FEM and it's applications

An overall view of what FEM is.

1. FEM & It’s Applications

2. Concept today

3. Numerical Method Finite Element Method Boundary Element Method Finite Difference Method Finite Volume Method Meshles s Method

4. What is FEM ? Many physical phenomena in engineering and science can be described in terms of partial differential equations . In general, solving these equations by classical analytical methods for arbitrary shapes is almost impossible. The finite element method (FEM) is a numerical approach by which these partial differential equations can be solved approximately . From an engineering standpoint, the FEM is a method for solving engineering problems such as stress analysis, heat transfer, fluid flow and electromagnetics by computer simulation . Millions of engineers and scientists worldwide use the FEM to predict the behaviour of structural, mechanical, thermal, electrical and chemical systems for both design and performance analyses .

5. Its popularity can be gleaned by the fact that over $1 billion is spent annually in the United States on FEM software and computer time. A 1991 bibliography (Noor, 1991) lists nearly 400 finite element books in English and other languages. A web search (in 2006) for the phrase ‘finite element’ using the Google search engine yielded over 14 million pages of results. Mackerle (http://ohio.ikp.liu.se/fe) lists 578 finite element books published between 1967 and 2005.

6. Basic Approach of FEM Consider a plate with a hole as shown in Figure 1.1 for which we wish to find the temperature distribution . It is straightforward to write a heat balance equation for each point in the plate.

7. However, the solution of the resulting partial differential equation for a complicated geometry, such as an engine block, is impossible by classical methods like separation of variables. Numerical methods such as finite difference methods are also quite awkward for arbitrary shapes; software developers have not marketed finite difference programs that can deal with the complicated geometries that are commonplace in engineering. Similarly, stress analysis requires the solution of partial differential equations that are very difficult to solve by analytical methods except for very simple shapes, such as rectangles, and engineering problems seldom have such simple shapes.

8. The basic idea of FEM is to divide the body into finite elements , often just called elements , connected by nodes , and obtain an approximate solution as shown in Figure 1.1. This is called the finite element mesh and the process of making the mesh is called mesh generation . The FEM provides a systematic methodology by which the solution, in the case of our example, The temperature field , can be determined by a computer program .

9. FEA Mesh (Q8, 493 elements)

10. For linear problems, the solution is determined by solving a system of linear equations; the number of unknowns (which are the nodal temperatures) is equal to the number of nodes. To obtain a reasonably accurate solution, thousands of nodes are usually needed, so computers are essential for solving these equations. Generally, the accuracy of the solution improves as the number of elements (and nodes) increases, but the computer time, and hence the cost, also increases. The finite element program determines the temperature at each node and the heat flow through each element. The results are usually presented as computer visualizations, such as contour plots, although selected results are often output on monitors. This information is then used in the engineering design process.

11. The same basic approach is used in other types of problems: In stress analysis, the field variables are the displacements; In chemical systems, the field variables are material concentrations; and In electro-magnetics, the potential field. The same type of mesh is used to represent the geometry of the structure or component and to develop the finite element equations, and for a linear system, the nodal values are obtained by solving large systems (from 10 3 to 10 6 equations are common today, and in special applications,10 9 ) of linear algebraic equations.

12. The preponderance of finite element analyses in engineering design is today still linear FEM. In heat conduction, linearity requires that the conductance be independent of temperature. In stress analysis, linear FEM is applicable only if the material behaviour is linear elastic and the displacements are small. In stress analysis, for most analyses of operational loads, linear analysis is adequate as it is usually undesirable to have operational loads that can lead to nonlinear material behaviour or large displacements. For the simulation of extreme loads, such as crash loads and drop tests of electronic components, nonlinear analysis is required.

13. Development of FEM The FEM was developed in the 1950s in the aerospace industry. The major players were Boeing and Bell Aerospace (long vanished) in the United States and Rolls Royce in the United Kingdom. M.J.Turner,R.W. Clough, H.C. Martin and L.J. Topp published one of the first papers that laid out the major ideas in 1956 (Turner et al., 1956) It established the procedures of element matrix assembly and element formulations, but did not use the term ‘finite elements’ . The second author of this paper, Ray Clough, was a professor at Berkeley, who was at Boeing for a summer job. Subsequently, he wrote a paper that first used the term ‘finite elements’, and he was given much credit as one of the founders of the method.

14. This research coincided with the rapid growth of computer power, and the method quickly became widely used in the nuclear power, defence, automotive and aeronautics industries. Much of the academic community first viewed FEM very skeptically , and some of the most prestigious journals refused to publish papers on FEM. It is interesting that for many years the FEM lacked a theoretical basis, i.e. there was no mathematical proof that finite element solutions give the right answer. In the late 1960s, the field aroused the interest of many mathematicians, who showed that for linear problems, finite element solutions converge to the correct solution of the partial differential equation (provided that certain aspects of the problem are sufficiently smooth). In other words, it has been shown that as the number of elements increases, the solutions improve and tend in the limit to the exact solution of the partial differential equations.

15. E. Wilson developed one of the first finite element programs that was widely used. Its dissemination was hastened by the fact that it was ‘freeware’, which was very common in the early 1960s, as the commercial value of software was not widely recognized at that time. The program was limited to two-dimensional stress analysis. Then in 1965, NASA funded a project to develop a general-purpose finite element program by a group in California led by Dick MacNeal . This program, which came to be known as NASTRAN, included a large array of capabilities, such as two- and three-dimensional stress analyses, beam and shell elements, for analyzing complex structures, such as airframes, and analysis of vibrations and time-dependent response to dynamic loads. The initial program was put in the public domain, but it had many bugs.

16. Shortly after the completion of the program, Dick MacNeal and Bruce McCormick started a software firm that fixed most of the bugs and marketed the program to industry. By 1990, the program was the workhorse of most large industrial firms and the company, MacNeal-Schwendler , was a $100 million company. At about the same time, John Swanson developed a finite element program at Westinghouse Electric Corp. for the analysis of nuclear reactors. In 1969, Swanson left Westinghouse to market a program called ANSYS. The program had both linear and nonlinear capabilities, and it was soon widely adopted by many companies. In 1996, ANSYS went public, and it now (in 2018) has a capitalization of $15.116 billion.

17. Another nonlinear software package of more recent vintage is LS-DYNA. This program was first developed at Livermore National Laboratory by John Hallquist . ABAQUS was developed by a company called HKS, which was founded in 1978. The program was initially focused on nonlinear applications, but gradually linear capabilities were also added. The program was widely used by researchers because HKS introduced gateways to the program, so that users could add new material models and elements. In 2005, the company was sold to Dassault Systemes for $413 million and it now (in 2018) has a capitalization of $33.960 Billion.

18. In many industrial projects, the finite element database becomes a key component of product development because it is used for a large number of different analyses, although in many cases, the mesh has to be tailored for specific applications. The finite element database interfaces with the CAD database and is often generated from the CAD database. Unfortunately, in today’s environment, the two are substantially different. Therefore, finite element systems contain translators, which generate finite element meshes from CAD databases; they can also generate finite element meshes from digitization's of surface data. The need for two databases causes substantial headaches and is one of the major bottlenecks in computerized analysis today, as often the two are not compatible.

19. The availability of a wide range of analysis capabilities in one program makes possible analyses of many Complex real-life problems. For example, the flow around a car and through the engine compartment can be obtained by a fluid solver, called computational fluid dynamics (CFD) solver. This enables the designers to predict the drag factor and the lift of the shape and the flow in the engine compartment. The flow in the engine compartment is then used as a basis for heat transfer calculations on the engine block and radiator. These yield temperature distributions, which are combined with the loads, to obtain a stress analysis of the engine.

20. Similarly, in the design of a computer or microdevice, the temperatures in the components can be determined through a combination of fluid analysis (for the air flowing around the components) and heat conduction analysis. The resulting temperatures can then be used to determine the stresses in the components, such as at solder joints, that are crucial to the life of the component. The same finite element model, with some modifications, can be used to determine the electromagnetic fields in various situations. These are of importance for assessing operability when the component is exposed to various electromagnetic fields.

21. In aircraft design, loads from CFD calculations and wind tunnel tests are used to predict loads on the airframe. A finite element model is then used with thousands of load cases, which include loads in various maneuvers such as banking, landing, take-off and so on, to determine the stresses in the airframe. Almost all of these are linear analyses; only determining the ultimate load capacity of an airframe requires a nonlinear analysis. It is interesting that in the 1980s a famous professor predicted that by 1990 wind tunnels would be used only to store computer output. He was wrong on two counts: Printed computer output almost completely disappeared, but wind tunnels are still needed because turbulent flow is so difficult to compute that complete reliance on computer simulation is not feasible.

22. Manufacturing processes are also simulated by finite elements. Thus, the solidification of castings is simulated to ensure good quality of the product. In the design of sheet metal for applications such as cars and washing machines, the forming process is simulated to insure that the part can be formed and to check that after spring-back (when the part is released from the die) the part still conforms to specifications. Thus, FEA has led to tremendous reductions in design cycle time, and effective use of this tool is crucial to remaining competitive in many industries.

23. A question that may occur to you is: Why has this tremendous change taken place?

24. As can be seen from Figure 1.2, the increase in computational power has been linear on a log scale, indicating a geometric progression in speed. This geometric progression was first publicized by Moore, a founder of Intel, in the 1990s. He noticed that the number of transistors that could be packed on a chip, and hence the speed of computers, doubled every 18 months. This came to be known as Moore’s law, and remarkably, it still holds.

25. General Description of The Finite Element Method In the finite element method, the actual continuum or body of matter, such as a solid, liquid, or gas, is represented as an assemblage of subdivisions called finite elements. These elements are considered to be interconnected at specified joints called nodes or nodal points. The nodes usually lie on the element boundaries where adjacent elements are considered to be connected. Since the actual variation of the field variable (e.g., displacement, stress, temperature, pressure, or velocity) inside the continuum is not known, we assume that the variation of the field variable inside a finite element can be approximated by a simple function. These approximating functions (also called interpolation models) are defined in terms of the values of the field variables at the nodes. When field equations (like equilibrium equations) for the whole continuum are written, the new unknowns will be the nodal values of the field variable. By solving the field equations, which are generally in the form of matrix equations, the nodal values of the field variable will be known. Once these are known, the approximating functions define the field variable throughout the assemblage of elements.

26. The solution of a general continuum problem by the finite element method always follows an orderly step-by-step process. With reference to static structural problems, the step-by-step procedure can be stated as follows: Step ( i ): Discretization of the structure The first step in the finite element method is to divide the structure or solution region into subdivisions or elements. Hence, the structure is to be modelled with suitable finite elements. The number, type, size, and arrangement of the elements are to be decided.

27. Step (ii): Selection of a proper interpolation or displacement model Since the displacement solution of a complex structure under any specified load conditions cannot be predicted exactly, we assume some suitable solution within an element to approximate the unknown solution. The assumed solution must be simple from a computational standpoint, but it should satisfy certain convergence requirements. In general, the solution or the interpolation model is taken in the form of a polynomial.

28. Step (iii): Derivation of element stiffness matrices and load vectors From the assumed displacement model, the stiffness matrix [K (e) ] and the load vector of element e are to be derived by using either equilibrium conditions or a suitable variational principle. Step (iv): Assemblage of element equations to obtain the overall equilibrium equations Since the structure is composed of several finite elements, the individual element stiffness matrices and load vectors are to be assembled in a suitable manner and the overall equilibrium equations have to be formulated as where [K] is the assembled stiffness matrix, is the vector of nodal displacements, and is the vector of nodal forces for the complete structure.

29. Step (v): Solution for the unknown nodal displacements The overall equilibrium equations have to be modified to account for the boundary conditions of the problem. After the incorporation of the boundary conditions, the equilibrium equations can be expressed as For linear problems, the vector can be solved very easily. However, for nonlinear problems, the solution has to be obtained in a sequence of steps, with each step involving the modification of the stiffness matrix [K] and/or the load vector .

30. Step (vi): Computation of element strains and stresses From the known nodal displacements (I), if required, the element strains and stresses can be computed by using the necessary equations of solid or structural mechanics. The terminology used in the previous six steps has to be modified if we want to extend the concept to other fields. For example, we have to use the term continuum or domain in place of structure, field variable in place of displacement, characteristic matrix in place of stiffness matrix, and element resultants in place of element strains.

31. Fundamental Concepts (1) Many engineering phenomena can be expressed by “governing equations” and “boundary conditions” Elastic problems L (  )  f  Thermal problems Fluid f low Electrostatics etc. B (  )  g  Bound a ry Conditions Governing Equation (Differential equation)

32. Fundamental Concepts (2) Example: Vert i cal machining center Geometry is very complex! Elastic defo r mation Thermal behavior etc. A set of simultaneous algebraic equations FEM Governing Equation: L (  )  f  B (  )  g  [ K ] { u }  { F } Boundary Conditions: Approximate! Y o u kn o w all the equations, but yo u canno t s ol ve i t b y hand

33. Fundamental [ K ] { u }  { F } Concepts (3) { u }  [ K ]  1 { F } Property Action Behavior Unknown Property [ K ] Behavior { u } Action { F } Elas t ic stiffness dis p l a c e ment force Thermal conductivity temperature heat source Fluid v i scosity vel o city body fo rc e Electrosta t ic d i alectri permittivity e l ectric potent i al charge

34. Fundamental Concepts (4) It is very difficult to make the algebraic equations for the entire domain Divide the domain into a number of small, si m ple elements A field quantity is interpolated by a polynomial over an element Adjacent elements share the DOF at connecting nodes Finite element: Small piece of structure

35. Fundamental Concepts (6) Solve the equations, obtaining unk n own variabless at nodes. { u }  [ K ]  1 { F } [ K ] { u }  { F }

36. Concepts - Summary - FEM uses the concept of piecewise polynomial interpolation. - By connecting elements together, the field quantity becomes interpolated over the entire structure in piecewise fashion. - A set of simultaneous algebraic equations at nodes. K x = F [ K ] { u }  { F } K: Stiffness matrix x: Displacement F: Load K Property Action Behavior x F

37. Typical FEA Procedure by Commercial Software User Build a FE model Computer Conduct numerical analysis See results User Postprocess Process Preprocess

38. Preprocess (1) [1] Select analysis type - S tructural Static Anal y sis - Mo d al Anal y sis - Transient D y namic Anal y sis - B u c kl i n g An a l y sis - C ontact - S tead y -state The r mal Ana l y sis - Transient Thermal Anal y sis Linear Truss 2-D [2] Select element type Beam Quadratic 3-D Shell Plate Solid E ,  ,  ,  , " [3] Material properties

39. Preprocess (2) [4] M a ke nodes [5] Build elements connectivity by assigning [6] Apply boundary and loads conditions

40. Process and Postprocess [7] Process - Solve the boundary v alue problem [8] Postprocess - See the results Displacement Stress Strain Natural frequency Temperature Time history

41. Responsibility of the user Fancy, colorful contours can 200 m m be produced by any model, good or bad!! 1 ms pressure pulse BC: Hinged supports Lo a d: Pressure pulse Unkno w n : Lateral mid point displacement in the time domain Re s ults obtained from ten reputable FEM codes and by use r s rega r ded as expert.* * R. D. C o ok, F i nite Element M odel i n g f o r Stress A nalysis, Jo h n Wiley & S o ns, 1995 Displacement (mm) Time (ms)

42. Errors Inherent in FEM Formulation Approximated Domain - Geometry is simplified. domain FEM - F ield quantity is assumed to be a polynomial over an eleme nt . ( which is not true) True deform a tion Quadratic element C u bic element L i near element FEM - Use very simple integration techniques (Gauss Quadrature) f(x)  1     1  1   1 A r ea: f ( x ) dx  f f  3   3      x -1 1

43. Errors Inherent in Computing - The computer carries only a finite number of digit s . 2  1.4142135 6 ,   3.14159265 e . g.) - Numeric a l Diffic u l ties e.g.) Very large stiffness difference k 1 » k 2 , k 2 � O P P [ ( k 1  k 2 )  k 2 ] u 2  P  u 2  � k 2 O

44. Mistakes by Users - Elements are of the wrong type e.g) Shell elements are used where solid elements are needed - D istorted elements - Supports are insuffi c ient to prevent a ll rigid-body m otions - I nconsistent units (e.g. E=200 GPa, Force = 100 lbs) - Too large stiffness differences  Numerical difficulties

45. Types of Finite Elements 1-D (Line) Element (Spring, truss, beam, pipe, etc.) 2-D (Plane) Element (Membrane, plate, shell, etc.) 3-D (Solid) Element (3-D fields - temperature, displacement, stress, flow velocity)

46. Typical 3-D Solid Elements Tetrahedron: linear ( 4 nodes) quadratic ( 10 nodes) Hexahedron (brick): linear ( 8 nodes) quadratic ( 20 nodes) Penta: linear ( 6 nodes) quadratic ( 15 nodes) Avoid using the linear (4-node) tetrahedron element in 3-D stress analysis (Inaccurate! However, it is OK for static deformation or vibration analysis).

47. Substructures (Superelements) Substructuring is a process of analyzing a large structure as a collection of (natural) components. The FE models for these component s ar e calle d substructures or superelements (SE). Physical Meaning: A finite element model of a portion of structure. Mathematical Meaning: Boundary matrices which are load and stiffness matrices reduced (condensed) from the interior points to the exterior or boundary points.

48. Advantages of Using Substructures/Superelements:  Large problems (which will otherwise exceed your computer capabilities) Less CPU time per run once the superelements have been processed (i.e., matrices have been saved) Components may be modeled by different groups Partial redesign requires only partial reanalysis (reduced cost) Efficient for problems with local nonlinearities (such as confined plastic deformations) which can be placed in one superelement (residual structure) Exact for static stress analysis      Disadvantages:  Increased overhead for file management  Matrix condensation for dynamic problems introduce new approximations  ...

49. I . Spring Element One Spring Element x i j f i u i u j f j k Two nodes: Nodal displacements: Nodal forces: Spring constant (stiffness): Spring force-displacement relationship: i, j u i , u j (in, m, mm) f i , f j (lb, Newton) k (lb/in, N/m, N/mm) F  k  with   u j  u i Linear N onlinear F k  k  F /  (> 0) is the force needed to produce a unit stretch.

50. Consider the equilibrium of forces for the spring. we have At node i, f i  and at node j,  F   k ( u j  u i )  k u i  ku j f j  F  k ( u j  u i )   k u i  ku j In matrix form,  k   u i  f i k           k  u f k    j   j  or, ku  f where k = (element) stiffness matrix u = (element nodal) displacement vector f = (element nodal) force vector Note that k is symmetric. Is k singular or nonsingular? That is, can we solve the equation? If not, why?

51. Spring System x k 2 k 1 1 u 1, F 1 2 u 2, F 2 3 u 3, F 3 For element 1, k 1  k 1   u 1  1  f 1          k  1 k u f  1 1   2   2  element 2,  k 2   u 2  2 k 2  f 1      u   f 2    k k    3    2 2 2 m where f i is the (internal) force acting on m ( i = 1, 2). local node i of element Assemble the stiffness matrix for the whole syste m : Consider the equilibrium of forces at node 1, F 1  f 1 at node 2, 1 1 2 F 2  f 2  and node 3, F 3  f 2 f 1 2

52. That is, F 1  k 1 u 1  k 1 u 2 F 2   k 1 u 1  ( k 1  k 2 ) u 2  k 2 u 3 F 3   k 2 u 2  k 2 u 3 In matrix form, k 1  k 1  k 2   u 1      F 1       k 2  u 2    F 2           k 2 k 2    u 3   F 3  or KU  F K is the stiffness matrix (structure matrix) for the spring system. An alte r native way of assembling the whole stiffness matri x : “Enlarging” the stiffness matrices for elements 1 and 2, we have 1 k 1  k 1   u 1   f 1       1     k 1     k 1   u 2    f 2         u 3      k 2   u 1      2      k 2   u 2    f 1     2   k 2 k 2    u 3   f 2    k 1 k 1  k 

53. Adding the two matrix equations ( superposition ), we have 1 k 1  k 1   u 1   f 1       2   1   k 1 k 1  k 2  k 2   u 2    f 2  f 1      2    k 2 k 2    u 3  f 2   This is the same equation we derived by using the force equilibrium concept. Boundary and load conditions : Assuming, we have  u 1  and F 2  F 3  P k 1  k 1     F 1          k 1 k 1  k 2  k 2   u 2    P        which reduces to  k 2 k 2    u 3   P   k 1  k 2  k 2   u 2   P          k 2 k 2   u 3   P   and F 1  Unknowns are  k 1 u 2  u 2    U  and the reaction force F (if desired). 1  u 3 

54. Solving the equations, we obtain the displacements  u 2  2 P / k 1         u 3   2 P / k 1  P / k 2  and the reaction force F 1   2 P Checking the Results  Deformed shape of the structure  Balance of the external forces  Order of magnitudes of the numbers

55. II. Bar Element Consider a uniform prismatic bar: u j u i f i f j i j L A E u  u ( x )    ( x )    ( x ) length cross-sectional area elastic modulus displacement strain stress Strain-displacement relation : du   (1) dx Stress-strain relation :   E  (2) x A, E L

56. Stiffness Matrix --- Direct Method Assuming that the displacement u is varying linearly along the axis of the bar, i.e., u ( x )   1  x  u  x u (3)    L  i j L we have u  u  L j i    (  = elongation) (4) L E  L   E   (5) We also have F A   ( F = force in bar) (6) Thus, (5) and (6) lead to E A  F   k  (7) L EA L where k  is the stiffness of the bar. The bar is acting like a spring in this case and we conclude that element stiffness matrix is

57. EA EA       k  k    L EA L k      EA   k k      L L  or 1  1  E A  k  (8)     1 1 L  This can be verified by considering the equilibrium of the forces at the two nodes. Element equilibrium equation is 1  1   u i  f i   EA   (9)         u j   f j    1 1 L Degree of Freedom (dof) Number of components of the displacement vector at a node. For 1-D bar element: one dof at each node. Physical Meaning of the Coefficients in k The j th column of k (here j = 1 or 2) represents the forces applied to the bar to maintain a deformed shape with unit displacement at node j and zero displacement at the other node.

58. Equation Solving Direct Methods (Gauss Elimination):  Solution time proportional to N B 2 ( N is the dimension of the matrix, B the bandwidth)  Suitable for small to medium problems, or slender structures (small bandwidth)  Easy to handle multiple load cases Iterative Methods:  Solution time is unknown beforehand  Reduced storage requirement  Suitable for large problems, or bulky structures (large bandwidth, converge faster)  Need solving again for different load cases

59. Stress Calculation The stress in an element is determined by the following relation,   x      x      y     E   y  EBd (39)      xy    xy  where B is the strain-nodal displacement matrix and d is the nodal displacement vector which is known for each element once the global FE equation has been solved. Stresses can be evaluated at any point inside the element (such as the center) or at the nodes. Contour plots are usually used in FEA software packages (during post-process) for users to visually inspect the s t ress results. The von Mises Stres s : The von Mises stress is the effective or equivalent stress for 2-D and 3-D stress analysis. For a ductile material, the stress level is considered to be safe, if  e   Y where  e material. is the von Mises stress and  Y the yield stress of the This is a generalization of the 1-D (experimental) result to 2-D and 3-D situation s .

60. The von Mises stress is defined by 1 2 2 2  e  (    )  (    )  (    ) (40) 1 2 2 3 3 1 2 in which  1 ,  2 and  3 are the three principle stresses at the considered point in a structure. For 2-D problems, the two principle stresses in the plane are determined by 2         P x y x y 2  1    xy       2  2   (41) 2     P x y x y 2  2    xy     2  2 Thus, we can also express the von Mises stress in terms of the stress components in the xy coordinate system. For plane stress conditions, we have,  e  (  x   y )  3 (  x  y   xy ) 2 2 (42) Averaged Stresses : Stresses are usually averaged at nodes in FEA software packages to provide more accurate stress values. This option should be turned off at nodes between two materials or other geometry discontinuity locations where stress discontinuity does exist.

61. FEA Stress Plot (Q8, 493 elements)

62. Discussions 1) Know the behaviors of each type of element s : T3 an d Q4 : T6 an d Q8 : linear displacement, constant strain and stress; quadratic displacement, linear strain and stress. 2) Choose the right type of elements for a given proble m : When in doubt, use higher order elements or a finer mesh. 3) Avoid elements with large aspect ratios and corner angle s : Aspect ratio = L max L min / where L max and L min are the largest and smallest characteristic lengths of an element, respectively. E le m e nt s wit h B a d Shapes E le m e nt s wit h Nic e Shapes

63. 4) Connect the elements properly : Don’t leave unintended gaps or free elements in FE models. A C D B I m prope r connection s (gap s alon g A B an d C D )

64. Nature of Finite Element Solutions  FE Model – A mathematical model of the real structure, based on many approximations.  Real Structure -- Infinite number of nodes (physical points or particles), thus infinite number of DOF’s.  FE Model – finite number of nodes, thus finite number of DOF’s. Ö Displacement field is controlled (or constrained) by the values at a limited number of nodes. Recall that on an element : 4 u   N  u    1 Stiffening Effect:  FE Model is stiffer than the real structure.  In general, displacement results are smaller in magnitudes than the exact values.

65. Hence, FEM solution of displacement provides a lower bound of the exact solution.  ( Di s p lacement ) E x act S o l u ti o n FEM S o l u ti o n s N o . o f DO F’s The FEM solution approaches the exact solution from below. This is true for displacement based FEA!

66. Numerical Error Error  Mistakes in FEM (modeling or solution). Type of Errors:  Modeling Error (beam, plate … theories)  Discretization Error (finite, piecewise …)  Numerical Error ( in solving FE equations) Example (numerical error): u 1 u 2 P x 2 k 2 1 k 1 F E Equations: k  k u P    1     1 1         k k  k   u    1 1 2 2 and Det K  k 1 k 2 . Th e syste m wil l b e singular if k 2 is small compared with k 1 .

67. 2 1 k u 2 u 1 2 k  k u 1 2 1 k u 2  k 1 u u 1 2 k  k 1 2 u 1  Large difference in stiffness of different parts in FE model may cause ill-conditioning in FE equations. Hence giving results with large errors.  Ill-conditioned system of equations can lead to large changes in solution with small changes in input (right hand side vector). P / k 1 k 2 > > k 1 (two line apart ) : Ö S y stem well con d itio n ed. u  u  P 1 P/ k 1 k 2 << k 1 (two lines clos e ) : Ö S y s t em ill-condi t io n ed. u  k 1 1 2 u  u  P 1

68. Conve r gence of FE Solut i ons The selection of the approximation functions in each element will be made to satisfy the necessary conditions to ensure the convergence of the method, i.e., that the approximate solution will eventually get closer and closer to the exact solution as we refine the mesh, i.e. as we divide the domain into more and more elements. The two necessary conditions for a finite element method to converge are the following:

69. As the mesh in an FE model is “refined” repeatedly, the FE solution will converge to the exact solution of the mathematical model of the problem (the model based on bar, beam, plane stress/strain, plate, shell, or 3-D elasticity theories or assumptions). Type of Refinements : h-refinemen t : reduce the size of the element ( “ h” refers to the typical size of the elements); Increase the order of the polynomials on an element (linear to quadratic, etc.; “ h ” refers to the highest order in a polynomial); re-arrange the nodes in the mesh; Combination of the h- and p-refinements (better results!). p-refinemen t : r-refinemen t : hp-refinemen t :

70. A daptivity (h-, p-, and hp-Methods)  Future of FE applications  Automatic refinement of FE meshes until converged results are obtained  User’s responsibility reduced: only need to generate a goo d initia l mesh Error Indicators: Define,  --- element by element stress field (discontinuous),  * --- averaged or smooth stress (continuous),  *  E =  - --- the error stress field. Compute strain energy, M U   U , 1 σ T E  1 σ dV ; U   V i i i 2 i  1 1 M   U * , σ * T E  1 σ * dV ; U * U *   i 2 i i  1 V i 1 M U   U σ T E  1 σ , U   V i dV ; E E i E i E E 2 i  1 where M is the total number of elements, V i is the volume of the element i .

71. One error indicator --- the relative energy error: 1 / 2  U    (    1 ) E   .  U  U E  The indicator  is computed after each FE solution. of the FE model continues until, say   0.05. => converged FE solution. Refinement

72. Advantages of the FEM Can readily handle very complex geometry: - The heart and power of the FEM Can handle a wide - Solid mechanics - Fluids variety of engineering p r oblems - Dynamics - Heat problems - Electrostatic problems Can handle complex restraints - Indeterminate structures can be solved. Can handle complex loading - Nodal load (point loads) - Element load (pressure, thermal, inertial forces) - Time or frequency dependent loading 15

73. Disadvantages of the FEM A general closed-form solutio n , whic h woul d permit to examine system response to changes in various parameters, is not produced. one The FEM obtains only "approximate" solutions. The FEM has "inherent" errors. Mistakes by users can be fatal. 16

74. Applications of Finite Element Method

75. Examples Crash Anal y sis for a Car (from LS-DYNA3D )

76. Modeling of gear coupling Can Drop Test

77. FEA of an Unloader Trolley

78. Crack Growth Analysis In this pressure vessel example, FEA software allows for the prediction of crack growth along arbitrary paths that do not correspond to element boundaries.

79. Thank You !!!