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- 1. MCE 565 Wave Motion & Vibration in Continuous Media Spring 2005 Professor M. H. Sadd Introduction to Finite Element Methods
- 2. Need for Computational Methods • Solutions Using Either Strength of Materials or Theory of Elasticity Are Normally Accomplished for Regions and Loadings With Relatively Simple Geometry • Many Applicaitons Involve Cases with Complex Shape, Boundary Conditions and Material Behavior • Therefore a Gap Exists Between What Is Needed in Applications and What Can Be Solved by Analytical Closed- form Methods • This Has Lead to the Development of Several Numerical/Computational Schemes Including: Finite Difference, Finite Element and Boundary Element Methods
- 3. Introduction to Finite Element Analysis The finite element method is a computational scheme to solve field problems in engineering and science. The technique has very wide application, and has been used on problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations, electrical and magnetic fields, etc. The fundamental concept involves dividing the body under study into a finite number of pieces (subdomains) called elements (see Figure). Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or approximation functions. This approximated variation is quantified in terms of solution values at special element locations called nodes. Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution. Because element size, shape and approximating scheme can be varied to suit the problem, the method can accurately simulate solutions to problems of complex geometry and loading and thus this technique has become a very useful and practical tool.
- 4. Advantages of Finite Element Analysis - Models Bodies of Complex Shape - Can Handle General Loading/Boundary Conditions - Models Bodies Composed of Composite and Multiphase Materials - Model is Easily Refined for Improved Accuracy by Varying Element Size and Type (Approximation Scheme) - Time Dependent and Dynamic Effects Can Be Included - Can Handle a Variety Nonlinear Effects Including Material Behavior, Large Deformations, Boundary Conditions, Etc.
- 5. Basic Concept of the Finite Element Method Any continuous solution field such as stress, displacement, temperature, pressure, etc. can be approximated by a discrete model composed of a set of piecewise continuous functions defined over a finite number of subdomains. Exact Analytical Solution x T Approximate Piecewise Linear Solution x T One-Dimensional Temperature Distribution
- 6. Two-Dimensional Discretization -1 -0.5 0 0.5 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 -3 -2 -1 0 1 2 x y u(x,y) Approximate Piecewise Linear Representation
- 7. Discretization Concepts x T Exact Temperature Distribution, T(x) Finite Element Discretization Linear Interpolation Model (Four Elements) Quadratic Interpolation Model (Two Elements) T1 T2 T2 T3 T3 T4 T4 T5 T1 T2 T3 T4 T5 Piecewise Linear Approximation T x T1 T2 T3 T3 T4 T5 T T1 T2 T3 T4 T5 Piecewise Quadratic Approximation x Temperature Continuous but with Discontinuous Temperature Gradients Temperature and Temperature Gradients Continuous
- 8. Common Types of Elements One-Dimensional Elements Line Rods, Beams, Trusses, Frames Two-Dimensional Elements Triangular, Quadrilateral Plates, Shells, 2-D Continua Three-Dimensional Elements Tetrahedral, Rectangular Prism (Brick) 3-D Continua
- 9. Discretization Examples One-Dimensional Frame Elements Two-Dimensional Triangular Elements Three-Dimensional Brick Elements
- 10. Basic Steps in the Finite Element Method Time Independent Problems - Domain Discretization - Select Element Type (Shape and Approximation) - Derive Element Equations (Variational and Energy Methods) - Assemble Element Equations to Form Global System [K]{U} = {F} [K] = Stiffness or Property Matrix {U} = Nodal Displacement Vector {F} = Nodal Force Vector - Incorporate Boundary and Initial Conditions - Solve Assembled System of Equations for Unknown Nodal Displacements and Secondary Unknowns of Stress and Strain Values
- 11. Common Sources of Error in FEA • Domain Approximation • Element Interpolation/Approximation • Numerical Integration Errors (Including Spatial and Time Integration) • Computer Errors (Round-Off, Etc., )
- 12. Measures of Accuracy in FEA Accuracy Error = |(Exact Solution)-(FEM Solution)| Convergence Limit of Error as: Number of Elements (h-convergence) or Approximation Order (p-convergence) Increases Ideally, Error → 0 as Number of Elements or Approximation Order → ∞
- 13. Two-Dimensional Discretization Refinement (Discretization with 228 Elements) (Discretization with 912 Elements) (Triangular Element) (Node) • • •
- 14. One Dimensional Examples Static Case 1 2 u1 u2 Bar Element Uniaxial Deformation of Bars Using Strength of Materials Theory Beam Element Deflection of Elastic Beams Using Euler-Bernouli Theory 1 2 w1 w2 θ2 θ1 dx du au qcuau dx d , :ionSpecificatCondtionsBoundary 0)( :EquationalDifferenti =−+− )(,,, :ionSpecificatCondtionsBoundary )()( :EquationalDifferenti 2 2 2 2 2 2 2 2 dx wd b dx d dx wd b dx dw w xf dx wd b dx d =−
- 15. Two Dimensional Examples u1 u2 1 2 3 u3 v1 v2 v3 1 2 3 φ1 φ2 φ3 Triangular Element Scalar-Valued, Two-Dimensional Field Problems Triangular Element Vector/Tensor-Valued, Two- Dimensional Field Problems yx n y n xdn d yxf yx ∂ φ∂ + ∂ φ∂ = φ φ = ∂ φ∂ + ∂ φ∂ , :ionSpecificatCondtionsBoundary ),( :EquationialDifferentExample 2 2 2 2 yxy yxx y x n y v C x u Cn x v y u CT n x v y u Cn y v C x u CT F y v x u y E v F y v x u x E u ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = =+ ∂ ∂ + ∂ ∂ ∂ ∂ ν− +∇µ =+ ∂ ∂ + ∂ ∂ ∂ ∂ ν− +∇µ 221266 661211 2 2 ConditonsBoundary 0 )1(2 0 )1(2 entsDisplacemofTermsinEquationsFieldElasticity
- 16. Development of Finite Element Equation • The Finite Element Equation Must Incorporate the Appropriate Physics of the Problem • For Problems in Structural Solid Mechanics, the Appropriate Physics Comes from Either Strength of Materials or Theory of Elasticity • FEM Equations are Commonly Developed Using Direct, Variational- Virtual Work or Weighted Residual Methods Variational-Virtual Work Method Based on the concept of virtual displacements, leads to relations between internal and external virtual work and to minimization of system potential energy for equilibrium Weighted Residual Method Starting with the governing differential equation, special mathematical operations develop the “weak form” that can be incorporated into a FEM equation. This method is particularly suited for problems that have no variational statement. For stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that found by variational methods. Direct Method Based on physical reasoning and limited to simple cases, this method is worth studying because it enhances physical understanding of the process
- 17. Simple Element Equation Example Direct Stiffness Derivation 1 2 k u1 u2 F1 F2 }{}]{[ rmMatrix Foinor 2NodeatmEquilibriu 1NodeatmEquilibriu 2 1 2 1 212 211 FuK F F u u kk kk kukuF kukuF = = − − +−=⇒ −=⇒ Stiffness Matrix Nodal Force Vector
- 18. Common Approximation Schemes One-Dimensional Examples Linear Quadratic Cubic Polynomial Approximation Most often polynomials are used to construct approximation functions for each element. Depending on the order of approximation, different numbers of element parameters are needed to construct the appropriate function. Special Approximation For some cases (e.g. infinite elements, crack or other singular elements) the approximation function is chosen to have special properties as determined from theoretical considerations
- 19. One-Dimensional Bar Element ⇒δ++=δσ ∫∫ ΩΩ udVfuPuPedV jjii }]{[:LawStrain-Stress }]{[}{ ][ )(:Strain }{][)(:ionApproximat dB dBd N dN EEe dx d ux dx d dx du e uxu k kk k kk ==σ ==ψ== =ψ= ∑ ∑ ⇒+ = ∫∫ L TT j iT L TT fdxA P P dxEA 00 ][}{}{}{][][}{ NδdδddBBδd ∫∫ += L T L T fdxAdxEA 00 ][}{}{][][ NPdBB VectorentDisplacemNodal}{ VectorLoading][}{ MatrixStiffness][][][ 0 0 = = =+ = == ∫ ∫ j i L T j i L T u u fdxA P P dxEAK d NF BB }{}]{[ FdK =
- 20. One-Dimensional Bar Element A = Cross-sectional Area E = Elastic Modulus f(x) = Distributed Loading dVuFdSuTdVe i V i S i n iij V ij t δ+δ=δσ ∫∫∫ Virtual Strain Energy = Virtual Work Done by Surface and Body Forces For One-Dimensional Case ∫∫ ΩΩ δ++=δσ udVfuPuPedV jjii Ω (i) (j) Axial Deformation of an Elastic Bar Typical Bar Element dx du AEP i i −= dx du AEP j j −= iu ju L x (Two Degrees of Freedom)
- 21. Linear Approximation Scheme [ ] VectorentDisplacemNodal}{ MatrixFunctionionApproximat][ }]{[1 )()( 1 2 1 2 1 21 2211 21 12 1 212 11 21 = = = −= ψψ= ψ+ψ= + −= − +=⇒ += = ⇒+= d N dN ntDisplacemeElasticeApproximat u u L x L x u u u uxux u L x u L x x L uu uu Laau au xaau x (local coordinate system) (1) (2) iu ju L x (1) (2) u(x) x (1) (2) ψ1(x) ψ2(x) 1 ψk(x) – Lagrange Interpolation Functions
- 22. Element Equation Linear Approximation Scheme, Constant Properties VectorentDisplacemNodal}{ 1 1 2 ][}{ 11 1111 1 1 ][][][][][ 2 1 2 1 0 2 1 0 2 1 00 = = + = − + =+ = − − = − − === ∫∫ ∫∫ u u LAf P P dx L x L x Af P P fdxA P P L AE L LL L LAEdxAEdxEAK o L o L T L T L T d NF BBBB + = − − ⇒= 1 1 211 11 }{}]{[ 2 1 2 1 LAf P P u u L AE o FdK
- 23. Quadratic Approximation Scheme [ ] }]{[ )()()( 42 3 2 1 321 332211 2 3213 2 3212 11 2 321 dN ntDisplacemeElasticeApproximat = ψψψ= ψ+ψ+ψ= ++= ++= = ⇒++= u u u u uxuxuxu LaLaau L a L aau au xaxaau x (1) (3) 1u 3u (2) 2u L u(x) x (1) (3)(2) x (1) (3)(2) 1 ψ1(x) ψ3(x) ψ2(x) = − −− − 3 2 1 3 2 1 781 8168 187 3 F F F u u u L AE EquationElement
- 24. Lagrange Interpolation Functions Using Natural or Normalized Coordinates 11 ≤ξ≤− ξ (1) (2) )1( 2 1 )1( 2 1 2 1 ξ+=ψ ξ−=ψ )1( 2 1 )1)(1( )1( 2 1 3 2 1 ξ+ξ=ψ ξ+ξ−=ψ ξ−ξ−=ψ )1)( 3 1 )( 3 1 ( 16 9 ) 3 1 )(1)(1( 16 27 ) 3 1 )(1)(1( 16 27 ) 3 1 )( 3 1 )(1( 16 9 4 3 2 1 ξ+ξ−ξ+−=ψ ξ+ξ+ξ−=ψ ξ−ξ+ξ−=ψ ξ−ξ+ξ−−=ψ ξ (1) (2) (3) ξ (1) (2) (3) (4) ≠ = =ξψ ji ji ji ,0 ,1 )(
- 25. Simple Example P A1,E1,L1 A2,E2,L2 (1) (3)(2) 1 2 = − − 0000 011 011 1ElementEquationGlobal )1( 2 )1( 1 3 2 1 1 11 P P U U U L EA = − − )2( 2 )2( 1 3 2 1 2 22 0 110 110 000 2ElementEquationGlobal P P U U U L EA = += − −+− − 3 2 1 )2( 2 )2( 1 )1( 2 )1( 1 3 2 1 2 22 2 22 2 22 2 22 1 11 1 11 1 11 1 11 0 0 EquationSystemGlobalAssembled P P P P PP P U U U L EA L EA L EA L EA L EA L EA L EA L EA 0 LoadingedDistributZeroTake =f
- 26. Simple Example Continued P A1,E1,L1 A2,E2,L2 (1) (3)(2) 1 2 0 0 ConditionsBoundary )2( 1 )1( 2 )2( 2 1 =+ = = PP PP U = − −+− − P P U U L EA L EA L EA L EA L EA L EA L EA L EA 0 0 0 0 EquationSystemGlobalReduced )1( 1 3 2 2 22 2 22 2 22 2 22 1 11 1 11 1 11 1 11 = − −+ PU U L EA L EA L EA L EA L EA 0 3 2 2 22 2 22 2 22 2 22 1 11 LEA ,,Properties mFor Unifor = − − PU U L AE 0 11 12 3 2 PP AE PL U AE PL U −===⇒ )1( 132 , 2 ,Solving
- 27. One-Dimensional Beam Element Deflection of an Elastic Beam 2 2423 1 1211 2 2 2 4 2 2 2 3 1 2 2 2 1 2 2 1 ,,, , , dx dw uwu dx dw uwu dx wd EIQ dx wd EI dx d Q dx wd EIQ dx wd EI dx d Q −=θ==−=θ== −= −= = = I = Section Moment of Inertia E = Elastic Modulus f(x) = Distributed Loading Ω (1) (2) Typical Beam Element 1w L 2w 1θ 2θ 1M 2M 1V 2V x Virtual Strain Energy = Virtual Work Done by Surface and Body Forces ⇒δ++++=δσ ∫∫ ΩΩ wdVfwQuQuQuQedV 44332211 ∫∫ ++++= L T L dVfwQuQuQuQdxEI 0 44332211 0 ][}{][][ NdBB T (Four Degrees of Freedom)
- 28. Beam Approximation Functions To approximate deflection and slope at each node requires approximation of the form 3 4 2 321)( xcxcxccxw +++= Evaluating deflection and slope at each node allows the determination of ci thus leading to FunctionsionApproximatCubicHermitethearewhere ,)()()()()( 44332211 i uxuxuxuxxw φ φ+φ+φ+φ=
- 29. Beam Element Equation ∫∫ ++++= L T L dVfwQuQuQuQdxEI 0 44332211 0 ][}{][][ NdBB T = 4 3 2 1 }{ u u u u d ][ ][ ][ 4321 dx d dx d dx d dx d dx d φφφφ == N B − − − −−− == ∫ 22 22 30 233 3636 323 3636 2 ][][][ LLLL LL LLLL LL L EI dxEI L BBK T − + = − − − −−− L LfL Q Q Q Q u u u u LLLL LL LLLL LL L EI 6 6 12 233 3636 323 3636 2 4 3 2 1 4 3 2 1 22 22 3 − = φ φ φ φ = ∫∫ L LfL dxfdxf LL T 6 6 12 ][ 0 4 3 2 1 0 N
- 30. FEA Beam Problem f a b Uniform EI + − −= − − − −−− 0 0 0 0 6 6 12 000000 000000 00/2/3/1/3 00/3/6/3/6 00/1/3/2/3 00/3/6/3/6 2 )1( 4 )1( 3 )1( 2 )1( 1 6 5 4 3 2 1 22 2323 22 2323 Q Q Q Q a a fa U U U U U U aaaa aaaa aaaa aaaa EI 1Element = − − − −−− )2( 4 )2( 3 )2( 2 )2( 1 6 5 4 3 2 1 22 2323 22 2323 0 0 /2/3/1/300 /3/6/3/600 /1/3/2/300 /3/6/3/600 000000 000000 2 Q Q Q Q U U U U U U bbbb bbbb bbbb bbbb EI 2Element (1) (3)(2) 1 2
- 31. FEA Beam Problem + + + − −= ⋅⋅⋅⋅⋅ ⋅⋅⋅⋅ +⋅⋅⋅ −−−+⋅⋅ ⋅ −−− )2( 4 )2( 3 )2( 2 )1( 4 )2( 1 )1( 3 )1( 2 )1( 1 6 5 4 3 2 1 23 2 232233 2 2323 0 0 6 6 12 /2 /3/6 /1/3/2/2 /3/6/3/3/6/6 00/1/3/2 00/3/6/3/6 2 Q Q QQ QQ Q Q a a fa U U U U U U a aa aaba aababa aaa aaaa EI SystemAssembledGlobal 0,0,0 )2( 4 )2( 3 )1( 12 )1( 11 ===θ=== QQUwU ConditionsBoundary 0,0 )2( 2 )1( 4 )2( 1 )1( 3 =+=+ QQQQ ConditionsMatching + −= ⋅⋅⋅ ⋅⋅ +⋅ −−−+ 0 0 0 0 0 0 6 12 /2 /3/6 /1/3/2/2 /3/6/3/3/6/6 2 4 3 2 1 23 2 332233 afa U U U U a aa aaba aababa EI SystemReduced Solve System for Primary Unknowns U1 ,U2 ,U3 ,U4 Nodal Forces Q1 and Q2 Can Then Be Determined (1) (3)(2) 1 2
- 32. Special Features of Beam FEA Analytical Solution Gives Cubic Deflection Curve Analytical Solution Gives Quartic Deflection Curve FEA Using Hermit Cubic Interpolation Will Yield Results That Match Exactly With Cubic Analytical Solutions
- 33. Truss Element Generalization of Bar Element With Arbitrary Orientation x y k=AE/L θ=θ= cos,sin cs
- 34. Frame Element Generalization of Bar and Beam Element with Arbitrary Orientation Ω (1) (2) 1w L 2w 1θ 2θ 1M 2M 1V 2V 2P1P 1u 2u = θ θ − −−− − − − − 4 3 2 2 1 1 2 2 2 1 1 1 22 2323 22 2323 46 0 26 0 612 0 612 0 0000 26 0 46 0 612 0 612 0 0000 Q Q P Q Q P w u w u L EI L EI L EI L EI L EI L EI L EI L EI L AE L AE L EI L EI L EI L EI L EI L EI L EI L EI L AE L AE Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation
- 35. Some Standard FEA References Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, 1982, 1995. Beer, G. and Watson, J.O., Introduction to Finite and Boundary Element Methods for Engineers, John Wiley, 1993 Bickford, W.B., A First Course in the Finite Element Method, Irwin, 1990. Burnett, D.S., Finite Element Analysis, Addison-Wesley, 1987. Chandrupatla, T.R. and Belegundu, A.D., Introduction to Finite Elements in Engineering, Prentice-Hall, 2002. Cook, R.D., Malkus, D.S. and Plesha, M.E., Concepts and Applications of Finite Element Analysis, 3rd Ed., John Wiley, 1989. Desai, C.S., Elementary Finite Element Method, Prentice-Hall, 1979. Fung, Y.C. and Tong, P., Classical and Computational Solid Mechanics, World Scientific, 2001. Grandin, H., Fundamentals of the Finite Element Method, Macmillan, 1986. Huebner, K.H., Thorton, E.A. and Byrom, T.G., The Finite Element Method for Engineers, 3rd Ed., John Wiley, 1994. Knight, C.E., The Finite Element Method in Mechanical Design, PWS-KENT, 1993. Logan, D.L., A First Course in the Finite Element Method, 2nd Ed., PWS Engineering, 1992. Moaveni, S., Finite Element Analysis – Theory and Application with ANSYS, 2nd Ed., Pearson Education, 2003. Pepper, D.W. and Heinrich, J.C., The Finite Element Method: Basic Concepts and Applications, Hemisphere, 1992. Pao, Y.C., A First Course in Finite Element Analysis, Allyn and Bacon, 1986. Rao, S.S., Finite Element Method in Engineering, 3rd Ed., Butterworth-Heinemann, 1998. Reddy, J.N., An Introduction to the Finite Element Method, McGraw-Hill, 1993. Ross, C.T.F., Finite Element Methods in Engineering Science, Prentice-Hall, 1993. Stasa, F.L., Applied Finite Element Analysis for Engineers, Holt, Rinehart and Winston, 1985. Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, Fourth Edition, McGraw-Hill, 1977, 1989.