Lecture 1: Course Introduction.

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1 Lecture : Course Introduction. What is the Finite Element Method (FEM)? a numerical method for solving problems of engineering and mathematical physics. (Logan Pg. #). In MECH 40 we are concerned with problems arising in applied mechanics. Mechanics: That science, or branch of applied mathematics, which treats of the action of forces on bodies. Applied Mechanics: the practical application of the laws of matter and motion to the construction of machines and structures of all kinds. The FEM is a numerical tool that factors into the design of machines and structures. In the modern engineering paradigm: CAD/CAM & Solids Modelling. Computational Fluid Dynamics. Virtual prototyping. The FEM is equally applicable to fluid flow, heat transfer, solid mechanics.

2 Lecture : Course Introduction. COMPUTATIONAL MECHANICS. Solids & Structures (statics & dynamics).. Fluid Flow 3. Heat Transfer 4. Coupled Systems (Mass transport ie: Fuel cells) There are several ways to obtain approimate solutions to the DE s that arise in these fields of study. Finite Element Method. Finite Difference Method. Finite Volume Method. Computational Fluid Dynamics Boundary Element Method.

3 Lecture : Course Introduction. In terms of the UVic Mech Eng curriculum: Mechanics Theoretical Applied Computational MECH 4 MECH 0 MECH 30 MECH 330 MECH 34 MECH 395 MECH 40 A means for the solution of the governing equations that are developed through application of theory and idealizations. Applied mechanics generally includes some idealization (ie: small displacements).

4 Lecture : Course Introduction What is the objective of MECH 40? Vector calculus. Vibrations. Elasticity. Linear algebra. Object oriented programming.???

5 Lecture : Course Introduction Why is MECH 40 a part of a valuable part of a B.Eng. degree? By its very definition the FEM can NOT give an eact solution (see comments to come on weak form formulations). FE modelling is a tool to establish the best approimation possible. The best approimation can only be identified by evaluation of the FEM output. Without knowledge of the FEM s limitations and the assumptions inherent in element equations the best guess can be a terrible approimation. MECH 40 will show that the FEM requires the user to eert a great deal of creative choice. Variational formulations choosing a functional. Weighted Residual Formulations choosing a weak form. Direct Stiffness Method choosing a displacement field.

6 Lecture : FEM Introduction. Brief History. In historical discussions, FEM is often replaced with MSA. The evolution of the current FEM/MSA occurred in distinct stages. 930 s: aeronautic engineers begin to put aeroelasticity problems in discrete, or matri, form. British National Physics Laboratory (NPL). Concerned with flutter and wing vibration. Divide an airscrew blade into 0 segments and define screw vibration in terms of ten DOF. Use an iterative procedure to solve a matri set of equations. The epressions Mass, Force, & Stiffness matrices were coined. WWII interrupted the development of MSA. Timelines were short and eperimentation ruled the day. Publication restrictions prevented an evolution of the technique.

7 Lecture : FEM Introduction : Boeing s analysis of the Delta Wing. Engineers couldn t model the wing using D beams and struts. Placed restrictions on the displacement on the nodes of a triangular piece. Related the displacement field to stresses and then to nodal loads. Method was referred to as the Direct Stiffness Method (DSM). In parallel, British Engineers developed a rectangular element : Direct Stiffness Method (DSM). Method of relating forces and displacements algebraically (matri notation). Solution to these systems was fostered by a growth in computing power. UNIVAC (95): bit word capacity. A lot of work conducted in the solution process. 960 s & 970 s the FEM is born. Clough (Boeing) coined the phrase Finite Element referring to use of tangible rectangular and triangular elements to solve plane stress problems. Sparked by developments in computing power. The assembly process is mentioned in print.

8 Lecture : FEM Introduction Most general purpose structural FE packages use element equations derived in or prior to the 970 s using DSM. However, DSM is a natural fit for only structural problems. Weighted Residuals Formulations & Variational Methods (which is called the work-energy method when applied in structural analysis) bridge the gap between structural analysis and multiphysics applications. Multi-physics: Fluid Flow, Mass Transport, Electromagnetic fields. 000 and beyond: Bio-engineering (non-linear elasticity); large displacement or finite rotation problems (non-linear structural FEA); MEMS; two-phase flows;

9 Lecture : FEM Introduction The FEM as a whole includes DSM, variational, and weighted residual formulations Grey Matter Bits and Bytes 950 s 960 s Logan Ch#,3,4, & 5 Logan Ch# 4,6,9,3,4, & 6

10 Lecture : FEM Introduction Recovery & Interpretation MECH 40 will provide knowledge of how an FE code or package can generate a solution. MECH 40 will focus on the importance of the human presence in the FE analysis loop. MECH 40 will begin by looking at structural analysis but will conclude by etending the FE technique to other classes of problems.

11 Lecture : FEM Introduction. Introduction to Matri Notation. The FEM translates differential equations into a matri set of algebraic equations. duˆ ε = and σ = Eε dˆ FEM AE dˆ ˆ f L = dˆ fˆ

12 Lecture : FEM Introduction The state of the structural system is defined by a matri of displacements (generalized displacements), D. { y z Ny Nz} D = d d d d d d The eternal factors acting on the system are given by a force (generalized force) matri, F. { y z Ny Nz} F = f f f f f f However, cartesian coordinate systems are used to define the sense of each displacement and force term. T T Force is a vector with the sense of components being set by some -y coordinate system f = f f f { y z} T Node 35

13 Lecture : FEM Introduction.3 Role of the Computer. The assembly process takes advantage of computing power. We will see in Chapters and 3 how assembly occurs. Assembly can be automated. It only requires knowledge of where an element sits within the finite element model. When working by hand we can use a connectivity table to help us perform assembly. Modern FE packages will handle assemblies of thousands of elements. The FE procedure has the advantage of producing a banded set of equations. Banded equations are less epensive to solve.

14 Lecture : FEM Introduction.4 General Steps to the Finite Element Method (p.6-3). ) (Discretize and) Set the Element Type. ) Set the Displacement Function. 3) Define the Governing DE(s). 4) Derive the Element Equations*. 5) Assemble the Element Equations to Produce the Global Equations & Apply BC s to Reduce the Global System. 6) Solve for the Unknowns. 7) Recovery. 8) Interpret. *Step 4 has wildly differing looks depending on whether we apply a DSM, Variational, or Weighted Residual (WR) formulation.

15 Lecture : The Generalized FEM. For now, we preview a WR formulation to emphasize the multiphysics nature of the FEM, and give some meaning to FEM terminology. d du( ) = d d The governing DE is the actual problem: find u()..0.0 The global domain of the problem [ u] =.0 du =.0 ; d =.0 = The conditions that must hold at the boundaries: the BC s.

16 Lecture : The Generalized FEM u ( ) node node node 3 node 4 node 5 u element domain element domain element 3 domain element 4 domain () u ( ) u Step : (Discretize and) Set the Element Type. We will use line segments to model the D function u(). Step : Set the Displacement Function. For this problem the displacement is u. ( ) ( ) ( ) () u = u φ + u φ φ ( ) =.0 =.5 3 =.50 4 =.75 5 =.0 0 φ ( ) =.0 =.5 0 =.0 =.5

17 Lecture : The Generalized FEM. Step 3: Define the Governing DE(s). The problem defines the governing DE in this case. Step 4: Derive the Element Equations. We are using a WR formulation. In Step 4 we are only concerned with satisfying the governing DE within the element. The WR formulation forces the approimate elemental solution to be a good one. How does one define a good approimation? Each eisting WR method uses a different criterion for the measure of goodness. Collocation WR Method Least-Squares WR method Galerkin WR Method used eclusively in MECH 40.

18 Lecture : The Generalized FEM. Step 4: Derive the Element Equations. (cont d ) Element ( ) d du d d ()( ) φ = d d du d d ( ) 0 () ( ) φ d = ( ) 0 du ( ) k (, ) u + k (, ) u = + f = F d () () () (),, du ( ) k (, ) u + k (, ) u = + f = F d () () () (),, () () k (), k, u F () () = () k u, k, F

19 Lecture : The Generalized FEM Step 4: Derive the Element Equations. (cont d ). The evaluation of the residual equations for elements,, 3, and 4 produces 4 sets of element equations. () () k (), k, u F () () = () k u, k, F Element ( ) (3) (3) k (3), k, u3 F3 (3) (3) = (3) k u, k, 4 F4 Element 3 ( ) 3 4 () () () k, k, u F () () = () k u, k, 3 F3 Element ( ) 3 (4) (4) (4) k, k, u4 F4 (4) (4) = (4) k u, k, 5 F5 Element 4 ( ) 4 5

20 Lecture : The Generalized FEM Step 5: Assemble the Element Equations to form the Global System. The assembly process (or superposition process as in Logan.4) produces a set of global equations. The global equations form a boundary value problem () du d d 5 () () k (), k, u f () () () () k () (), k, + k, k, 0 0 u f + f 0 () () (3) (3) () (3) 0 k, k, + k, k, 0 u3 = f3 + f (3) (3) (4) (4) (3) (4) 0 0 k u, k, + k, k, 4 f4 + f 0 4 (4) (4) (4) (4) k u, k, 5 f5 du Banded matri structure makes numerical solution inepensive. The structure is a result of the choice of node numbering scheme.

21 Lecture : The Generalized FEM Step 5: Applying the Boundary Conditions to Reduce the Global System. (cont d ) Recall the conditions that eisted at the boundaries of the global domain (the BCs): u =.0 du d =.0 = This is a condition on a flu term. A flu is a rate of change of the desired function, u(). This natural boundary condition is satisfied by the actual/true function.

22 Lecture : The Generalized FEM Step 5: Applying the Boundary Conditions to Reduce the Global System. (cont d ) () du d d 5 () () k (), k, u f () () () () k () (), k, + k, k, 0 0 u f + f 0 () () (3) (3) () (3) 0 k, k, + k, k, 0 u3 = f3 + f (3) (3) (4) (4) (3) (4) 0 0 k u, k, + k, k, 4 f4 + f 0 4 (4) (4) (4) (4) k u, k, 5 f5 du.0 0.5

23 Lecture : The Generalized FEM Step 6: Solve for the remaining unknowns. () () k (), k, f () () () () k () (), k, + k, k, 0 0 u f + f () () (3) (3) () (3) 0 k, k, + k, k, 0 u3 = f3 + f3 (3) (3) (4) (4) (3) (4) 0 0 k u, k, + k, k, 4 f4 + f4 (4) (4) (4) k u, k, 5 f5 Reduction () du d () () () k () (), + k, k, 0 0 () u f + f 0 k, () () (3) (3) k () (3), k, + k, k, 0 u 3 f3 f = + (3) (3) (4) (4) (3) (4) 0 k u, k, k, k.0 +, 4 f4 + f4 0 0 (4) (4) (4) 0 0 k u, k, 5 f

24 Lecture : The Generalized FEM Step 7: Recovery. Knowing the values of the function at the 5 key points, we can recover some information about other values of interest. For eample, what is the approimate value of the flu term at the point =.? The point =. lies inside the first element () u = u φ + u φ ( ) ( ) ( ) φ ( ) 0 φ ( ) =.0 =.5 du d. ( uφ + uφ) d =. d u u =.0 =.5

25 Lecture : The Generalized FEM Step 8: Interpret the results. This case study illustrates the importance of the FEM user in the solution process: Interpretation includes looking at: Convergence. Continuity and completeness conditions What if you had wanted to know something about du ( )? d u() flu

MECh300H Introduction to Finite Element Methods. Finite Element Analysis (F.E.A.) of 1-D Problems

MECh300H Introduction to Finite Element Methods Finite Element Analysis (F.E.A.) of -D Problems Historical Background Hrenikoff, 94 frame work method Courant, 943 piecewise polynomial interpolation Turner,