The Finite Element Method - TAMU Mechanics

1 JN ReddyThe Finite Element MethodRead: Chapters 1 and 2 GENERAL introduction Engineering and analysis Simulation of a physical process Examples mathematical modeldevelopment Approximate solutions and methods of approximation The basic features of the Finite Element Method Examples Finite Element discretization Terminology Steps involved in the Finite Element model developmentCONTENTSJN ReddyIntroduction: 2 INTRODUCTORY REMARKS What we do as engineers? Knowing the fundamentals associated with each engineering problem you set out to tackle, not only makes you a better engineer but also empowers you as an engineer. develop mathematical models, conduct physical experiments, carry out numerical simulations to help designer, and design and build systems to achieve a (1) functionalityin (2) most economical ReddyIntroduction: 3 INTRODUCTORY REMARKSE ngineering is the discipline, art, and professionof acquiring and applying technical, scientific,and mathematical knowledge to design andimplementmaterials,structures,machine s,devices,systems,andprocessesthatsafely realize a desired a problem-solving discipline, and solution requires an understanding of the phenomena that occurs in the system or ReddyIntroduction.

2 4 AnalysisAnalysisis an aid to design and manufacturing, and not an end in involves the following steps: identifying the problem and nature of the response to be determined, selecting the mathematical model ( , governing equations), solving the problem with a solution Method ( , FEM), and evaluating the results in light of the design ReddyIntroduction: 5 MODELING OF APhysical ProcessPhysical SystemAssumptions concerningthe systemLaws of physics(conservationprinciples)Mathemati cal Model(BVP, IVP)Numerical SimulationsComputationaldeviceNumerical Method (FEM,FDM,BEM,etc.)BVP Boundary value problems (equilibrium problems)IVP Initial value problems (time-dependent problems)FEM Finite Element Method FDM Finite Difference MethodBEM Boundary Element MethodJN ReddyEXAMPLES OF MATHEMATICAL MODEL DEVELOPMENTR ectangular fins(a)Body from which heat has to be extracted(b)Lateral surface and right end areexposed to ambient temperature, T ahL()PT T -Convection,, P= perimeterxyz()xqA()xxqA +la()020() ()()(),xxxxxxhhAAqAqAP x T TrxddTAqP T Tr Aqkdxdx ++ + ---+= ---+==-Objective: Determine heat flow in a heat exchanger finBasic Concepts.

3 63D to 2D2D to 1 DJN ReddyEXAMPLE OF ENGINEERING MODEL DEVELOPMENT (continued)()hddTAkPTTrAdxdx -+-= ,,,huTT a Ak c PrA fdduacufdxdx =- = == -+= ()000()(),,,xxxdAAfA xAfAdxdudduEAEfdxdxdx +-+ = += == - += , force per unit lengthLx xf x()xxA +()xA fDetermine: Axial deformation of a barBasic Concepts: 7JN ReddyAPPROXIMATE SOLUTIONS point boundary aat)()(in)()()(PuubdxduaLxfuxcdxduxadxd= += = + 0,00 Model ProblemBasic Concepts: 8E, Ab=k L u(L)u0Px, ucElastic deformation of a bar()fxaEA=Heat transfer in a finuninsulated barakAcP ==()fxxL0,uTT u T =- =JN ReddyENGINEERING EXAMPLES OF THE MODEL PROBLEM IN 1-DBasic Concepts: 9 Flow of viscous fluid through a channel00in( , )xdvdfbdydy = = ( )( ), horizontal velocitypressure gradient, /(constant)() fluid viscosityshear stresselocity of the top surfacexuy v yfdpdxyQUv =====0U ybx2bCouette flowPoiseuille flowJN ReddyExact and Approximate SolutionsApproximation of the actual solution over the entire domainActual solutionApproximate solutionAn exact solutionsatisfies (a) the differential equation at every point of the domain and (b) boundary conditions on the boundary.

4 An approximate solutionsatisfies the differential equation as well as the specified boundary conditions in some acceptable sense (to be made clearer shortly). We seek the approximate solution as a linear combination of unknown parametersciand known functions01()()()()NNiiiuxu xcxx= = + ff0andiffBasic Concepts: 10JN ReddyDetermining Approximate Solutions(continued)Then ciare determined such that the residual, R(x), is zero in some sense. 0 NNdudaxcxuf xdxdx -+-= ()()()0 NNduda( x )c( x )uf ( x )R( x )dxdx -+- will result in a non-zero function on the left side of the equality:1. Suppose that is selected to satisfy the boundary conditionsexactly. Then substitution of uN(x) into the differentialequationifBasic Concepts: 11JN ReddyMETHODS OF APPROXIMATION1.

5 One sense in which the residual, R(x), can be made zero is to require it to be zero at selected number of points. The number of points should be equal to the number of unknowns in the approximate solutionThis way of determining ciis known as the Collocation Method . We obtain Nalgebraic equations in Nunknown C s NixRi,,2,1,0 ==)(010()()()( ) and( ) are functions to be selected to satisfy thespecified boundary conditions andare parameters to be determined such that the residual iszero insome =+ Basic Concepts: 12JN ReddyMethods of Approximation(Continued)2. Another approach in which the residual, R(x), can be made zero is in a least-squares sense; , minimize the integral of the square of the residual with respect to C s.

6 02),,,(00221= = = dxcRRcJdxRcccJLiiLNorMinimize This Method is known as the least-Squares Method . We obtain Nalgebraic equations in Nunknown C s 00= dxcRRLiBasic Concepts: 13JN Reddy3. Yet, another approach in which the residual, R(x), can be made zero is in a weighted-residual sense00,1,2,,whereare linearly independent set of functionsLiiRdx iN =y =y This Method is known as the Weighted-Residual Method . We obtain Nalgebraic equations in Nunknown C s. In general, weight functions are not the same as the approximation functions . Various special cases arePetrov-Galerkin Method :Galerkin Method :iiiiy fy=fi i Basic Concepts: 14 methods of Approximation(Continued)JN ReddyWEIGHTED-INTEGRAL FORMULATIONSin the Numerical Solution of Differential approximation methods discussed earlier can be viewed as special cases of the weighted-residual methods of approximation.

7 In particular, we have Collocation Method Least-squares Method Petrov-Galerkin Method Galerkin Method =-()()iixxxyd = ()iiRxcy ()()iixxyf()()iixx=yfBasic Concepts: 15JN ReddyIntroduction: 164. Another approach in which the governing equation is castin a weak-formand the weight function is taken the sameas the approximation function is known as the Ritz Method : methods of Approximation(Continued)(, ) (),1,2,,iNiBuiNff== The Ritz Method is the most commonly used Method forall commercial software. In this Method , satisfies onlythe specified essential (geometric) boundary conditionswhile satisfies the homogeneous form of the specifiedessential boundary conditions. The specified natural (force)boundary condition are included in the weak ReddyIntroduction: 17 WORKING EXAMPLE000, 0(0),xLdduafxLdxdxduuuaPdx= --=<< ==For a weighted-residual Method :0fifsatisfy the actual specified BCssatisfy the homogeneous form of the actualspecified BCsFor the Ritz Method :0fifsatisfy the actual specified essential BCssatisfy the homogeneous form of the actualspecified essential BCsJN ReddyIntroduction: 18 WORKING EXAMPLE (CONTINUED)00000100002(),(),()xLiixLdPua PuxdxadaxLxdxffffff==== =+== =-For a weighted-residual Method :For the Ritz (or weak-form Galerkin) Method .

8 00000(), ()iuff==00,()iiuxxff = =JN ReddyBASIC FEATURES OF THE Finite Element Method (FEM) Divide whole into parts ( Finite Element mesh) Set up the `problem over a typical part(derive a set of relationships between primary and secondary variables) Assemble the parts to obtain the solution to the wholeBasic Concepts: 19JN Reddy11 111 1,,NN Niiiiiiii iNN Nii iii imxmymzXY Zmm m== === === = 1. See the simplicity in the complicated (see the geometric shapes that are simple to identify the center of mass).2. Determine the center of mass of each Put the parts together to obtain the required 1: Determine the center of massof a 3D machine partBasic Concepts: 20JN ReddyF(x)xbaActual functionExample 2: Determine the integral of a function =badxxFI)(Basic Concepts: 21JN ReddyBasic Concepts: 22F(x)xActual functionApproximationEXAMPLE 2(continued) + + + bxx,xbaxxx,xbaxxa,xbaxF3333222211)(1xa=4 xb=2x3x +++==11iiiixxiixxiidxxbadxxFI)()(321 IIII++ 1I2I3 IJN ReddyBasic Concepts: 23 EXAMPLE 2(continued) Refined meshF(x)xActual function1xa=7bx=ApproximationJN ReddyApproximation of a curved surface with a plane(Triangular Element )(represents the solution)(temperature profile)DomainBasic Concepts.

9 24JN ReddyBasic Concepts: 25 (b) Finite Element meshh Domain, Boundary, Boundary flux e Domain, Finite Element DiscretizationElements Nodes (a) Given domain(c) Typical Element with boundary fluxes(d) Discretized domain e JN ReddySome Examples of Real-World Finite Element DiscretizationsIntroduction: 26JN ReddyFEM Terminology ElementA geometric sub-domain of the region being simulated, with the property that it allows a unique (1) representation of its geometry and (2) derivation of the approximation (interpolation) functions. NodeA geometric location in the Element which plays a role in the derivation of the interpolation functions and it is the point at which solution is sought.

10 MeshA collection of elements (or nodes) that replaces the actual domain. Weak FormAn integral statement equivalent to the governing equations and naturalboundary conditions. More to Concepts: 27JN ReddyFEM Terminology (continued) Finite Element ModelA set of algebraic equations relating the nodal values of the primary variables ( , displacements) to the nodal values of the secondary variables ( , forces) in an Element . Finite Element modelis NOT the same as the Finite Element is only one Finite Element Method but there can be more than one Finite Element model of a problem (depending on the approximate Method used to derive the algebraic equations). Numerical SimulationEvaluation of the mathematical model ( , solution of the governing equations) using a numerical Method and Concepts: 28JN ReddyMajor Steps of Finite Element Model Development Begin with the governing equations of the problem Develop its weak form over a typical Finite Element Approximatethe solution over each Finite Element Obtain algebraic relations among the quantities of interest over each Finite Element ( , Finite Element model)Basic Concepts: 29JN ReddyMajor Steps of Finite Element Model DevelopmentFinite Element Model DevelopmentGoverning (Differential) EquationsEngineering ProblemFormulationVirtual work statementsWeak Form