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ARTICLE DE FOND FINITE ELEMENT ANALYSIS ISSUES AND TRENDS BY IN SOLID MECHANICS: NADER G. ZAMANI T he area of Finite Element Analysis (FEA) has become a standard tool in the numerical solution of the field equations governing physical problems. These field equations arise in diverse areas such as: solid mechanics, fluid dynamics, heat transfer, electrostatics, and electromagnetism. Generally speaking, these are coupled, nonlinear, and time dependent partial differential equations which describe some form of conservation law. Depending of the nature of the field, the conservation of momentum, energy, and charge are normally taken into account. The concepts behind these field equations have been known to the science community since the early 1800s. These concepts are attributed to prominent physicists/mathematicians such as Euler, Lagrange, Laplace, Bernoulli, and Fourier to name a few. Some of the approximate numerical schemes which are the basis of the FEA approach are due to well known scientists such as Raleigh, Ritz, and Galerkin. The breakthrough, however, came about due to the development of high speed digital computers. At that point, the early numerical schemes were altered and modified making them efficient, accurate, and feasible for implementation on computers. The first comprehensive numerical solution which embraced FEA concepts in its modern form is attributed to Courant [1] where he used piecewise linear polynomials on a triangular mesh to solve the Laplace equation. Although this work set the wheel in motion, it was not until the early 1960 where serious development in FEA started. It is not surprising that this activity coincided with the development of high speed computers in the private sector. One of the early projects along this path was the development of the NASTRAN program which was created by NASA for structural analysis [2]. This code that still exists, has gone through continuous updating and improvements and prob- SUMMARY This expository paper discusses the area of Finite Element Analysis (FEA) as pertaining to the subject of solid mechanics. FEA as a computational tool has evolved rapidly in the past fifty years and continues to do so with technological advances in the computer industry. The paper briefly presents a historical background together with the current status of the field, and the future trends. ably is the most widely used FEA software for structures. Due to the NASA connection, the NASTRAN code is in the public domain and the source code can be acquired at no cost to the user. APPROPRIATE ELEMENT SELECTION To avoid total generality, the rest of this expository article focuses on the FEA formulation for solid mechanics applications. Other areas such as fluids, heat transfer, and electromagnetism follow the same track. In structural analysis, the primary variable of interest is the displacement vector. Once the displacements are determined, strains can be computed, and based on the material response, the stresses are evaluated. For the sake of illustration, assume a linear behavior in kinematics and constitutive law. Depending on the topological nature of the structure, the three most common elements are solid, shell, and beam elements which are symbolically displayed in Fig. 1. The number of nodes is one of the factors determining the accuracy of the results [3]. Fig. 1 (a) solid, (b) shell, (c) beam elements If the topology is one dimensional (or a composite of onedimensional parts) such as the frame of building, or a communications tower, the beam elements have to be used. On the other hand, if the topology is two-dimensional such as a pressure vessel or aircraft wing, the shell elements covering surfaces are the most appropriate. Finally, for a bulky object with no specific topological characteristics, solid elements are commonly used In principle; every structure can be modeled with solid elements, but the demands on resources make it impractical even with today’s computing power. All these elements have to be modified in one form or other to be able to handle special situations such as: material incompressibility, material plasticity and visoelasticity. N.G. Zamani <zamani@uwindsor. ca>, Department of Mechanical, Automotive and Materials Engineering University of Windsor, Windsor, Ontario, Canada LINEAR VS. NONLINEAR RESPONSE In a finite element analysis, there are three sources of nonlinearity. These are labeled as geometric, material, and contact [4]. For the special case of strictly linear problems, LA PHYSIQUE AU CANADA / Vol. 64, No. 2 ( avr. à juin (printemps) 2008 ) C 55 FINITE ELEMENT ANALYSIS (ZAMANI) the details of the code may vary, but all FEA codes basically give the same results. This is assuming that the same elements are used and the same numerical integration algorithm is employed. The minor differences are due only to code implementation. A geometric nonlinearity refers to the case of large displacements, large rotations, and large strains. These are considered to be mild nonlinearities which can easily be handled with a good iterative solver. All nonlinearities require an iteration approach for the numerical solution. Such algorithms are variations of the Newton-Raphson method or its secant implementation. For a mnemonic of this behavior, see Fig. 2(a). The material response is also known as the constitutive law. This represents the relationship between the stress and the strain (or force and deflection). Most materials display a linear response in a very narrow range. To give the reader a better idea, consider the stretching of a rubber band. For small forces, the relationship between the applied force and the resulting stretch is linear. However, very quickly this linearity is lost and a rather complicated path is traversed. This is an example of a nonlinear elastic response. The situation in plasticity is considerably more complicated but falls into the category of nonlinear constitutive response. Material nonlinearity is also considered to be a mild nonlinearity. For a mnemonic of this behavior, see Fig. 2(b). Presently, the majority of commercial FEA codes are capable of handling both geometric and material nonlinearities. The degree of their performance (in terms of efficiency and accuracy) varies from code to code. Furthermore, some codes have a vast database of material properties which could be preferred by the users. Fig. 2 (a) nonlinear geometry, (b) nonlinear material, (c) nonlinear contact The most severe type of nonlinearity is generally due to contact condition. Basically, any type of metal forming application such as forging, stamping, and casting require contact calculations, see Fig. 2(c). The mathematical tools for handling contact algorithms involve the Lagrange multiplier and/or constrained optimization. A poor formulation often leads to lack of convergence and other numerical difficulties. STATIC VS. DYNAMIC RESPONSE Technically speaking, all loads are dynamic (time dependent) in a real world environment. The main issue is whether the inertia effect (mass times acceleration) is significant compared to other loads. In a nutshell, this has to do with the duration of the applied load compared to the natural periods (inverse of the 56 C PHYSICS IN natural frequencies) of the structure [5]. For example, an impact load on many occasions leads to a substantial inertia effect. In terms of dynamic analysis, commercial FEA packages give the user several options for carrying out the calculations. This is schematically represented by Fig 3. For linear problems, the modal superposition is generally available. The user can select the number of modes and therefore control the accuracy of the results. One can also use the full history analysis by numerically integrating the equations of motion in time. The term full time history analysis refers to the fact that the governing field equation is a time dependent differential equation. Therefore, the unknowns are also time dependent. This system of differential equations has to be solved numerically by an approximate integration it time. Clearly, in nonlinear problems, this is the default approach. A variety of integration routines is available but the most common ones are the central differencing and the Newmark method. The former is conditionally stable whereas the latter is unconditionally stable. Fig. 3 (a) Static response for a slowly varying load, (b) Dynamic response for a fast varying load IMPLICIT VS. EXPLICIT FORMULATION There are two finite element methodologies in solid mechanics. These are known as the Explicit and Implicit methodologies [6]. The term Explicit refers to the fact that when numerical integration in time is carried out, a predicted entity can be written directly in terms of the past values without actually solving a system of algebraic equations. Whereas in the Implicit approach, in order to calculate a predicted value, one is bound to solve a system of equations and therefore more computation is involved. Both are designed to solve (integrate) the equations of motion in time. The equation of motion for the linear case can be stated as [M]{ẍ}+[C]{ẋ}+[K]{x}={F(t)}. Here [M], [C], and [K] are the mass, damping and stiffness matrices respectively. The vector represents the displacement vector and {F(t)} is the vector of applied loads. The methodology used depends on the nature of the application being considered. Generally speaking, short duration events such as metal forming, crashworthiness, and detonation require explicit codes. In such codes, the mass matrix is approximated to become diagonal, and the central differencing method is used for time integration. The stiffness matrix is not stored in its entirety at every time step and no iterations are carried at each time step. However, the conditional stability of central differencing requires an extremely small time step selection. There are very few explicit FEA codes and they require consid- CANADA / VOL. 64, NO. 2 ( Apr.-June. (Spring) 2008 ) FINITE ELEMENT ANALYSIS (ZAMANI)AA erable computing resources. The majority of the existing commercial FEA codes however are based on an implicit formulation. This is not surprising as the bulk of the design problems in engineering and product developments can satisfactorily be handled with the implicit FEA formulation. There is another important difference between the implicit and explicit codes. In nonlinear problems, implicit codes require substantial iteration steps. If the conditions are not realistic, the solution usually diverges and the user is informed. However, in explicit calculations, since no iteration is involved, the software always arrives at a solution. The difficulty is that this may not be the solution to the problem under consideration. It is worth mentioning that problems which ordinarily can be solved with an implicit code can also be solved with an explicit one. However, the extremely small time step will dictate an unreasonable solution time. The remedy is referred to as the mass scaling option that is available in explicit codes. In this option, the density of the material is artificially changed to result in an attainable run time. One should carefully check the energy history to make sure that the results are not contaminated by non-physical effects. MESH ADEQUACY AND REFINEMENT One of the most common questions when dealing with finite element analysis is how small a mesh should be used for a desirable accuracy. The user should be reminded that one cannot provide an answer without performing a mesh convergence study. Basically, making a single run regardless of how small the elements are; provides no information on the accuracy. The key is in making a sequence of runs with decreasing element size and comparing the differences in the results. Of course the refinement should be performed in the critical regions and the comparison of the results should also be made in the critical locations. When the percentage change is to the user’s satisfaction, the mesh is assumed to be satisfactory. It is well known that displacements converge faster that the stresses, however, the latter entities are more important. Therefore, the convergence should be based on the stress variable. The strategy above is known as “h” refinement. There are two other strategies known as the “r” and “p” methods [7]. In the “p” method, the mesh is fixed but the degree of the approximating polynomial is increasing. Although the “p” strategy displays promising results in linear problems, it is not available in most commercial codes. The final refinement strategy is the so called “r” method. There, the number of nodes (and elements) is fixed. However, their locations are adaptively changed to reduce the error estimate. This method has also been implemented for the Boundary Element Method for linear problems [8]. Currently, most commercial FEA codes have an adaptive (automatic) mesh refinement capability for solving linear problems. Sophisticated error estimators are used to perform the refinement strategies [9]. SOURCES OF ERROR IN AN FEA CALCULATION Understanding the sources of error in a finite element calculation is vital to obtaining good results [10]. In this section, these sources are briefly described. The most obvious source is the mathematical model that is expected to represent a physical phenomenon. This source is beyond the control of the typical user. Engineers and physicists are primarily responsible in arriving at an accurate model. The second source is the approximation of the physical domain with the finite element model. If the boundaries of the domain are curved surfaces, finite elements may only approximately represent this domain as shown in Fig. 4(a). The use of higher order elements can reduce this error. Mesh refinement will also improve the error. The interpolation error is displayed in Fig. 4(b). The nature of the shape functions dictates how well the finite element functional variation approximates the exact solution. Higher order elements approximate the exact solution more accurately. The error in numerical integration is also a critical factor in controlling the error. This is symbolically displayed in Fig. 4(c) where the area under a curve represented by an integral is approximated by the area of the trapezoid. There are however circumstances where intentionally some error is introduced in the integration process. This eliminates the possibility of unrealistically stiff structures [11]. The final source of error is the mathematical round-off which could dramatically affect the results. There are different reasons for this undesirable effect. Among the reasons are the single precision calculations, extreme mesh transition, and hard/soft regions being present [12]. Fig. 4 (a) physical domain approximated by the finite element domain, (b) exact solution approximated by the finite element solution, (c) area under curve approximated by area under line OPTIMIZATION The primary role of a commercial finite element package is to perform analysis. However, the ability to perform analysis naturally leads to the idea of optimization. In this situation, the objective function, constraints and design variables are defined first. A sequence of analysis is performed which systematically updates the design variables such that the objective function is optimized [13]. The optimization calculations can be based on the gradient methods or more recent approaches such as the Genetic algorithm [14]. Most recent commercial codes have an optimization module. To give a concrete example of how optimization is used, consider the design of a loaded part to have a LA PHYSIQUE AU CANADA / Vol. 64, No. 2 ( avr. à juin (printemps) 2008 ) C 57 FINITE ELEMENT ANALYSIS (ZAMANI) minimum weight, where the von Mises stress is to remain below the yield strength of the material. VECTORIZATION FOR MULTIPROCESSING The mainframe supercomputers appeared in the market about twenty-five years ago. This prompted the FEA software companies to revise and reexamine their codes to run efficiently on these machines. It mainly consisted of vectorizing their codes to utilize the multiprocessor nature of the supercomputers. The multiprocessing capability has recently been introduced in the personal computer (PC) market. Currently, the major FEA software has separate installations which allow them to use a number of processors. Naturally, the licensing cost for such versions is more expensive than for a single processor version. PRE AND POSTPROCESSING CAPABILITIES It was not very long ago that commercial FEA software relied solely on third party pre- and pos-processors. The finite element software companies primary put their efforts on the solver module. This caused a great deal of inconvenience for the user who needed to invest additional time to train in separate software. This was particularly troublesome when FEA software were marketed to operate on the personal computers. The turn around solution was achieved by two approaches. Some FE software where modified to have a proprietary preand post-processor written from scratch to handle the needs, while others incorporated third party codes and integrated them with the solver module. This allowed the user to seamlessly perform the pre- and post-processing, and run the finite element analysis simultaneously. Currently, all commercial FEA software packages have their own pre- and post-processors. They also have the flexibility of transferring data to and from third party software. Mesh generation still remains a challenging issue in a preprocessor. This is particularly the case when the geometry under consideration is complicated. As an example, one can visualize the meshing of a full automobile engine block. Creating a free mesh using tetrahedron elements is now feasible regardless of the complexity of the geometry. However, if all hexahedral elements are needed, the situation is not completely satisfactory. Mesh generation remains an active research area in applied mathematics. FEA AND CAD SOFTWARE INTEGRATION A large number of general purpose commercial FEA software packages has been developed and made available in the public domain since early 1970s. This is also the case with CAD packages which are widely used in industry. Clearly, the spectrum of the CAD software is rather wide depending on their capabilities. These packages are traditionally used by the so called designers who are not formally trained in physics or engineering. Their experience is gained by on-the-job training and they usually act as the interface between the production (fabrication) and the engineering divisions. The global trend is the elimination of such positions and replacing them with qualified engineers or physicists. This has prompted the integration of FEA modules in standalone CAD packages. The numbers of CAD and FEA software packages have dwindled in the past decade and now there are a handful of fully integrated CAD/FEA packages which are referred to as CAE software. In this context, CAE also embraces Computational Fluid Dynamics (CFD) modules. Therefore, the analysis capabilities are seamlessly integrated with CAD features. The cycle does not end at this stage and in most cases are directly linked to the Computer Aided Manufacturing (CAM) area which is the end of the product development cycle. It is expected that this tend will continue with only a few fully integrated CAE software packages handling the entire design process. CLOSING REMARKS One of the important points that is being raised in this expository article is to emphasize that not all FEA software packages are the same. The user should clearly identify the needs for his/her analysis. The decision should also factor the CAD requirements. The cost of the software has a direct link to the capabilities of the acquisition. Another factor which should be seriously taken into account is the type and level of the technical support available for the CAE software. One should not assume that the software’s documentation is sufficient and well enough written for an average reader. This could be a major issue, as training courses can be extremely expensive, and in some cases not even available. Online searches and sharing information with other users can be of great value to decide which software fits the users’ needs. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. R. Courant. Bull. Am. Math. Soc., 49,1 (1943). R.H. MacNeal, NASTRAN Theoretical Manual, NASA-SP, 221, 3 (1976). R.D. Cook, Finite Element Modelling for Stress Analysis, John Wiley and Sons (1995). K.J. Bathe, Finite Element Procedures, Prentice Hall (1996). T.J. Hughes, Linear Static and Dynamic Finite Element Analysis, Prentice Hall (1987) T. Belytschko, W.K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley and Sons (2000). J.N. Reddy, Introduction to the Finite Element Mathod, McGraw Hill (2006) N.G. Zamani and W. Sun, IJNME, 44, 3 (1991). B. Szabo and I. Babuska, Finite Element Analysis, John Wiley and Sons (1991). P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland (1978). O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, McGraw Hill (1988). G. Strang and G. Fix, An Analysis of the Finite Element Method, Prentice Hall (1973). S. Moaveni, Finite Element Analysis, Theory and Applications with ANSYS, Prentice Hall (2008). G. Lindfield and J. Penny, Numerical Methods Using MATLAB, Prentice Hall (2000). 58 C PHYSICS IN CANADA / VOL. 64, NO. 2 ( Apr.-June. (Spring) 2008 )