function can be represented by a linear combination of algebraic polynomials. In which the approximation of functions are algebraic polynomials and values of the solutions are represented by undetermined parameters at
a finite number of pre selected points termed as nodes, in the interior and on the boundary of the clement from interpolation theory one finds that the order (or degree) of the interpolation function depends on the number node in the elements.
5.2 Element types considered Element type - shape, size & configuration should be chosen from the start. Elements numbers should be chosen so that geometry in entirety should be represented. For system description, element selection relies on body type & independent-coordinate s number. Figure 5.1: Various types of elements used in FEM Elements of One-dimension are assumed having two nodes, one at each end. e.g. in a rod distribution of temperature, under axial-load bar deformation & in a pipe flow distribution of pressure. When two spatial coordinates are used to describe the configuration, two-dimensional elements may be used. Basic element for 2D analysis is the triangular element. By combining two or four triangular elements a quadrilateral element can be formed.
Three independent spatial-coordinates can describe material properties & geometry of threedimensional elements. The basic three- dimensional element is the tetrahedron element. To produce an element of type hexahedron, five tetrahedrons can be assembled. Many three dimensional problems description can be done by one or two independent coordinates. Using an axis-symmetric or ring element, such problems can be idealized. Figure 5.2: Types of 3D elements used in FEM Linear elements are Finite elements having sides which are straight. Higher order elements have curved sides. On a wide variety of two-and three-dimensional problems, these elements are effective and can capture variations in stress such as occur near fillets, holes, etc.
5.3 Interpolation models The basic idea of the finite element method is piecewise approximation that is the solution of problem can be achieved when the region of interest is divided into small regions by approximation of solution on every sub region by a simple function. It is an important & necessary step to get every element s solution by selecting a simple function. Within an element, interpolation models or approximating/interpolation functions can represent solution behavior Following reasons describe most widely used polynomial types of interpolation functions in literature. a) Equations having polynomial functions can be formulated & computerized easily. More impotantly, performing polynomials integration & differentiation is easier. b) Results accuracy improvement may be made possible by increasing polynomial order. In theoretical sense, exact solution corresponds to an infinite order polynomial. But as an approximation, only finite order polynomials we use in practice. Trigonometric functions seldom use these properties possessed by them in the finite element analysis. It will consider only polynomial type of interpolation functions. The element is termed as a linear element if the order of interpolation polynomial is one. If in the element, no. of nodes is 2, 3, & 4 in l, 2, and 3 dimensions then it is a simples element. For elements of higher order, part from primary nodes, some interior (secondary) nodes are covered. It is done for matching the no. of generalized coordinates (constants) with no. of nodal degrees of freedom in the interpretation polynomial. In final result, to get same accuracy, less no. of elements of higher order, in general, are needed, even though time of computation is not reduced. Generally, the required effort is reduced by reduction in elements number. Cases in which the gradient of the field variable expected to vary rapidly, the elements whose order is higher are useful especially. In cases like this, good results are not yielded by simplex elements where gradient is approximated. Decreased effort in preparation of data combined with accuracy is widely used., in several practical applications, of higher order elements.
Two methods can be used to improve the discretization of the domain (or region), in case the order of the interpolation polynomial is fixed. Change in total elements is not required for altering nodes location in r-method, the first described. Number of elements increases in h- method, the second method. In p-method polynomial order increase, accuracy can be improved. By using straight-sided elements, there is no satisfactory modeling where boundaries involved are curved in problems. To achieve the purpose, development of iso parametric components took place. For defining geometry/shape, same interpolation functions are used in isoparametric-elements. For the shape of each element, a natural/local coordinate is introduced & then iso-parametric element-equations can be derived. Then natural coordinates are used to express shape/interpolation functions. Using shape functions in geometry, a mapping procedure is its representation. Using it, rectangle/triangle like regular shapes could be converted into Cartesian systems. Problems with boundaries having curve can be represented using this concept. Currently, problems like shell analysis & 3D use iso-parametric elements extensively. Expression of (x) in1-dimensional element is given below assuming variation of polynomial type for field (x) : Φ(x) =α 1 +α 2 x+α 3 x 2 +..+α m x n.. 4.1 In a similar manner, expression of interpolation functions polynomial forms in 1-D & 2-D is given below: Φ(x,y) = α 1 +α 2 x+ α 3 y+ α 4 x 2 + α 5 y 2 +α 6 xy+ + α m y n 4.2 Φ(x,y,z) = α 1 +α 2 x+ α 3 y+ α 4 z+ α 5 x 2 + α 6 y 2 + α 7 z 2 + α 8 xy+ α 9 yz+ α 10 xz+ + α m z n 4.3 5.4 Requirements of Convergence
As the size of the element is reduced in succession approximate solutions sequence is obtained by FEM since it is numerical technique. If the following convergence requirements are satisfied by interpolation polynomial, exact solution will be provided by the sequence. a) Partial derivatives & field variable of Φ in field variable in the functional 1(Φ) should have continuity at interfaces or boundaries of elment. b) Field variable Φ & all its uniform states with partial derivatives should have been represented in polynomial. c) Within the elements, the field variable must be continuous. By choosing continuous functions as interpolation models, this requirement is easily satisfied. Complete Elements are those which satisfy condition 2. Compatible or confirming are those whose polynomials satisfy the requirements I and 3. 5.5 Engineering applications of FEM
Aircraft structure is the original user of FEM. But it is widely used in problems using boundary values. FEM s specific applications are in the following fields:, a) Problems related to transient/propagation. b) Problems using Eigen values c) Problems related to time independent/steady state/equilibrium. In equilibrium problems: Finding of velocity/pressure distribution is required if problem is related to fluid mechanics. Finding of stress-distribution or steady-state displacement is required if problem is related to heat transfer. Apart from steady-state configuration if determination of some parameters critical values is required then Eigen value problems are equilibrium problems extensions.. Time will not appear explicitly in such problems. It need to find the buckling loads or natural frequencies and mode shapes if it is structural problem or solid mechanics, resonance characteristics if it is a electrical circuit problem and stability of laminar flows if it is a fluid mechanics problem. Problems related to transient or propagation dependent upon time. In solid mechanics area, it can find the response of a body under force which varies with time under sudden heating or cooling in the area of heat transfer. 5.6 Steps involved in FEM
In the finite element method the actual continuum or body of matter like gas, liquid, or solid representation of finite elements is sub domains assembly. These elements, called nodes, are assumed as interconnected on particular joints. Usual positions of the nodes are on the element boundaries where adjacent elements are to be connected. The actual variation of the field variable (like displacements, stress, temperature etc.) inside the continuum is not known, It will approximate the field variable with simple function. These approximating functions are defined by values of field variable at the nodes. Using these, equilibrium equations for the whole continuum are developed in terms of the unknown nodal values of the field variables which are in the form of matrix equations can be solved to find the unknown field variables. Once these are known, the field variable is defined by approximating functions during element s assembly. Finite Element Analysis steps are described below: Step 1: Structure discretization. Divide the structure or continuum or solution region into finite elements that is, using finite elements structure modeling is prepared. Number, type, size and arrangement of the elements are to be decided. Mesh generation programs called Preprocessors can be used for this purpose. Step 2: Each element's properties formulation. This means determining nodal loads associated with all element deformation are allowed. Step 3: Selecting proper displacement model or interpolation. Exact prediction is not possible irrespective of any load condition for complex structure s displacement solution. To approximate the unknown solution, a proper solution can be assumed.. There should be simple assumed solution from computational view point, but certain convergence should be satisfied and compatibility requirements. Generally the solution or interpolation model is in the form of a polynomial. Step 4: To derive load vector and element stiffness matrix An element e and its load factor {F 1 } & stiffness matrix {k} need derivation from the assumed displacement model by using either equilibrium conditions or a suitable variation principle.
Step 5: Assembling the equations of element As the structure is composed of several finite elements, suitable assembly of load vectors & formulation of l equilibrium equations is done as KQ = F, where K is structural or global stiffness matrix Q is vector of nodal displacements F is nodal forces vector Step 6: solution of nodal displacements which are unknown The equilibrium equations have to be modified using the boundary conditions of the problem. After incorporating the boundary conditions the equilibrium equations are expressed as KQ=F. For linear problems the displacement vector Q can be calculated comfortably, but we can obtain solution of non-linear problems by solving a number of steps in sequence. Here in we modify load vector F or stiffness matrix K in every step. Step7: Computation of Element's computation of strains &stresses The strains &stresses are computed with the help of known displacement Q using necessary of solid and structural mechanics. Step 8: Presentation of results Using output interpolation programs called Postprocessors the results can be framed and displayed in vector or raster forms as desired. 5.7 Classification
Finite element analysis is done principally with commercially purchased software. Finite element packages may include pre-processors.this preprocessor may be utilized for creating structure s geometry. The preprocessor can also be imported from software generated CAD files. The Finite Element Analysis Software contains modules for creating element mesh, for analyzing problem definition or for reviewing analysis results. Output may be in plotted form or in printed form. Examples are output parameters graphs, definition plots or stresses computer maps Many user- oriented general-purpose finite element packages such as ANSYS, MSC- NASTRAN, NISA etc., have the capability of combining the lamination theory with finite element codes. Many of these packages are capable of calculating in plane as well as interlaminar stresses incorporating more than one failure criterion, and contain a library of plate, shell, and solid elements with orthotropic material properties. In the present context, the finite element analysis for the blower is carried out by packages, ANSYS 12.1 ANSYS can perform the following analysis types: a) Linear static analysis b) Modal analysis c) Harmonic analysis d) Transient analysis e) Spectrum analysis f) Eigen Buckling analysis g) Substructuring analysis In thesis, Static, Modal and Harmonic analyses are performed. 5.8 Element types used in FEM for this thesis:
In present problem, element type solid 46 is used for composite blower and solid45 is used for Aluminum blower. [A] SOLID 46 Element Description (from ANSYS library) 3-D 8-Node Layered Structural Solid 8-node structural solid (SOLID46) is a layered version designed for layered solids or thick shells. Different material layers of up to 250 are permitted by the element. If requirement of layers is more than 250 then availability of a user-input constitutive matrix option is there. As a substitute stacking of elements may also be done. At each node, the element has three degrees of freedom translations in nodal x/y/z directions. Figure 5.3: SOLID46 Geometry x o = Element x-axis if ESYS is not supplied. x = Element x-axis if ESYS is supplied.
[B] SOLID 45 3-D Structural Solid (from ANSYS library) 3-D 8-Node Structural Solid Figure 5.4: SOLID 45 Geometry For the 3-D modeling of solid structures, SOLID45 is used. Eight nodes having three degrees of freedom at each node defines the element. Large strain capabilities, large deflection, stress stiffening, swelling, creep and plasticity properties are held by the element. Availability of reduced integration option having hourglass control is there. The geometry, node locations, and the coordinate system for this element are shown in FIGURE. Orthotropic material properties and eight nodes define the element. The element coordinate directions are corresponded by orthotropic material directions. The element coordinate system orientation is as described in Coordinate systems.