: What is Finite Element Analysis (FEA)?

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Magnus Laurence Lucas
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1 Q: What is Finite Element Analysis (FEA)? A1: It is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations (ODE), which are then numerically integrated [and solved] using standard techniques (Wikipedia entry) To elaborate further: The equations are numerically integrated (they are always stated in integral, or weak form) in order to generate an approximate algebraic equation. It is this characteristic, that gives the Finite Element Method (FEM) its great advantage over other techniques. The algebraic equation is then solved using some (more or less) standard direct or iterative matrix solution algorithm Prior to FEA, engineers and scientists were limited to analytical solutions and more limited numerical techniques. Note that even today, although the Finite Element Method (FEM) dominates the numerical analysis of PDE S and ODE s, it is not the only method still in use! 1

2 Other Techniques: Boundary Element Method (BEM): Generally used for radiation-type problems (or far field problems -see the Helmholtz Equation). Although it can be applied successfully to structural problems, it results in non-sparse matrix systems (we will come back to this concept). Another drawback is that it has difficulty with nonlinearity. Finite Difference Method (FDM): The differential equation is discretized directly, turning quantities like df/dx into (f2-f1)/(x2-x1),etc. This puts severe restrictions on allowable grids or meshes and restricts the possible solution accuracy (FDM solutions are equivalent to FEM solutions with linear shape functions over the same grid spacings). Heterogeneous materials are another difficulty. Finite Volume Method (FVM): Closely related to FEM. Same basic idea except that volume integrals are replaced by surface integrals via the divergence theorem. This only works for Eulerian grids and so finds wide usage in Computation Fluid Dynamics (CFD) 2

3 Other Techniques (Cont.): Legacy Structural Methods : FEA started life in applications to structural problems (engineering) and first emerged in its currently recognizable form in the 1950 s in the Aerospace industry. Before that, aerospace engineers struggled to design aircraft that were both strong and lightweight (and notably, wouldn t induce excessive flutter). Up until the 1930 s, engineers relied on tables, experiment, and FDM. These tools were not up to the challenge of meeting aircraft design challenges in a reliable and timely manner. Then, between 1934 and 1938, Duncan and Collar* invented a matrix technique which involved: 1. Identifying simple structural members (initially, just made of lines or squares in two dimensions) for which analytical (matrix) solutions for various boundary conditions and loading types were known 2. Connecting multiple instances of these together to approximate a larger part or assembly *W. J. Duncan and A. R. Collar, A Method for the Solution of Oscillation Problems By Matrices, Philosophical Magazine Series 7, vol. 17, no. 115, pp ,

4 The last two properties of the Matrix Structural Analysis (MSA) developed by Duncan and Collar are key ingredients in today s Finite Element Method especially as it applies to the analysis of structures which is focus of this course. These are so important, in fact, that we ll incorporate them into our own definition, which combines this aspect with the definition on slide 1. So, let s return to the question and provide a slightly more satisfactory answer: Q: What is Finite Element Analysis (FEA)? A2: It is an approximate numerical technique of solving partial differential equations by converting them to weak (integral) form and assuming a piecewise solution (usually, but not necessarily) in polynomial form. Key properties of the technique are: 1. Complete approximate solutions of anyconservative differential equation are provided over simple polyhedral domains through the use of energy principles, and: 2. The technique is equipped with rules for connecting these polyhedral domains together to approximate more complicated geometries even allowing for certain types of discontinuity 4

5 In studying the Finite Element Method (FEM), one finds that the subject can be approached on several levels and from the perspective of several disciplines. This reflects not just the colorful and somewhat complex history of the subject, but also it s broad applicability. The student can probably already sense this. Note that we started by asking What is FEA?, went to Wikipedia for an answer, and found it strangely unsatisfying. Not only did it not really answer the question, but seemed to come from a perspective far removed from structural analysis. Let there be no mistake: The analysis of engineering structures IS the context in which FEM was born, and happens to be the focus of our course. Thus, it may help to pause for a moment and survey the landscape of modern FEM and identify where our course fits into this vast landscape The categories we ll use are loosely based on those used by C. Felippa in his course taught at the University of Colorado at Boulder* * 5

6 The study AND practice of FEM tends to be approached from the following broad areas Mathematics Theoretical Applied Computational Variational Calculus and energy methods Differential Equations Functional Analysis Numerical Solutions an Approximate Methods Analytical and approximate modeling (continuum modeling) Matrix Methods Convergence of discrete systems/hardware enhancement Solution Algorithms This is our path Engineering Mechanics Theoretical Applied Computational Numerical Solutions an Approximate Methods Analytical and approximate modeling Structural model problems Thermal model problems Other (fluid, electromagnetic, etc.) Combined: Multiphysics Solution Algorithms Parallelization, etc. 6

7 We can elaborate a little further Engineering Mechanics Theoretical Applied Computational Numerical Solutions an Approximate Methods Analytical and approximate modeling Structural model problems Thermal model problems Other (fluid, electromagnetic, etc.) Combined: Multiphysics Solution Algorithms Parallelization, etc. Structural Model Problems Practical applications (implementation) Analytical Numerical Methods MAE 323 is practice-oriented. There is a theoretical component, but only as an aid to understanding how FEM is applied 7

8 Basically, there are two main super-disciplines from which one can drill down to the FEM. These are mathematics and engineering mechanics (a possible third path could be through the physical sciences, but their pathways intersect ours). A recurring theme is that under each super-category, there is always a theoretical, applied, and computational treatment. Textbooks can usually be easily classified using these categories So, we are going to approach FEM from the path of structural analysis. Specifically, we will look mainly at linear problems in elastostatics and then only from a practical standpoint. We will not look at problems in structural dynamics or nonlinear structural problems. Furthermore, as this is an undergraduate junior-level, design-oriented course, we will focus mainly on FEM practice. That is to say on building, solving, and validating FE solutions to structural problems. 8

9 On-Line Resources Students will have access to (and the course will liberally draw from) the following resources: Cornell University ANSYS Course: ANSYS, Inc. Academic Portal (you have to register first): C.A. Felippa s FEM course at the University of Colorado at Boulder: PADT, Inc: 9

AS 5850 Finite Element Analysis

AS 5850 Finite Element Analysis INTRODUCTION Prof. IIT Madras Goal of engineering computations Perform analysis and design of physical systems and processes subjected to imposed conditions (or loads) and